src/HOL/Probability/Lebesgue_Integration.thy
 author haftmann Sun Mar 16 18:09:04 2014 +0100 (2014-03-16) changeset 56166 9a241bc276cd parent 56154 f0a927235162 child 56193 c726ecfb22b6 permissions -rw-r--r--
normalising simp rules for compound operators
```     1 (*  Title:      HOL/Probability/Lebesgue_Integration.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 header {*Lebesgue Integration*}
```
```     7
```
```     8 theory Lebesgue_Integration
```
```     9   imports Measure_Space Borel_Space
```
```    10 begin
```
```    11
```
```    12 lemma tendsto_real_max:
```
```    13   fixes x y :: real
```
```    14   assumes "(X ---> x) net"
```
```    15   assumes "(Y ---> y) net"
```
```    16   shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
```
```    17 proof -
```
```    18   have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
```
```    19     by (auto split: split_max simp: field_simps)
```
```    20   show ?thesis
```
```    21     unfolding *
```
```    22     by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
```
```    23 qed
```
```    24
```
```    25 lemma measurable_sets2[intro]:
```
```    26   assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
```
```    27   and "A \<in> sets M'" "B \<in> sets M''"
```
```    28   shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
```
```    29 proof -
```
```    30   have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
```
```    31     by auto
```
```    32   then show ?thesis using assms by (auto intro: measurable_sets)
```
```    33 qed
```
```    34
```
```    35 section "Simple function"
```
```    36
```
```    37 text {*
```
```    38
```
```    39 Our simple functions are not restricted to positive real numbers. Instead
```
```    40 they are just functions with a finite range and are measurable when singleton
```
```    41 sets are measurable.
```
```    42
```
```    43 *}
```
```    44
```
```    45 definition "simple_function M g \<longleftrightarrow>
```
```    46     finite (g ` space M) \<and>
```
```    47     (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
```
```    48
```
```    49 lemma simple_functionD:
```
```    50   assumes "simple_function M g"
```
```    51   shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
```
```    52 proof -
```
```    53   show "finite (g ` space M)"
```
```    54     using assms unfolding simple_function_def by auto
```
```    55   have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
```
```    56   also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
```
```    57   finally show "g -` X \<inter> space M \<in> sets M" using assms
```
```    58     by (auto simp del: UN_simps simp: simple_function_def)
```
```    59 qed
```
```    60
```
```    61 lemma simple_function_measurable2[intro]:
```
```    62   assumes "simple_function M f" "simple_function M g"
```
```    63   shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
```
```    64 proof -
```
```    65   have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
```
```    66     by auto
```
```    67   then show ?thesis using assms[THEN simple_functionD(2)] by auto
```
```    68 qed
```
```    69
```
```    70 lemma simple_function_indicator_representation:
```
```    71   fixes f ::"'a \<Rightarrow> ereal"
```
```    72   assumes f: "simple_function M f" and x: "x \<in> space M"
```
```    73   shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
```
```    74   (is "?l = ?r")
```
```    75 proof -
```
```    76   have "?r = (\<Sum>y \<in> f ` space M.
```
```    77     (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
```
```    78     by (auto intro!: setsum_cong2)
```
```    79   also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
```
```    80     using assms by (auto dest: simple_functionD simp: setsum_delta)
```
```    81   also have "... = f x" using x by (auto simp: indicator_def)
```
```    82   finally show ?thesis by auto
```
```    83 qed
```
```    84
```
```    85 lemma simple_function_notspace:
```
```    86   "simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
```
```    87 proof -
```
```    88   have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
```
```    89   hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
```
```    90   have "?h -` {0} \<inter> space M = space M" by auto
```
```    91   thus ?thesis unfolding simple_function_def by auto
```
```    92 qed
```
```    93
```
```    94 lemma simple_function_cong:
```
```    95   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
```
```    96   shows "simple_function M f \<longleftrightarrow> simple_function M g"
```
```    97 proof -
```
```    98   have "f ` space M = g ` space M"
```
```    99     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
```
```   100     using assms by (auto intro!: image_eqI)
```
```   101   thus ?thesis unfolding simple_function_def using assms by simp
```
```   102 qed
```
```   103
```
```   104 lemma simple_function_cong_algebra:
```
```   105   assumes "sets N = sets M" "space N = space M"
```
```   106   shows "simple_function M f \<longleftrightarrow> simple_function N f"
```
```   107   unfolding simple_function_def assms ..
```
```   108
```
```   109 lemma borel_measurable_simple_function[measurable_dest]:
```
```   110   assumes "simple_function M f"
```
```   111   shows "f \<in> borel_measurable M"
```
```   112 proof (rule borel_measurableI)
```
```   113   fix S
```
```   114   let ?I = "f ` (f -` S \<inter> space M)"
```
```   115   have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
```
```   116   have "finite ?I"
```
```   117     using assms unfolding simple_function_def
```
```   118     using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto
```
```   119   hence "?U \<in> sets M"
```
```   120     apply (rule sets.finite_UN)
```
```   121     using assms unfolding simple_function_def by auto
```
```   122   thus "f -` S \<inter> space M \<in> sets M" unfolding * .
```
```   123 qed
```
```   124
```
```   125 lemma simple_function_borel_measurable:
```
```   126   fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
```
```   127   assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
```
```   128   shows "simple_function M f"
```
```   129   using assms unfolding simple_function_def
```
```   130   by (auto intro: borel_measurable_vimage)
```
```   131
```
```   132 lemma simple_function_eq_borel_measurable:
```
```   133   fixes f :: "'a \<Rightarrow> ereal"
```
```   134   shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
```
```   135   using simple_function_borel_measurable[of f] borel_measurable_simple_function[of M f]
```
```   136   by (fastforce simp: simple_function_def)
```
```   137
```
```   138 lemma simple_function_const[intro, simp]:
```
```   139   "simple_function M (\<lambda>x. c)"
```
```   140   by (auto intro: finite_subset simp: simple_function_def)
```
```   141 lemma simple_function_compose[intro, simp]:
```
```   142   assumes "simple_function M f"
```
```   143   shows "simple_function M (g \<circ> f)"
```
```   144   unfolding simple_function_def
```
```   145 proof safe
```
```   146   show "finite ((g \<circ> f) ` space M)"
```
```   147     using assms unfolding simple_function_def by (auto simp: image_comp [symmetric])
```
```   148 next
```
```   149   fix x assume "x \<in> space M"
```
```   150   let ?G = "g -` {g (f x)} \<inter> (f`space M)"
```
```   151   have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
```
```   152     (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
```
```   153   show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
```
```   154     using assms unfolding simple_function_def *
```
```   155     by (rule_tac sets.finite_UN) auto
```
```   156 qed
```
```   157
```
```   158 lemma simple_function_indicator[intro, simp]:
```
```   159   assumes "A \<in> sets M"
```
```   160   shows "simple_function M (indicator A)"
```
```   161 proof -
```
```   162   have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
```
```   163     by (auto simp: indicator_def)
```
```   164   hence "finite ?S" by (rule finite_subset) simp
```
```   165   moreover have "- A \<inter> space M = space M - A" by auto
```
```   166   ultimately show ?thesis unfolding simple_function_def
```
```   167     using assms by (auto simp: indicator_def [abs_def])
```
```   168 qed
```
```   169
```
```   170 lemma simple_function_Pair[intro, simp]:
```
```   171   assumes "simple_function M f"
```
```   172   assumes "simple_function M g"
```
```   173   shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
```
```   174   unfolding simple_function_def
```
```   175 proof safe
```
```   176   show "finite (?p ` space M)"
```
```   177     using assms unfolding simple_function_def
```
```   178     by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
```
```   179 next
```
```   180   fix x assume "x \<in> space M"
```
```   181   have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
```
```   182       (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
```
```   183     by auto
```
```   184   with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
```
```   185     using assms unfolding simple_function_def by auto
```
```   186 qed
```
```   187
```
```   188 lemma simple_function_compose1:
```
```   189   assumes "simple_function M f"
```
```   190   shows "simple_function M (\<lambda>x. g (f x))"
```
```   191   using simple_function_compose[OF assms, of g]
```
```   192   by (simp add: comp_def)
```
```   193
```
```   194 lemma simple_function_compose2:
```
```   195   assumes "simple_function M f" and "simple_function M g"
```
```   196   shows "simple_function M (\<lambda>x. h (f x) (g x))"
```
```   197 proof -
```
```   198   have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
```
```   199     using assms by auto
```
```   200   thus ?thesis by (simp_all add: comp_def)
```
```   201 qed
```
```   202
```
```   203 lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
```
```   204   and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
```
```   205   and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
```
```   206   and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
```
```   207   and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
```
```   208   and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
```
```   209   and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
```
```   210
```
```   211 lemma simple_function_setsum[intro, simp]:
```
```   212   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
```
```   213   shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
```
```   214 proof cases
```
```   215   assume "finite P" from this assms show ?thesis by induct auto
```
```   216 qed auto
```
```   217
```
```   218 lemma
```
```   219   fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
```
```   220   shows simple_function_ereal[intro, simp]: "simple_function M (\<lambda>x. ereal (f x))"
```
```   221   by (auto intro!: simple_function_compose1[OF sf])
```
```   222
```
```   223 lemma
```
```   224   fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
```
```   225   shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
```
```   226   by (auto intro!: simple_function_compose1[OF sf])
```
```   227
```
```   228 lemma borel_measurable_implies_simple_function_sequence:
```
```   229   fixes u :: "'a \<Rightarrow> ereal"
```
```   230   assumes u: "u \<in> borel_measurable M"
```
```   231   shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
```
```   232              (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
```
```   233 proof -
```
```   234   def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
```
```   235   { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
```
```   236     proof (split split_if, intro conjI impI)
```
```   237       assume "\<not> real j \<le> u x"
```
```   238       then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
```
```   239          by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
```
```   240       moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
```
```   241         by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
```
```   242       ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
```
```   243         unfolding real_of_nat_le_iff by auto
```
```   244     qed auto }
```
```   245   note f_upper = this
```
```   246
```
```   247   have real_f:
```
```   248     "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))"
```
```   249     unfolding f_def by auto
```
```   250
```
```   251   let ?g = "\<lambda>j x. real (f x j) / 2^j :: ereal"
```
```   252   show ?thesis
```
```   253   proof (intro exI[of _ ?g] conjI allI ballI)
```
```   254     fix i
```
```   255     have "simple_function M (\<lambda>x. real (f x i))"
```
```   256     proof (intro simple_function_borel_measurable)
```
```   257       show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
```
```   258         using u by (auto simp: real_f)
```
```   259       have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
```
```   260         using f_upper[of _ i] by auto
```
```   261       then show "finite ((\<lambda>x. real (f x i))`space M)"
```
```   262         by (rule finite_subset) auto
```
```   263     qed
```
```   264     then show "simple_function M (?g i)"
```
```   265       by (auto intro: simple_function_ereal simple_function_div)
```
```   266   next
```
```   267     show "incseq ?g"
```
```   268     proof (intro incseq_ereal incseq_SucI le_funI)
```
```   269       fix x and i :: nat
```
```   270       have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
```
```   271       proof ((split split_if)+, intro conjI impI)
```
```   272         assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
```
```   273         then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
```
```   274           by (cases "u x") (auto intro!: le_natfloor)
```
```   275       next
```
```   276         assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
```
```   277         then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
```
```   278           by (cases "u x") auto
```
```   279       next
```
```   280         assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
```
```   281         have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
```
```   282           by simp
```
```   283         also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
```
```   284         proof cases
```
```   285           assume "0 \<le> u x" then show ?thesis
```
```   286             by (intro le_mult_natfloor)
```
```   287         next
```
```   288           assume "\<not> 0 \<le> u x" then show ?thesis
```
```   289             by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
```
```   290         qed
```
```   291         also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
```
```   292           by (simp add: ac_simps)
```
```   293         finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
```
```   294       qed simp
```
```   295       then show "?g i x \<le> ?g (Suc i) x"
```
```   296         by (auto simp: field_simps)
```
```   297     qed
```
```   298   next
```
```   299     fix x show "(SUP i. ?g i x) = max 0 (u x)"
```
```   300     proof (rule SUP_eqI)
```
```   301       fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
```
```   302         by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
```
```   303                                      mult_nonpos_nonneg mult_nonneg_nonneg)
```
```   304     next
```
```   305       fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
```
```   306       have "\<And>i. 0 \<le> ?g i x" by (auto simp: divide_nonneg_pos)
```
```   307       from order_trans[OF this *] have "0 \<le> y" by simp
```
```   308       show "max 0 (u x) \<le> y"
```
```   309       proof (cases y)
```
```   310         case (real r)
```
```   311         with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
```
```   312         from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
```
```   313         then have "\<exists>p. max 0 (u x) = ereal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
```
```   314         then guess p .. note ux = this
```
```   315         obtain m :: nat where m: "p < real m" using reals_Archimedean2 ..
```
```   316         have "p \<le> r"
```
```   317         proof (rule ccontr)
```
```   318           assume "\<not> p \<le> r"
```
```   319           with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
```
```   320           obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: inverse_eq_divide field_simps)
```
```   321           then have "r * 2^max N m < p * 2^max N m - 1" by simp
```
```   322           moreover
```
```   323           have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
```
```   324             using *[of "max N m"] m unfolding real_f using ux
```
```   325             by (cases "0 \<le> u x") (simp_all add: max_def mult_nonneg_nonneg split: split_if_asm)
```
```   326           then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
```
```   327             by (metis real_natfloor_gt_diff_one less_le_trans)
```
```   328           ultimately show False by auto
```
```   329         qed
```
```   330         then show "max 0 (u x) \<le> y" using real ux by simp
```
```   331       qed (insert `0 \<le> y`, auto)
```
```   332     qed
```
```   333   qed (auto simp: divide_nonneg_pos)
```
```   334 qed
```
```   335
```
```   336 lemma borel_measurable_implies_simple_function_sequence':
```
```   337   fixes u :: "'a \<Rightarrow> ereal"
```
```   338   assumes u: "u \<in> borel_measurable M"
```
```   339   obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
```
```   340     "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
```
```   341   using borel_measurable_implies_simple_function_sequence[OF u] by auto
```
```   342
```
```   343 lemma simple_function_induct[consumes 1, case_names cong set mult add, induct set: simple_function]:
```
```   344   fixes u :: "'a \<Rightarrow> ereal"
```
```   345   assumes u: "simple_function M u"
```
```   346   assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
```
```   347   assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
```
```   348   assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)"
```
```   349   assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
```
```   350   shows "P u"
```
```   351 proof (rule cong)
```
```   352   from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x"
```
```   353   proof eventually_elim
```
```   354     fix x assume x: "x \<in> space M"
```
```   355     from simple_function_indicator_representation[OF u x]
```
```   356     show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
```
```   357   qed
```
```   358 next
```
```   359   from u have "finite (u ` space M)"
```
```   360     unfolding simple_function_def by auto
```
```   361   then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
```
```   362   proof induct
```
```   363     case empty show ?case
```
```   364       using set[of "{}"] by (simp add: indicator_def[abs_def])
```
```   365   qed (auto intro!: add mult set simple_functionD u)
```
```   366 next
```
```   367   show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
```
```   368     apply (subst simple_function_cong)
```
```   369     apply (rule simple_function_indicator_representation[symmetric])
```
```   370     apply (auto intro: u)
```
```   371     done
```
```   372 qed fact
```
```   373
```
```   374 lemma simple_function_induct_nn[consumes 2, case_names cong set mult add]:
```
```   375   fixes u :: "'a \<Rightarrow> ereal"
```
```   376   assumes u: "simple_function M u" and nn: "\<And>x. 0 \<le> u x"
```
```   377   assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
```
```   378   assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
```
```   379   assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
```
```   380   assumes add: "\<And>u v. simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> simple_function M v \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
```
```   381   shows "P u"
```
```   382 proof -
```
```   383   show ?thesis
```
```   384   proof (rule cong)
```
```   385     fix x assume x: "x \<in> space M"
```
```   386     from simple_function_indicator_representation[OF u x]
```
```   387     show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
```
```   388   next
```
```   389     show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
```
```   390       apply (subst simple_function_cong)
```
```   391       apply (rule simple_function_indicator_representation[symmetric])
```
```   392       apply (auto intro: u)
```
```   393       done
```
```   394   next
```
```   395     from u nn have "finite (u ` space M)" "\<And>x. x \<in> u ` space M \<Longrightarrow> 0 \<le> x"
```
```   396       unfolding simple_function_def by auto
```
```   397     then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
```
```   398     proof induct
```
```   399       case empty show ?case
```
```   400         using set[of "{}"] by (simp add: indicator_def[abs_def])
```
```   401     qed (auto intro!: add mult set simple_functionD u setsum_nonneg
```
```   402        simple_function_setsum)
```
```   403   qed fact
```
```   404 qed
```
```   405
```
```   406 lemma borel_measurable_induct[consumes 2, case_names cong set mult add seq, induct set: borel_measurable]:
```
```   407   fixes u :: "'a \<Rightarrow> ereal"
```
```   408   assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
```
```   409   assumes cong: "\<And>f g. f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P g \<Longrightarrow> P f"
```
```   410   assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
```
```   411   assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
```
```   412   assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
```
```   413   assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow>  (\<And>i x. 0 \<le> U i x) \<Longrightarrow>  (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> P (SUP i. U i)"
```
```   414   shows "P u"
```
```   415   using u
```
```   416 proof (induct rule: borel_measurable_implies_simple_function_sequence')
```
```   417   fix U assume U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and
```
```   418     sup: "\<And>x. (SUP i. U i x) = max 0 (u x)" and nn: "\<And>i x. 0 \<le> U i x"
```
```   419   have u_eq: "u = (SUP i. U i)"
```
```   420     using nn u sup by (auto simp: max_def)
```
```   421
```
```   422   from U have "\<And>i. U i \<in> borel_measurable M"
```
```   423     by (simp add: borel_measurable_simple_function)
```
```   424
```
```   425   show "P u"
```
```   426     unfolding u_eq
```
```   427   proof (rule seq)
```
```   428     fix i show "P (U i)"
```
```   429       using `simple_function M (U i)` nn
```
```   430       by (induct rule: simple_function_induct_nn)
```
```   431          (auto intro: set mult add cong dest!: borel_measurable_simple_function)
```
```   432   qed fact+
```
```   433 qed
```
```   434
```
```   435 lemma simple_function_If_set:
```
```   436   assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
```
```   437   shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
```
```   438 proof -
```
```   439   def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
```
```   440   show ?thesis unfolding simple_function_def
```
```   441   proof safe
```
```   442     have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
```
```   443     from finite_subset[OF this] assms
```
```   444     show "finite (?IF ` space M)" unfolding simple_function_def by auto
```
```   445   next
```
```   446     fix x assume "x \<in> space M"
```
```   447     then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
```
```   448       then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
```
```   449       else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
```
```   450       using sets.sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
```
```   451     have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
```
```   452       unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
```
```   453     show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
```
```   454   qed
```
```   455 qed
```
```   456
```
```   457 lemma simple_function_If:
```
```   458   assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
```
```   459   shows "simple_function M (\<lambda>x. if P x then f x else g x)"
```
```   460 proof -
```
```   461   have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
```
```   462   with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
```
```   463 qed
```
```   464
```
```   465 lemma simple_function_subalgebra:
```
```   466   assumes "simple_function N f"
```
```   467   and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
```
```   468   shows "simple_function M f"
```
```   469   using assms unfolding simple_function_def by auto
```
```   470
```
```   471 lemma simple_function_comp:
```
```   472   assumes T: "T \<in> measurable M M'"
```
```   473     and f: "simple_function M' f"
```
```   474   shows "simple_function M (\<lambda>x. f (T x))"
```
```   475 proof (intro simple_function_def[THEN iffD2] conjI ballI)
```
```   476   have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
```
```   477     using T unfolding measurable_def by auto
```
```   478   then show "finite ((\<lambda>x. f (T x)) ` space M)"
```
```   479     using f unfolding simple_function_def by (auto intro: finite_subset)
```
```   480   fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
```
```   481   then have "i \<in> f ` space M'"
```
```   482     using T unfolding measurable_def by auto
```
```   483   then have "f -` {i} \<inter> space M' \<in> sets M'"
```
```   484     using f unfolding simple_function_def by auto
```
```   485   then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
```
```   486     using T unfolding measurable_def by auto
```
```   487   also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
```
```   488     using T unfolding measurable_def by auto
```
```   489   finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
```
```   490 qed
```
```   491
```
```   492 section "Simple integral"
```
```   493
```
```   494 definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>S") where
```
```   495   "integral\<^sup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
```
```   496
```
```   497 syntax
```
```   498   "_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>S _. _ \<partial>_" [60,61] 110)
```
```   499
```
```   500 translations
```
```   501   "\<integral>\<^sup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)"
```
```   502
```
```   503 lemma simple_integral_cong:
```
```   504   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
```
```   505   shows "integral\<^sup>S M f = integral\<^sup>S M g"
```
```   506 proof -
```
```   507   have "f ` space M = g ` space M"
```
```   508     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
```
```   509     using assms by (auto intro!: image_eqI)
```
```   510   thus ?thesis unfolding simple_integral_def by simp
```
```   511 qed
```
```   512
```
```   513 lemma simple_integral_const[simp]:
```
```   514   "(\<integral>\<^sup>Sx. c \<partial>M) = c * (emeasure M) (space M)"
```
```   515 proof (cases "space M = {}")
```
```   516   case True thus ?thesis unfolding simple_integral_def by simp
```
```   517 next
```
```   518   case False hence "(\<lambda>x. c) ` space M = {c}" by auto
```
```   519   thus ?thesis unfolding simple_integral_def by simp
```
```   520 qed
```
```   521
```
```   522 lemma simple_function_partition:
```
```   523   assumes f: "simple_function M f" and g: "simple_function M g"
```
```   524   shows "integral\<^sup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * (emeasure M) A)"
```
```   525     (is "_ = setsum _ (?p ` space M)")
```
```   526 proof-
```
```   527   let ?sub = "\<lambda>x. ?p ` (f -` {x} \<inter> space M)"
```
```   528   let ?SIGMA = "Sigma (f`space M) ?sub"
```
```   529
```
```   530   have [intro]:
```
```   531     "finite (f ` space M)"
```
```   532     "finite (g ` space M)"
```
```   533     using assms unfolding simple_function_def by simp_all
```
```   534
```
```   535   { fix A
```
```   536     have "?p ` (A \<inter> space M) \<subseteq>
```
```   537       (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
```
```   538       by auto
```
```   539     hence "finite (?p ` (A \<inter> space M))"
```
```   540       by (rule finite_subset) auto }
```
```   541   note this[intro, simp]
```
```   542   note sets = simple_function_measurable2[OF f g]
```
```   543
```
```   544   { fix x assume "x \<in> space M"
```
```   545     have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
```
```   546     with sets have "(emeasure M) (f -` {f x} \<inter> space M) = setsum (emeasure M) (?sub (f x))"
```
```   547       by (subst setsum_emeasure) (auto simp: disjoint_family_on_def) }
```
```   548   hence "integral\<^sup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * (emeasure M) A)"
```
```   549     unfolding simple_integral_def using f sets
```
```   550     by (subst setsum_Sigma[symmetric])
```
```   551        (auto intro!: setsum_cong setsum_ereal_right_distrib)
```
```   552   also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * (emeasure M) A)"
```
```   553   proof -
```
```   554     have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
```
```   555     have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
```
```   556       = (\<lambda>x. (f x, ?p x)) ` space M"
```
```   557     proof safe
```
```   558       fix x assume "x \<in> space M"
```
```   559       thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
```
```   560         by (auto intro!: image_eqI[of _ _ "?p x"])
```
```   561     qed auto
```
```   562     thus ?thesis
```
```   563       apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
```
```   564       apply (rule_tac x="xa" in image_eqI)
```
```   565       by simp_all
```
```   566   qed
```
```   567   finally show ?thesis .
