src/HOL/Probability/Lebesgue_Integration.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 43941 481566bc20e4 child 44568 e6f291cb5810 permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     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 Borel_Space
```
```    10 begin
```
```    11
```
```    12 lemma real_ereal_1[simp]: "real (1::ereal) = 1"
```
```    13   unfolding one_ereal_def by simp
```
```    14
```
```    15 lemma ereal_indicator_pos[simp,intro]: "0 \<le> (indicator A x ::ereal)"
```
```    16   unfolding indicator_def by auto
```
```    17
```
```    18 lemma tendsto_real_max:
```
```    19   fixes x y :: real
```
```    20   assumes "(X ---> x) net"
```
```    21   assumes "(Y ---> y) net"
```
```    22   shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
```
```    23 proof -
```
```    24   have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
```
```    25     by (auto split: split_max simp: field_simps)
```
```    26   show ?thesis
```
```    27     unfolding *
```
```    28     by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
```
```    29 qed
```
```    30
```
```    31 lemma (in measure_space) measure_Union:
```
```    32   assumes "finite S" "S \<subseteq> sets M" "\<And>A B. A \<in> S \<Longrightarrow> B \<in> S \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}"
```
```    33   shows "setsum \<mu> S = \<mu> (\<Union>S)"
```
```    34 proof -
```
```    35   have "setsum \<mu> S = \<mu> (\<Union>i\<in>S. i)"
```
```    36     using assms by (intro measure_setsum[OF `finite S`]) (auto simp: disjoint_family_on_def)
```
```    37   also have "\<dots> = \<mu> (\<Union>S)" by (auto intro!: arg_cong[where f=\<mu>])
```
```    38   finally show ?thesis .
```
```    39 qed
```
```    40
```
```    41 lemma (in sigma_algebra) measurable_sets2[intro]:
```
```    42   assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
```
```    43   and "A \<in> sets M'" "B \<in> sets M''"
```
```    44   shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
```
```    45 proof -
```
```    46   have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
```
```    47     by auto
```
```    48   then show ?thesis using assms by (auto intro: measurable_sets)
```
```    49 qed
```
```    50
```
```    51 lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
```
```    52 proof
```
```    53   assume "\<forall>n. f n \<le> f (Suc n)" then show "incseq f" by (auto intro!: incseq_SucI)
```
```    54 qed (auto simp: incseq_def)
```
```    55
```
```    56 lemma borel_measurable_real_floor:
```
```    57   "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
```
```    58   unfolding borel.borel_measurable_iff_ge
```
```    59 proof (intro allI)
```
```    60   fix a :: real
```
```    61   { fix x have "a \<le> real \<lfloor>x\<rfloor> \<longleftrightarrow> real \<lceil>a\<rceil> \<le> x"
```
```    62       using le_floor_eq[of "\<lceil>a\<rceil>" x] ceiling_le_iff[of a "\<lfloor>x\<rfloor>"]
```
```    63       unfolding real_eq_of_int by simp }
```
```    64   then have "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} = {real \<lceil>a\<rceil>..}" by auto
```
```    65   then show "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} \<in> sets borel" by auto
```
```    66 qed
```
```    67
```
```    68 lemma measure_preservingD2:
```
```    69   "f \<in> measure_preserving A B \<Longrightarrow> f \<in> measurable A B"
```
```    70   unfolding measure_preserving_def by auto
```
```    71
```
```    72 lemma measure_preservingD3:
```
```    73   "f \<in> measure_preserving A B \<Longrightarrow> f \<in> space A \<rightarrow> space B"
```
```    74   unfolding measure_preserving_def measurable_def by auto
```
```    75
```
```    76 lemma measure_preservingD:
```
```    77   "T \<in> measure_preserving A B \<Longrightarrow> X \<in> sets B \<Longrightarrow> measure A (T -` X \<inter> space A) = measure B X"
```
```    78   unfolding measure_preserving_def by auto
```
```    79
```
```    80 lemma (in sigma_algebra) borel_measurable_real_natfloor[intro, simp]:
```
```    81   assumes "f \<in> borel_measurable M"
```
```    82   shows "(\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
```
```    83 proof -
```
```    84   have "\<And>x. real (natfloor (f x)) = max 0 (real \<lfloor>f x\<rfloor>)"
```
```    85     by (auto simp: max_def natfloor_def)
```
```    86   with borel_measurable_max[OF measurable_comp[OF assms borel_measurable_real_floor] borel_measurable_const]
```
```    87   show ?thesis by (simp add: comp_def)
```
```    88 qed
```
```    89
```
```    90 lemma (in measure_space) AE_not_in:
```
```    91   assumes N: "N \<in> null_sets" shows "AE x. x \<notin> N"
```
```    92   using N by (rule AE_I') auto
```
```    93
```
```    94 lemma sums_If_finite:
```
```    95   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```    96   assumes finite: "finite {r. P r}"
```
```    97   shows "(\<lambda>r. if P r then f r else 0) sums (\<Sum>r\<in>{r. P r}. f r)" (is "?F sums _")
```
```    98 proof cases
```
```    99   assume "{r. P r} = {}" hence "\<And>r. \<not> P r" by auto
```
```   100   thus ?thesis by (simp add: sums_zero)
```
```   101 next
```
```   102   assume not_empty: "{r. P r} \<noteq> {}"
```
```   103   have "?F sums (\<Sum>r = 0..< Suc (Max {r. P r}). ?F r)"
```
```   104     by (rule series_zero)
```
```   105        (auto simp add: Max_less_iff[OF finite not_empty] less_eq_Suc_le[symmetric])
```
```   106   also have "(\<Sum>r = 0..< Suc (Max {r. P r}). ?F r) = (\<Sum>r\<in>{r. P r}. f r)"
```
```   107     by (subst setsum_cases)
```
```   108        (auto intro!: setsum_cong simp: Max_ge_iff[OF finite not_empty] less_Suc_eq_le)
```
```   109   finally show ?thesis .
```
```   110 qed
```
```   111
```
```   112 lemma sums_single:
```
```   113   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   114   shows "(\<lambda>r. if r = i then f r else 0) sums f i"
```
```   115   using sums_If_finite[of "\<lambda>r. r = i" f] by simp
```
```   116
```
```   117 section "Simple function"
```
```   118
```
```   119 text {*
```
```   120
```
```   121 Our simple functions are not restricted to positive real numbers. Instead
```
```   122 they are just functions with a finite range and are measurable when singleton
```
```   123 sets are measurable.
```
```   124
```
```   125 *}
```
```   126
```
```   127 definition "simple_function M g \<longleftrightarrow>
```
```   128     finite (g ` space M) \<and>
```
```   129     (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
```
```   130
```
```   131 lemma (in sigma_algebra) simple_functionD:
```
```   132   assumes "simple_function M g"
```
```   133   shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
```
```   134 proof -
```
```   135   show "finite (g ` space M)"
```
```   136     using assms unfolding simple_function_def by auto
```
```   137   have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
```
```   138   also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
```
```   139   finally show "g -` X \<inter> space M \<in> sets M" using assms
```
```   140     by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
```
```   141 qed
```
```   142
```
```   143 lemma (in sigma_algebra) simple_function_measurable2[intro]:
```
```   144   assumes "simple_function M f" "simple_function M g"
```
```   145   shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
```
```   146 proof -
```
```   147   have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
```
```   148     by auto
```
```   149   then show ?thesis using assms[THEN simple_functionD(2)] by auto
```
```   150 qed
```
```   151
```
```   152 lemma (in sigma_algebra) simple_function_indicator_representation:
```
```   153   fixes f ::"'a \<Rightarrow> ereal"
```
```   154   assumes f: "simple_function M f" and x: "x \<in> space M"
```
```   155   shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
```
```   156   (is "?l = ?r")
```
```   157 proof -
```
```   158   have "?r = (\<Sum>y \<in> f ` space M.
```
```   159     (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
```
```   160     by (auto intro!: setsum_cong2)
```
```   161   also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
```
```   162     using assms by (auto dest: simple_functionD simp: setsum_delta)
```
```   163   also have "... = f x" using x by (auto simp: indicator_def)
```
```   164   finally show ?thesis by auto
```
```   165 qed
```
```   166
```
```   167 lemma (in measure_space) simple_function_notspace:
```
```   168   "simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
```
```   169 proof -
```
```   170   have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
```
```   171   hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
```
```   172   have "?h -` {0} \<inter> space M = space M" by auto
```
```   173   thus ?thesis unfolding simple_function_def by auto
```
```   174 qed
```
```   175
```
```   176 lemma (in sigma_algebra) simple_function_cong:
```
```   177   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
```
```   178   shows "simple_function M f \<longleftrightarrow> simple_function M g"
```
```   179 proof -
```
```   180   have "f ` space M = g ` space M"
```
```   181     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
```
```   182     using assms by (auto intro!: image_eqI)
```
```   183   thus ?thesis unfolding simple_function_def using assms by simp
```
```   184 qed
```
```   185
```
```   186 lemma (in sigma_algebra) simple_function_cong_algebra:
```
```   187   assumes "sets N = sets M" "space N = space M"
```
```   188   shows "simple_function M f \<longleftrightarrow> simple_function N f"
```
```   189   unfolding simple_function_def assms ..
```
```   190
```
```   191 lemma (in sigma_algebra) borel_measurable_simple_function:
```
```   192   assumes "simple_function M f"
```
```   193   shows "f \<in> borel_measurable M"
```
```   194 proof (rule borel_measurableI)
```
```   195   fix S
```
```   196   let ?I = "f ` (f -` S \<inter> space M)"
```
```   197   have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
```
```   198   have "finite ?I"
```
```   199     using assms unfolding simple_function_def
```
```   200     using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto
```
```   201   hence "?U \<in> sets M"
```
```   202     apply (rule finite_UN)
```
```   203     using assms unfolding simple_function_def by auto
```
```   204   thus "f -` S \<inter> space M \<in> sets M" unfolding * .
```
```   205 qed
```
```   206
```
```   207 lemma (in sigma_algebra) simple_function_borel_measurable:
```
```   208   fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
```
```   209   assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
```
```   210   shows "simple_function M f"
```
```   211   using assms unfolding simple_function_def
```
```   212   by (auto intro: borel_measurable_vimage)
```
```   213
```
```   214 lemma (in sigma_algebra) simple_function_eq_borel_measurable:
```
```   215   fixes f :: "'a \<Rightarrow> ereal"
```
```   216   shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
```
```   217   using simple_function_borel_measurable[of f]
```
```   218     borel_measurable_simple_function[of f]
```
```   219   by (fastsimp simp: simple_function_def)
```
```   220
```
```   221 lemma (in sigma_algebra) simple_function_const[intro, simp]:
```
```   222   "simple_function M (\<lambda>x. c)"
```
```   223   by (auto intro: finite_subset simp: simple_function_def)
```
```   224 lemma (in sigma_algebra) simple_function_compose[intro, simp]:
```
```   225   assumes "simple_function M f"
```
```   226   shows "simple_function M (g \<circ> f)"
```
```   227   unfolding simple_function_def
```
```   228 proof safe
```
```   229   show "finite ((g \<circ> f) ` space M)"
```
```   230     using assms unfolding simple_function_def by (auto simp: image_compose)
```
```   231 next
```
```   232   fix x assume "x \<in> space M"
```
```   233   let ?G = "g -` {g (f x)} \<inter> (f`space M)"
```
```   234   have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
```
```   235     (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
```
```   236   show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
```
```   237     using assms unfolding simple_function_def *
```
```   238     by (rule_tac finite_UN) (auto intro!: finite_UN)
```
```   239 qed
```
```   240
```
```   241 lemma (in sigma_algebra) simple_function_indicator[intro, simp]:
```
```   242   assumes "A \<in> sets M"
```
```   243   shows "simple_function M (indicator A)"
```
```   244 proof -
```
```   245   have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
```
```   246     by (auto simp: indicator_def)
```
```   247   hence "finite ?S" by (rule finite_subset) simp
```
```   248   moreover have "- A \<inter> space M = space M - A" by auto
```
```   249   ultimately show ?thesis unfolding simple_function_def
```
```   250     using assms by (auto simp: indicator_def_raw)
```
```   251 qed
```
```   252
```
```   253 lemma (in sigma_algebra) simple_function_Pair[intro, simp]:
```
```   254   assumes "simple_function M f"
```
```   255   assumes "simple_function M g"
```
```   256   shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
```
```   257   unfolding simple_function_def
```
```   258 proof safe
```
```   259   show "finite (?p ` space M)"
```
```   260     using assms unfolding simple_function_def
```
```   261     by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
```
```   262 next
```
```   263   fix x assume "x \<in> space M"
```
```   264   have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
```
```   265       (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
```
```   266     by auto
```
```   267   with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
```
```   268     using assms unfolding simple_function_def by auto
```
```   269 qed
```
```   270
```
```   271 lemma (in sigma_algebra) simple_function_compose1:
```
```   272   assumes "simple_function M f"
```
```   273   shows "simple_function M (\<lambda>x. g (f x))"
```
```   274   using simple_function_compose[OF assms, of g]
```
```   275   by (simp add: comp_def)
```
```   276
```
```   277 lemma (in sigma_algebra) simple_function_compose2:
```
```   278   assumes "simple_function M f" and "simple_function M g"
```
```   279   shows "simple_function M (\<lambda>x. h (f x) (g x))"
```
```   280 proof -
```
```   281   have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
```
```   282     using assms by auto
```
```   283   thus ?thesis by (simp_all add: comp_def)
```
```   284 qed
```
```   285
```
```   286 lemmas (in sigma_algebra) simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
```
```   287   and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
```
```   288   and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
```
```   289   and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
```
```   290   and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
```
```   291   and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
```
```   292   and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
```
```   293
```
```   294 lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
```
```   295   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
```
```   296   shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
```
```   297 proof cases
```
```   298   assume "finite P" from this assms show ?thesis by induct auto
```
```   299 qed auto
```
```   300
```
```   301 lemma (in sigma_algebra)
```
```   302   fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
```
```   303   shows simple_function_ereal[intro, simp]: "simple_function M (\<lambda>x. ereal (f x))"
```
```   304   by (auto intro!: simple_function_compose1[OF sf])
```
```   305
```
```   306 lemma (in sigma_algebra)
```
```   307   fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
```
```   308   shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
```
```   309   by (auto intro!: simple_function_compose1[OF sf])
```
```   310
```
```   311 lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
```
```   312   fixes u :: "'a \<Rightarrow> ereal"
```
```   313   assumes u: "u \<in> borel_measurable M"
```
```   314   shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
```
```   315              (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
```
```   316 proof -
```
```   317   def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
```
```   318   { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
```
```   319     proof (split split_if, intro conjI impI)
```
```   320       assume "\<not> real j \<le> u x"
```
```   321       then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
```
```   322          by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
```
```   323       moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
```
```   324         by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
```
```   325       ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
```
```   326         unfolding real_of_nat_le_iff by auto
```
```   327     qed auto }
```
```   328   note f_upper = this
```
```   329
```
```   330   have real_f:
```
```   331     "\<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)))"
```
```   332     unfolding f_def by auto
```
```   333
```
```   334   let "?g j x" = "real (f x j) / 2^j :: ereal"
```
```   335   show ?thesis
```
```   336   proof (intro exI[of _ ?g] conjI allI ballI)
```
```   337     fix i
```
```   338     have "simple_function M (\<lambda>x. real (f x i))"
```
```   339     proof (intro simple_function_borel_measurable)
```
```   340       show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
```
```   341         using u by (auto intro!: measurable_If simp: real_f)
```
```   342       have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
```
```   343         using f_upper[of _ i] by auto
```
```   344       then show "finite ((\<lambda>x. real (f x i))`space M)"
```
```   345         by (rule finite_subset) auto
```
```   346     qed
```
```   347     then show "simple_function M (?g i)"
```
```   348       by (auto intro: simple_function_ereal simple_function_div)
```
```   349   next
```
```   350     show "incseq ?g"
```
```   351     proof (intro incseq_ereal incseq_SucI le_funI)
```
```   352       fix x and i :: nat
```
```   353       have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
```
```   354       proof ((split split_if)+, intro conjI impI)
```
```   355         assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
```
```   356         then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
```
```   357           by (cases "u x") (auto intro!: le_natfloor)
```
```   358       next
```
```   359         assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
```
```   360         then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
```
```   361           by (cases "u x") auto
```
```   362       next
```
```   363         assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
```
```   364         have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
```
```   365           by simp
```
```   366         also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
```
```   367         proof cases
```
```   368           assume "0 \<le> u x" then show ?thesis
```
```   369             by (intro le_mult_natfloor) (cases "u x", auto intro!: mult_nonneg_nonneg)
```
```   370         next
```
```   371           assume "\<not> 0 \<le> u x" then show ?thesis
```
```   372             by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
```
```   373         qed
```
```   374         also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
```
```   375           by (simp add: ac_simps)
```
```   376         finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
```
```   377       qed simp
```
```   378       then show "?g i x \<le> ?g (Suc i) x"
```
```   379         by (auto simp: field_simps)
```
```   380     qed
```
```   381   next
```
```   382     fix x show "(SUP i. ?g i x) = max 0 (u x)"
```
```   383     proof (rule ereal_SUPI)
```
```   384       fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
```
```   385         by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
```
```   386                                      mult_nonpos_nonneg mult_nonneg_nonneg)
```
```   387     next
```
```   388       fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
```
```   389       have "\<And>i. 0 \<le> ?g i x" by (auto simp: divide_nonneg_pos)
```
```   390       from order_trans[OF this *] have "0 \<le> y" by simp
```
```   391       show "max 0 (u x) \<le> y"
```
```   392       proof (cases y)
```
```   393         case (real r)
```
```   394         with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
```
```   395         from real_arch_lt[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
```
```   396         then have "\<exists>p. max 0 (u x) = ereal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
```
```   397         then guess p .. note ux = this
```
```   398         obtain m :: nat where m: "p < real m" using real_arch_lt ..
```
```   399         have "p \<le> r"
```
```   400         proof (rule ccontr)
```
```   401           assume "\<not> p \<le> r"
```
```   402           with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
```
```   403           obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: inverse_eq_divide field_simps)
```
```   404           then have "r * 2^max N m < p * 2^max N m - 1" by simp
```
```   405           moreover
```
```   406           have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
```
```   407             using *[of "max N m"] m unfolding real_f using ux
```
```   408             by (cases "0 \<le> u x") (simp_all add: max_def mult_nonneg_nonneg split: split_if_asm)
```
```   409           then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
```
```   410             by (metis real_natfloor_gt_diff_one less_le_trans)
```
```   411           ultimately show False by auto
```
```   412         qed
```
```   413         then show "max 0 (u x) \<le> y" using real ux by simp
```
```   414       qed (insert `0 \<le> y`, auto)
```
```   415     qed
```
```   416   qed (auto simp: divide_nonneg_pos)
```
```   417 qed
```
```   418
```
```   419 lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
```
```   420   fixes u :: "'a \<Rightarrow> ereal"
```
```   421   assumes u: "u \<in> borel_measurable M"
```
```   422   obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
```
```   423     "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
```
```   424   using borel_measurable_implies_simple_function_sequence[OF u] by auto
```
```   425
```
```   426 lemma (in sigma_algebra) simple_function_If_set:
```
```   427   assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
```
```   428   shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
```
```   429 proof -
```
```   430   def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
```
```   431   show ?thesis unfolding simple_function_def
```
```   432   proof safe
```
```   433     have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
```
```   434     from finite_subset[OF this] assms
```
```   435     show "finite (?IF ` space M)" unfolding simple_function_def by auto
```
```   436   next
```
```   437     fix x assume "x \<in> space M"
```
```   438     then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
```
```   439       then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
```
```   440       else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
```