```
```   568 qed
```
```   569
```
```   570 lemma simple_integral_add[simp]:
```
```   571   assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
```
```   572   shows "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = integral\<^sup>S M f + integral\<^sup>S M g"
```
```   573 proof -
```
```   574   { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
```
```   575     assume "x \<in> space M"
```
```   576     hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
```
```   577         "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
```
```   578       by auto }
```
```   579   with assms show ?thesis
```
```   580     unfolding
```
```   581       simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
```
```   582       simple_function_partition[OF f g]
```
```   583       simple_function_partition[OF g f]
```
```   584     by (subst (3) Int_commute)
```
```   585        (auto simp add: ereal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
```
```   586 qed
```
```   587
```
```   588 lemma simple_integral_setsum[simp]:
```
```   589   assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
```
```   590   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
```
```   591   shows "(\<integral>\<^sup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>S M (f i))"
```
```   592 proof cases
```
```   593   assume "finite P"
```
```   594   from this assms show ?thesis
```
```   595     by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
```
```   596 qed auto
```
```   597
```
```   598 lemma simple_integral_mult[simp]:
```
```   599   assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
```
```   600   shows "(\<integral>\<^sup>Sx. c * f x \<partial>M) = c * integral\<^sup>S M f"
```
```   601 proof -
```
```   602   note mult = simple_function_mult[OF simple_function_const[of _ c] f(1)]
```
```   603   { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
```
```   604     assume "x \<in> space M"
```
```   605     hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
```
```   606       by auto }
```
```   607   with assms show ?thesis
```
```   608     unfolding simple_function_partition[OF mult f(1)]
```
```   609               simple_function_partition[OF f(1) mult]
```
```   610     by (subst setsum_ereal_right_distrib)
```
```   611        (auto intro!: ereal_0_le_mult setsum_cong simp: mult_assoc)
```
```   612 qed
```
```   613
```
```   614 lemma simple_integral_mono_AE:
```
```   615   assumes f: "simple_function M f" and g: "simple_function M g"
```
```   616   and mono: "AE x in M. f x \<le> g x"
```
```   617   shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
```
```   618 proof -
```
```   619   let ?S = "\<lambda>x. (g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
```
```   620   have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
```
```   621     "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
```
```   622   show ?thesis
```
```   623     unfolding *
```
```   624       simple_function_partition[OF f g]
```
```   625       simple_function_partition[OF g f]
```
```   626   proof (safe intro!: setsum_mono)
```
```   627     fix x assume "x \<in> space M"
```
```   628     then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
```
```   629     show "the_elem (f`?S x) * (emeasure M) (?S x) \<le> the_elem (g`?S x) * (emeasure M) (?S x)"
```
```   630     proof (cases "f x \<le> g x")
```
```   631       case True then show ?thesis
```
```   632         using * assms(1,2)[THEN simple_functionD(2)]
```
```   633         by (auto intro!: ereal_mult_right_mono)
```
```   634     next
```
```   635       case False
```
```   636       obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "(emeasure M) N = 0"
```
```   637         using mono by (auto elim!: AE_E)
```
```   638       have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
```
```   639       moreover have "?S x \<in> sets M" using assms
```
```   640         by (rule_tac sets.Int) (auto intro!: simple_functionD)
```
```   641       ultimately have "(emeasure M) (?S x) \<le> (emeasure M) N"
```
```   642         using `N \<in> sets M` by (auto intro!: emeasure_mono)
```
```   643       moreover have "0 \<le> (emeasure M) (?S x)"
```
```   644         using assms(1,2)[THEN simple_functionD(2)] by auto
```
```   645       ultimately have "(emeasure M) (?S x) = 0" using `(emeasure M) N = 0` by auto
```
```   646       then show ?thesis by simp
```
```   647     qed
```
```   648   qed
```
```   649 qed
```
```   650
```
```   651 lemma simple_integral_mono:
```
```   652   assumes "simple_function M f" and "simple_function M g"
```
```   653   and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
```
```   654   shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
```
```   655   using assms by (intro simple_integral_mono_AE) auto
```
```   656
```
```   657 lemma simple_integral_cong_AE:
```
```   658   assumes "simple_function M f" and "simple_function M g"
```
```   659   and "AE x in M. f x = g x"
```
```   660   shows "integral\<^sup>S M f = integral\<^sup>S M g"
```
```   661   using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
```
```   662
```
```   663 lemma simple_integral_cong':
```
```   664   assumes sf: "simple_function M f" "simple_function M g"
```
```   665   and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0"
```
```   666   shows "integral\<^sup>S M f = integral\<^sup>S M g"
```
```   667 proof (intro simple_integral_cong_AE sf AE_I)
```
```   668   show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact
```
```   669   show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
```
```   670     using sf[THEN borel_measurable_simple_function] by auto
```
```   671 qed simp
```
```   672
```
```   673 lemma simple_integral_indicator:
```
```   674   assumes "A \<in> sets M"
```
```   675   assumes f: "simple_function M f"
```
```   676   shows "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
```
```   677     (\<Sum>x \<in> f ` space M. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
```
```   678 proof (cases "A = space M")
```
```   679   case True
```
```   680   then have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) = integral\<^sup>S M f"
```
```   681     by (auto intro!: simple_integral_cong)
```
```   682   with True show ?thesis by (simp add: simple_integral_def)
```
```   683 next
```
```   684   assume "A \<noteq> space M"
```
```   685   then obtain x where x: "x \<in> space M" "x \<notin> A" using sets.sets_into_space[OF assms(1)] by auto
```
```   686   have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
```
```   687   proof safe
```
```   688     fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
```
```   689   next
```
```   690     fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
```
```   691       using sets.sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
```
```   692   next
```
```   693     show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
```
```   694   qed
```
```   695   have *: "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
```
```   696     (\<Sum>x \<in> f ` space M \<union> {0}. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
```
```   697     unfolding simple_integral_def I
```
```   698   proof (rule setsum_mono_zero_cong_left)
```
```   699     show "finite (f ` space M \<union> {0})"
```
```   700       using assms(2) unfolding simple_function_def by auto
```
```   701     show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
```
```   702       using sets.sets_into_space[OF assms(1)] by auto
```
```   703     have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
```
```   704       by (auto simp: image_iff)
```
```   705     thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
```
```   706       i * (emeasure M) (f -` {i} \<inter> space M \<inter> A) = 0" by auto
```
```   707   next
```
```   708     fix x assume "x \<in> f`A \<union> {0}"
```
```   709     hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
```
```   710       by (auto simp: indicator_def split: split_if_asm)
```
```   711     thus "x * (emeasure M) (?I -` {x} \<inter> space M) =
```
```   712       x * (emeasure M) (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
```
```   713   qed
```
```   714   show ?thesis unfolding *
```
```   715     using assms(2) unfolding simple_function_def
```
```   716     by (auto intro!: setsum_mono_zero_cong_right)
```
```   717 qed
```
```   718
```
```   719 lemma simple_integral_indicator_only[simp]:
```
```   720   assumes "A \<in> sets M"
```
```   721   shows "integral\<^sup>S M (indicator A) = emeasure M A"
```
```   722 proof cases
```
```   723   assume "space M = {}" hence "A = {}" using sets.sets_into_space[OF assms] by auto
```
```   724   thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
```
```   725 next
```
```   726   assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::ereal}" by auto
```
```   727   thus ?thesis
```
```   728     using simple_integral_indicator[OF assms simple_function_const[of _ 1]]
```
```   729     using sets.sets_into_space[OF assms]
```
```   730     by (auto intro!: arg_cong[where f="(emeasure M)"])
```
```   731 qed
```
```   732
```
```   733 lemma simple_integral_null_set:
```
```   734   assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets M"
```
```   735   shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0"
```
```   736 proof -
```
```   737   have "AE x in M. indicator N x = (0 :: ereal)"
```
```   738     using `N \<in> null_sets M` by (auto simp: indicator_def intro!: AE_I[of _ _ N])
```
```   739   then have "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^sup>Sx. 0 \<partial>M)"
```
```   740     using assms apply (intro simple_integral_cong_AE) by auto
```
```   741   then show ?thesis by simp
```
```   742 qed
```
```   743
```
```   744 lemma simple_integral_cong_AE_mult_indicator:
```
```   745   assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<in> sets M"
```
```   746   shows "integral\<^sup>S M f = (\<integral>\<^sup>Sx. f x * indicator S x \<partial>M)"
```
```   747   using assms by (intro simple_integral_cong_AE) auto
```
```   748
```
```   749 lemma simple_integral_cmult_indicator:
```
```   750   assumes A: "A \<in> sets M"
```
```   751   shows "(\<integral>\<^sup>Sx. c * indicator A x \<partial>M) = c * (emeasure M) A"
```
```   752   using simple_integral_mult[OF simple_function_indicator[OF A]]
```
```   753   unfolding simple_integral_indicator_only[OF A] by simp
```
```   754
```
```   755 lemma simple_integral_positive:
```
```   756   assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x"
```
```   757   shows "0 \<le> integral\<^sup>S M f"
```
```   758 proof -
```
```   759   have "integral\<^sup>S M (\<lambda>x. 0) \<le> integral\<^sup>S M f"
```
```   760     using simple_integral_mono_AE[OF _ f ae] by auto
```
```   761   then show ?thesis by simp
```
```   762 qed
```
```   763
```
```   764 section "Continuous positive integration"
```
```   765
```
```   766 definition positive_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>P") where
```
```   767   "integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^sup>S M g)"
```
```   768
```
```   769 syntax
```
```   770   "_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>+ _. _ \<partial>_" [60,61] 110)
```
```   771
```
```   772 translations
```
```   773   "\<integral>\<^sup>+ x. f \<partial>M" == "CONST positive_integral M (%x. f)"
```
```   774
```
```   775 lemma positive_integral_positive:
```
```   776   "0 \<le> integral\<^sup>P M f"
```
```   777   by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
```
```   778
```
```   779 lemma positive_integral_not_MInfty[simp]: "integral\<^sup>P M f \<noteq> -\<infinity>"
```
```   780   using positive_integral_positive[of M f] by auto
```
```   781
```
```   782 lemma positive_integral_def_finite:
```
```   783   "integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^sup>S M g)"
```
```   784     (is "_ = SUPR ?A ?f")
```
```   785   unfolding positive_integral_def
```
```   786 proof (safe intro!: antisym SUP_least)
```
```   787   fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
```
```   788   let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
```
```   789   note gM = g(1)[THEN borel_measurable_simple_function]
```
```   790   have \<mu>_G_pos: "0 \<le> (emeasure M) ?G" using gM by auto
```
```   791   let ?g = "\<lambda>y x. if g x = \<infinity> then y else max 0 (g x)"
```
```   792   from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
```
```   793     apply (safe intro!: simple_function_max simple_function_If)
```
```   794     apply (force simp: max_def le_fun_def split: split_if_asm)+
```
```   795     done
```
```   796   show "integral\<^sup>S M g \<le> SUPR ?A ?f"
```
```   797   proof cases
```
```   798     have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
```
```   799     assume "(emeasure M) ?G = 0"
```
```   800     with gM have "AE x in M. x \<notin> ?G"
```
```   801       by (auto simp add: AE_iff_null intro!: null_setsI)
```
```   802     with gM g show ?thesis
```
```   803       by (intro SUP_upper2[OF g0] simple_integral_mono_AE)
```
```   804          (auto simp: max_def intro!: simple_function_If)
```
```   805   next
```
```   806     assume \<mu>_G: "(emeasure M) ?G \<noteq> 0"
```
```   807     have "SUPR ?A (integral\<^sup>S M) = \<infinity>"
```
```   808     proof (intro SUP_PInfty)
```
```   809       fix n :: nat
```
```   810       let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)"
```
```   811       have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>_G \<mu>_G_pos by (auto simp: ereal_divide_eq)
```
```   812       then have "?g ?y \<in> ?A" by (rule g_in_A)
```
```   813       have "real n \<le> ?y * (emeasure M) ?G"
```
```   814         using \<mu>_G \<mu>_G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps)
```
```   815       also have "\<dots> = (\<integral>\<^sup>Sx. ?y * indicator ?G x \<partial>M)"
```
```   816         using `0 \<le> ?y` `?g ?y \<in> ?A` gM
```
```   817         by (subst simple_integral_cmult_indicator) auto
```
```   818       also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
```
```   819         by (intro simple_integral_mono) auto
```
```   820       finally show "\<exists>i\<in>?A. real n \<le> integral\<^sup>S M i"
```
```   821         using `?g ?y \<in> ?A` by blast
```
```   822     qed
```
```   823     then show ?thesis by simp
```
```   824   qed
```
```   825 qed (auto intro: SUP_upper)
```
```   826
```
```   827 lemma positive_integral_mono_AE:
```
```   828   assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^sup>P M u \<le> integral\<^sup>P M v"
```
```   829   unfolding positive_integral_def
```
```   830 proof (safe intro!: SUP_mono)
```
```   831   fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
```
```   832   from ae[THEN AE_E] guess N . note N = this
```
```   833   then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in)
```
```   834   let ?n = "\<lambda>x. n x * indicator (space M - N) x"
```
```   835   have "AE x in M. n x \<le> ?n x" "simple_function M ?n"
```
```   836     using n N ae_N by auto
```
```   837   moreover
```
```   838   { fix x have "?n x \<le> max 0 (v x)"
```
```   839     proof cases
```
```   840       assume x: "x \<in> space M - N"
```
```   841       with N have "u x \<le> v x" by auto
```
```   842       with n(2)[THEN le_funD, of x] x show ?thesis
```
```   843         by (auto simp: max_def split: split_if_asm)
```
```   844     qed simp }
```
```   845   then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
```
```   846   moreover have "integral\<^sup>S M n \<le> integral\<^sup>S M ?n"
```
```   847     using ae_N N n by (auto intro!: simple_integral_mono_AE)
```
```   848   ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^sup>S M n \<le> integral\<^sup>S M m"
```
```   849     by force
```
```   850 qed
```
```   851
```
```   852 lemma positive_integral_mono:
```
```   853   "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^sup>P M u \<le> integral\<^sup>P M v"
```
```   854   by (auto intro: positive_integral_mono_AE)
```
```   855
```
```   856 lemma positive_integral_cong_AE:
```
```   857   "AE x in M. u x = v x \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
```
```   858   by (auto simp: eq_iff intro!: positive_integral_mono_AE)
```
```   859
```
```   860 lemma positive_integral_cong:
```
```   861   "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
```
```   862   by (auto intro: positive_integral_cong_AE)
```
```   863
```
```   864 lemma positive_integral_eq_simple_integral:
```
```   865   assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^sup>P M f = integral\<^sup>S M f"
```
```   866 proof -
```
```   867   let ?f = "\<lambda>x. f x * indicator (space M) x"
```
```   868   have f': "simple_function M ?f" using f by auto
```
```   869   with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
```
```   870     by (auto simp: fun_eq_iff max_def split: split_indicator)
```
```   871   have "integral\<^sup>P M ?f \<le> integral\<^sup>S M ?f" using f'
```
```   872     by (force intro!: SUP_least simple_integral_mono simp: le_fun_def positive_integral_def)
```
```   873   moreover have "integral\<^sup>S M ?f \<le> integral\<^sup>P M ?f"
```
```   874     unfolding positive_integral_def
```
```   875     using f' by (auto intro!: SUP_upper)
```
```   876   ultimately show ?thesis
```
```   877     by (simp cong: positive_integral_cong simple_integral_cong)
```
```   878 qed
```
```   879
```
```   880 lemma positive_integral_eq_simple_integral_AE:
```
```   881   assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^sup>P M f = integral\<^sup>S M f"
```
```   882 proof -
```
```   883   have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max)
```
```   884   with f have "integral\<^sup>P M f = integral\<^sup>S M (\<lambda>x. max 0 (f x))"
```
```   885     by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
```
```   886              add: positive_integral_eq_simple_integral)
```
```   887   with assms show ?thesis
```
```   888     by (auto intro!: simple_integral_cong_AE split: split_max)
```
```   889 qed
```
```   890
```
```   891 lemma positive_integral_SUP_approx:
```
```   892   assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
```
```   893   and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
```
```   894   shows "integral\<^sup>S M u \<le> (SUP i. integral\<^sup>P M (f i))" (is "_ \<le> ?S")
```
```   895 proof (rule ereal_le_mult_one_interval)
```
```   896   have "0 \<le> (SUP i. integral\<^sup>P M (f i))"
```
```   897     using f(3) by (auto intro!: SUP_upper2 positive_integral_positive)
```
```   898   then show "(SUP i. integral\<^sup>P M (f i)) \<noteq> -\<infinity>" by auto
```
```   899   have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
```
```   900     using u(3) by auto
```
```   901   fix a :: ereal assume "0 < a" "a < 1"
```
```   902   hence "a \<noteq> 0" by auto
```
```   903   let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}"
```
```   904   have B: "\<And>i. ?B i \<in> sets M"
```
```   905     using f `simple_function M u` by auto
```
```   906
```
```   907   let ?uB = "\<lambda>i x. u x * indicator (?B i) x"
```
```   908
```
```   909   { fix i have "?B i \<subseteq> ?B (Suc i)"
```
```   910     proof safe
```
```   911       fix i x assume "a * u x \<le> f i x"
```
```   912       also have "\<dots> \<le> f (Suc i) x"
```
```   913         using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
```
```   914       finally show "a * u x \<le> f (Suc i) x" .