```   441       using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
```
```   442     have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
```
```   443       unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
```
```   444     show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
```
```   445   qed
```
```   446 qed
```
```   447
```
```   448 lemma (in sigma_algebra) simple_function_If:
```
```   449   assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
```
```   450   shows "simple_function M (\<lambda>x. if P x then f x else g x)"
```
```   451 proof -
```
```   452   have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
```
```   453   with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
```
```   454 qed
```
```   455
```
```   456 lemma (in measure_space) simple_function_restricted:
```
```   457   fixes f :: "'a \<Rightarrow> ereal" assumes "A \<in> sets M"
```
```   458   shows "simple_function (restricted_space A) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator A x)"
```
```   459     (is "simple_function ?R f \<longleftrightarrow> simple_function M ?f")
```
```   460 proof -
```
```   461   interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
```
```   462   have f: "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
```
```   463   proof cases
```
```   464     assume "A = space M"
```
```   465     then have "f`A = ?f`space M" by (fastsimp simp: image_iff)
```
```   466     then show ?thesis by simp
```
```   467   next
```
```   468     assume "A \<noteq> space M"
```
```   469     then obtain x where x: "x \<in> space M" "x \<notin> A"
```
```   470       using sets_into_space `A \<in> sets M` by auto
```
```   471     have *: "?f`space M = f`A \<union> {0}"
```
```   472     proof (auto simp add: image_iff)
```
```   473       show "\<exists>x\<in>space M. f x = 0 \<or> indicator A x = 0"
```
```   474         using x by (auto intro!: bexI[of _ x])
```
```   475     next
```
```   476       fix x assume "x \<in> A"
```
```   477       then show "\<exists>y\<in>space M. f x = f y * indicator A y"
```
```   478         using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
```
```   479     next
```
```   480       fix x
```
```   481       assume "indicator A x \<noteq> (0::ereal)"
```
```   482       then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
```
```   483       moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
```
```   484       ultimately show "f x = 0" by auto
```
```   485     qed
```
```   486     then show ?thesis by auto
```
```   487   qed
```
```   488   then show ?thesis
```
```   489     unfolding simple_function_eq_borel_measurable
```
```   490       R.simple_function_eq_borel_measurable
```
```   491     unfolding borel_measurable_restricted[OF `A \<in> sets M`]
```
```   492     using assms(1)[THEN sets_into_space]
```
```   493     by (auto simp: indicator_def)
```
```   494 qed
```
```   495
```
```   496 lemma (in sigma_algebra) simple_function_subalgebra:
```
```   497   assumes "simple_function N f"
```
```   498   and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
```
```   499   shows "simple_function M f"
```
```   500   using assms unfolding simple_function_def by auto
```
```   501
```
```   502 lemma (in measure_space) simple_function_vimage:
```
```   503   assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
```
```   504     and f: "simple_function M' f"
```
```   505   shows "simple_function M (\<lambda>x. f (T x))"
```
```   506 proof (intro simple_function_def[THEN iffD2] conjI ballI)
```
```   507   interpret T: sigma_algebra M' by fact
```
```   508   have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
```
```   509     using T unfolding measurable_def by auto
```
```   510   then show "finite ((\<lambda>x. f (T x)) ` space M)"
```
```   511     using f unfolding simple_function_def by (auto intro: finite_subset)
```
```   512   fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
```
```   513   then have "i \<in> f ` space M'"
```
```   514     using T unfolding measurable_def by auto
```
```   515   then have "f -` {i} \<inter> space M' \<in> sets M'"
```
```   516     using f unfolding simple_function_def by auto
```
```   517   then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
```
```   518     using T unfolding measurable_def by auto
```
```   519   also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
```
```   520     using T unfolding measurable_def by auto
```
```   521   finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
```
```   522 qed
```
```   523
```
```   524 section "Simple integral"
```
```   525
```
```   526 definition simple_integral_def:
```
```   527   "integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * measure M (f -` {x} \<inter> space M))"
```
```   528
```
```   529 syntax
```
```   530   "_simple_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> ereal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
```
```   531
```
```   532 translations
```
```   533   "\<integral>\<^isup>S x. f \<partial>M" == "CONST integral\<^isup>S M (%x. f)"
```
```   534
```
```   535 lemma (in measure_space) simple_integral_cong:
```
```   536   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
```
```   537   shows "integral\<^isup>S M f = integral\<^isup>S M g"
```
```   538 proof -
```
```   539   have "f ` space M = g ` space M"
```
```   540     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
```
```   541     using assms by (auto intro!: image_eqI)
```
```   542   thus ?thesis unfolding simple_integral_def by simp
```
```   543 qed
```
```   544
```
```   545 lemma (in measure_space) simple_integral_cong_measure:
```
```   546   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
```
```   547     and "simple_function M f"
```
```   548   shows "integral\<^isup>S N f = integral\<^isup>S M f"
```
```   549 proof -
```
```   550   interpret v: measure_space N
```
```   551     by (rule measure_space_cong) fact+
```
```   552   from simple_functionD[OF `simple_function M f`] assms show ?thesis
```
```   553     by (auto intro!: setsum_cong simp: simple_integral_def)
```
```   554 qed
```
```   555
```
```   556 lemma (in measure_space) simple_integral_const[simp]:
```
```   557   "(\<integral>\<^isup>Sx. c \<partial>M) = c * \<mu> (space M)"
```
```   558 proof (cases "space M = {}")
```
```   559   case True thus ?thesis unfolding simple_integral_def by simp
```
```   560 next
```
```   561   case False hence "(\<lambda>x. c) ` space M = {c}" by auto
```
```   562   thus ?thesis unfolding simple_integral_def by simp
```
```   563 qed
```
```   564
```
```   565 lemma (in measure_space) simple_function_partition:
```
```   566   assumes f: "simple_function M f" and g: "simple_function M g"
```
```   567   shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
```
```   568     (is "_ = setsum _ (?p ` space M)")
```
```   569 proof-
```
```   570   let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
```
```   571   let ?SIGMA = "Sigma (f`space M) ?sub"
```
```   572
```
```   573   have [intro]:
```
```   574     "finite (f ` space M)"
```
```   575     "finite (g ` space M)"
```
```   576     using assms unfolding simple_function_def by simp_all
```
```   577
```
```   578   { fix A
```
```   579     have "?p ` (A \<inter> space M) \<subseteq>
```
```   580       (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
```
```   581       by auto
```
```   582     hence "finite (?p ` (A \<inter> space M))"
```
```   583       by (rule finite_subset) auto }
```
```   584   note this[intro, simp]
```
```   585   note sets = simple_function_measurable2[OF f g]
```
```   586
```
```   587   { fix x assume "x \<in> space M"
```
```   588     have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
```
```   589     with sets have "\<mu> (f -` {f x} \<inter> space M) = setsum \<mu> (?sub (f x))"
```
```   590       by (subst measure_Union) auto }
```
```   591   hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
```
```   592     unfolding simple_integral_def using f sets
```
```   593     by (subst setsum_Sigma[symmetric])
```
```   594        (auto intro!: setsum_cong setsum_ereal_right_distrib)
```
```   595   also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
```
```   596   proof -
```
```   597     have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
```
```   598     have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
```
```   599       = (\<lambda>x. (f x, ?p x)) ` space M"
```
```   600     proof safe
```
```   601       fix x assume "x \<in> space M"
```
```   602       thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
```
```   603         by (auto intro!: image_eqI[of _ _ "?p x"])
```
```   604     qed auto
```
```   605     thus ?thesis
```
```   606       apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
```
```   607       apply (rule_tac x="xa" in image_eqI)
```
```   608       by simp_all
```
```   609   qed
```
```   610   finally show ?thesis .
```
```   611 qed
```
```   612
```
```   613 lemma (in measure_space) simple_integral_add[simp]:
```
```   614   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"
```
```   615   shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g"
```
```   616 proof -
```
```   617   { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
```
```   618     assume "x \<in> space M"
```
```   619     hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
```
```   620         "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
```
```   621       by auto }
```
```   622   with assms show ?thesis
```
```   623     unfolding
```
```   624       simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
```
```   625       simple_function_partition[OF f g]
```
```   626       simple_function_partition[OF g f]
```
```   627     by (subst (3) Int_commute)
```
```   628        (auto simp add: ereal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
```
```   629 qed
```
```   630
```
```   631 lemma (in measure_space) simple_integral_setsum[simp]:
```
```   632   assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
```
```   633   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
```
```   634   shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))"
```
```   635 proof cases
```
```   636   assume "finite P"
```
```   637   from this assms show ?thesis
```
```   638     by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
```
```   639 qed auto
```
```   640
```
```   641 lemma (in measure_space) simple_integral_mult[simp]:
```
```   642   assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
```
```   643   shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f"
```
```   644 proof -
```
```   645   note mult = simple_function_mult[OF simple_function_const[of c] f(1)]
```
```   646   { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
```
```   647     assume "x \<in> space M"
```
```   648     hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
```
```   649       by auto }
```
```   650   with assms show ?thesis
```
```   651     unfolding simple_function_partition[OF mult f(1)]
```
```   652               simple_function_partition[OF f(1) mult]
```
```   653     by (subst setsum_ereal_right_distrib)
```
```   654        (auto intro!: ereal_0_le_mult setsum_cong simp: mult_assoc)
```
```   655 qed
```
```   656
```
```   657 lemma (in measure_space) simple_integral_mono_AE:
```
```   658   assumes f: "simple_function M f" and g: "simple_function M g"
```
```   659   and mono: "AE x. f x \<le> g x"
```
```   660   shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
```
```   661 proof -
```
```   662   let "?S x" = "(g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
```
```   663   have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
```
```   664     "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
```
```   665   show ?thesis
```
```   666     unfolding *
```
```   667       simple_function_partition[OF f g]
```
```   668       simple_function_partition[OF g f]
```
```   669   proof (safe intro!: setsum_mono)
```
```   670     fix x assume "x \<in> space M"
```
```   671     then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
```
```   672     show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
```
```   673     proof (cases "f x \<le> g x")
```
```   674       case True then show ?thesis
```
```   675         using * assms(1,2)[THEN simple_functionD(2)]
```
```   676         by (auto intro!: ereal_mult_right_mono)
```
```   677     next
```
```   678       case False
```
```   679       obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
```
```   680         using mono by (auto elim!: AE_E)
```
```   681       have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
```
```   682       moreover have "?S x \<in> sets M" using assms
```
```   683         by (rule_tac Int) (auto intro!: simple_functionD)
```
```   684       ultimately have "\<mu> (?S x) \<le> \<mu> N"
```
```   685         using `N \<in> sets M` by (auto intro!: measure_mono)
```
```   686       moreover have "0 \<le> \<mu> (?S x)"
```
```   687         using assms(1,2)[THEN simple_functionD(2)] by auto
```
```   688       ultimately have "\<mu> (?S x) = 0" using `\<mu> N = 0` by auto
```
```   689       then show ?thesis by simp
```
```   690     qed
```
```   691   qed
```
```   692 qed
```
```   693
```
```   694 lemma (in measure_space) simple_integral_mono:
```
```   695   assumes "simple_function M f" and "simple_function M g"
```
```   696   and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
```
```   697   shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
```
```   698   using assms by (intro simple_integral_mono_AE) auto
```
```   699
```
```   700 lemma (in measure_space) simple_integral_cong_AE:
```
```   701   assumes "simple_function M f" and "simple_function M g"
```
```   702   and "AE x. f x = g x"
```
```   703   shows "integral\<^isup>S M f = integral\<^isup>S M g"
```
```   704   using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
```
```   705
```
```   706 lemma (in measure_space) simple_integral_cong':
```
```   707   assumes sf: "simple_function M f" "simple_function M g"
```
```   708   and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
```
```   709   shows "integral\<^isup>S M f = integral\<^isup>S M g"
```
```   710 proof (intro simple_integral_cong_AE sf AE_I)
```
```   711   show "\<mu> {x\<in>space M. f x \<noteq> g x} = 0" by fact
```
```   712   show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
```
```   713     using sf[THEN borel_measurable_simple_function] by auto
```
```   714 qed simp
```
```   715
```
```   716 lemma (in measure_space) simple_integral_indicator:
```
```   717   assumes "A \<in> sets M"
```
```   718   assumes "simple_function M f"
```
```   719   shows "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
```
```   720     (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
```
```   721 proof cases
```
```   722   assume "A = space M"
```
```   723   moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = integral\<^isup>S M f"
```
```   724     by (auto intro!: simple_integral_cong)
```
```   725   moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
```
```   726   ultimately show ?thesis by (simp add: simple_integral_def)
```
```   727 next
```
```   728   assume "A \<noteq> space M"
```
```   729   then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
```
```   730   have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
```
```   731   proof safe
```
```   732     fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
```
```   733   next
```
```   734     fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
```
```   735       using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
```
```   736   next
```
```   737     show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
```
```   738   qed
```
```   739   have *: "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
```
```   740     (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
```
```   741     unfolding simple_integral_def I
```
```   742   proof (rule setsum_mono_zero_cong_left)
```
```   743     show "finite (f ` space M \<union> {0})"
```
```   744       using assms(2) unfolding simple_function_def by auto
```
```   745     show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
```
```   746       using sets_into_space[OF assms(1)] by auto
```
```   747     have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
```
```   748       by (auto simp: image_iff)
```
```   749     thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
```
```   750       i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
```
```   751   next
```
```   752     fix x assume "x \<in> f`A \<union> {0}"
```
```   753     hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
```
```   754       by (auto simp: indicator_def split: split_if_asm)
```
```   755     thus "x * \<mu> (?I -` {x} \<inter> space M) =
```
```   756       x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
```
```   757   qed
```
```   758   show ?thesis unfolding *
```
```   759     using assms(2) unfolding simple_function_def
```
```   760     by (auto intro!: setsum_mono_zero_cong_right)
```
```   761 qed
```
```   762
```
```   763 lemma (in measure_space) simple_integral_indicator_only[simp]:
```
```   764   assumes "A \<in> sets M"
```
```   765   shows "integral\<^isup>S M (indicator A) = \<mu> A"
```
```   766 proof cases
```
```   767   assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
```
```   768   thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
```
```   769 next
```
```   770   assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::ereal}" by auto
```
```   771   thus ?thesis
```
```   772     using simple_integral_indicator[OF assms simple_function_const[of 1]]
```
```   773     using sets_into_space[OF assms]
```
```   774     by (auto intro!: arg_cong[where f="\<mu>"])
```
```   775 qed
```
```   776
```
```   777 lemma (in measure_space) simple_integral_null_set:
```
```   778   assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets"
```
```   779   shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0"
```
```   780 proof -
```
```   781   have "AE x. indicator N x = (0 :: ereal)"
```
```   782     using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
```
```   783   then have "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^isup>Sx. 0 \<partial>M)"
```
```   784     using assms apply (intro simple_integral_cong_AE) by auto
```
```   785   then show ?thesis by simp
```
```   786 qed
```
```   787
```
```   788 lemma (in measure_space) simple_integral_cong_AE_mult_indicator:
```
```   789   assumes sf: "simple_function M f" and eq: "AE x. x \<in> S" and "S \<in> sets M"
```
```   790   shows "integral\<^isup>S M f = (\<integral>\<^isup>Sx. f x * indicator S x \<partial>M)"
```
```   791   using assms by (intro simple_integral_cong_AE) auto
```
```   792
```
```   793 lemma (in measure_space) simple_integral_restricted:
```
```   794   assumes "A \<in> sets M"
```
```   795   assumes sf: "simple_function M (\<lambda>x. f x * indicator A x)"
```
```   796   shows "integral\<^isup>S (restricted_space A) f = (\<integral>\<^isup>Sx. f x * indicator A x \<partial>M)"
```
```   797     (is "_ = integral\<^isup>S M ?f")
```
```   798   unfolding simple_integral_def
```
```   799 proof (simp, safe intro!: setsum_mono_zero_cong_left)
```
```   800   from sf show "finite (?f ` space M)"
```
```   801     unfolding simple_function_def by auto
```
```   802 next
```
```   803   fix x assume "x \<in> A"
```
```   804   then show "f x \<in> ?f ` space M"
```
```   805     using sets_into_space `A \<in> sets M` by (auto intro!: image_eqI[of _ _ x])
```
```   806 next
```
```   807   fix x assume "x \<in> space M" "?f x \<notin> f`A"
```
```   808   then have "x \<notin> A" by (auto simp: image_iff)
```
```   809   then show "?f x * \<mu> (?f -` {?f x} \<inter> space M) = 0" by simp
```
```   810 next
```
```   811   fix x assume "x \<in> A"
```
```   812   then have "f x \<noteq> 0 \<Longrightarrow>
```
```   813     f -` {f x} \<inter> A = ?f -` {f x} \<inter> space M"
```
```   814     using `A \<in> sets M` sets_into_space
```
```   815     by (auto simp: indicator_def split: split_if_asm)
```
```   816   then show "f x * \<mu> (f -` {f x} \<inter> A) =
```
```   817     f x * \<mu> (?f -` {f x} \<inter> space M)"
```
```   818     unfolding ereal_mult_cancel_left by auto
```
```   819 qed
```
```   820
```
```   821 lemma (in measure_space) simple_integral_subalgebra:
```
```   822   assumes N: "measure_space N" and [simp]: "space N = space M" "measure N = measure M"
```
```   823   shows "integral\<^isup>S N = integral\<^isup>S M"
```
```   824   unfolding simple_integral_def_raw by simp
```
```   825
```
```   826 lemma (in measure_space) simple_integral_vimage:
```
```   827   assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
```
```   828     and f: "simple_function M' f"
```
```   829   shows "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
```
```   830 proof -
```
```   831   interpret T: measure_space M' by (rule measure_space_vimage[OF T])
```
```   832   show "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
```
```   833     unfolding simple_integral_def
```
```   834   proof (intro setsum_mono_zero_cong_right ballI)
```
```   835     show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
```
```   836       using T unfolding measurable_def measure_preserving_def by auto
```
```   837     show "finite (f ` space M')"
```
```   838       using f unfolding simple_function_def by auto
```
```   839   next
```
```   840     fix i assume "i \<in> f ` space M' - (\<lambda>x. f (T x)) ` space M"
```
```   841     then have "T -` (f -` {i} \<inter> space M') \<inter> space M = {}" by (auto simp: image_iff)
```
```   842     with f[THEN T.simple_functionD(2), THEN measure_preservingD[OF T(2)], of "{i}"]
```
```   843     show "i * T.\<mu> (f -` {i} \<inter> space M') = 0" by simp
```
```   844   next
```
```   845     fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M"
```
```   846     then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
```
```   847       using T unfolding measurable_def measure_preserving_def by auto
```
```   848     with f[THEN T.simple_functionD(2), THEN measure_preservingD[OF T(2)], of "{i}"]
```
```   849     show "i * T.\<mu> (f -` {i} \<inter> space M') = i * \<mu> ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
```
```   850       by auto
```
```   851   qed
```
```   852 qed
```
```   853
```
```   854 lemma (in measure_space) simple_integral_cmult_indicator:
```
```   855   assumes A: "A \<in> sets M"
```
```   856   shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * \<mu> A"
```
```   857   using simple_integral_mult[OF simple_function_indicator[OF A]]
```
```   858   unfolding simple_integral_indicator_only[OF A] by simp
```
```   859
```
```   860 lemma (in measure_space) simple_integral_positive:
```
```   861   assumes f: "simple_function M f" and ae: "AE x. 0 \<le> f x"
```
```   862   shows "0 \<le> integral\<^isup>S M f"
```
```   863 proof -
```
```   864   have "integral\<^isup>S M (\<lambda>x. 0) \<le> integral\<^isup>S M f"
```
```   865     using simple_integral_mono_AE[OF _ f ae] by auto
```
```   866   then show ?thesis by simp
```
```   867 qed
```
```   868
```
```   869 section "Continuous positive integration"
```
```   870
```
```   871 definition positive_integral_def:
```
```   872   "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^isup>S M g)"
```
```   873
```
```   874 syntax
```
```   875   "_positive_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> ereal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
```
```   876
```
```   877 translations
```
```   878   "\<integral>\<^isup>+ x. f \<partial>M" == "CONST integral\<^isup>P M (%x. f)"
```
```   879
```
```   880 lemma (in measure_space) positive_integral_cong_measure:
```
```   881   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
```
```   882   shows "integral\<^isup>P N f = integral\<^isup>P M f"
```
```   883   unfolding positive_integral_def
```
```   884   unfolding simple_function_cong_algebra[OF assms(2,3), symmetric]
```
```   885   using AE_cong_measure[OF assms]
```
```   886   using simple_integral_cong_measure[OF assms]
```
```   887   by (auto intro!: SUP_cong)
```
```   888
```
```   889 lemma (in measure_space) positive_integral_positive:
```
```   890   "0 \<le> integral\<^isup>P M f"
```
```   891   by (auto intro!: le_SUPI2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
```
```   892
```
```   893 lemma (in measure_space) positive_integral_def_finite:
```
```   894   "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^isup>S M g)"
```
```   895     (is "_ = SUPR ?A ?f")
```
```   896   unfolding positive_integral_def
```
```   897 proof (safe intro!: antisym SUP_leI)
```
```   898   fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
```
```   899   let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
```
```   900   note gM = g(1)[THEN borel_measurable_simple_function]
```
```   901   have \<mu>G_pos: "0 \<le> \<mu> ?G" using gM by auto
```
```   902   let "?g y x" = "if g x = \<infinity> then y else max 0 (g x)"
```
```   903   from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
```
```   904     apply (safe intro!: simple_function_max simple_function_If)
```
```   905     apply (force simp: max_def le_fun_def split: split_if_asm)+
```
```   906     done
```
```   907   show "integral\<^isup>S M g \<le> SUPR ?A ?f"
```
```   908   proof cases
```
```   909     have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
```
```   910     assume "\<mu> ?G = 0"
```
```   911     with gM have "AE x. x \<notin> ?G" by (simp add: AE_iff_null_set)
```
```   912     with gM g show ?thesis
```
```   913       by (intro le_SUPI2[OF g0] simple_integral_mono_AE)
```
```   914          (auto simp: max_def intro!: simple_function_If)
```
```   915   next
```
```   916     assume \<mu>G: "\<mu> ?G \<noteq> 0"
```
```   917     have "SUPR ?A (integral\<^isup>S M) = \<infinity>"
```
```   918     proof (intro SUP_PInfty)
```
```   919       fix n :: nat
```
```   920       let ?y = "ereal (real n) / (if \<mu> ?G = \<infinity> then 1 else \<mu> ?G)"
```