```
```   915     qed }
```
```   916   note B_mono = this
```
```   917
```
```   918   note B_u = sets.Int[OF u(1)[THEN simple_functionD(2)] B]
```
```   919
```
```   920   let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n"
```
```   921   have measure_conv: "\<And>i. (emeasure M) (u -` {i} \<inter> space M) = (SUP n. (emeasure M) (?B' i n))"
```
```   922   proof -
```
```   923     fix i
```
```   924     have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
```
```   925     have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
```
```   926     have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
```
```   927     proof safe
```
```   928       fix x i assume x: "x \<in> space M"
```
```   929       show "x \<in> (\<Union>i. ?B' (u x) i)"
```
```   930       proof cases
```
```   931         assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
```
```   932       next
```
```   933         assume "u x \<noteq> 0"
```
```   934         with `a < 1` u_range[OF `x \<in> space M`]
```
```   935         have "a * u x < 1 * u x"
```
```   936           by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
```
```   937         also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def)
```
```   938         finally obtain i where "a * u x < f i x" unfolding SUP_def
```
```   939           by (auto simp add: less_SUP_iff)
```
```   940         hence "a * u x \<le> f i x" by auto
```
```   941         thus ?thesis using `x \<in> space M` by auto
```
```   942       qed
```
```   943     qed
```
```   944     then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp
```
```   945   qed
```
```   946
```
```   947   have "integral\<^sup>S M u = (SUP i. integral\<^sup>S M (?uB i))"
```
```   948     unfolding simple_integral_indicator[OF B `simple_function M u`]
```
```   949   proof (subst SUPR_ereal_setsum, safe)
```
```   950     fix x n assume "x \<in> space M"
```
```   951     with u_range show "incseq (\<lambda>i. u x * (emeasure M) (?B' (u x) i))" "\<And>i. 0 \<le> u x * (emeasure M) (?B' (u x) i)"
```
```   952       using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
```
```   953   next
```
```   954     show "integral\<^sup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))"
```
```   955       using measure_conv u_range B_u unfolding simple_integral_def
```
```   956       by (auto intro!: setsum_cong SUPR_ereal_cmult[symmetric])
```
```   957   qed
```
```   958   moreover
```
```   959   have "a * (SUP i. integral\<^sup>S M (?uB i)) \<le> ?S"
```
```   960     apply (subst SUPR_ereal_cmult[symmetric])
```
```   961   proof (safe intro!: SUP_mono bexI)
```
```   962     fix i
```
```   963     have "a * integral\<^sup>S M (?uB i) = (\<integral>\<^sup>Sx. a * ?uB i x \<partial>M)"
```
```   964       using B `simple_function M u` u_range
```
```   965       by (subst simple_integral_mult) (auto split: split_indicator)
```
```   966     also have "\<dots> \<le> integral\<^sup>P M (f i)"
```
```   967     proof -
```
```   968       have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
```
```   969       show ?thesis using f(3) * u_range `0 < a`
```
```   970         by (subst positive_integral_eq_simple_integral[symmetric])
```
```   971            (auto intro!: positive_integral_mono split: split_indicator)
```
```   972     qed
```
```   973     finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>P M (f i)"
```
```   974       by auto
```
```   975   next
```
```   976     fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
```
```   977       by (intro simple_integral_positive) (auto split: split_indicator)
```
```   978   qed (insert `0 < a`, auto)
```
```   979   ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp
```
```   980 qed
```
```   981
```
```   982 lemma incseq_positive_integral:
```
```   983   assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>P M (f i))"
```
```   984 proof -
```
```   985   have "\<And>i x. f i x \<le> f (Suc i) x"
```
```   986     using assms by (auto dest!: incseq_SucD simp: le_fun_def)
```
```   987   then show ?thesis
```
```   988     by (auto intro!: incseq_SucI positive_integral_mono)
```
```   989 qed
```
```   990
```
```   991 text {* Beppo-Levi monotone convergence theorem *}
```
```   992 lemma positive_integral_monotone_convergence_SUP:
```
```   993   assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
```
```   994   shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
```
```   995 proof (rule antisym)
```
```   996   show "(SUP j. integral\<^sup>P M (f j)) \<le> (\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M)"
```
```   997     by (auto intro!: SUP_least SUP_upper positive_integral_mono)
```
```   998 next
```
```   999   show "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^sup>P M (f j))"
```
```  1000     unfolding positive_integral_def_finite[of _ "\<lambda>x. SUP i. f i x"]
```
```  1001   proof (safe intro!: SUP_least)
```
```  1002     fix g assume g: "simple_function M g"
```
```  1003       and *: "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
```
```  1004     then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
```
```  1005       using f by (auto intro!: SUP_upper2)
```
```  1006     with * show "integral\<^sup>S M g \<le> (SUP j. integral\<^sup>P M (f j))"
```
```  1007       by (intro  positive_integral_SUP_approx[OF f g _ g'])
```
```  1008          (auto simp: le_fun_def max_def)
```
```  1009   qed
```
```  1010 qed
```
```  1011
```
```  1012 lemma positive_integral_monotone_convergence_SUP_AE:
```
```  1013   assumes f: "\<And>i. AE x in M. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
```
```  1014   shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
```
```  1015 proof -
```
```  1016   from f have "AE x in M. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
```
```  1017     by (simp add: AE_all_countable)
```
```  1018   from this[THEN AE_E] guess N . note N = this
```
```  1019   let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0"
```
```  1020   have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N])
```
```  1021   then have "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. ?f i x) \<partial>M)"
```
```  1022     by (auto intro!: positive_integral_cong_AE)
```
```  1023   also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>M))"
```
```  1024   proof (rule positive_integral_monotone_convergence_SUP)
```
```  1025     show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
```
```  1026     { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
```
```  1027         using f N(3) by (intro measurable_If_set) auto
```
```  1028       fix x show "0 \<le> ?f i x"
```
```  1029         using N(1) by auto }
```
```  1030   qed
```
```  1031   also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. f i x \<partial>M))"
```
```  1032     using f_eq by (force intro!: arg_cong[where f="SUPR UNIV"] positive_integral_cong_AE ext)
```
```  1033   finally show ?thesis .
```
```  1034 qed
```
```  1035
```
```  1036 lemma positive_integral_monotone_convergence_SUP_AE_incseq:
```
```  1037   assumes f: "incseq f" "\<And>i. AE x in M. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
```
```  1038   shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
```
```  1039   using f[unfolded incseq_Suc_iff le_fun_def]
```
```  1040   by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
```
```  1041      auto
```
```  1042
```
```  1043 lemma positive_integral_monotone_convergence_simple:
```
```  1044   assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
```
```  1045   shows "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
```
```  1046   using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
```
```  1047     f(3)[THEN borel_measurable_simple_function] f(2)]
```
```  1048   by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
```
```  1049
```
```  1050 lemma positive_integral_max_0:
```
```  1051   "(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>P M f"
```
```  1052   by (simp add: le_fun_def positive_integral_def)
```
```  1053
```
```  1054 lemma positive_integral_cong_pos:
```
```  1055   assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
```
```  1056   shows "integral\<^sup>P M f = integral\<^sup>P M g"
```
```  1057 proof -
```
```  1058   have "integral\<^sup>P M (\<lambda>x. max 0 (f x)) = integral\<^sup>P M (\<lambda>x. max 0 (g x))"
```
```  1059   proof (intro positive_integral_cong)
```
```  1060     fix x assume "x \<in> space M"
```
```  1061     from assms[OF this] show "max 0 (f x) = max 0 (g x)"
```
```  1062       by (auto split: split_max)
```
```  1063   qed
```
```  1064   then show ?thesis by (simp add: positive_integral_max_0)
```
```  1065 qed
```
```  1066
```
```  1067 lemma SUP_simple_integral_sequences:
```
```  1068   assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
```
```  1069   and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
```
```  1070   and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
```
```  1071   shows "(SUP i. integral\<^sup>S M (f i)) = (SUP i. integral\<^sup>S M (g i))"
```
```  1072     (is "SUPR _ ?F = SUPR _ ?G")
```
```  1073 proof -
```
```  1074   have "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
```
```  1075     using f by (rule positive_integral_monotone_convergence_simple)
```
```  1076   also have "\<dots> = (\<integral>\<^sup>+x. (SUP i. g i x) \<partial>M)"
```
```  1077     unfolding eq[THEN positive_integral_cong_AE] ..
```
```  1078   also have "\<dots> = (SUP i. ?G i)"
```
```  1079     using g by (rule positive_integral_monotone_convergence_simple[symmetric])
```
```  1080   finally show ?thesis by simp
```
```  1081 qed
```
```  1082
```
```  1083 lemma positive_integral_const[simp]:
```
```  1084   "0 \<le> c \<Longrightarrow> (\<integral>\<^sup>+ x. c \<partial>M) = c * (emeasure M) (space M)"
```
```  1085   by (subst positive_integral_eq_simple_integral) auto
```
```  1086
```
```  1087 lemma positive_integral_linear:
```
```  1088   assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
```
```  1089   and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
```
```  1090   shows "(\<integral>\<^sup>+ x. a * f x + g x \<partial>M) = a * integral\<^sup>P M f + integral\<^sup>P M g"
```
```  1091     (is "integral\<^sup>P M ?L = _")
```
```  1092 proof -
```
```  1093   from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
```
```  1094   note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
```
```  1095   from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
```
```  1096   note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
```
```  1097   let ?L' = "\<lambda>i x. a * u i x + v i x"
```
```  1098
```
```  1099   have "?L \<in> borel_measurable M" using assms by auto
```
```  1100   from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
```
```  1101   note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
```
```  1102
```
```  1103   have inc: "incseq (\<lambda>i. a * integral\<^sup>S M (u i))" "incseq (\<lambda>i. integral\<^sup>S M (v i))"
```
```  1104     using u v `0 \<le> a`
```
```  1105     by (auto simp: incseq_Suc_iff le_fun_def
```
```  1106              intro!: add_mono ereal_mult_left_mono simple_integral_mono)
```
```  1107   have pos: "\<And>i. 0 \<le> integral\<^sup>S M (u i)" "\<And>i. 0 \<le> integral\<^sup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^sup>S M (u i)"
```
```  1108     using u v `0 \<le> a` by (auto simp: simple_integral_positive)
```
```  1109   { fix i from pos[of i] have "a * integral\<^sup>S M (u i) \<noteq> -\<infinity>" "integral\<^sup>S M (v i) \<noteq> -\<infinity>"
```
```  1110       by (auto split: split_if_asm) }
```
```  1111   note not_MInf = this
```
```  1112
```
```  1113   have l': "(SUP i. integral\<^sup>S M (l i)) = (SUP i. integral\<^sup>S M (?L' i))"
```
```  1114   proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
```
```  1115     show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
```
```  1116       using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
```
```  1117       by (auto intro!: add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg)
```
```  1118     { fix x
```
```  1119       { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
```
```  1120           by auto }
```
```  1121       then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
```
```  1122         using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
```
```  1123         by (subst SUPR_ereal_cmult[symmetric, OF u(6) `0 \<le> a`])
```
```  1124            (auto intro!: SUPR_ereal_add
```
```  1125                  simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg) }
```
```  1126     then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
```
```  1127       unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
```
```  1128       by (intro AE_I2) (auto split: split_max simp add: ereal_add_nonneg_nonneg)
```
```  1129   qed
```
```  1130   also have "\<dots> = (SUP i. a * integral\<^sup>S M (u i) + integral\<^sup>S M (v i))"
```
```  1131     using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPR UNIV"] ext)
```
```  1132   finally have "(\<integral>\<^sup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^sup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+x. max 0 (g x) \<partial>M)"
```
```  1133     unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
```
```  1134     unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
```
```  1135     apply (subst SUPR_ereal_cmult[symmetric, OF pos(1) `0 \<le> a`])
```
```  1136     apply (subst SUPR_ereal_add[symmetric, OF inc not_MInf]) .
```
```  1137   then show ?thesis by (simp add: positive_integral_max_0)
```
```  1138 qed
```
```  1139
```
```  1140 lemma positive_integral_cmult:
```
```  1141   assumes f: "f \<in> borel_measurable M" "0 \<le> c"
```
```  1142   shows "(\<integral>\<^sup>+ x. c * f x \<partial>M) = c * integral\<^sup>P M f"
```
```  1143 proof -
```
```  1144   have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
```
```  1145     by (auto split: split_max simp: ereal_zero_le_0_iff)
```
```  1146   have "(\<integral>\<^sup>+ x. c * f x \<partial>M) = (\<integral>\<^sup>+ x. c * max 0 (f x) \<partial>M)"
```
```  1147     by (simp add: positive_integral_max_0)
```
```  1148   then show ?thesis
```
```  1149     using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
```
```  1150     by (auto simp: positive_integral_max_0)
```
```  1151 qed
```
```  1152
```
```  1153 lemma positive_integral_multc:
```
```  1154   assumes "f \<in> borel_measurable M" "0 \<le> c"
```
```  1155   shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>P M f * c"
```
```  1156   unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
```
```  1157
```
```  1158 lemma positive_integral_indicator[simp]:
```
```  1159   "A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A"
```
```  1160   by (subst positive_integral_eq_simple_integral)
```
```  1161      (auto simp: simple_integral_indicator)
```
```  1162
```
```  1163 lemma positive_integral_cmult_indicator:
```
```  1164   "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A"
```
```  1165   by (subst positive_integral_eq_simple_integral)
```
```  1166      (auto simp: simple_function_indicator simple_integral_indicator)
```
```  1167
```
```  1168 lemma positive_integral_indicator':
```
```  1169   assumes [measurable]: "A \<inter> space M \<in> sets M"
```
```  1170   shows "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = emeasure M (A \<inter> space M)"
```
```  1171 proof -
```
```  1172   have "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = (\<integral>\<^sup>+ x. indicator (A \<inter> space M) x \<partial>M)"
```
```  1173     by (intro positive_integral_cong) (simp split: split_indicator)
```
```  1174   also have "\<dots> = emeasure M (A \<inter> space M)"
```
```  1175     by simp
```
```  1176   finally show ?thesis .
```
```  1177 qed
```
```  1178
```
```  1179 lemma positive_integral_add:
```
```  1180   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
```
```  1181   and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
```
```  1182   shows "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = integral\<^sup>P M f + integral\<^sup>P M g"
```
```  1183 proof -
```
```  1184   have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
```
```  1185     using assms by (auto split: split_max simp: ereal_add_nonneg_nonneg)
```
```  1186   have "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = (\<integral>\<^sup>+ x. max 0 (f x + g x) \<partial>M)"
```
```  1187     by (simp add: positive_integral_max_0)
```
```  1188   also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
```
```  1189     unfolding ae[THEN positive_integral_cong_AE] ..
```
```  1190   also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (g x) \<partial>M)"
```
```  1191     using positive_integral_linear[of "\<lambda>x. max 0 (f x)" _ 1 "\<lambda>x. max 0 (g x)"] f g
```
```  1192     by auto
```
```  1193   finally show ?thesis
```
```  1194     by (simp add: positive_integral_max_0)
```
```  1195 qed
```
```  1196
```
```  1197 lemma positive_integral_setsum:
```
```  1198   assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x in M. 0 \<le> f i x"
```
```  1199   shows "(\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>P M (f i))"
```
```  1200 proof cases
```
```  1201   assume f: "finite P"
```
```  1202   from assms have "AE x in M. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
```
```  1203   from f this assms(1) show ?thesis
```
```  1204   proof induct
```
```  1205     case (insert i P)
```
```  1206     then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x"
```
```  1207       "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x in M. 0 \<le> (\<Sum>i\<in>P. f i x)"
```
```  1208       by (auto intro!: setsum_nonneg)
```
```  1209     from positive_integral_add[OF this]
```
```  1210     show ?case using insert by auto
```
```  1211   qed simp
```
```  1212 qed simp
```
```  1213
```
```  1214 lemma positive_integral_Markov_inequality:
```
```  1215   assumes u: "u \<in> borel_measurable M" "AE x in M. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c"
```
```  1216   shows "(emeasure M) ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
```
```  1217     (is "(emeasure M) ?A \<le> _ * ?PI")
```
```  1218 proof -
```
```  1219   have "?A \<in> sets M"
```
```  1220     using `A \<in> sets M` u by auto
```
```  1221   hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)"
```
```  1222     using positive_integral_indicator by simp
```
```  1223   also have "\<dots> \<le> (\<integral>\<^sup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
```
```  1224     by (auto intro!: positive_integral_mono_AE
```
```  1225       simp: indicator_def ereal_zero_le_0_iff)
```
```  1226   also have "\<dots> = c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
```
```  1227     using assms
```
```  1228     by (auto intro!: positive_integral_cmult simp: ereal_zero_le_0_iff)
```
```  1229   finally show ?thesis .
```
```  1230 qed
```
```  1231
```
```  1232 lemma positive_integral_noteq_infinite:
```
```  1233   assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
```
```  1234   and "integral\<^sup>P M g \<noteq> \<infinity>"
```
```  1235   shows "AE x in M. g x \<noteq> \<infinity>"
```
```  1236 proof (rule ccontr)
```
```  1237   assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)"
```
```  1238   have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0"
```
```  1239     using c g by (auto simp add: AE_iff_null)
```
```  1240   moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
```
```  1241   ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
```
```  1242   then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
```
```  1243   also have "\<dots> \<le> (\<integral>\<^sup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
```
```  1244     using g by (subst positive_integral_cmult_indicator) auto
```
```  1245   also have "\<dots> \<le> integral\<^sup>P M g"
```
```  1246     using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
```
```  1247   finally show False using `integral\<^sup>P M g \<noteq> \<infinity>` by auto
```
```  1248 qed
```
```  1249
```
```  1250 lemma positive_integral_diff:
```
```  1251   assumes f: "f \<in> borel_measurable M"
```
```  1252   and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
```
```  1253   and fin: "integral\<^sup>P M g \<noteq> \<infinity>"
```
```  1254   and mono: "AE x in M. g x \<le> f x"
```
```  1255   shows "(\<integral>\<^sup>+ x. f x - g x \<partial>M) = integral\<^sup>P M f - integral\<^sup>P M g"
```
```  1256 proof -
```
```  1257   have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x"
```
```  1258     using assms by (auto intro: ereal_diff_positive)
```
```  1259   have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto
```
```  1260   { fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
```
```  1261       by (cases rule: ereal2_cases[of a b]) auto }
```
```  1262   note * = this
```
```  1263   then have "AE x in M. f x = f x - g x + g x"
```
```  1264     using mono positive_integral_noteq_infinite[OF g fin] assms by auto
```
```  1265   then have **: "integral\<^sup>P M f = (\<integral>\<^sup>+x. f x - g x \<partial>M) + integral\<^sup>P M g"
```
```  1266     unfolding positive_integral_add[OF diff g, symmetric]
```
```  1267     by (rule positive_integral_cong_AE)
```
```  1268   show ?thesis unfolding **
```
```  1269     using fin positive_integral_positive[of M g]
```
```  1270     by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. f x - g x \<partial>M" "integral\<^sup>P M g"]) auto
```
```  1271 qed
```
```  1272
```
```  1273 lemma positive_integral_suminf:
```
```  1274   assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x"
```
```  1275   shows "(\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>P M (f i))"
```
```  1276 proof -
```
```  1277   have all_pos: "AE x in M. \<forall>i. 0 \<le> f i x"
```
```  1278     using assms by (auto simp: AE_all_countable)
```
```  1279   have "(\<Sum>i. integral\<^sup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>P M (f i))"
```
```  1280     using positive_integral_positive by (rule suminf_ereal_eq_SUPR)
```
```  1281   also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)"
```
```  1282     unfolding positive_integral_setsum[OF f] ..
```
```  1283   also have "\<dots> = \<integral>\<^sup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
```
```  1284     by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
```
```  1285        (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
```
```  1286   also have "\<dots> = \<integral>\<^sup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
```
```  1287     by (intro positive_integral_cong_AE) (auto simp: suminf_ereal_eq_SUPR)
```
```  1288   finally show ?thesis by simp
```
```  1289 qed
```
```  1290
```
```  1291 text {* Fatou's lemma: convergence theorem on limes inferior *}
```
```  1292 lemma positive_integral_lim_INF:
```
```  1293   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1294   assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x"
```
```  1295   shows "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
```
```  1296 proof -
```
```  1297   have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
```
```  1298   have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
```
```  1299     (SUP n. \<integral>\<^sup>+ x. (INF i:{n..}. u i x) \<partial>M)"
```
```  1300     unfolding liminf_SUPR_INFI using pos u
```
```  1301     by (intro positive_integral_monotone_convergence_SUP_AE)
```
```  1302        (elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono)
```
```  1303   also have "\<dots> \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
```
```  1304     unfolding liminf_SUPR_INFI
```
```  1305     by (auto intro!: SUP_mono exI INF_greatest positive_integral_mono INF_lower)
```
```  1306   finally show ?thesis .
```
```  1307 qed
```
```  1308
```
```  1309 lemma positive_integral_null_set:
```
```  1310   assumes "N \<in> null_sets M" shows "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = 0"
```
```  1311 proof -
```
```  1312   have "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
```
```  1313   proof (intro positive_integral_cong_AE AE_I)
```
```  1314     show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
```
```  1315       by (auto simp: indicator_def)
```
```  1316     show "(emeasure M) N = 0" "N \<in> sets M"
```
```  1317       using assms by auto
```
```  1318   qed
```
```  1319   then show ?thesis by simp
```
```  1320 qed
```
```  1321
```
```  1322 lemma positive_integral_0_iff:
```
```  1323   assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x"
```
```  1324   shows "integral\<^sup>P M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0"
```
```  1325     (is "_ \<longleftrightarrow> (emeasure M) ?A = 0")
```
```  1326 proof -
```
```  1327   have u_eq: "(\<integral>\<^sup>+ x. u x * indicator ?A x \<partial>M) = integral\<^sup>P M u"
```
```  1328     by (auto intro!: positive_integral_cong simp: indicator_def)
```
```  1329   show ?thesis
```
```  1330   proof
```
```  1331     assume "(emeasure M) ?A = 0"
```
```  1332     with positive_integral_null_set[of ?A M u] u
```
```  1333     show "integral\<^sup>P M u = 0" by (simp add: u_eq null_sets_def)
```
```  1334   next
```
```  1335     { fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r"
```
```  1336       then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def)
```
```  1337       then have "0 \<le> r" by (auto simp add: ereal_zero_less_0_iff) }
```
```  1338     note gt_1 = this
```
```  1339     assume *: "integral\<^sup>P M u = 0"
```
```  1340     let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}"
```
```  1341     have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))"
```
```  1342     proof -
```
```  1343       { fix n :: nat
```
```  1344         from positive_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"]
```
```  1345         have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
```
```  1346         moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto
```
```  1347         ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto }
```
```  1348       thus ?thesis by simp
```
```  1349     qed
```
```  1350     also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)"
```
```  1351     proof (safe intro!: SUP_emeasure_incseq)
```
```  1352       fix n show "?M n \<inter> ?A \<in> sets M"
```
```  1353         using u by (auto intro!: sets.Int)
```
```  1354     next
```
```  1355       show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
```
```  1356       proof (safe intro!: incseq_SucI)
```
```  1357         fix n :: nat and x
```
```  1358         assume *: "1 \<le> real n * u x"
```
```  1359         also from gt_1[OF *] have "real n * u x \<le> real (Suc n) * u x"
```
```  1360           using `0 \<le> u x` by (auto intro!: ereal_mult_right_mono)
```
```  1361         finally show "1 \<le> real (Suc n) * u x" by auto
```
```  1362       qed
```
```  1363     qed
```
```  1364     also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}"
```
```  1365     proof (safe intro!: arg_cong[where f="(emeasure M)"] dest!: gt_1)
```
```  1366       fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
```
```  1367       show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
```
```  1368       proof (cases "u x")
```
```  1369         case (real r) with `0 < u x` have "0 < r" by auto
```
```  1370         obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
```
```  1371         hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
```
```  1372         hence "1 \<le> real j * r" using real `0 < r` by auto
```
```  1373         thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
```
```  1374       qed (insert `0 < u x`, auto)
```
```  1375     qed auto
```
```  1376     finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp
```
```  1377     moreover
```
```  1378     from pos have "AE x in M. \<not> (u x < 0)" by auto
```
```  1379     then have "(emeasure M) {x\<in>space M. u x < 0} = 0"
```
```  1380       using AE_iff_null[of M] u by auto
```
```  1381     moreover have "(emeasure M) {x\<in>space M. u x \<noteq> 0} = (emeasure M) {x\<in>space M. u x < 0} + (emeasure M) {x\<in>space M. 0 < u x}"
```
```  1382       using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"])
```
```  1383     ultimately show "(emeasure M) ?A = 0" by simp
```
```  1384   qed
```
```  1385 qed
```
```  1386
```
```  1387 lemma positive_integral_0_iff_AE:
```
```  1388   assumes u: "u \<in> borel_measurable M"
```
```  1389   shows "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)"
```
```  1390 proof -
```
```  1391   have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
```
```  1392     using u by auto
```
```  1393   from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
```
```  1394   have "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)"
```
```  1395     unfolding positive_integral_max_0
```
```  1396     using AE_iff_null[OF sets] u by auto
```
```  1397   also have "\<dots> \<longleftrightarrow> (AE x in M. u x \<le> 0)" by (auto split: split_max)
```
```  1398   finally show ?thesis .
```
```  1399 qed
```
```  1400
```
```  1401 lemma AE_iff_positive_integral:
```
```  1402   "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> integral\<^sup>P M (indicator {x. \<not> P x}) = 0"
```
```  1403   by (subst positive_integral_0_iff_AE) (auto simp: one_ereal_def zero_ereal_def
```
```  1404     sets.sets_Collect_neg indicator_def[abs_def] measurable_If)
```
```  1405
```
```  1406 lemma positive_integral_const_If:
```
```  1407   "(\<integral>\<^sup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)"
```
```  1408   by (auto intro!: positive_integral_0_iff_AE[THEN iffD2])
```
```  1409
```
```  1410 lemma positive_integral_subalgebra:
```
```  1411   assumes f: "f \<in> borel_measurable N" "\<And>x. 0 \<le> f x"
```
```  1412   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
```
```  1413   shows "integral\<^sup>P N f = integral\<^sup>P M f"
```
```  1414 proof -
```
```  1415   have [simp]: "\<And>f :: 'a \<Rightarrow> ereal. f \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable M"
```
```  1416     using N by (auto simp: measurable_def)
```
```  1417   have [simp]: "\<And>P. (AE x in N. P x) \<Longrightarrow> (AE x in M. P x)"
```
```  1418     using N by (auto simp add: eventually_ae_filter null_sets_def)
```
```  1419   have [simp]: "\<And>A. A \<in> sets N \<Longrightarrow> A \<in> sets M"
```
```  1420     using N by auto
```
```  1421   from f show ?thesis
```
```  1422     apply induct
```
```  1423     apply (simp_all add: positive_integral_add positive_integral_cmult positive_integral_monotone_convergence_SUP N)
```
```  1424     apply (auto intro!: positive_integral_cong cong: positive_integral_cong simp: N(2)[symmetric])
```
```  1425     done
```
```  1426 qed
```
```  1427
```
```  1428 lemma positive_integral_nat_function:
```
```  1429   fixes f :: "'a \<Rightarrow> nat"
```
```  1430   assumes "f \<in> measurable M (count_space UNIV)"
```
```  1431   shows "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<Sum>t. emeasure M {x\<in>space M. t < f x})"
```
```  1432 proof -
```
```  1433   def F \<equiv> "\<lambda>i. {x\<in>space M. i < f x}"
```
```  1434   with assms have [measurable]: "\<And>i. F i \<in> sets M"
```
```  1435     by auto
```
```  1436
```
```  1437   { fix x assume "x \<in> space M"
```
```  1438     have "(\<lambda>i. if i < f x then 1 else 0) sums (of_nat (f x)::real)"
```
```  1439       using sums_If_finite[of "\<lambda>i. i < f x" "\<lambda>_. 1::real"] by simp
```
```  1440     then have "(\<lambda>i. ereal(if i < f x then 1 else 0)) sums (ereal(of_nat(f x)))"
```
```  1441       unfolding sums_ereal .
```
```  1442     moreover have "\<And>i. ereal (if i < f x then 1 else 0) = indicator (F i) x"
```
```  1443       using `x \<in> space M` by (simp add: one_ereal_def F_def)
```
```  1444     ultimately have "ereal(of_nat(f x)) = (\<Sum>i. indicator (F i) x)"
```
```  1445       by (simp add: sums_iff) }
```
```  1446   then have "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. indicator (F i) x) \<partial>M)"
```
```  1447     by (simp cong: positive_integral_cong)
```
```  1448   also have "\<dots> = (\<Sum>i. emeasure M (F i))"
```
```  1449     by (simp add: positive_integral_suminf)
```
```  1450   finally show ?thesis
```
```  1451     by (simp add: F_def)
```
```  1452 qed
```
```  1453
```
```  1454 section "Lebesgue Integral"
```
```  1455
```
```  1456 definition integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
```
```  1457   "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
```
```  1458     (\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
```
```  1459
```
```  1460 lemma borel_measurable_integrable[measurable_dest]:
```
```  1461   "integrable M f \<Longrightarrow> f \<in> borel_measurable M"
```
```  1462   by (auto simp: integrable_def)
```
```  1463
```
```  1464 lemma integrableD[dest]:
```
```  1465   assumes "integrable M f"
```
```  1466   shows "f \<in> borel_measurable M" "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
```
```  1467   using assms unfolding integrable_def by auto
```
```  1468
```
```  1469 definition lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("integral\<^sup>L") where
```
```  1470   "integral\<^sup>L M f = real ((\<integral>\<^sup>+ x. ereal (f x) \<partial>M)) - real ((\<integral>\<^sup>+ x. ereal (- f x) \<partial>M))"
```
```  1471
```
```  1472 syntax
```
```  1473   "_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
```
```  1474
```
```  1475 translations
```
```  1476   "\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (%x. f)"
```
```  1477
```
```  1478 lemma integrableE:
```
```  1479   assumes "integrable M f"
```
```  1480   obtains r q where
```
```  1481     "(\<integral>\<^sup>+x. ereal (f x)\<partial>M) = ereal r"
```
```  1482     "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M) = ereal q"
```
```  1483     "f \<in> borel_measurable M" "integral\<^sup>L M f = r - q"
```
```  1484   using assms unfolding integrable_def lebesgue_integral_def
```
```  1485   using positive_integral_positive[of M "\<lambda>x. ereal (f x)"]
```
```  1486   using positive_integral_positive[of M "\<lambda>x. ereal (-f x)"]
```
```  1487   by (cases rule: ereal2_cases[of "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^sup>+x. ereal (f x)\<partial>M)"]) auto
```
```  1488
```
```  1489 lemma integral_cong:
```
```  1490   assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
```
```  1491   shows "integral\<^sup>L M f = integral\<^sup>L M g"
```
```  1492   using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
```
```  1493
```
```  1494 lemma integral_cong_AE:
```
```  1495   assumes cong: "AE x in M. f x = g x"
```
```  1496   shows "integral\<^sup>L M f = integral\<^sup>L M g"
```
```  1497 proof -
```
```  1498   have *: "AE x in M. ereal (f x) = ereal (g x)"
```
```  1499     "AE x in M. ereal (- f x) = ereal (- g x)" using cong by auto
```
```  1500   show ?thesis
```
```  1501     unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
```
```  1502 qed
```
```  1503
```
```  1504 lemma integrable_cong_AE:
```
```  1505   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1506   assumes "AE x in M. f x = g x"
```
```  1507   shows "integrable M f = integrable M g"
```
```  1508 proof -
```
```  1509   have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (g x) \<partial>M)"
```
```  1510     "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (- g x) \<partial>M)"
```
```  1511     using assms by (auto intro!: positive_integral_cong_AE)
```
```  1512   with assms show ?thesis
```
```  1513     by (auto simp: integrable_def)
```
```  1514 qed
```
```  1515
```
```  1516 lemma integrable_cong:
```
```  1517   "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
```
```  1518   by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
```
```  1519
```
```  1520 lemma integral_mono_AE:
```
```  1521   assumes fg: "integrable M f" "integrable M g"
```
```  1522   and mono: "AE t in M. f t \<le> g t"
```
```  1523   shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
```
```  1524 proof -
```
```  1525   have "AE x in M. ereal (f x) \<le> ereal (g x)"
```
```  1526     using mono by auto
```
```  1527   moreover have "AE x in M. ereal (- g x) \<le> ereal (- f x)"
```
```  1528     using mono by auto
```
```  1529   ultimately show ?thesis using fg
```
```  1530     by (auto intro!: add_mono positive_integral_mono_AE real_of_ereal_positive_mono
```
```  1531              simp: positive_integral_positive lebesgue_integral_def algebra_simps)
```
```  1532 qed
```
```  1533
```
```  1534 lemma integral_mono:
```
```  1535   assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
```
```  1536   shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
```
```  1537   using assms by (auto intro: integral_mono_AE)
```
```  1538
```
```  1539 lemma positive_integral_eq_integral:
```
```  1540   assumes f: "integrable M f"
```
```  1541   assumes nonneg: "AE x in M. 0 \<le> f x"
```
```  1542   shows "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = integral\<^sup>L M f"
```
```  1543 proof -
```
```  1544   have "(\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
```
```  1545     using nonneg by (intro positive_integral_cong_AE) (auto split: split_max)
```
```  1546   with f positive_integral_positive show ?thesis
```
```  1547     by (cases "\<integral>\<^sup>+ x. ereal (f x) \<partial>M")
```
```  1548        (auto simp add: lebesgue_integral_def positive_integral_max_0 integrable_def)
```
```  1549 qed
```
```  1550
```
```  1551 lemma integral_eq_positive_integral:
```
```  1552   assumes f: "\<And>x. 0 \<le> f x"
```
```  1553   shows "integral\<^sup>L M f = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
```
```  1554 proof -
```
```  1555   { fix x have "max 0 (ereal (- f x)) = 0" using f[of x] by (simp split: split_max) }
```
```  1556   then have "0 = (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)" by simp
```
```  1557   also have "\<dots> = (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
```
```  1558   finally show ?thesis
```
```  1559     unfolding lebesgue_integral_def by simp
```
```  1560 qed
```
```  1561
```
```  1562 lemma integral_minus[intro, simp]:
```
```  1563   assumes "integrable M f"
```
```  1564   shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^sup>L M f"
```
```  1565   using assms by (auto simp: integrable_def lebesgue_integral_def)
```
```  1566
```
```  1567 lemma integral_minus_iff[simp]:
```
```  1568   "integrable M (\<lambda>x. - f x) \<longleftrightarrow> integrable M f"
```
```  1569 proof
```
```  1570   assume "integrable M (\<lambda>x. - f x)"
```
```  1571   then have "integrable M (\<lambda>x. - (- f x))"
```
```  1572     by (rule integral_minus)
```
```  1573   then show "integrable M f" by simp
```
```  1574 qed (rule integral_minus)
```