```   921       have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>G \<mu>G_pos by (auto simp: ereal_divide_eq)
```
```   922       then have "?g ?y \<in> ?A" by (rule g_in_A)
```
```   923       have "real n \<le> ?y * \<mu> ?G"
```
```   924         using \<mu>G \<mu>G_pos by (cases "\<mu> ?G") (auto simp: field_simps)
```
```   925       also have "\<dots> = (\<integral>\<^isup>Sx. ?y * indicator ?G x \<partial>M)"
```
```   926         using `0 \<le> ?y` `?g ?y \<in> ?A` gM
```
```   927         by (subst simple_integral_cmult_indicator) auto
```
```   928       also have "\<dots> \<le> integral\<^isup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
```
```   929         by (intro simple_integral_mono) auto
```
```   930       finally show "\<exists>i\<in>?A. real n \<le> integral\<^isup>S M i"
```
```   931         using `?g ?y \<in> ?A` by blast
```
```   932     qed
```
```   933     then show ?thesis by simp
```
```   934   qed
```
```   935 qed (auto intro: le_SUPI)
```
```   936
```
```   937 lemma (in measure_space) positive_integral_mono_AE:
```
```   938   assumes ae: "AE x. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
```
```   939   unfolding positive_integral_def
```
```   940 proof (safe intro!: SUP_mono)
```
```   941   fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
```
```   942   from ae[THEN AE_E] guess N . note N = this
```
```   943   then have ae_N: "AE x. x \<notin> N" by (auto intro: AE_not_in)
```
```   944   let "?n x" = "n x * indicator (space M - N) x"
```
```   945   have "AE x. n x \<le> ?n x" "simple_function M ?n"
```
```   946     using n N ae_N by auto
```
```   947   moreover
```
```   948   { fix x have "?n x \<le> max 0 (v x)"
```
```   949     proof cases
```
```   950       assume x: "x \<in> space M - N"
```
```   951       with N have "u x \<le> v x" by auto
```
```   952       with n(2)[THEN le_funD, of x] x show ?thesis
```
```   953         by (auto simp: max_def split: split_if_asm)
```
```   954     qed simp }
```
```   955   then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
```
```   956   moreover have "integral\<^isup>S M n \<le> integral\<^isup>S M ?n"
```
```   957     using ae_N N n by (auto intro!: simple_integral_mono_AE)
```
```   958   ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^isup>S M n \<le> integral\<^isup>S M m"
```
```   959     by force
```
```   960 qed
```
```   961
```
```   962 lemma (in measure_space) positive_integral_mono:
```
```   963   "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
```
```   964   by (auto intro: positive_integral_mono_AE)
```
```   965
```
```   966 lemma (in measure_space) positive_integral_cong_AE:
```
```   967   "AE x. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
```
```   968   by (auto simp: eq_iff intro!: positive_integral_mono_AE)
```
```   969
```
```   970 lemma (in measure_space) positive_integral_cong:
```
```   971   "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
```
```   972   by (auto intro: positive_integral_cong_AE)
```
```   973
```
```   974 lemma (in measure_space) positive_integral_eq_simple_integral:
```
```   975   assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
```
```   976 proof -
```
```   977   let "?f x" = "f x * indicator (space M) x"
```
```   978   have f': "simple_function M ?f" using f by auto
```
```   979   with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
```
```   980     by (auto simp: fun_eq_iff max_def split: split_indicator)
```
```   981   have "integral\<^isup>P M ?f \<le> integral\<^isup>S M ?f" using f'
```
```   982     by (force intro!: SUP_leI simple_integral_mono simp: le_fun_def positive_integral_def)
```
```   983   moreover have "integral\<^isup>S M ?f \<le> integral\<^isup>P M ?f"
```
```   984     unfolding positive_integral_def
```
```   985     using f' by (auto intro!: le_SUPI)
```
```   986   ultimately show ?thesis
```
```   987     by (simp cong: positive_integral_cong simple_integral_cong)
```
```   988 qed
```
```   989
```
```   990 lemma (in measure_space) positive_integral_eq_simple_integral_AE:
```
```   991   assumes f: "simple_function M f" "AE x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
```
```   992 proof -
```
```   993   have "AE x. f x = max 0 (f x)" using f by (auto split: split_max)
```
```   994   with f have "integral\<^isup>P M f = integral\<^isup>S M (\<lambda>x. max 0 (f x))"
```
```   995     by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
```
```   996              add: positive_integral_eq_simple_integral)
```
```   997   with assms show ?thesis
```
```   998     by (auto intro!: simple_integral_cong_AE split: split_max)
```
```   999 qed
```
```  1000
```
```  1001 lemma (in measure_space) positive_integral_SUP_approx:
```
```  1002   assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
```
```  1003   and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
```
```  1004   shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S")
```
```  1005 proof (rule ereal_le_mult_one_interval)
```
```  1006   have "0 \<le> (SUP i. integral\<^isup>P M (f i))"
```
```  1007     using f(3) by (auto intro!: le_SUPI2 positive_integral_positive)
```
```  1008   then show "(SUP i. integral\<^isup>P M (f i)) \<noteq> -\<infinity>" by auto
```
```  1009   have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
```
```  1010     using u(3) by auto
```
```  1011   fix a :: ereal assume "0 < a" "a < 1"
```
```  1012   hence "a \<noteq> 0" by auto
```
```  1013   let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
```
```  1014   have B: "\<And>i. ?B i \<in> sets M"
```
```  1015     using f `simple_function M u` by (auto simp: borel_measurable_simple_function)
```
```  1016
```
```  1017   let "?uB i x" = "u x * indicator (?B i) x"
```
```  1018
```
```  1019   { fix i have "?B i \<subseteq> ?B (Suc i)"
```
```  1020     proof safe
```
```  1021       fix i x assume "a * u x \<le> f i x"
```
```  1022       also have "\<dots> \<le> f (Suc i) x"
```
```  1023         using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
```
```  1024       finally show "a * u x \<le> f (Suc i) x" .
```
```  1025     qed }
```
```  1026   note B_mono = this
```
```  1027
```
```  1028   note B_u = Int[OF u(1)[THEN simple_functionD(2)] B]
```
```  1029
```
```  1030   let "?B' i n" = "(u -` {i} \<inter> space M) \<inter> ?B n"
```
```  1031   have measure_conv: "\<And>i. \<mu> (u -` {i} \<inter> space M) = (SUP n. \<mu> (?B' i n))"
```
```  1032   proof -
```
```  1033     fix i
```
```  1034     have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
```
```  1035     have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
```
```  1036     have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
```
```  1037     proof safe
```
```  1038       fix x i assume x: "x \<in> space M"
```
```  1039       show "x \<in> (\<Union>i. ?B' (u x) i)"
```
```  1040       proof cases
```
```  1041         assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
```
```  1042       next
```
```  1043         assume "u x \<noteq> 0"
```
```  1044         with `a < 1` u_range[OF `x \<in> space M`]
```
```  1045         have "a * u x < 1 * u x"
```
```  1046           by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
```
```  1047         also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def SUPR_apply)
```
```  1048         finally obtain i where "a * u x < f i x" unfolding SUPR_def
```
```  1049           by (auto simp add: less_Sup_iff)
```
```  1050         hence "a * u x \<le> f i x" by auto
```
```  1051         thus ?thesis using `x \<in> space M` by auto
```
```  1052       qed
```
```  1053     qed
```
```  1054     then show "?thesis i" using continuity_from_below[OF 1 2] by simp
```
```  1055   qed
```
```  1056
```
```  1057   have "integral\<^isup>S M u = (SUP i. integral\<^isup>S M (?uB i))"
```
```  1058     unfolding simple_integral_indicator[OF B `simple_function M u`]
```
```  1059   proof (subst SUPR_ereal_setsum, safe)
```
```  1060     fix x n assume "x \<in> space M"
```
```  1061     with u_range show "incseq (\<lambda>i. u x * \<mu> (?B' (u x) i))" "\<And>i. 0 \<le> u x * \<mu> (?B' (u x) i)"
```
```  1062       using B_mono B_u by (auto intro!: measure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
```
```  1063   next
```
```  1064     show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (?B' i n))"
```
```  1065       using measure_conv u_range B_u unfolding simple_integral_def
```
```  1066       by (auto intro!: setsum_cong SUPR_ereal_cmult[symmetric])
```
```  1067   qed
```
```  1068   moreover
```
```  1069   have "a * (SUP i. integral\<^isup>S M (?uB i)) \<le> ?S"
```
```  1070     apply (subst SUPR_ereal_cmult[symmetric])
```
```  1071   proof (safe intro!: SUP_mono bexI)
```
```  1072     fix i
```
```  1073     have "a * integral\<^isup>S M (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x \<partial>M)"
```
```  1074       using B `simple_function M u` u_range
```
```  1075       by (subst simple_integral_mult) (auto split: split_indicator)
```
```  1076     also have "\<dots> \<le> integral\<^isup>P M (f i)"
```
```  1077     proof -
```
```  1078       have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
```
```  1079       show ?thesis using f(3) * u_range `0 < a`
```
```  1080         by (subst positive_integral_eq_simple_integral[symmetric])
```
```  1081            (auto intro!: positive_integral_mono split: split_indicator)
```
```  1082     qed
```
```  1083     finally show "a * integral\<^isup>S M (?uB i) \<le> integral\<^isup>P M (f i)"
```
```  1084       by auto
```
```  1085   next
```
```  1086     fix i show "0 \<le> \<integral>\<^isup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
```
```  1087       by (intro simple_integral_positive) (auto split: split_indicator)
```
```  1088   qed (insert `0 < a`, auto)
```
```  1089   ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp
```
```  1090 qed
```
```  1091
```
```  1092 lemma (in measure_space) incseq_positive_integral:
```
```  1093   assumes "incseq f" shows "incseq (\<lambda>i. integral\<^isup>P M (f i))"
```
```  1094 proof -
```
```  1095   have "\<And>i x. f i x \<le> f (Suc i) x"
```
```  1096     using assms by (auto dest!: incseq_SucD simp: le_fun_def)
```
```  1097   then show ?thesis
```
```  1098     by (auto intro!: incseq_SucI positive_integral_mono)
```
```  1099 qed
```
```  1100
```
```  1101 text {* Beppo-Levi monotone convergence theorem *}
```
```  1102 lemma (in measure_space) positive_integral_monotone_convergence_SUP:
```
```  1103   assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
```
```  1104   shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
```
```  1105 proof (rule antisym)
```
```  1106   show "(SUP j. integral\<^isup>P M (f j)) \<le> (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
```
```  1107     by (auto intro!: SUP_leI le_SUPI positive_integral_mono)
```
```  1108 next
```
```  1109   show "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^isup>P M (f j))"
```
```  1110     unfolding positive_integral_def_finite[of "\<lambda>x. SUP i. f i x"]
```
```  1111   proof (safe intro!: SUP_leI)
```
```  1112     fix g assume g: "simple_function M g"
```
```  1113       and "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
```
```  1114     moreover then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
```
```  1115       using f by (auto intro!: le_SUPI2)
```
```  1116     ultimately show "integral\<^isup>S M g \<le> (SUP j. integral\<^isup>P M (f j))"
```
```  1117       by (intro  positive_integral_SUP_approx[OF f g _ g'])
```
```  1118          (auto simp: le_fun_def max_def SUPR_apply)
```
```  1119   qed
```
```  1120 qed
```
```  1121
```
```  1122 lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE:
```
```  1123   assumes f: "\<And>i. AE x. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
```
```  1124   shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
```
```  1125 proof -
```
```  1126   from f have "AE x. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
```
```  1127     by (simp add: AE_all_countable)
```
```  1128   from this[THEN AE_E] guess N . note N = this
```
```  1129   let "?f i x" = "if x \<in> space M - N then f i x else 0"
```
```  1130   have f_eq: "AE x. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ N])
```
```  1131   then have "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP i. ?f i x) \<partial>M)"
```
```  1132     by (auto intro!: positive_integral_cong_AE)
```
```  1133   also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. ?f i x \<partial>M))"
```
```  1134   proof (rule positive_integral_monotone_convergence_SUP)
```
```  1135     show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
```
```  1136     { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
```
```  1137         using f N(3) by (intro measurable_If_set) auto
```
```  1138       fix x show "0 \<le> ?f i x"
```
```  1139         using N(1) by auto }
```
```  1140   qed
```
```  1141   also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. f i x \<partial>M))"
```
```  1142     using f_eq by (force intro!: arg_cong[where f="SUPR UNIV"] positive_integral_cong_AE ext)
```
```  1143   finally show ?thesis .