```  1575
```
```  1576 lemma integral_of_positive_diff:
```
```  1577   assumes integrable: "integrable M u" "integrable M v"
```
```  1578   and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
```
```  1579   shows "integrable M f" and "integral\<^sup>L M f = integral\<^sup>L M u - integral\<^sup>L M v"
```
```  1580 proof -
```
```  1581   let ?f = "\<lambda>x. max 0 (ereal (f x))"
```
```  1582   let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
```
```  1583   let ?u = "\<lambda>x. max 0 (ereal (u x))"
```
```  1584   let ?v = "\<lambda>x. max 0 (ereal (v x))"
```
```  1585
```
```  1586   from borel_measurable_diff[of u M v] integrable
```
```  1587   have f_borel: "?f \<in> borel_measurable M" and
```
```  1588     mf_borel: "?mf \<in> borel_measurable M" and
```
```  1589     v_borel: "?v \<in> borel_measurable M" and
```
```  1590     u_borel: "?u \<in> borel_measurable M" and
```
```  1591     "f \<in> borel_measurable M"
```
```  1592     by (auto simp: f_def[symmetric] integrable_def)
```
```  1593
```
```  1594   have "(\<integral>\<^sup>+ x. ereal (u x - v x) \<partial>M) \<le> integral\<^sup>P M ?u"
```
```  1595     using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
```
```  1596   moreover have "(\<integral>\<^sup>+ x. ereal (v x - u x) \<partial>M) \<le> integral\<^sup>P M ?v"
```
```  1597     using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
```
```  1598   ultimately show f: "integrable M f"
```
```  1599     using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
```
```  1600     by (auto simp: integrable_def f_def positive_integral_max_0)
```
```  1601
```
```  1602   have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
```
```  1603     unfolding f_def using pos by (simp split: split_max)
```
```  1604   then have "(\<integral>\<^sup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^sup>+ x. ?v x + ?f x \<partial>M)" by simp
```
```  1605   then have "real (integral\<^sup>P M ?u + integral\<^sup>P M ?mf) =
```
```  1606       real (integral\<^sup>P M ?v + integral\<^sup>P M ?f)"
```
```  1607     using positive_integral_add[OF u_borel _ mf_borel]
```
```  1608     using positive_integral_add[OF v_borel _ f_borel]
```
```  1609     by auto
```
```  1610   then show "integral\<^sup>L M f = integral\<^sup>L M u - integral\<^sup>L M v"
```
```  1611     unfolding positive_integral_max_0
```
```  1612     unfolding pos[THEN integral_eq_positive_integral]
```
```  1613     using integrable f by (auto elim!: integrableE)
```
```  1614 qed
```
```  1615
```
```  1616 lemma integral_linear:
```
```  1617   assumes "integrable M f" "integrable M g" and "0 \<le> a"
```
```  1618   shows "integrable M (\<lambda>t. a * f t + g t)"
```
```  1619   and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^sup>L M f + integral\<^sup>L M g" (is ?EQ)
```
```  1620 proof -
```
```  1621   let ?f = "\<lambda>x. max 0 (ereal (f x))"
```
```  1622   let ?g = "\<lambda>x. max 0 (ereal (g x))"
```
```  1623   let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
```
```  1624   let ?mg = "\<lambda>x. max 0 (ereal (- g x))"
```
```  1625   let ?p = "\<lambda>t. max 0 (a * f t) + max 0 (g t)"
```
```  1626   let ?n = "\<lambda>t. max 0 (- (a * f t)) + max 0 (- g t)"
```
```  1627
```
```  1628   from assms have linear:
```
```  1629     "(\<integral>\<^sup>+ x. ereal a * ?f x + ?g x \<partial>M) = ereal a * integral\<^sup>P M ?f + integral\<^sup>P M ?g"
```
```  1630     "(\<integral>\<^sup>+ x. ereal a * ?mf x + ?mg x \<partial>M) = ereal a * integral\<^sup>P M ?mf + integral\<^sup>P M ?mg"
```
```  1631     by (auto intro!: positive_integral_linear simp: integrable_def)
```
```  1632
```
```  1633   have *: "(\<integral>\<^sup>+x. ereal (- ?p x) \<partial>M) = 0" "(\<integral>\<^sup>+x. ereal (- ?n x) \<partial>M) = 0"
```
```  1634     using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
```
```  1635   have **: "\<And>x. ereal a * ?f x + ?g x = max 0 (ereal (?p x))"
```
```  1636            "\<And>x. ereal a * ?mf x + ?mg x = max 0 (ereal (?n x))"
```
```  1637     using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
```
```  1638
```
```  1639   have "integrable M ?p" "integrable M ?n"
```
```  1640       "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
```
```  1641     using linear assms unfolding integrable_def ** *
```
```  1642     by (auto simp: positive_integral_max_0)
```
```  1643   note diff = integral_of_positive_diff[OF this]
```
```  1644
```
```  1645   show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
```
```  1646   from assms linear show ?EQ
```
```  1647     unfolding diff(2) ** positive_integral_max_0
```
```  1648     unfolding lebesgue_integral_def *
```
```  1649     by (auto elim!: integrableE simp: field_simps)
```
```  1650 qed
```
```  1651
```
```  1652 lemma integral_add[simp, intro]:
```
```  1653   assumes "integrable M f" "integrable M g"
```
```  1654   shows "integrable M (\<lambda>t. f t + g t)"
```
```  1655   and "(\<integral> t. f t + g t \<partial>M) = integral\<^sup>L M f + integral\<^sup>L M g"
```
```  1656   using assms integral_linear[where a=1] by auto
```
```  1657
```
```  1658 lemma integral_zero[simp, intro]:
```
```  1659   shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
```
```  1660   unfolding integrable_def lebesgue_integral_def
```
```  1661   by auto
```
```  1662
```
```  1663 lemma lebesgue_integral_uminus:
```
```  1664     "(\<integral>x. - f x \<partial>M) = - integral\<^sup>L M f"
```
```  1665   unfolding lebesgue_integral_def by simp
```
```  1666
```
```  1667 lemma lebesgue_integral_cmult_nonneg:
```
```  1668   assumes f: "f \<in> borel_measurable M" and "0 \<le> c"
```
```  1669   shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^sup>L M f"
```
```  1670 proof -
```
```  1671   { have "real (ereal c * integral\<^sup>P M (\<lambda>x. max 0 (ereal (f x)))) =
```
```  1672       real (integral\<^sup>P M (\<lambda>x. ereal c * max 0 (ereal (f x))))"
```
```  1673       using f `0 \<le> c` by (subst positive_integral_cmult) auto
```
```  1674     also have "\<dots> = real (integral\<^sup>P M (\<lambda>x. max 0 (ereal (c * f x))))"
```
```  1675       using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def zero_le_mult_iff)
```
```  1676     finally have "real (integral\<^sup>P M (\<lambda>x. ereal (c * f x))) = c * real (integral\<^sup>P M (\<lambda>x. ereal (f x)))"
```
```  1677       by (simp add: positive_integral_max_0) }
```
```  1678   moreover
```
```  1679   { have "real (ereal c * integral\<^sup>P M (\<lambda>x. max 0 (ereal (- f x)))) =
```
```  1680       real (integral\<^sup>P M (\<lambda>x. ereal c * max 0 (ereal (- f x))))"
```
```  1681       using f `0 \<le> c` by (subst positive_integral_cmult) auto
```
```  1682     also have "\<dots> = real (integral\<^sup>P M (\<lambda>x. max 0 (ereal (- c * f x))))"
```
```  1683       using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def mult_le_0_iff)
```
```  1684     finally have "real (integral\<^sup>P M (\<lambda>x. ereal (- c * f x))) = c * real (integral\<^sup>P M (\<lambda>x. ereal (- f x)))"
```
```  1685       by (simp add: positive_integral_max_0) }
```
```  1686   ultimately show ?thesis
```
```  1687     by (simp add: lebesgue_integral_def field_simps)
```
```  1688 qed
```
```  1689
```
```  1690 lemma lebesgue_integral_cmult:
```
```  1691   assumes f: "f \<in> borel_measurable M"
```
```  1692   shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^sup>L M f"
```
```  1693 proof (cases rule: linorder_le_cases)
```
```  1694   assume "0 \<le> c" with f show ?thesis by (rule lebesgue_integral_cmult_nonneg)
```
```  1695 next
```
```  1696   assume "c \<le> 0"
```
```  1697   with lebesgue_integral_cmult_nonneg[OF f, of "-c"]
```
```  1698   show ?thesis
```
```  1699     by (simp add: lebesgue_integral_def)
```
```  1700 qed
```
```  1701
```
```  1702 lemma lebesgue_integral_multc:
```
```  1703   "f \<in> borel_measurable M \<Longrightarrow> (\<integral>x. f x * c \<partial>M) = integral\<^sup>L M f * c"
```
```  1704   using lebesgue_integral_cmult[of f M c] by (simp add: ac_simps)
```
```  1705
```
```  1706 lemma integral_multc:
```
```  1707   "integrable M f \<Longrightarrow> (\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
```
```  1708   by (simp add: lebesgue_integral_multc)
```
```  1709
```
```  1710 lemma integral_cmult[simp, intro]:
```
```  1711   assumes "integrable M f"
```
```  1712   shows "integrable M (\<lambda>t. a * f t)" (is ?P)
```
```  1713   and "(\<integral> t. a * f t \<partial>M) = a * integral\<^sup>L M f" (is ?I)
```
```  1714 proof -
```
```  1715   have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^sup>L M f"
```
```  1716   proof (cases rule: le_cases)
```
```  1717     assume "0 \<le> a" show ?thesis
```
```  1718       using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
```
```  1719       by simp
```
```  1720   next
```
```  1721     assume "a \<le> 0" hence "0 \<le> - a" by auto
```
```  1722     have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
```
```  1723     show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
```
```  1724         integral_minus(1)[of M "\<lambda>t. - a * f t"]
```
```  1725       unfolding * integral_zero by simp
```
```  1726   qed
```
```  1727   thus ?P ?I by auto
```
```  1728 qed
```
```  1729
```
```  1730 lemma integral_diff[simp, intro]:
```
```  1731   assumes f: "integrable M f" and g: "integrable M g"
```
```  1732   shows "integrable M (\<lambda>t. f t - g t)"
```
```  1733   and "(\<integral> t. f t - g t \<partial>M) = integral\<^sup>L M f - integral\<^sup>L M g"
```
```  1734   using integral_add[OF f integral_minus(1)[OF g]]
```
```  1735   unfolding integral_minus(2)[OF g]
```
```  1736   by auto
```
```  1737
```
```  1738 lemma integral_indicator[simp, intro]:
```
```  1739   assumes "A \<in> sets M" and "(emeasure M) A \<noteq> \<infinity>"
```
```  1740   shows "integral\<^sup>L M (indicator A) = real (emeasure M A)" (is ?int)
```
```  1741   and "integrable M (indicator A)" (is ?able)
```
```  1742 proof -
```
```  1743   from `A \<in> sets M` have *:
```
```  1744     "\<And>x. ereal (indicator A x) = indicator A x"
```
```  1745     "(\<integral>\<^sup>+x. ereal (- indicator A x) \<partial>M) = 0"
```
```  1746     by (auto split: split_indicator simp: positive_integral_0_iff_AE one_ereal_def)
```
```  1747   show ?int ?able
```
```  1748     using assms unfolding lebesgue_integral_def integrable_def
```
```  1749     by (auto simp: *)
```
```  1750 qed
```
```  1751
```
```  1752 lemma integral_cmul_indicator:
```
```  1753   assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> (emeasure M) A \<noteq> \<infinity>"
```
```  1754   shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
```
```  1755   and "(\<integral>x. c * indicator A x \<partial>M) = c * real ((emeasure M) A)" (is ?I)
```
```  1756 proof -
```
```  1757   show ?P
```
```  1758   proof (cases "c = 0")
```
```  1759     case False with assms show ?thesis by simp
```
```  1760   qed simp
```
```  1761
```
```  1762   show ?I
```
```  1763   proof (cases "c = 0")
```
```  1764     case False with assms show ?thesis by simp
```
```  1765   qed simp
```
```  1766 qed
```
```  1767
```
```  1768 lemma integral_setsum[simp, intro]:
```
```  1769   assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f n)"
```
```  1770   shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^sup>L M (f i))" (is "?int S")
```
```  1771     and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
```
```  1772 proof -
```
```  1773   have "?int S \<and> ?I S"
```
```  1774   proof (cases "finite S")
```
```  1775     assume "finite S"
```
```  1776     from this assms show ?thesis by (induct S) simp_all
```
```  1777   qed simp
```
```  1778   thus "?int S" and "?I S" by auto
```
```  1779 qed
```
```  1780
```
```  1781 lemma integrable_bound:
```
```  1782   assumes "integrable M f" and f: "AE x in M. \<bar>g x\<bar> \<le> f x"
```
```  1783   assumes borel: "g \<in> borel_measurable M"
```
```  1784   shows "integrable M g"
```
```  1785 proof -
```
```  1786   have "(\<integral>\<^sup>+ x. ereal (g x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal \<bar>g x\<bar> \<partial>M)"
```
```  1787     by (auto intro!: positive_integral_mono)
```
```  1788   also have "\<dots> \<le> (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
```
```  1789     using f by (auto intro!: positive_integral_mono_AE)
```
```  1790   also have "\<dots> < \<infinity>"
```
```  1791     using `integrable M f` unfolding integrable_def by auto
```
```  1792   finally have pos: "(\<integral>\<^sup>+ x. ereal (g x) \<partial>M) < \<infinity>" .
```
```  1793
```
```  1794   have "(\<integral>\<^sup>+ x. ereal (- g x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (\<bar>g x\<bar>) \<partial>M)"
```
```  1795     by (auto intro!: positive_integral_mono)
```
```  1796   also have "\<dots> \<le> (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
```
```  1797     using f by (auto intro!: positive_integral_mono_AE)
```
```  1798   also have "\<dots> < \<infinity>"
```
```  1799     using `integrable M f` unfolding integrable_def by auto
```
```  1800   finally have neg: "(\<integral>\<^sup>+ x. ereal (- g x) \<partial>M) < \<infinity>" .
```
```  1801
```
```  1802   from neg pos borel show ?thesis
```
```  1803     unfolding integrable_def by auto
```
```  1804 qed
```
```  1805
```
```  1806 lemma integrable_abs:
```
```  1807   assumes f[measurable]: "integrable M f"
```
```  1808   shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
```
```  1809 proof -
```
```  1810   from assms have *: "(\<integral>\<^sup>+x. ereal (- \<bar>f x\<bar>)\<partial>M) = 0"
```
```  1811     "\<And>x. ereal \<bar>f x\<bar> = max 0 (ereal (f x)) + max 0 (ereal (- f x))"
```
```  1812     by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
```
```  1813   with assms show ?thesis
```
```  1814     by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
```
```  1815 qed
```
```  1816
```
```  1817 lemma integral_subalgebra:
```
```  1818   assumes borel: "f \<in> borel_measurable N"
```
```  1819   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
```
```  1820   shows "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
```
```  1821     and "integral\<^sup>L N f = integral\<^sup>L M f" (is ?I)
```
```  1822 proof -
```
```  1823   have "(\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>N) = (\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>M)"
```
```  1824        "(\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>N) = (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)"
```
```  1825     using borel by (auto intro!: positive_integral_subalgebra N)
```
```  1826   moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
```
```  1827     using assms unfolding measurable_def by auto
```
```  1828   ultimately show ?P ?I
```
```  1829     by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
```
```  1830 qed
```
```  1831
```
```  1832 lemma lebesgue_integral_nonneg:
```
```  1833   assumes ae: "(AE x in M. 0 \<le> f x)" shows "0 \<le> integral\<^sup>L M f"
```
```  1834 proof -
```
```  1835   have "(\<integral>\<^sup>+x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^sup>+x. 0 \<partial>M)"
```
```  1836     using ae by (intro positive_integral_cong_AE) (auto simp: max_def)
```
```  1837   then show ?thesis
```
```  1838     by (auto simp: lebesgue_integral_def positive_integral_max_0
```
```  1839              intro!: real_of_ereal_pos positive_integral_positive)
```
```  1840 qed
```
```  1841
```
```  1842 lemma integrable_abs_iff:
```
```  1843   "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
```
```  1844   by (auto intro!: integrable_bound[where g=f] integrable_abs)
```
```  1845
```
```  1846 lemma integrable_max:
```
```  1847   assumes int: "integrable M f" "integrable M g"
```
```  1848   shows "integrable M (\<lambda> x. max (f x) (g x))"
```
```  1849 proof (rule integrable_bound)
```
```  1850   show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
```
```  1851     using int by (simp add: integrable_abs)
```
```  1852   show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
```
```  1853     using int unfolding integrable_def by auto
```
```  1854 qed auto
```
```  1855
```
```  1856 lemma integrable_min:
```
```  1857   assumes int: "integrable M f" "integrable M g"
```
```  1858   shows "integrable M (\<lambda> x. min (f x) (g x))"
```
```  1859 proof (rule integrable_bound)
```
```  1860   show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
```
```  1861     using int by (simp add: integrable_abs)
```
```  1862   show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
```
```  1863     using int unfolding integrable_def by auto
```
```  1864 qed auto
```
```  1865
```
```  1866 lemma integral_triangle_inequality:
```
```  1867   assumes "integrable M f"
```
```  1868   shows "\<bar>integral\<^sup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
```
```  1869 proof -
```
```  1870   have "\<bar>integral\<^sup>L M f\<bar> = max (integral\<^sup>L M f) (- integral\<^sup>L M f)" by auto
```
```  1871   also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
```
```  1872       using assms integral_minus(2)[of M f, symmetric]
```
```  1873       by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
```
```  1874   finally show ?thesis .
```
```  1875 qed
```
```  1876
```
```  1877 lemma integrable_nonneg:
```
```  1878   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+ x. f x \<partial>M) \<noteq> \<infinity>"
```
```  1879   shows "integrable M f"
```
```  1880   unfolding integrable_def
```
```  1881 proof (intro conjI f)
```
```  1882   have "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) = 0"
```
```  1883     using f by (subst positive_integral_0_iff_AE) auto
```
```  1884   then show "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>" by simp
```
```  1885 qed
```
```  1886
```
```  1887 lemma integral_positive:
```
```  1888   assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1889   shows "0 \<le> integral\<^sup>L M f"
```
```  1890 proof -
```
```  1891   have "0 = (\<integral>x. 0 \<partial>M)" by auto
```
```  1892   also have "\<dots> \<le> integral\<^sup>L M f"
```
```  1893     using assms by (rule integral_mono[OF integral_zero(1)])
```
```  1894   finally show ?thesis .
```
```  1895 qed
```
```  1896
```
```  1897 lemma integral_monotone_convergence_pos:
```
```  1898   assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
```
```  1899     and pos: "\<And>i. AE x in M. 0 \<le> f i x"
```
```  1900     and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
```
```  1901     and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
```
```  1902     and u: "u \<in> borel_measurable M"
```
```  1903   shows "integrable M u"
```
```  1904   and "integral\<^sup>L M u = x"
```
```  1905 proof -
```
```  1906   have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = (SUP n. (\<integral>\<^sup>+ x. ereal (f n x) \<partial>M))"
```
```  1907   proof (subst positive_integral_monotone_convergence_SUP_AE[symmetric])
```
```  1908     fix i
```
```  1909     from mono pos show "AE x in M. ereal (f i x) \<le> ereal (f (Suc i) x) \<and> 0 \<le> ereal (f i x)"
```
```  1910       by eventually_elim (auto simp: mono_def)
```
```  1911     show "(\<lambda>x. ereal (f i x)) \<in> borel_measurable M"
```
```  1912       using i by auto
```
```  1913   next
```
```  1914     show "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = \<integral>\<^sup>+ x. (SUP i. ereal (f i x)) \<partial>M"
```
```  1915       apply (rule positive_integral_cong_AE)
```
```  1916       using lim mono
```
```  1917       by eventually_elim (simp add: SUP_eq_LIMSEQ[THEN iffD2])
```
```  1918   qed
```
```  1919   also have "\<dots> = ereal x"
```
```  1920     using mono i unfolding positive_integral_eq_integral[OF i pos]
```
```  1921     by (subst SUP_eq_LIMSEQ) (auto simp: mono_def intro!: integral_mono_AE ilim)
```
```  1922   finally have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = ereal x" .