```
```  1144 qed
```
```  1145
```
```  1146 lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE_incseq:
```
```  1147   assumes f: "incseq f" "\<And>i. AE x. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
```
```  1148   shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
```
```  1149   using f[unfolded incseq_Suc_iff le_fun_def]
```
```  1150   by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
```
```  1151      auto
```
```  1152
```
```  1153 lemma (in measure_space) positive_integral_monotone_convergence_simple:
```
```  1154   assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
```
```  1155   shows "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
```
```  1156   using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
```
```  1157     f(3)[THEN borel_measurable_simple_function] f(2)]
```
```  1158   by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
```
```  1159
```
```  1160 lemma positive_integral_max_0:
```
```  1161   "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M) = integral\<^isup>P M f"
```
```  1162   by (simp add: le_fun_def positive_integral_def)
```
```  1163
```
```  1164 lemma (in measure_space) positive_integral_cong_pos:
```
```  1165   assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
```
```  1166   shows "integral\<^isup>P M f = integral\<^isup>P M g"
```
```  1167 proof -
```
```  1168   have "integral\<^isup>P M (\<lambda>x. max 0 (f x)) = integral\<^isup>P M (\<lambda>x. max 0 (g x))"
```
```  1169   proof (intro positive_integral_cong)
```
```  1170     fix x assume "x \<in> space M"
```
```  1171     from assms[OF this] show "max 0 (f x) = max 0 (g x)"
```
```  1172       by (auto split: split_max)
```
```  1173   qed
```
```  1174   then show ?thesis by (simp add: positive_integral_max_0)
```
```  1175 qed
```
```  1176
```
```  1177 lemma (in measure_space) SUP_simple_integral_sequences:
```
```  1178   assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
```
```  1179   and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
```
```  1180   and eq: "AE x. (SUP i. f i x) = (SUP i. g i x)"
```
```  1181   shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))"
```
```  1182     (is "SUPR _ ?F = SUPR _ ?G")
```
```  1183 proof -
```
```  1184   have "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
```
```  1185     using f by (rule positive_integral_monotone_convergence_simple)
```
```  1186   also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. g i x) \<partial>M)"
```
```  1187     unfolding eq[THEN positive_integral_cong_AE] ..
```
```  1188   also have "\<dots> = (SUP i. ?G i)"
```
```  1189     using g by (rule positive_integral_monotone_convergence_simple[symmetric])
```
```  1190   finally show ?thesis by simp
```
```  1191 qed
```
```  1192
```
```  1193 lemma (in measure_space) positive_integral_const[simp]:
```
```  1194   "0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
```
```  1195   by (subst positive_integral_eq_simple_integral) auto
```
```  1196
```
```  1197 lemma (in measure_space) positive_integral_vimage:
```
```  1198   assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
```
```  1199   and f: "f \<in> borel_measurable M'"
```
```  1200   shows "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
```
```  1201 proof -
```
```  1202   interpret T: measure_space M' by (rule measure_space_vimage[OF T])
```
```  1203   from T.borel_measurable_implies_simple_function_sequence'[OF f]
```
```  1204   guess f' . note f' = this
```
```  1205   let "?f i x" = "f' i (T x)"
```
```  1206   have inc: "incseq ?f" using f' by (force simp: le_fun_def incseq_def)
```
```  1207   have sup: "\<And>x. (SUP i. ?f i x) = max 0 (f (T x))"
```
```  1208     using f'(4) .
```
```  1209   have sf: "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
```
```  1210     using simple_function_vimage[OF T(1) measure_preservingD2[OF T(2)] f'(1)] .
```
```  1211   show "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
```
```  1212     using
```
```  1213       T.positive_integral_monotone_convergence_simple[OF f'(2,5,1)]
```
```  1214       positive_integral_monotone_convergence_simple[OF inc f'(5) sf]
```
```  1215     by (simp add: positive_integral_max_0 simple_integral_vimage[OF T f'(1)] f')
```
```  1216 qed
```
```  1217
```
```  1218 lemma (in measure_space) positive_integral_linear:
```
```  1219   assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
```
```  1220   and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
```
```  1221   shows "(\<integral>\<^isup>+ x. a * f x + g x \<partial>M) = a * integral\<^isup>P M f + integral\<^isup>P M g"
```
```  1222     (is "integral\<^isup>P M ?L = _")
```
```  1223 proof -
```
```  1224   from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
```
```  1225   note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
```
```  1226   from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
```
```  1227   note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
```
```  1228   let "?L' i x" = "a * u i x + v i x"
```
```  1229
```
```  1230   have "?L \<in> borel_measurable M" using assms by auto
```
```  1231   from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
```
```  1232   note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
```
```  1233
```
```  1234   have inc: "incseq (\<lambda>i. a * integral\<^isup>S M (u i))" "incseq (\<lambda>i. integral\<^isup>S M (v i))"
```
```  1235     using u v `0 \<le> a`
```
```  1236     by (auto simp: incseq_Suc_iff le_fun_def
```
```  1237              intro!: add_mono ereal_mult_left_mono simple_integral_mono)
```
```  1238   have pos: "\<And>i. 0 \<le> integral\<^isup>S M (u i)" "\<And>i. 0 \<le> integral\<^isup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^isup>S M (u i)"
```
```  1239     using u v `0 \<le> a` by (auto simp: simple_integral_positive)
```
```  1240   { fix i from pos[of i] have "a * integral\<^isup>S M (u i) \<noteq> -\<infinity>" "integral\<^isup>S M (v i) \<noteq> -\<infinity>"
```
```  1241       by (auto split: split_if_asm) }
```
```  1242   note not_MInf = this
```
```  1243
```
```  1244   have l': "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
```
```  1245   proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
```
```  1246     show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
```
```  1247       using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
```
```  1248       by (auto intro!: add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg)
```
```  1249     { fix x
```
```  1250       { 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]
```
```  1251           by auto }
```
```  1252       then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
```
```  1253         using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
```
```  1254         by (subst SUPR_ereal_cmult[symmetric, OF u(6) `0 \<le> a`])
```
```  1255            (auto intro!: SUPR_ereal_add
```
```  1256                  simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg) }
```
```  1257     then show "AE x. (SUP i. l i x) = (SUP i. ?L' i x)"
```
```  1258       unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
```
```  1259       by (intro AE_I2) (auto split: split_max simp add: ereal_add_nonneg_nonneg)
```
```  1260   qed
```
```  1261   also have "\<dots> = (SUP i. a * integral\<^isup>S M (u i) + integral\<^isup>S M (v i))"
```
```  1262     using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPR UNIV"] ext)
```
```  1263   finally have "(\<integral>\<^isup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^isup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+x. max 0 (g x) \<partial>M)"
```
```  1264     unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
```
```  1265     unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
```
```  1266     apply (subst SUPR_ereal_cmult[symmetric, OF pos(1) `0 \<le> a`])
```
```  1267     apply (subst SUPR_ereal_add[symmetric, OF inc not_MInf]) .
```
```  1268   then show ?thesis by (simp add: positive_integral_max_0)
```
```  1269 qed
```
```  1270
```
```  1271 lemma (in measure_space) positive_integral_cmult:
```
```  1272   assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
```
```  1273   shows "(\<integral>\<^isup>+ x. c * f x \<partial>M) = c * integral\<^isup>P M f"
```
```  1274 proof -
```
```  1275   have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
```
```  1276     by (auto split: split_max simp: ereal_zero_le_0_iff)
```
```  1277   have "(\<integral>\<^isup>+ x. c * f x \<partial>M) = (\<integral>\<^isup>+ x. c * max 0 (f x) \<partial>M)"
```
```  1278     by (simp add: positive_integral_max_0)
```
```  1279   then show ?thesis
```
```  1280     using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" "\<lambda>x. 0"] f
```
```  1281     by (auto simp: positive_integral_max_0)
```
```  1282 qed
```
```  1283
```
```  1284 lemma (in measure_space) positive_integral_multc:
```
```  1285   assumes "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
```
```  1286   shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c"
```
```  1287   unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
```
```  1288
```
```  1289 lemma (in measure_space) positive_integral_indicator[simp]:
```
```  1290   "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x\<partial>M) = \<mu> A"
```
```  1291   by (subst positive_integral_eq_simple_integral)
```
```  1292      (auto simp: simple_function_indicator simple_integral_indicator)
```
```  1293
```
```  1294 lemma (in measure_space) positive_integral_cmult_indicator:
```
```  1295   "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
```
```  1296   by (subst positive_integral_eq_simple_integral)
```
```  1297      (auto simp: simple_function_indicator simple_integral_indicator)
```
```  1298
```
```  1299 lemma (in measure_space) positive_integral_add:
```
```  1300   assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
```
```  1301   and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
```
```  1302   shows "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = integral\<^isup>P M f + integral\<^isup>P M g"
```
```  1303 proof -
```
```  1304   have ae: "AE x. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
```
```  1305     using assms by (auto split: split_max simp: ereal_add_nonneg_nonneg)
```
```  1306   have "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = (\<integral>\<^isup>+ x. max 0 (f x + g x) \<partial>M)"
```
```  1307     by (simp add: positive_integral_max_0)
```
```  1308   also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
```
```  1309     unfolding ae[THEN positive_integral_cong_AE] ..
```
```  1310   also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+ x. max 0 (g x) \<partial>M)"
```
```  1311     using positive_integral_linear[of "\<lambda>x. max 0 (f x)" 1 "\<lambda>x. max 0 (g x)"] f g
```
```  1312     by auto
```
```  1313   finally show ?thesis
```
```  1314     by (simp add: positive_integral_max_0)
```
```  1315 qed
```
```  1316
```
```  1317 lemma (in measure_space) positive_integral_setsum:
```
```  1318   assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x. 0 \<le> f i x"
```
```  1319   shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))"
```
```  1320 proof cases
```
```  1321   assume f: "finite P"
```
```  1322   from assms have "AE x. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
```
```  1323   from f this assms(1) show ?thesis
```
```  1324   proof induct
```
```  1325     case (insert i P)
```
```  1326     then have "f i \<in> borel_measurable M" "AE x. 0 \<le> f i x"
```
```  1327       "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x. 0 \<le> (\<Sum>i\<in>P. f i x)"
```
```  1328       by (auto intro!: borel_measurable_ereal_setsum setsum_nonneg)
```
```  1329     from positive_integral_add[OF this]
```
```  1330     show ?case using insert by auto
```
```  1331   qed simp
```
```  1332 qed simp
```
```  1333
```
```  1334 lemma (in measure_space) positive_integral_Markov_inequality:
```
```  1335   assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" "c \<noteq> \<infinity>"
```
```  1336   shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
```
```  1337     (is "\<mu> ?A \<le> _ * ?PI")
```
```  1338 proof -
```
```  1339   have "?A \<in> sets M"
```
```  1340     using `A \<in> sets M` u by auto
```
```  1341   hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
```
```  1342     using positive_integral_indicator by simp
```
```  1343   also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
```
```  1344     by (auto intro!: positive_integral_mono_AE
```
```  1345       simp: indicator_def ereal_zero_le_0_iff)
```
```  1346   also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
```
```  1347     using assms
```
```  1348     by (auto intro!: positive_integral_cmult borel_measurable_indicator simp: ereal_zero_le_0_iff)
```
```  1349   finally show ?thesis .
```
```  1350 qed
```
```  1351
```
```  1352 lemma (in measure_space) positive_integral_noteq_infinite:
```
```  1353   assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
```
```  1354   and "integral\<^isup>P M g \<noteq> \<infinity>"
```
```  1355   shows "AE x. g x \<noteq> \<infinity>"
```
```  1356 proof (rule ccontr)
```
```  1357   assume c: "\<not> (AE x. g x \<noteq> \<infinity>)"
```
```  1358   have "\<mu> {x\<in>space M. g x = \<infinity>} \<noteq> 0"
```
```  1359     using c g by (simp add: AE_iff_null_set)
```
```  1360   moreover have "0 \<le> \<mu> {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
```
```  1361   ultimately have "0 < \<mu> {x\<in>space M. g x = \<infinity>}" by auto
```
```  1362   then have "\<infinity> = \<infinity> * \<mu> {x\<in>space M. g x = \<infinity>}" by auto
```
```  1363   also have "\<dots> \<le> (\<integral>\<^isup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
```
```  1364     using g by (subst positive_integral_cmult_indicator) auto
```
```  1365   also have "\<dots> \<le> integral\<^isup>P M g"
```
```  1366     using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
```
```  1367   finally show False using `integral\<^isup>P M g \<noteq> \<infinity>` by auto
```
```  1368 qed
```
```  1369
```
```  1370 lemma (in measure_space) positive_integral_diff:
```
```  1371   assumes f: "f \<in> borel_measurable M"
```
```  1372   and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
```
```  1373   and fin: "integral\<^isup>P M g \<noteq> \<infinity>"
```
```  1374   and mono: "AE x. g x \<le> f x"
```
```  1375   shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g"
```
```  1376 proof -
```
```  1377   have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x. 0 \<le> f x - g x"
```
```  1378     using assms by (auto intro: ereal_diff_positive)
```
```  1379   have pos_f: "AE x. 0 \<le> f x" using mono g by auto
```
```  1380   { fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
```
```  1381       by (cases rule: ereal2_cases[of a b]) auto }
```
```  1382   note * = this
```
```  1383   then have "AE x. f x = f x - g x + g x"
```
```  1384     using mono positive_integral_noteq_infinite[OF g fin] assms by auto
```
```  1385   then have **: "integral\<^isup>P M f = (\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g"
```
```  1386     unfolding positive_integral_add[OF diff g, symmetric]
```
```  1387     by (rule positive_integral_cong_AE)
```
```  1388   show ?thesis unfolding **
```
```  1389     using fin positive_integral_positive[of g]
```
```  1390     by (cases rule: ereal2_cases[of "\<integral>\<^isup>+ x. f x - g x \<partial>M" "integral\<^isup>P M g"]) auto
```
```  1391 qed
```
```  1392
```
```  1393 lemma (in measure_space) positive_integral_suminf:
```
```  1394   assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> f i x"
```
```  1395   shows "(\<integral>\<^isup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^isup>P M (f i))"
```
```  1396 proof -
```
```  1397   have all_pos: "AE x. \<forall>i. 0 \<le> f i x"
```
```  1398     using assms by (auto simp: AE_all_countable)
```
```  1399   have "(\<Sum>i. integral\<^isup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^isup>P M (f i))"
```
```  1400     using positive_integral_positive by (rule suminf_ereal_eq_SUPR)
```
```  1401   also have "\<dots> = (SUP n. \<integral>\<^isup>+x. (\<Sum>i<n. f i x) \<partial>M)"
```
```  1402     unfolding positive_integral_setsum[OF f] ..
```
```  1403   also have "\<dots> = \<integral>\<^isup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
```
```  1404     by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
```
```  1405        (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
```
```  1406   also have "\<dots> = \<integral>\<^isup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
```
```  1407     by (intro positive_integral_cong_AE) (auto simp: suminf_ereal_eq_SUPR)
```
```  1408   finally show ?thesis by simp
```
```  1409 qed
```
```  1410
```
```  1411 text {* Fatou's lemma: convergence theorem on limes inferior *}
```
```  1412 lemma (in measure_space) positive_integral_lim_INF:
```
```  1413   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
```
```  1414   assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> u i x"
```
```  1415   shows "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
```
```  1416 proof -
```
```  1417   have pos: "AE x. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
```
```  1418   have "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
```
```  1419     (SUP n. \<integral>\<^isup>+ x. (INF i:{n..}. u i x) \<partial>M)"
```
```  1420     unfolding liminf_SUPR_INFI using pos u
```
```  1421     by (intro positive_integral_monotone_convergence_SUP_AE)
```
```  1422        (elim AE_mp, auto intro!: AE_I2 intro: le_INFI INF_subset)
```
```  1423   also have "\<dots> \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
```
```  1424     unfolding liminf_SUPR_INFI
```
```  1425     by (auto intro!: SUP_mono exI le_INFI positive_integral_mono INF_leI)
```
```  1426   finally show ?thesis .
```
```  1427 qed
```
```  1428
```
```  1429 lemma (in measure_space) measure_space_density:
```
```  1430   assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x"
```
```  1431     and M'[simp]: "M' = (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)\<rparr>)"
```
```  1432   shows "measure_space M'"
```
```  1433 proof -
```
```  1434   interpret M': sigma_algebra M' by (intro sigma_algebra_cong) auto
```
```  1435   show ?thesis
```
```  1436   proof
```
```  1437     have pos: "\<And>A. AE x. 0 \<le> u x * indicator A x"
```
```  1438       using u by (auto simp: ereal_zero_le_0_iff)
```
```  1439     then show "positive M' (measure M')" unfolding M'
```
```  1440       using u(1) by (auto simp: positive_def intro!: positive_integral_positive)
```
```  1441     show "countably_additive M' (measure M')"
```
```  1442     proof (intro countably_additiveI)
```
```  1443       fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M'"
```
```  1444       then have *: "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
```
```  1445         using u by (auto intro: borel_measurable_indicator)
```
```  1446       assume disj: "disjoint_family A"
```
```  1447       have "(\<Sum>n. measure M' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. u x * indicator (A n) x) \<partial>M)"
```
```  1448         unfolding M' using u(1) *
```
```  1449         by (simp add: positive_integral_suminf[OF _ pos, symmetric])
```
```  1450       also have "\<dots> = (\<integral>\<^isup>+ x. u x * (\<Sum>n. indicator (A n) x) \<partial>M)" using u
```
```  1451         by (intro positive_integral_cong_AE)
```
```  1452            (elim AE_mp, auto intro!: AE_I2 suminf_cmult_ereal)
```
```  1453       also have "\<dots> = (\<integral>\<^isup>+ x. u x * indicator (\<Union>n. A n) x \<partial>M)"
```
```  1454         unfolding suminf_indicator[OF disj] ..
```
```  1455       finally show "(\<Sum>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
```
```  1456         unfolding M' by simp
```
```  1457     qed
```
```  1458   qed
```
```  1459 qed
```
```  1460
```
```  1461 lemma (in measure_space) positive_integral_null_set:
```
```  1462   assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
```
```  1463 proof -
```
```  1464   have "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
```
```  1465   proof (intro positive_integral_cong_AE AE_I)
```
```  1466     show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
```
```  1467       by (auto simp: indicator_def)
```
```  1468     show "\<mu> N = 0" "N \<in> sets M"
```
```  1469       using assms by auto
```
```  1470   qed
```
```  1471   then show ?thesis by simp
```
```  1472 qed
```
```  1473
```
```  1474 lemma (in measure_space) positive_integral_translated_density:
```
```  1475   assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
```
```  1476   assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
```
```  1477     and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
```
```  1478   shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
```
```  1479 proof -
```
```  1480   from measure_space_density[OF f M']
```
```  1481   interpret T: measure_space M' .
```
```  1482   have borel[simp]:
```
```  1483     "borel_measurable M' = borel_measurable M"
```
```  1484     "simple_function M' = simple_function M"
```
```  1485     unfolding measurable_def simple_function_def_raw by (auto simp: M')
```
```  1486   from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess G . note G = this
```
```  1487   note G' = borel_measurable_simple_function[OF this(1)] simple_functionD[OF G(1)]
```
```  1488   note G'(2)[simp]
```
```  1489   { fix P have "AE x. P x \<Longrightarrow> AE x in M'. P x"
```
```  1490       using positive_integral_null_set[of _ f]
```
```  1491       unfolding T.almost_everywhere_def almost_everywhere_def
```
```  1492       by (auto simp: M') }
```
```  1493   note ac = this
```
```  1494   from G(4) g(2) have G_M': "AE x in M'. (SUP i. G i x) = g x"
```
```  1495     by (auto intro!: ac split: split_max)
```
```  1496   { fix i
```
```  1497     let "?I y x" = "indicator (G i -` {y} \<inter> space M) x"
```
```  1498     { fix x assume *: "x \<in> space M" "0 \<le> f x" "0 \<le> g x"
```
```  1499       then have [simp]: "G i ` space M \<inter> {y. G i x = y \<and> x \<in> space M} = {G i x}" by auto
```
```  1500       from * G' G have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * (\<Sum>y\<in>G i`space M. (y * ?I y x))"
```
```  1501         by (subst setsum_ereal_right_distrib) (auto simp: ac_simps)
```
```  1502       also have "\<dots> = f x * G i x"
```
```  1503         by (simp add: indicator_def if_distrib setsum_cases)
```
```  1504       finally have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * G i x" . }
```
```  1505     note to_singleton = this
```
```  1506     have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
```
```  1507       using G T.positive_integral_eq_simple_integral by simp
```
```  1508     also have "\<dots> = (\<Sum>y\<in>G i`space M. y * (\<integral>\<^isup>+x. f x * ?I y x \<partial>M))"
```
```  1509       unfolding simple_integral_def M' by simp
```
```  1510     also have "\<dots> = (\<Sum>y\<in>G i`space M. (\<integral>\<^isup>+x. y * (f x * ?I y x) \<partial>M))"
```
```  1511       using f G' G by (auto intro!: setsum_cong positive_integral_cmult[symmetric])
```
```  1512     also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) \<partial>M)"
```
```  1513       using f G' G by (auto intro!: positive_integral_setsum[symmetric])
```
```  1514     finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
```
```  1515       using f g G' to_singleton by (auto intro!: positive_integral_cong_AE) }
```
```  1516   note [simp] = this
```
```  1517   have "integral\<^isup>P M' g = (SUP i. integral\<^isup>P M' (G i))" using G'(1) G_M'(1) G
```
```  1518     using T.positive_integral_monotone_convergence_SUP[symmetric, OF `incseq G`]
```
```  1519     by (simp cong: T.positive_integral_cong_AE)
```
```  1520   also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f x * G i x \<partial>M))" by simp
```
```  1521   also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. f x * G i x) \<partial>M)"
```
```  1522     using f G' G(2)[THEN incseq_SucD] G
```
```  1523     by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
```
```  1524        (auto simp: ereal_mult_left_mono le_fun_def ereal_zero_le_0_iff)
```
```  1525   also have "\<dots> = (\<integral>\<^isup>+x. f x * g x \<partial>M)" using f G' G g
```
```  1526     by (intro positive_integral_cong_AE)
```
```  1527        (auto simp add: SUPR_ereal_cmult split: split_max)
```
```  1528   finally show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)" .