```
```  1923   moreover have "(\<integral>\<^sup>+ x. ereal (- u x) \<partial>M) = 0"
```
```  1924   proof (subst positive_integral_0_iff_AE)
```
```  1925     show "(\<lambda>x. ereal (- u x)) \<in> borel_measurable M"
```
```  1926       using u by auto
```
```  1927     from mono pos[of 0] lim show "AE x in M. ereal (- u x) \<le> 0"
```
```  1928     proof eventually_elim
```
```  1929       fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) ----> u x"
```
```  1930       then show "ereal (- u x) \<le> 0"
```
```  1931         using incseq_le[of "\<lambda>n. f n x" "u x" 0] by (simp add: mono_def incseq_def)
```
```  1932     qed
```
```  1933   qed
```
```  1934   ultimately show "integrable M u" "integral\<^sup>L M u = x"
```
```  1935     by (auto simp: integrable_def lebesgue_integral_def u)
```
```  1936 qed
```
```  1937
```
```  1938 lemma integral_monotone_convergence:
```
```  1939   assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
```
```  1940   and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
```
```  1941   and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
```
```  1942   and u: "u \<in> borel_measurable M"
```
```  1943   shows "integrable M u"
```
```  1944   and "integral\<^sup>L M u = x"
```
```  1945 proof -
```
```  1946   have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
```
```  1947     using f by auto
```
```  1948   have 2: "AE x in M. mono (\<lambda>n. f n x - f 0 x)"
```
```  1949     using mono by (auto simp: mono_def le_fun_def)
```
```  1950   have 3: "\<And>n. AE x in M. 0 \<le> f n x - f 0 x"
```
```  1951     using mono by (auto simp: field_simps mono_def le_fun_def)
```
```  1952   have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
```
```  1953     using lim by (auto intro!: tendsto_diff)
```
```  1954   have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^sup>L M (f 0)"
```
```  1955     using f ilim by (auto intro!: tendsto_diff)
```
```  1956   have 6: "(\<lambda>x. u x - f 0 x) \<in> borel_measurable M"
```
```  1957     using f[of 0] u by auto
```
```  1958   note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5 6]
```
```  1959   have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
```
```  1960     using diff(1) f by (rule integral_add(1))
```
```  1961   with diff(2) f show "integrable M u" "integral\<^sup>L M u = x"
```
```  1962     by auto
```
```  1963 qed
```
```  1964
```
```  1965 lemma integral_0_iff:
```
```  1966   assumes "integrable M f"
```
```  1967   shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> (emeasure M) {x\<in>space M. f x \<noteq> 0} = 0"
```
```  1968 proof -
```
```  1969   have *: "(\<integral>\<^sup>+x. ereal (- \<bar>f x\<bar>) \<partial>M) = 0"
```
```  1970     using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
```
```  1971   have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
```
```  1972   hence "(\<lambda>x. ereal (\<bar>f x\<bar>)) \<in> borel_measurable M"
```
```  1973     "(\<integral>\<^sup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
```
```  1974   from positive_integral_0_iff[OF this(1)] this(2)
```
```  1975   show ?thesis unfolding lebesgue_integral_def *
```
```  1976     using positive_integral_positive[of M "\<lambda>x. ereal \<bar>f x\<bar>"]
```
```  1977     by (auto simp add: real_of_ereal_eq_0)
```
```  1978 qed
```
```  1979
```
```  1980 lemma positive_integral_PInf:
```
```  1981   assumes f: "f \<in> borel_measurable M"
```
```  1982   and not_Inf: "integral\<^sup>P M f \<noteq> \<infinity>"
```
```  1983   shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
```
```  1984 proof -
```
```  1985   have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^sup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
```
```  1986     using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
```
```  1987   also have "\<dots> \<le> integral\<^sup>P M (\<lambda>x. max 0 (f x))"
```
```  1988     by (auto intro!: positive_integral_mono simp: indicator_def max_def)
```
```  1989   finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^sup>P M f"
```
```  1990     by (simp add: positive_integral_max_0)
```
```  1991   moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)"
```
```  1992     by (rule emeasure_nonneg)
```
```  1993   ultimately show ?thesis
```
```  1994     using assms by (auto split: split_if_asm)
```
```  1995 qed
```
```  1996
```
```  1997 lemma positive_integral_PInf_AE:
```
```  1998   assumes "f \<in> borel_measurable M" "integral\<^sup>P M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
```
```  1999 proof (rule AE_I)
```
```  2000   show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
```
```  2001     by (rule positive_integral_PInf[OF assms])
```
```  2002   show "f -` {\<infinity>} \<inter> space M \<in> sets M"
```
```  2003     using assms by (auto intro: borel_measurable_vimage)
```
```  2004 qed auto
```
```  2005
```
```  2006 lemma simple_integral_PInf:
```
```  2007   assumes "simple_function M f" "\<And>x. 0 \<le> f x"
```
```  2008   and "integral\<^sup>S M f \<noteq> \<infinity>"
```
```  2009   shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
```
```  2010 proof (rule positive_integral_PInf)
```
```  2011   show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
```
```  2012   show "integral\<^sup>P M f \<noteq> \<infinity>"
```
```  2013     using assms by (simp add: positive_integral_eq_simple_integral)
```
```  2014 qed
```
```  2015
```
```  2016 lemma integral_real:
```
```  2017   "AE x in M. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^sup>P M f) - real (integral\<^sup>P M (\<lambda>x. - f x))"
```
```  2018   using assms unfolding lebesgue_integral_def
```
```  2019   by (subst (1 2) positive_integral_cong_AE) (auto simp add: ereal_real)
```
```  2020
```
```  2021 lemma (in finite_measure) lebesgue_integral_const[simp]:
```
```  2022   shows "integrable M (\<lambda>x. a)"
```
```  2023   and  "(\<integral>x. a \<partial>M) = a * measure M (space M)"
```
```  2024 proof -
```
```  2025   { fix a :: real assume "0 \<le> a"
```
```  2026     then have "(\<integral>\<^sup>+ x. ereal a \<partial>M) = ereal a * (emeasure M) (space M)"
```
```  2027       by (subst positive_integral_const) auto
```
```  2028     moreover
```
```  2029     from `0 \<le> a` have "(\<integral>\<^sup>+ x. ereal (-a) \<partial>M) = 0"
```
```  2030       by (subst positive_integral_0_iff_AE) auto
```
```  2031     ultimately have "integrable M (\<lambda>x. a)" by (auto simp: integrable_def) }
```
```  2032   note * = this
```
```  2033   show "integrable M (\<lambda>x. a)"
```
```  2034   proof cases
```
```  2035     assume "0 \<le> a" with * show ?thesis .
```
```  2036   next
```
```  2037     assume "\<not> 0 \<le> a"
```
```  2038     then have "0 \<le> -a" by auto
```
```  2039     from *[OF this] show ?thesis by simp
```
```  2040   qed
```
```  2041   show "(\<integral>x. a \<partial>M) = a * measure M (space M)"
```
```  2042     by (simp add: lebesgue_integral_def positive_integral_const_If emeasure_eq_measure)
```
```  2043 qed
```
```  2044
```
```  2045 lemma (in finite_measure) integrable_const_bound:
```
```  2046   assumes "AE x in M. \<bar>f x\<bar> \<le> B" and "f \<in> borel_measurable M" shows "integrable M f"
```
```  2047   by (auto intro: integrable_bound[where f="\<lambda>x. B"] lebesgue_integral_const assms)
```
```  2048
```
```  2049 lemma indicator_less[simp]:
```
```  2050   "indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
```
```  2051   by (simp add: indicator_def not_le)
```
```  2052
```
```  2053 lemma (in finite_measure) integral_less_AE:
```
```  2054   assumes int: "integrable M X" "integrable M Y"
```
```  2055   assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x"
```
```  2056   assumes gt: "AE x in M. X x \<le> Y x"
```
```  2057   shows "integral\<^sup>L M X < integral\<^sup>L M Y"
```
```  2058 proof -
```
```  2059   have "integral\<^sup>L M X \<le> integral\<^sup>L M Y"
```
```  2060     using gt int by (intro integral_mono_AE) auto
```
```  2061   moreover
```
```  2062   have "integral\<^sup>L M X \<noteq> integral\<^sup>L M Y"
```
```  2063   proof
```
```  2064     assume eq: "integral\<^sup>L M X = integral\<^sup>L M Y"
```
```  2065     have "integral\<^sup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^sup>L M (\<lambda>x. Y x - X x)"
```
```  2066       using gt by (intro integral_cong_AE) auto
```
```  2067     also have "\<dots> = 0"
```
```  2068       using eq int by simp
```
```  2069     finally have "(emeasure M) {x \<in> space M. Y x - X x \<noteq> 0} = 0"
```
```  2070       using int by (simp add: integral_0_iff)
```
```  2071     moreover
```
```  2072     have "(\<integral>\<^sup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^sup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
```
```  2073       using A by (intro positive_integral_mono_AE) auto
```
```  2074     then have "(emeasure M) A \<le> (emeasure M) {x \<in> space M. Y x - X x \<noteq> 0}"
```
```  2075       using int A by (simp add: integrable_def)
```
```  2076     ultimately have "emeasure M A = 0"
```
```  2077       using emeasure_nonneg[of M A] by simp
```
```  2078     with `(emeasure M) A \<noteq> 0` show False by auto
```
```  2079   qed
```
```  2080   ultimately show ?thesis by auto
```
```  2081 qed
```
```  2082
```
```  2083 lemma (in finite_measure) integral_less_AE_space:
```
```  2084   assumes int: "integrable M X" "integrable M Y"
```
```  2085   assumes gt: "AE x in M. X x < Y x" "(emeasure M) (space M) \<noteq> 0"
```
```  2086   shows "integral\<^sup>L M X < integral\<^sup>L M Y"
```
```  2087   using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
```
```  2088
```
```  2089 lemma integral_dominated_convergence:
```
```  2090   assumes u[measurable]: "\<And>i. integrable M (u i)" and bound: "\<And>j. AE x in M. \<bar>u j x\<bar> \<le> w x"
```
```  2091   and w[measurable]: "integrable M w"
```
```  2092   and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
```
```  2093   and [measurable]: "u' \<in> borel_measurable M"
```
```  2094   shows "integrable M u'"
```
```  2095   and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar> \<partial>M)) ----> 0" (is "?lim_diff")
```
```  2096   and "(\<lambda>i. integral\<^sup>L M (u i)) ----> integral\<^sup>L M u'" (is ?lim)
```
```  2097 proof -
```
```  2098   have all_bound: "AE x in M. \<forall>j. \<bar>u j x\<bar> \<le> w x"
```
```  2099     using bound by (auto simp: AE_all_countable)
```
```  2100   with u' have u'_bound: "AE x in M. \<bar>u' x\<bar> \<le> w x"
```
```  2101     by eventually_elim (auto intro: LIMSEQ_le_const2 tendsto_rabs)
```
```  2102
```
```  2103   from bound[of 0] have w_pos: "AE x in M. 0 \<le> w x"
```
```  2104     by eventually_elim auto
```
```  2105
```
```  2106   show "integrable M u'"
```
```  2107     by (rule integrable_bound) fact+
```
```  2108
```
```  2109   let ?diff = "\<lambda>n x. 2 * w x - \<bar>u n x - u' x\<bar>"
```
```  2110   have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)"
```
```  2111     using w u `integrable M u'` by (auto intro!: integrable_abs)
```
```  2112
```
```  2113   from u'_bound all_bound
```
```  2114   have diff_less_2w: "AE x in M. \<forall>j. \<bar>u j x - u' x\<bar> \<le> 2 * w x"
```
```  2115   proof (eventually_elim, intro allI)
```
```  2116     fix x j assume *: "\<bar>u' x\<bar> \<le> w x" "\<forall>j. \<bar>u j x\<bar> \<le> w x"
```
```  2117     then have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
```
```  2118     also have "\<dots> \<le> w x + w x"
```
```  2119       using * by (intro add_mono) auto
```
```  2120     finally show "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp
```
```  2121   qed
```
```  2122
```
```  2123   have PI_diff: "\<And>n. (\<integral>\<^sup>+ x. ereal (?diff n x) \<partial>M) =
```
```  2124     (\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) - (\<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
```
```  2125     using diff w diff_less_2w w_pos
```
```  2126     by (subst positive_integral_diff[symmetric])
```
```  2127        (auto simp: integrable_def intro!: positive_integral_cong_AE)
```
```  2128
```
```  2129   have "integrable M (\<lambda>x. 2 * w x)"
```
```  2130     using w by auto
```
```  2131   hence I2w_fin: "(\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
```
```  2132     borel_2w: "(\<lambda>x. ereal (2 * w x)) \<in> borel_measurable M"
```
```  2133     unfolding integrable_def by auto
```
```  2134
```
```  2135   have "limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
```
```  2136   proof cases
```
```  2137     assume eq_0: "(\<integral>\<^sup>+ x. max 0 (ereal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
```
```  2138     { fix n
```
```  2139       have "?f n \<le> ?wx" (is "integral\<^sup>P M ?f' \<le> _")
```
```  2140         using diff_less_2w unfolding positive_integral_max_0
```
```  2141         by (intro positive_integral_mono_AE) auto
```
```  2142       then have "?f n = 0"
```
```  2143         using positive_integral_positive[of M ?f'] eq_0 by auto }
```
```  2144     then show ?thesis by (simp add: Limsup_const)
```
```  2145   next
```
```  2146     assume neq_0: "(\<integral>\<^sup>+ x. max 0 (ereal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
```
```  2147     have "0 = limsup (\<lambda>n. 0 :: ereal)" by (simp add: Limsup_const)
```
```  2148     also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
```
```  2149       by (simp add: Limsup_mono  positive_integral_positive)
```
```  2150     finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" .
```
```  2151     have "?wx = (\<integral>\<^sup>+ x. liminf (\<lambda>n. max 0 (ereal (?diff n x))) \<partial>M)"
```
```  2152       using u'
```
```  2153     proof (intro positive_integral_cong_AE, eventually_elim)
```
```  2154       fix x assume u': "(\<lambda>i. u i x) ----> u' x"
```
```  2155       show "max 0 (ereal (2 * w x)) = liminf (\<lambda>n. max 0 (ereal (?diff n x)))"
```
```  2156         unfolding ereal_max_0
```
```  2157       proof (rule lim_imp_Liminf[symmetric], unfold lim_ereal)
```
```  2158         have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
```
```  2159           using u' by (safe intro!: tendsto_intros)
```
```  2160         then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
```
```  2161           by (auto intro!: tendsto_real_max)
```
```  2162       qed (rule trivial_limit_sequentially)
```
```  2163     qed
```
```  2164     also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^sup>+ x. max 0 (ereal (?diff n x)) \<partial>M)"
```
```  2165       using w u unfolding integrable_def
```
```  2166       by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
```
```  2167     also have "\<dots> = (\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) -
```
```  2168         limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
```
```  2169       unfolding PI_diff positive_integral_max_0
```
```  2170       using positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"]
```
```  2171       by (subst liminf_ereal_cminus) auto
```
```  2172     finally show ?thesis
```
```  2173       using neq_0 I2w_fin positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"] pos
```
```  2174       unfolding positive_integral_max_0
```
```  2175       by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"])
```
```  2176          auto
```
```  2177   qed
```
```  2178
```
```  2179   have "liminf ?f \<le> limsup ?f"
```
```  2180     by (intro Liminf_le_Limsup trivial_limit_sequentially)
```
```  2181   moreover
```
```  2182   { have "0 = liminf (\<lambda>n. 0 :: ereal)" by (simp add: Liminf_const)
```
```  2183     also have "\<dots> \<le> liminf ?f"
```
```  2184       by (simp add: Liminf_mono positive_integral_positive)
```
```  2185     finally have "0 \<le> liminf ?f" . }
```
```  2186   ultimately have liminf_limsup_eq: "liminf ?f = ereal 0" "limsup ?f = ereal 0"
```
```  2187     using `limsup ?f = 0` by auto
```
```  2188   have "\<And>n. (\<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = ereal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
```
```  2189     using diff positive_integral_positive[of M]
```
```  2190     by (subst integral_eq_positive_integral[of _ M]) (auto simp: ereal_real integrable_def)
```
```  2191   then show ?lim_diff
```
```  2192     using Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
```
```  2193     by simp
```
```  2194
```
```  2195   show ?lim
```
```  2196   proof (rule LIMSEQ_I)
```
```  2197     fix r :: real assume "0 < r"
```
```  2198     from LIMSEQ_D[OF `?lim_diff` this]
```
```  2199     obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r"
```
```  2200       using diff by (auto simp: integral_positive)
```
```  2201
```
```  2202     show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^sup>L M (u n) - integral\<^sup>L M u') < r"
```
```  2203     proof (safe intro!: exI[of _ N])
```
```  2204       fix n assume "N \<le> n"
```
```  2205       have "\<bar>integral\<^sup>L M (u n) - integral\<^sup>L M u'\<bar> = \<bar>(\<integral>x. u n x - u' x \<partial>M)\<bar>"
```
```  2206         using u `integrable M u'` by auto
```
```  2207       also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'`
```
```  2208         by (rule_tac integral_triangle_inequality) auto
```
```  2209       also note N[OF `N \<le> n`]
```
```  2210       finally show "norm (integral\<^sup>L M (u n) - integral\<^sup>L M u') < r" by simp
```
```  2211     qed
```
```  2212   qed
```
```  2213 qed
```
```  2214
```
```  2215 lemma integral_sums:
```
```  2216   assumes integrable[measurable]: "\<And>i. integrable M (f i)"
```
```  2217   and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
```
```  2218   and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
```
```  2219   shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
```
```  2220   and "(\<lambda>i. integral\<^sup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
```
```  2221 proof -
```
```  2222   have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
```
```  2223     using summable unfolding summable_def by auto
```
```  2224   from bchoice[OF this]
```
```  2225   obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
```
```  2226   then have w_borel: "w \<in> borel_measurable M" unfolding sums_def
```
```  2227     by (rule borel_measurable_LIMSEQ) auto
```
```  2228
```
```  2229   let ?w = "\<lambda>y. if y \<in> space M then w y else 0"
```
```  2230
```
```  2231   obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
```
```  2232     using sums unfolding summable_def ..
```
```  2233
```
```  2234   have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i = 0..<n. f i x)"
```
```  2235     using integrable by auto
```
```  2236
```
```  2237   have 2: "\<And>j. AE x in M. \<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x"
```
```  2238     using AE_space
```
```  2239   proof eventually_elim
```
```  2240     fix j x assume [simp]: "x \<in> space M"
```
```  2241     have "\<bar>\<Sum>i = 0..< j. f i x\<bar> \<le> (\<Sum>i = 0..< j. \<bar>f i x\<bar>)" by (rule setsum_abs)
```
```  2242     also have "\<dots> \<le> w x" using w[of x] series_pos_le[of "\<lambda>i. \<bar>f i x\<bar>"] unfolding sums_iff by auto
```
```  2243     finally show "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp
```
```  2244   qed
```
```  2245
```
```  2246   have 3: "integrable M ?w"
```
```  2247   proof (rule integral_monotone_convergence(1))
```
```  2248     let ?F = "\<lambda>n y. (\<Sum>i = 0..<n. \<bar>f i y\<bar>)"
```
```  2249     let ?w' = "\<lambda>n y. if y \<in> space M then ?F n y else 0"
```
```  2250     have "\<And>n. integrable M (?F n)"
```
```  2251       using integrable by (auto intro!: integrable_abs)
```
```  2252     thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
```
```  2253     show "AE x in M. mono (\<lambda>n. ?w' n x)"
```
```  2254       by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
```
```  2255     show "AE x in M. (\<lambda>n. ?w' n x) ----> ?w x"
```
```  2256         using w by (simp_all add: tendsto_const sums_def)
```
```  2257     have *: "\<And>n. integral\<^sup>L M (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
```
```  2258       using integrable by (simp add: integrable_abs cong: integral_cong)
```
```  2259     from abs_sum
```
```  2260     show "(\<lambda>i. integral\<^sup>L M (?w' i)) ----> x" unfolding * sums_def .
```
```  2261   qed (simp add: w_borel measurable_If_set)
```
```  2262
```
```  2263   from summable[THEN summable_rabs_cancel]
```
```  2264   have 4: "AE x in M. (\<lambda>n. \<Sum>i = 0..<n. f i x) ----> (\<Sum>i. f i x)"
```
```  2265     by (auto intro: summable_sumr_LIMSEQ_suminf)
```
```  2266
```
```  2267   note int = integral_dominated_convergence(1,3)[OF 1 2 3 4
```
```  2268     borel_measurable_suminf[OF integrableD(1)[OF integrable]]]
```
```  2269
```
```  2270   from int show "integrable M ?S" by simp
```
```  2271
```
```  2272   show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF integrable]
```
```  2273     using int(2) by simp
```
```  2274 qed
```
```  2275
```
```  2276 lemma integrable_mult_indicator:
```
```  2277   "A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. f x * indicator A x)"
```
```  2278   by (rule integrable_bound[where f="\<lambda>x. \<bar>f x\<bar>"])
```
```  2279      (auto intro: integrable_abs split: split_indicator)
```
```  2280
```
```  2281 lemma tendsto_integral_at_top:
```
```  2282   fixes M :: "real measure"
```
```  2283   assumes M: "sets M = sets borel" and f[measurable]: "integrable M f"
```
```  2284   shows "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
```
```  2285 proof -
```
```  2286   have M_measure[simp]: "borel_measurable M = borel_measurable borel"
```
```  2287     using M by (simp add: sets_eq_imp_space_eq measurable_def)
```
```  2288   { fix f assume f: "integrable M f" "\<And>x. 0 \<le> f x"
```
```  2289     then have [measurable]: "f \<in> borel_measurable borel"
```
```  2290       by (simp add: integrable_def)
```
```  2291     have "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
```
```  2292     proof (rule tendsto_at_topI_sequentially)
```
```  2293       have "\<And>j. AE x in M. \<bar>f x * indicator {.. j} x\<bar> \<le> f x"
```
```  2294         using f(2) by (intro AE_I2) (auto split: split_indicator)
```
```  2295       have int: "\<And>n. integrable M (\<lambda>x. f x * indicator {.. n} x)"
```
```  2296         by (rule integrable_mult_indicator) (auto simp: M f)
```
```  2297       show "(\<lambda>n. \<integral> x. f x * indicator {..real n} x \<partial>M) ----> integral\<^sup>L M f"
```
```  2298       proof (rule integral_dominated_convergence)
```
```  2299         { fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
```
```  2300             by (rule eventually_sequentiallyI[of "natceiling x"])
```
```  2301                (auto split: split_indicator simp: natceiling_le_eq) }
```
```  2302         from filterlim_cong[OF refl refl this]
```
```  2303         show "AE x in M. (\<lambda>n. f x * indicator {..real n} x) ----> f x"
```
```  2304           by (simp add: tendsto_const)
```
```  2305       qed (fact+, simp)
```
```  2306       show "mono (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
```
```  2307         by (intro monoI integral_mono int) (auto split: split_indicator intro: f)
```
```  2308     qed }
```
```  2309   note nonneg = this
```
```  2310   let ?P = "\<lambda>y. \<integral> x. max 0 (f x) * indicator {..y} x \<partial>M"
```
```  2311   let ?N = "\<lambda>y. \<integral> x. max 0 (- f x) * indicator {..y} x \<partial>M"
```
```  2312   let ?p = "integral\<^sup>L M (\<lambda>x. max 0 (f x))"
```
```  2313   let ?n = "integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
```
```  2314   have "(?P ---> ?p) at_top" "(?N ---> ?n) at_top"
```
```  2315     by (auto intro!: nonneg integrable_max f)
```
```  2316   note tendsto_diff[OF this]
```
```  2317   also have "(\<lambda>y. ?P y - ?N y) = (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
```
```  2318     by (subst integral_diff(2)[symmetric])
```
```  2319        (auto intro!: integrable_mult_indicator integrable_max f integral_cong ext
```
```  2320              simp: M split: split_max)
```
```  2321   also have "?p - ?n = integral\<^sup>L M f"
```
```  2322     by (subst integral_diff(2)[symmetric])
```
```  2323        (auto intro!: integrable_max f integral_cong ext simp: M split: split_max)
```
```  2324   finally show ?thesis .
```
```  2325 qed
```
```  2326
```
```  2327 lemma integral_monotone_convergence_at_top:
```
```  2328   fixes M :: "real measure"
```
```  2329   assumes M: "sets M = sets borel"
```
```  2330   assumes nonneg: "AE x in M. 0 \<le> f x"
```
```  2331   assumes borel: "f \<in> borel_measurable borel"
```
```  2332   assumes int: "\<And>y. integrable M (\<lambda>x. f x * indicator {.. y} x)"
```
```  2333   assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> x) at_top"
```
```  2334   shows "integrable M f" "integral\<^sup>L M f = x"
```
```  2335 proof -
```
```  2336   from nonneg have "AE x in M. mono (\<lambda>n::nat. f x * indicator {..real n} x)"
```
```  2337     by (auto split: split_indicator intro!: monoI)
```
```  2338   { fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
```
```  2339       by (rule eventually_sequentiallyI[of "natceiling x"])
```
```  2340          (auto split: split_indicator simp: natceiling_le_eq) }
```
```  2341   from filterlim_cong[OF refl refl this]
```
```  2342   have "AE x in M. (\<lambda>i. f x * indicator {..real i} x) ----> f x"
```
```  2343     by (simp add: tendsto_const)
```
```  2344   have "(\<lambda>i. \<integral> x. f x * indicator {..real i} x \<partial>M) ----> x"
```
```  2345     using conv filterlim_real_sequentially by (rule filterlim_compose)
```
```  2346   have M_measure[simp]: "borel_measurable M = borel_measurable borel"
```
```  2347     using M by (simp add: sets_eq_imp_space_eq measurable_def)
```
```  2348   have "f \<in> borel_measurable M"
```
```  2349     using borel by simp
```
```  2350   show "integrable M f"
```
```  2351     by (rule integral_monotone_convergence) fact+
```
```  2352   show "integral\<^sup>L M f = x"
```
```  2353     by (rule integral_monotone_convergence) fact+
```
```  2354 qed
```
```  2355
```
```  2356
```
```  2357 section "Lebesgue integration on countable spaces"
```
```  2358
```
```  2359 lemma integral_on_countable:
```
```  2360   assumes f: "f \<in> borel_measurable M"
```
```  2361   and bij: "bij_betw enum S (f ` space M)"
```
```  2362   and enum_zero: "enum ` (-S) \<subseteq> {0}"
```
```  2363   and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> (emeasure M) (f -` {x} \<inter> space M) \<noteq> \<infinity>"
```
```  2364   and abs_summable: "summable (\<lambda>r. \<bar>enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))\<bar>)"
```
```  2365   shows "integrable M f"
```
```  2366   and "(\<lambda>r. enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))) sums integral\<^sup>L M f" (is ?sums)
```
```  2367 proof -
```
```  2368   let ?A = "\<lambda>r. f -` {enum r} \<inter> space M"
```
```  2369   let ?F = "\<lambda>r x. enum r * indicator (?A r) x"
```
```  2370   have enum_eq: "\<And>r. enum r * real ((emeasure M) (?A r)) = integral\<^sup>L M (?F r)"
```
```  2371     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2372
```
```  2373   { fix x assume "x \<in> space M"
```
```  2374     hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
```
```  2375     then obtain i where "i\<in>S" "enum i = f x" by auto
```
```  2376     have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
```
```  2377     proof cases
```
```  2378       fix j assume "j = i"
```
```  2379       thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
```
```  2380     next
```
```  2381       fix j assume "j \<noteq> i"
```
```  2382       show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
```
```  2383         by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
```
```  2384     qed
```
```  2385     hence F_abs: "\<And>j. \<bar>if j = i then f x else 0\<bar> = (if j = i then \<bar>f x\<bar> else 0)" by auto
```
```  2386     have "(\<lambda>i. ?F i x) sums f x"
```
```  2387          "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
```
```  2388       by (auto intro!: sums_single simp: F F_abs) }
```
```  2389   note F_sums_f = this(1) and F_abs_sums_f = this(2)
```
```  2390
```
```  2391   have int_f: "integral\<^sup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
```
```  2392     using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
```
```  2393
```
```  2394   { fix r
```
```  2395     have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
```
```  2396       by (auto simp: indicator_def intro!: integral_cong)
```
```  2397     also have "\<dots> = \<bar>enum r\<bar> * real ((emeasure M) (?A r))"
```
```  2398       using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2399     finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real ((emeasure M) (?A r))\<bar>"
```
```  2400       using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) }
```
```  2401   note int_abs_F = this
```
```  2402
```
```  2403   have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
```
```  2404     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2405
```
```  2406   have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
```
```  2407     using F_abs_sums_f unfolding sums_iff by auto
```
```  2408
```
```  2409   from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
```
```  2410   show ?sums unfolding enum_eq int_f by simp
```
```  2411
```
```  2412   from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
```
```  2413   show "integrable M f" unfolding int_f by simp
```
```  2414 qed
```
```  2415
```
```  2416 section {* Distributions *}
```
```  2417
```
```  2418 lemma positive_integral_distr':
```
```  2419   assumes T: "T \<in> measurable M M'"
```
```  2420   and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x"
```
```  2421   shows "integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
```
```  2422   using f
```
```  2423 proof induct
```
```  2424   case (cong f g)
```
```  2425   with T show ?case
```
```  2426     apply (subst positive_integral_cong[of _ f g])
```
```  2427     apply simp
```
```  2428     apply (subst positive_integral_cong[of _ "\<lambda>x. f (T x)" "\<lambda>x. g (T x)"])
```
```  2429     apply (simp add: measurable_def Pi_iff)
```
```  2430     apply simp
```
```  2431     done
```
```  2432 next
```
```  2433   case (set A)
```
```  2434   then have eq: "\<And>x. x \<in> space M \<Longrightarrow> indicator A (T x) = indicator (T -` A \<inter> space M) x"
```
```  2435     by (auto simp: indicator_def)
```
```  2436   from set T show ?case
```
```  2437     by (subst positive_integral_cong[OF eq])
```
```  2438        (auto simp add: emeasure_distr intro!: positive_integral_indicator[symmetric] measurable_sets)
```
```  2439 qed (simp_all add: measurable_compose[OF T] T positive_integral_cmult positive_integral_add
```
```  2440                    positive_integral_monotone_convergence_SUP le_fun_def incseq_def)
```
```  2441
```
```  2442 lemma positive_integral_distr:
```
```  2443   "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
```
```  2444   by (subst (1 2) positive_integral_max_0[symmetric])
```
```  2445      (simp add: positive_integral_distr')
```
```  2446
```
```  2447 lemma integral_distr:
```
```  2448   "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>L (distr M M' T) f = (\<integral> x. f (T x) \<partial>M)"
```
```  2449   unfolding lebesgue_integral_def
```
```  2450   by (subst (1 2) positive_integral_distr) auto
```
```  2451
```
```  2452 lemma integrable_distr_eq:
```
```  2453   "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integrable (distr M M' T) f \<longleftrightarrow> integrable M (\<lambda>x. f (T x))"
```
```  2454   unfolding integrable_def
```
```  2455   by (subst (1 2) positive_integral_distr) (auto simp: comp_def)
```
```  2456
```
```  2457 lemma integrable_distr:
```
```  2458   "T \<in> measurable M M' \<Longrightarrow> integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))"
```
```  2459   by (subst integrable_distr_eq[symmetric]) auto
```
```  2460
```
```  2461 section {* Lebesgue integration on @{const count_space} *}
```
```  2462
```
```  2463 lemma simple_function_count_space[simp]:
```
```  2464   "simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
```
```  2465   unfolding simple_function_def by simp
```
```  2466
```
```  2467 lemma positive_integral_count_space:
```
```  2468   assumes A: "finite {a\<in>A. 0 < f a}"
```
```  2469   shows "integral\<^sup>P (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
```
```  2470 proof -
```
```  2471   have *: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>count_space A) =
```
```  2472     (\<integral>\<^sup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)"
```
```  2473     by (auto intro!: positive_integral_cong
```
```  2474              simp add: indicator_def if_distrib setsum_cases[OF A] max_def le_less)
```
```  2475   also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^sup>+ x. f a * indicator {a} x \<partial>count_space A)"
```
```  2476     by (subst positive_integral_setsum)
```
```  2477        (simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le)
```
```  2478   also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
```
```  2479     by (auto intro!: setsum_cong simp: positive_integral_cmult_indicator one_ereal_def[symmetric])
```
```  2480   finally show ?thesis by (simp add: positive_integral_max_0)
```
```  2481 qed
```
```  2482
```
```  2483 lemma integrable_count_space:
```
```  2484   "finite X \<Longrightarrow> integrable (count_space X) f"
```
```  2485   by (auto simp: positive_integral_count_space integrable_def)
```
```  2486
```
```  2487 lemma positive_integral_count_space_finite:
```
```  2488     "finite A \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))"
```
```  2489   by (subst positive_integral_max_0[symmetric])
```
```  2490      (auto intro!: setsum_mono_zero_left simp: positive_integral_count_space less_le)
```
```  2491
```
```  2492 lemma lebesgue_integral_count_space_finite_support:
```
```  2493   assumes f: "finite {a\<in>A. f a \<noteq> 0}" shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)"
```
```  2494 proof -
```
```  2495   have *: "\<And>r::real. 0 < max 0 r \<longleftrightarrow> 0 < r" "\<And>x. max 0 (ereal x) = ereal (max 0 x)"
```
```  2496     "\<And>a. a \<in> A \<and> 0 < f a \<Longrightarrow> max 0 (f a) = f a"
```
```  2497     "\<And>a. a \<in> A \<and> f a < 0 \<Longrightarrow> max 0 (- f a) = - f a"
```
```  2498     "{a \<in> A. f a \<noteq> 0} = {a \<in> A. 0 < f a} \<union> {a \<in> A. f a < 0}"
```
```  2499     "({a \<in> A. 0 < f a} \<inter> {a \<in> A. f a < 0}) = {}"
```
```  2500     by (auto split: split_max)
```
```  2501   have "finite {a \<in> A. 0 < f a}" "finite {a \<in> A. f a < 0}"
```
```  2502     by (auto intro: finite_subset[OF _ f])
```
```  2503   then show ?thesis
```
```  2504     unfolding lebesgue_integral_def
```
```  2505     apply (subst (1 2) positive_integral_max_0[symmetric])
```
```  2506     apply (subst (1 2) positive_integral_count_space)
```
```  2507     apply (auto simp add: * setsum_negf setsum_Un
```
```  2508                 simp del: ereal_max)
```
```  2509     done
```
```  2510 qed
```
```  2511
```
```  2512 lemma lebesgue_integral_count_space_finite:
```
```  2513     "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
```
```  2514   apply (auto intro!: setsum_mono_zero_left
```
```  2515               simp: positive_integral_count_space_finite lebesgue_integral_def)
```
```  2516   apply (subst (1 2)  setsum_real_of_ereal[symmetric])
```
```  2517   apply (auto simp: max_def setsum_subtractf[symmetric] intro!: setsum_cong)
```
```  2518   done
```
```  2519
```
```  2520 lemma borel_measurable_count_space[simp, intro!]:
```
```  2521   "f \<in> borel_measurable (count_space A)"
```
```  2522   by simp
```
```  2523
```
```  2524 lemma lessThan_eq_empty_iff: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
```
```  2525   by auto
```
```  2526
```
```  2527 lemma emeasure_UN_countable:
```
```  2528   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I: "countable I"
```
```  2529   assumes disj: "disjoint_family_on X I"
```
```  2530   shows "emeasure M (UNION I X) = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
```
```  2531 proof cases
```
```  2532   assume "finite I" with sets disj show ?thesis
```
```  2533     by (subst setsum_emeasure[symmetric])
```
```  2534        (auto intro!: setsum_cong simp add: max_def subset_eq positive_integral_count_space_finite emeasure_nonneg)
```
```  2535 next
```
```  2536   assume f: "\<not> finite I"
```
```  2537   then have [intro]: "I \<noteq> {}" by auto
```
```  2538   from from_nat_into_inj_infinite[OF I f] from_nat_into[OF this] disj
```
```  2539   have disj2: "disjoint_family (\<lambda>i. X (from_nat_into I i))"
```
```  2540     unfolding disjoint_family_on_def by metis
```
```  2541
```
```  2542   from f have "bij_betw (from_nat_into I) UNIV I"
```
```  2543     using bij_betw_from_nat_into[OF I] by simp
```
```  2544   then have "(\<Union>i\<in>I. X i) = (\<Union>i. (X \<circ> from_nat_into I) i)"
```
```  2545     unfolding SUP_def image_comp [symmetric] by (simp add: bij_betw_def)
```
```  2546   then have "emeasure M (UNION I X) = emeasure M (\<Union>i. X (from_nat_into I i))"
```
```  2547     by simp
```
```  2548   also have "\<dots> = (\<Sum>i. emeasure M (X (from_nat_into I i)))"
```
```  2549     by (intro suminf_emeasure[symmetric] disj disj2) (auto intro!: sets from_nat_into[OF `I \<noteq> {}`])
```
```  2550   also have "\<dots> = (\<Sum>n. \<integral>\<^sup>+i. emeasure M (X i) * indicator {from_nat_into I n} i \<partial>count_space I)"
```
```  2551   proof (intro arg_cong[where f=suminf] ext)
```
```  2552     fix i
```
```  2553     have eq: "{a \<in> I. 0 < emeasure M (X a) * indicator {from_nat_into I i} a}
```
```  2554      = (if 0 < emeasure M (X (from_nat_into I i)) then {from_nat_into I i} else {})"
```
```  2555      using ereal_0_less_1
```
```  2556      by (auto simp: ereal_zero_less_0_iff indicator_def from_nat_into `I \<noteq> {}` simp del: ereal_0_less_1)
```
```  2557     have "(\<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I) =
```
```  2558       (if 0 < emeasure M (X (from_nat_into I i)) then emeasure M (X (from_nat_into I i)) else 0)"
```
```  2559       by (subst positive_integral_count_space) (simp_all add: eq)
```
```  2560     also have "\<dots> = emeasure M (X (from_nat_into I i))"
```
```  2561       by (simp add: less_le emeasure_nonneg)
```
```  2562     finally show "emeasure M (X (from_nat_into I i)) =
```
```  2563          \<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I" ..
```
```  2564   qed
```
```  2565   also have "\<dots> = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
```
```  2566     apply (subst positive_integral_suminf[symmetric])
```
```  2567     apply (auto simp: emeasure_nonneg intro!: positive_integral_cong)
```
```  2568   proof -
```
```  2569     fix x assume "x \<in> I"
```
```  2570     then have "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = (\<Sum>i\<in>{to_nat_on I x}. emeasure M (X x) * indicator {from_nat_into I i} x)"
```
```  2571       by (intro suminf_finite) (auto simp: indicator_def I f)
```
```  2572     also have "\<dots> = emeasure M (X x)"
```
```  2573       by (simp add: I f `x\<in>I`)
```
```  2574     finally show "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = emeasure M (X x)" .
```
```  2575   qed
```
```  2576   finally show ?thesis .
```
```  2577 qed
```
```  2578
```
```  2579 section {* Measures with Restricted Space *}
```
```  2580
```
```  2581 lemma positive_integral_restrict_space:
```
```  2582   assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "\<And>x. x \<in> space M - \<Omega> \<Longrightarrow> f x = 0"
```
```  2583   shows "positive_integral (restrict_space M \<Omega>) f = positive_integral M f"
```
```  2584 using f proof (induct rule: borel_measurable_induct)
```
```  2585   case (cong f g) then show ?case
```
```  2586     using positive_integral_cong[of M f g] positive_integral_cong[of "restrict_space M \<Omega>" f g]
```
```  2587       sets.sets_into_space[OF `\<Omega> \<in> sets M`]
```
```  2588     by (simp add: subset_eq space_restrict_space)
```
```  2589 next
```
```  2590   case (set A)
```
```  2591   then have "A \<subseteq> \<Omega>"
```
```  2592     unfolding indicator_eq_0_iff by (auto dest: sets.sets_into_space)
```
```  2593   with set `\<Omega> \<in> sets M` sets.sets_into_space[OF `\<Omega> \<in> sets M`] show ?case
```
```  2594     by (subst positive_integral_indicator')
```
```  2595        (auto simp add: sets_restrict_space_iff space_restrict_space
```
```  2596                   emeasure_restrict_space Int_absorb2
```
```  2597                 dest: sets.sets_into_space)
```
```  2598 next
```
```  2599   case (mult f c) then show ?case
```
```  2600     by (cases "c = 0") (simp_all add: measurable_restrict_space1 \<Omega> positive_integral_cmult)
```
```  2601 next
```
```  2602   case (add f g) then show ?case
```
```  2603     by (simp add: measurable_restrict_space1 \<Omega> positive_integral_add ereal_add_nonneg_eq_0_iff)
```
```  2604 next
```
```  2605   case (seq F) then show ?case
```
```  2606     by (auto simp add: SUP_eq_iff measurable_restrict_space1 \<Omega> positive_integral_monotone_convergence_SUP)
```
```  2607 qed
```
```  2608
```
```  2609 section {* Measure spaces with an associated density *}
```
```  2610
```
```  2611 definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
```
```  2612   "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
```
```  2613
```
```  2614 lemma
```
```  2615   shows sets_density[simp]: "sets (density M f) = sets M"
```
```  2616     and space_density[simp]: "space (density M f) = space M"
```
```  2617   by (auto simp: density_def)
```
```  2618
```
```  2619 (* FIXME: add conversion to simplify space, sets and measurable *)
```
```  2620 lemma space_density_imp[measurable_dest]:
```
```  2621   "\<And>x M f. x \<in> space (density M f) \<Longrightarrow> x \<in> space M" by auto
```
```  2622
```
```  2623 lemma
```
```  2624   shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'"
```
```  2625     and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'"
```
```  2626     and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u"
```
```  2627   unfolding measurable_def simple_function_def by simp_all
```
```  2628
```
```  2629 lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow>
```
```  2630   (AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'"
```
```  2631   unfolding density_def by (auto intro!: measure_of_eq positive_integral_cong_AE sets.space_closed)
```
```  2632
```
```  2633 lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))"
```
```  2634 proof -
```
```  2635   have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)"
```
```  2636     by (auto simp: indicator_def)
```
```  2637   then show ?thesis
```
```  2638     unfolding density_def by (simp add: positive_integral_max_0)
```
```  2639 qed
```
```  2640
```
```  2641 lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))"
```
```  2642   by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max)
```
```  2643
```
```  2644 lemma emeasure_density:
```
```  2645   assumes f[measurable]: "f \<in> borel_measurable M" and A[measurable]: "A \<in> sets M"
```
```  2646   shows "emeasure (density M f) A = (\<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
```
```  2647     (is "_ = ?\<mu> A")
```
```  2648   unfolding density_def
```
```  2649 proof (rule emeasure_measure_of_sigma)
```
```  2650   show "sigma_algebra (space M) (sets M)" ..
```
```  2651   show "positive (sets M) ?\<mu>"
```
```  2652     using f by (auto simp: positive_def intro!: positive_integral_positive)
```
```  2653   have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^sup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'")
```
```  2654     apply (subst positive_integral_max_0[symmetric])
```
```  2655     apply (intro ext positive_integral_cong_AE AE_I2)
```
```  2656     apply (auto simp: indicator_def)
```
```  2657     done
```
```  2658   show "countably_additive (sets M) ?\<mu>"
```
```  2659     unfolding \<mu>_eq
```
```  2660   proof (intro countably_additiveI)
```
```  2661     fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M"
```
```  2662     then have "\<And>i. A i \<in> sets M" by auto
```
```  2663     then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M"
```
```  2664       by (auto simp: set_eq_iff)
```
```  2665     assume disj: "disjoint_family A"
```
```  2666     have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^sup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)"
```
```  2667       using f * by (simp add: positive_integral_suminf)
```
```  2668     also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f
```
```  2669       by (auto intro!: suminf_cmult_ereal positive_integral_cong_AE)
```
```  2670     also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)"
```
```  2671       unfolding suminf_indicator[OF disj] ..
```
```  2672     finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp
```
```  2673   qed
```
```  2674 qed fact
```
```  2675
```
```  2676 lemma null_sets_density_iff:
```
```  2677   assumes f: "f \<in> borel_measurable M"
```
```  2678   shows "A \<in> null_sets (density M f) \<longleftrightarrow> A \<in> sets M \<and> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
```
```  2679 proof -
```
```  2680   { assume "A \<in> sets M"
```
```  2681     have eq: "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. max 0 (f x) * indicator A x \<partial>M)"
```
```  2682       apply (subst positive_integral_max_0[symmetric])
```
```  2683       apply (intro positive_integral_cong)
```
```  2684       apply (auto simp: indicator_def)
```
```  2685       done
```
```  2686     have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow>
```
```  2687       emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
```
```  2688       unfolding eq
```
```  2689       using f `A \<in> sets M`
```
```  2690       by (intro positive_integral_0_iff) auto
```
```  2691     also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
```
```  2692       using f `A \<in> sets M`
```
```  2693       by (intro AE_iff_measurable[OF _ refl, symmetric]) auto
```
```  2694     also have "(AE x in M. max 0 (f x) * indicator A x = 0) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
```
```  2695       by (auto simp add: indicator_def max_def split: split_if_asm)
```
```  2696     finally have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" . }
```
```  2697   with f show ?thesis
```
```  2698     by (simp add: null_sets_def emeasure_density cong: conj_cong)
```
```  2699 qed
```
```  2700
```
```  2701 lemma AE_density:
```
```  2702   assumes f: "f \<in> borel_measurable M"
```
```  2703   shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)"
```
```  2704 proof
```
```  2705   assume "AE x in density M f. P x"
```
```  2706   with f obtain N where "{x \<in> space M. \<not> P x} \<subseteq> N" "N \<in> sets M" and ae: "AE x in M. x \<in> N \<longrightarrow> f x \<le> 0"
```
```  2707     by (auto simp: eventually_ae_filter null_sets_density_iff)
```
```  2708   then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto
```
```  2709   with ae show "AE x in M. 0 < f x \<longrightarrow> P x"
```
```  2710     by (rule eventually_elim2) auto
```
```  2711 next
```
```  2712   fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x"
```
```  2713   then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M"
```
```  2714     by (auto simp: eventually_ae_filter)
```
```  2715   then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}"
```
```  2716     "N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N"
```
```  2717     using f by (auto simp: subset_eq intro!: sets.sets_Collect_neg AE_not_in)
```
```  2718   show "AE x in density M f. P x"
```
```  2719     using ae2
```
```  2720     unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f]
```
```  2721     by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *)
```
```  2722        (auto elim: eventually_elim2)
```
```  2723 qed
```
```  2724
```
```  2725 lemma positive_integral_density':
```
```  2726   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
```
```  2727   assumes g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
```
```  2728   shows "integral\<^sup>P (density M f) g = (\<integral>\<^sup>+ x. f x * g x \<partial>M)"
```
```  2729 using g proof induct
```
```  2730   case (cong u v)
```
```  2731   then show ?case
```
```  2732     apply (subst positive_integral_cong[OF cong(3)])
```
```  2733     apply (simp_all cong: positive_integral_cong)
```
```  2734     done
```
```  2735 next
```
```  2736   case (set A) then show ?case
```
```  2737     by (simp add: emeasure_density f)
```
```  2738 next
```
```  2739   case (mult u c)
```
```  2740   moreover have "\<And>x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps)
```
```  2741   ultimately show ?case
```
```  2742     using f by (simp add: positive_integral_cmult)
```
```  2743 next
```
```  2744   case (add u v)
```
```  2745   then have "\<And>x. f x * (v x + u x) = f x * v x + f x * u x"
```
```  2746     by (simp add: ereal_right_distrib)
```
```  2747   with add f show ?case
```
```  2748     by (auto simp add: positive_integral_add ereal_zero_le_0_iff intro!: positive_integral_add[symmetric])
```
```  2749 next
```
```  2750   case (seq U)
```
```  2751   from f(2) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)"
```
```  2752     by eventually_elim (simp add: SUPR_ereal_cmult seq)
```
```  2753   from seq f show ?case
```
```  2754     apply (simp add: positive_integral_monotone_convergence_SUP)
```
```  2755     apply (subst positive_integral_cong_AE[OF eq])
```
```  2756     apply (subst positive_integral_monotone_convergence_SUP_AE)
```
```  2757     apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono)
```
```  2758     done
```
```  2759 qed
```
```  2760
```
```  2761 lemma positive_integral_density:
```
```  2762   "f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow>
```
```  2763     integral\<^sup>P (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)"
```
```  2764   by (subst (1 2) positive_integral_max_0[symmetric])
```
```  2765      (auto intro!: positive_integral_cong_AE
```
```  2766            simp: measurable_If max_def ereal_zero_le_0_iff positive_integral_density')
```
```  2767
```
```  2768 lemma integral_density:
```
```  2769   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
```
```  2770     and g: "g \<in> borel_measurable M"
```
```  2771   shows "integral\<^sup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
```
```  2772     and "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
```
```  2773   unfolding lebesgue_integral_def integrable_def using f g
```
```  2774   by (auto simp: positive_integral_density)
```
```  2775
```
```  2776 lemma emeasure_restricted:
```
```  2777   assumes S: "S \<in> sets M" and X: "X \<in> sets M"
```
```  2778   shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
```
```  2779 proof -
```
```  2780   have "emeasure (density M (indicator S)) X = (\<integral>\<^sup>+x. indicator S x * indicator X x \<partial>M)"
```
```  2781     using S X by (simp add: emeasure_density)
```
```  2782   also have "\<dots> = (\<integral>\<^sup>+x. indicator (S \<inter> X) x \<partial>M)"
```
```  2783     by (auto intro!: positive_integral_cong simp: indicator_def)
```
```  2784   also have "\<dots> = emeasure M (S \<inter> X)"
```
```  2785     using S X by (simp add: sets.Int)
```
```  2786   finally show ?thesis .
```
```  2787 qed
```
```  2788
```
```  2789 lemma measure_restricted:
```
```  2790   "S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)"
```
```  2791   by (simp add: emeasure_restricted measure_def)
```
```  2792
```
```  2793 lemma (in finite_measure) finite_measure_restricted:
```
```  2794   "S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))"
```
```  2795   by default (simp add: emeasure_restricted)
```
```  2796
```
```  2797 lemma emeasure_density_const:
```
```  2798   "A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
```
```  2799   by (auto simp: positive_integral_cmult_indicator emeasure_density)
```
```  2800
```
```  2801 lemma measure_density_const:
```
```  2802   "A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A"
```
```  2803   by (auto simp: emeasure_density_const measure_def)
```
```  2804
```
```  2805 lemma density_density_eq:
```
```  2806    "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow>
```
```  2807    density (density M f) g = density M (\<lambda>x. f x * g x)"
```
```  2808   by (auto intro!: measure_eqI simp: emeasure_density positive_integral_density ac_simps)
```
```  2809
```
```  2810 lemma distr_density_distr:
```
```  2811   assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
```
```  2812     and inv: "\<forall>x\<in>space M. T' (T x) = x"
```
```  2813   assumes f: "f \<in> borel_measurable M'"
```
```  2814   shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L")
```
```  2815 proof (rule measure_eqI)
```
```  2816   fix A assume A: "A \<in> sets ?R"
```
```  2817   { fix x assume "x \<in> space M"
```
```  2818     with sets.sets_into_space[OF A]
```
```  2819     have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)"
```
```  2820       using T inv by (auto simp: indicator_def measurable_space) }
```
```  2821   with A T T' f show "emeasure ?R A = emeasure ?L A"
```
```  2822     by (simp add: measurable_comp emeasure_density emeasure_distr
```
```  2823                   positive_integral_distr measurable_sets cong: positive_integral_cong)
```
```  2824 qed simp
```
```  2825
```
```  2826 lemma density_density_divide:
```
```  2827   fixes f g :: "'a \<Rightarrow> real"
```
```  2828   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
```
```  2829   assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
```
```  2830   assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
```
```  2831   shows "density (density M f) (\<lambda>x. g x / f x) = density M g"
```
```  2832 proof -
```
```  2833   have "density M g = density M (\<lambda>x. f x * (g x / f x))"
```
```  2834     using f g ac by (auto intro!: density_cong measurable_If)
```
```  2835   then show ?thesis
```
```  2836     using f g by (subst density_density_eq) auto
```
```  2837 qed
```
```  2838
```
```  2839 section {* Point measure *}
```
```  2840
```
```  2841 definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
```
```  2842   "point_measure A f = density (count_space A) f"
```
```  2843
```
```  2844 lemma
```
```  2845   shows space_point_measure: "space (point_measure A f) = A"
```
```  2846     and sets_point_measure: "sets (point_measure A f) = Pow A"
```
```  2847   by (auto simp: point_measure_def)
```
```  2848
```
```  2849 lemma measurable_point_measure_eq1[simp]:
```
```  2850   "g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M"
```
```  2851   unfolding point_measure_def by simp
```
```  2852
```
```  2853 lemma measurable_point_measure_eq2_finite[simp]:
```
```  2854   "finite A \<Longrightarrow>
```
```  2855    g \<in> measurable M (point_measure A f) \<longleftrightarrow>
```
```  2856     (g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))"
```
```  2857   unfolding point_measure_def by (simp add: measurable_count_space_eq2)
```
```  2858
```
```  2859 lemma simple_function_point_measure[simp]:
```
```  2860   "simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)"
```
```  2861   by (simp add: point_measure_def)
```
```  2862
```
```  2863 lemma emeasure_point_measure:
```
```  2864   assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A"
```
```  2865   shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)"
```
```  2866 proof -
```
```  2867   have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}"
```
```  2868     using `X \<subseteq> A` by auto
```
```  2869   with A show ?thesis
```
```  2870     by (simp add: emeasure_density positive_integral_count_space ereal_zero_le_0_iff
```
```  2871                   point_measure_def indicator_def)
```
```  2872 qed
```
```  2873
```
```  2874 lemma emeasure_point_measure_finite:
```
```  2875   "finite A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
```
```  2876   by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
```
```  2877
```
```  2878 lemma emeasure_point_measure_finite2:
```
```  2879   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> (\<And>i. i \<in> X \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
```
```  2880   by (subst emeasure_point_measure)
```
```  2881      (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
```
```  2882
```
```  2883 lemma null_sets_point_measure_iff:
```
```  2884   "X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)"
```
```  2885  by (auto simp: AE_count_space null_sets_density_iff point_measure_def)
```
```  2886
```
```  2887 lemma AE_point_measure:
```
```  2888   "(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)"
```
```  2889   unfolding point_measure_def
```
```  2890   by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def)
```
```  2891
```
```  2892 lemma positive_integral_point_measure:
```
```  2893   "finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow>
```
```  2894     integral\<^sup>P (point_measure A f) g = (\<Sum>a|a\<in>A \<and> 0 < f a \<and> 0 < g a. f a * g a)"
```
```  2895   unfolding point_measure_def
```
```  2896   apply (subst density_max_0)
```
```  2897   apply (subst positive_integral_density)
```
```  2898   apply (simp_all add: AE_count_space positive_integral_density)
```
```  2899   apply (subst positive_integral_count_space )
```
```  2900   apply (auto intro!: setsum_cong simp: max_def ereal_zero_less_0_iff)
```
```  2901   apply (rule finite_subset)
```
```  2902   prefer 2
```
```  2903   apply assumption
```
```  2904   apply auto
```
```  2905   done
```
```  2906
```
```  2907 lemma positive_integral_point_measure_finite:
```
```  2908   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> g a) \<Longrightarrow>
```
```  2909     integral\<^sup>P (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
```
```  2910   by (subst positive_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le)
```
```  2911
```
```  2912 lemma lebesgue_integral_point_measure_finite:
```
```  2913   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> integral\<^sup>L (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
```
```  2914   by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
```
```  2915
```
```  2916 lemma integrable_point_measure_finite:
```
```  2917   "finite A \<Longrightarrow> integrable (point_measure A (\<lambda>x. ereal (f x))) g"
```
```  2918   unfolding point_measure_def
```
```  2919   apply (subst density_ereal_max_0)
```
```  2920   apply (subst integral_density)
```
```  2921   apply (auto simp: AE_count_space integrable_count_space)
```
```  2922   done
```
```  2923
```
```  2924 section {* Uniform measure *}
```
```  2925
```
```  2926 definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
```
```  2927
```
```  2928 lemma
```
```  2929   shows sets_uniform_measure[simp]: "sets (uniform_measure M A) = sets M"
```
```  2930     and space_uniform_measure[simp]: "space (uniform_measure M A) = space M"
```
```  2931   by (auto simp: uniform_measure_def)
```
```  2932
```
```  2933 lemma emeasure_uniform_measure[simp]:
```
```  2934   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
```
```  2935   shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
```
```  2936 proof -
```
```  2937   from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^sup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)"
```
```  2938     by (auto simp add: uniform_measure_def emeasure_density split: split_indicator
```
```  2939              intro!: positive_integral_cong)
```
```  2940   also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A"
```
```  2941     using A B
```
```  2942     by (subst positive_integral_cmult_indicator) (simp_all add: sets.Int emeasure_nonneg)
```
```  2943   finally show ?thesis .
```
```  2944 qed
```
```  2945
```
```  2946 lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
```
```  2947   using emeasure_notin_sets[of A M] by blast
```
```  2948
```
```  2949 lemma measure_uniform_measure[simp]:
```
```  2950   assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M"
```
```  2951   shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A"
```
```  2952   using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A
```
```  2953   by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def)
```
```  2954
```
```  2955 section {* Uniform count measure *}
```
```  2956
```
```  2957 definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
```
```  2958
```
```  2959 lemma
```
```  2960   shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
```
```  2961     and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
```
```  2962     unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)
```
```  2963
```
```  2964 lemma emeasure_uniform_count_measure:
```
```  2965   "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
```
```  2966   by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def)
```
```  2967
```
```  2968 lemma measure_uniform_count_measure:
```
```  2969   "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A"
```
```  2970   by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def)
```
```  2971
```
```  2972 end
```