```
```  1529 qed
```
```  1530
```
```  1531 lemma (in measure_space) positive_integral_0_iff:
```
```  1532   assumes u: "u \<in> borel_measurable M" and pos: "AE x. 0 \<le> u x"
```
```  1533   shows "integral\<^isup>P M u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
```
```  1534     (is "_ \<longleftrightarrow> \<mu> ?A = 0")
```
```  1535 proof -
```
```  1536   have u_eq: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
```
```  1537     by (auto intro!: positive_integral_cong simp: indicator_def)
```
```  1538   show ?thesis
```
```  1539   proof
```
```  1540     assume "\<mu> ?A = 0"
```
```  1541     with positive_integral_null_set[of ?A u] u
```
```  1542     show "integral\<^isup>P M u = 0" by (simp add: u_eq)
```
```  1543   next
```
```  1544     { fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r"
```
```  1545       then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def)
```
```  1546       then have "0 \<le> r" by (auto simp add: ereal_zero_less_0_iff) }
```
```  1547     note gt_1 = this
```
```  1548     assume *: "integral\<^isup>P M u = 0"
```
```  1549     let "?M n" = "{x \<in> space M. 1 \<le> real (n::nat) * u x}"
```
```  1550     have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
```
```  1551     proof -
```
```  1552       { fix n :: nat
```
```  1553         from positive_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"]
```
```  1554         have "\<mu> (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
```
```  1555         moreover have "0 \<le> \<mu> (?M n \<inter> ?A)" using u by auto
```
```  1556         ultimately have "\<mu> (?M n \<inter> ?A) = 0" by auto }
```
```  1557       thus ?thesis by simp
```
```  1558     qed
```
```  1559     also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
```
```  1560     proof (safe intro!: continuity_from_below)
```
```  1561       fix n show "?M n \<inter> ?A \<in> sets M"
```
```  1562         using u by (auto intro!: Int)
```
```  1563     next
```
```  1564       show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
```
```  1565       proof (safe intro!: incseq_SucI)
```
```  1566         fix n :: nat and x
```
```  1567         assume *: "1 \<le> real n * u x"
```
```  1568         also from gt_1[OF this] have "real n * u x \<le> real (Suc n) * u x"
```
```  1569           using `0 \<le> u x` by (auto intro!: ereal_mult_right_mono)
```
```  1570         finally show "1 \<le> real (Suc n) * u x" by auto
```
```  1571       qed
```
```  1572     qed
```
```  1573     also have "\<dots> = \<mu> {x\<in>space M. 0 < u x}"
```
```  1574     proof (safe intro!: arg_cong[where f="\<mu>"] dest!: gt_1)
```
```  1575       fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
```
```  1576       show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
```
```  1577       proof (cases "u x")
```
```  1578         case (real r) with `0 < u x` have "0 < r" by auto
```
```  1579         obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
```
```  1580         hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
```
```  1581         hence "1 \<le> real j * r" using real `0 < r` by auto
```
```  1582         thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
```
```  1583       qed (insert `0 < u x`, auto)
```
```  1584     qed auto
```
```  1585     finally have "\<mu> {x\<in>space M. 0 < u x} = 0" by simp
```
```  1586     moreover
```
```  1587     from pos have "AE x. \<not> (u x < 0)" by auto
```
```  1588     then have "\<mu> {x\<in>space M. u x < 0} = 0"
```
```  1589       using AE_iff_null_set u by auto
```
```  1590     moreover have "\<mu> {x\<in>space M. u x \<noteq> 0} = \<mu> {x\<in>space M. u x < 0} + \<mu> {x\<in>space M. 0 < u x}"
```
```  1591       using u by (subst measure_additive) (auto intro!: arg_cong[where f=\<mu>])
```
```  1592     ultimately show "\<mu> ?A = 0" by simp
```
```  1593   qed
```
```  1594 qed
```
```  1595
```
```  1596 lemma (in measure_space) positive_integral_0_iff_AE:
```
```  1597   assumes u: "u \<in> borel_measurable M"
```
```  1598   shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. u x \<le> 0)"
```
```  1599 proof -
```
```  1600   have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
```
```  1601     using u by auto
```
```  1602   from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
```
```  1603   have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. max 0 (u x) = 0)"
```
```  1604     unfolding positive_integral_max_0
```
```  1605     using AE_iff_null_set[OF sets] u by auto
```
```  1606   also have "\<dots> \<longleftrightarrow> (AE x. u x \<le> 0)" by (auto split: split_max)
```
```  1607   finally show ?thesis .
```
```  1608 qed
```
```  1609
```
```  1610 lemma (in measure_space) positive_integral_const_If:
```
```  1611   "(\<integral>\<^isup>+x. a \<partial>M) = (if 0 \<le> a then a * \<mu> (space M) else 0)"
```
```  1612   by (auto intro!: positive_integral_0_iff_AE[THEN iffD2])
```
```  1613
```
```  1614 lemma (in measure_space) positive_integral_restricted:
```
```  1615   assumes A: "A \<in> sets M"
```
```  1616   shows "integral\<^isup>P (restricted_space A) f = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
```
```  1617     (is "integral\<^isup>P ?R f = integral\<^isup>P M ?f")
```
```  1618 proof -
```
```  1619   interpret R: measure_space ?R
```
```  1620     by (rule restricted_measure_space) fact
```
```  1621   let "?I g x" = "g x * indicator A x :: ereal"
```
```  1622   show ?thesis
```
```  1623     unfolding positive_integral_def
```
```  1624     unfolding simple_function_restricted[OF A]
```
```  1625     unfolding AE_restricted[OF A]
```
```  1626   proof (safe intro!: SUPR_eq)
```
```  1627     fix g assume g: "simple_function M (?I g)" and le: "g \<le> max 0 \<circ> f"
```
```  1628     show "\<exists>j\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> ?I f}.
```
```  1629       integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M j"
```
```  1630     proof (safe intro!: bexI[of _ "?I g"])
```
```  1631       show "integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M (?I g)"
```
```  1632         using g A by (simp add: simple_integral_restricted)
```
```  1633       show "?I g \<le> max 0 \<circ> ?I f"
```
```  1634         using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
```
```  1635     qed fact
```
```  1636   next
```
```  1637     fix g assume g: "simple_function M g" and le: "g \<le> max 0 \<circ> ?I f"
```
```  1638     show "\<exists>i\<in>{g. simple_function M (?I g) \<and> g \<le> max 0 \<circ> f}.
```
```  1639       integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) i"
```
```  1640     proof (safe intro!: bexI[of _ "?I g"])
```
```  1641       show "?I g \<le> max 0 \<circ> f"
```
```  1642         using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
```
```  1643       from le have "\<And>x. g x \<le> ?I (?I g) x"
```
```  1644         by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
```
```  1645       then show "integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) (?I g)"
```
```  1646         using A g by (auto intro!: simple_integral_mono simp: simple_integral_restricted)
```
```  1647       show "simple_function M (?I (?I g))" using g A by auto
```
```  1648     qed
```
```  1649   qed
```
```  1650 qed
```
```  1651
```
```  1652 lemma (in measure_space) positive_integral_subalgebra:
```
```  1653   assumes f: "f \<in> borel_measurable N" "AE x in N. 0 \<le> f x"
```
```  1654   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
```
```  1655   and sa: "sigma_algebra N"
```
```  1656   shows "integral\<^isup>P N f = integral\<^isup>P M f"
```
```  1657 proof -
```
```  1658   interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
```
```  1659   from N.borel_measurable_implies_simple_function_sequence'[OF f(1)] guess fs . note fs = this
```
```  1660   note sf = simple_function_subalgebra[OF fs(1) N(1,2)]
```
```  1661   from N.positive_integral_monotone_convergence_simple[OF fs(2,5,1), symmetric]
```
```  1662   have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * N.\<mu> (fs i -` {x} \<inter> space M))"
```
```  1663     unfolding fs(4) positive_integral_max_0
```
```  1664     unfolding simple_integral_def `space N = space M` by simp
```
```  1665   also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * \<mu> (fs i -` {x} \<inter> space M))"
```
```  1666     using N N.simple_functionD(2)[OF fs(1)] unfolding `space N = space M` by auto
```
```  1667   also have "\<dots> = integral\<^isup>P M f"
```
```  1668     using positive_integral_monotone_convergence_simple[OF fs(2,5) sf, symmetric]
```
```  1669     unfolding fs(4) positive_integral_max_0
```
```  1670     unfolding simple_integral_def `space N = space M` by simp
```
```  1671   finally show ?thesis .
```
```  1672 qed
```
```  1673
```
```  1674 section "Lebesgue Integral"
```
```  1675
```
```  1676 definition integrable where
```
```  1677   "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
```
```  1678     (\<integral>\<^isup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
```
```  1679
```
```  1680 lemma integrableD[dest]:
```
```  1681   assumes "integrable M f"
```
```  1682   shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
```
```  1683   using assms unfolding integrable_def by auto
```
```  1684
```
```  1685 definition lebesgue_integral_def:
```
```  1686   "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. ereal (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. ereal (- f x) \<partial>M))"
```
```  1687
```
```  1688 syntax
```
```  1689   "_lebesgue_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
```
```  1690
```
```  1691 translations
```
```  1692   "\<integral> x. f \<partial>M" == "CONST integral\<^isup>L M (%x. f)"
```
```  1693
```
```  1694 lemma (in measure_space) integrableE:
```
```  1695   assumes "integrable M f"
```
```  1696   obtains r q where
```
```  1697     "(\<integral>\<^isup>+x. ereal (f x)\<partial>M) = ereal r"
```
```  1698     "(\<integral>\<^isup>+x. ereal (-f x)\<partial>M) = ereal q"
```
```  1699     "f \<in> borel_measurable M" "integral\<^isup>L M f = r - q"
```
```  1700   using assms unfolding integrable_def lebesgue_integral_def
```
```  1701   using positive_integral_positive[of "\<lambda>x. ereal (f x)"]
```
```  1702   using positive_integral_positive[of "\<lambda>x. ereal (-f x)"]
```
```  1703   by (cases rule: ereal2_cases[of "(\<integral>\<^isup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^isup>+x. ereal (f x)\<partial>M)"]) auto
```
```  1704
```
```  1705 lemma (in measure_space) integral_cong:
```
```  1706   assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
```
```  1707   shows "integral\<^isup>L M f = integral\<^isup>L M g"
```
```  1708   using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
```
```  1709
```
```  1710 lemma (in measure_space) integral_cong_measure:
```
```  1711   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
```
```  1712   shows "integral\<^isup>L N f = integral\<^isup>L M f"
```
```  1713   by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
```
```  1714
```
```  1715 lemma (in measure_space) integrable_cong_measure:
```
```  1716   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
```
```  1717   shows "integrable N f \<longleftrightarrow> integrable M f"
```
```  1718   using assms
```
```  1719   by (simp add: positive_integral_cong_measure[OF assms] integrable_def measurable_def)
```
```  1720
```
```  1721 lemma (in measure_space) integral_cong_AE:
```
```  1722   assumes cong: "AE x. f x = g x"
```
```  1723   shows "integral\<^isup>L M f = integral\<^isup>L M g"
```
```  1724 proof -
```
```  1725   have *: "AE x. ereal (f x) = ereal (g x)"
```
```  1726     "AE x. ereal (- f x) = ereal (- g x)" using cong by auto
```
```  1727   show ?thesis
```
```  1728     unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
```
```  1729 qed
```
```  1730
```
```  1731 lemma (in measure_space) integrable_cong_AE:
```
```  1732   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1733   assumes "AE x. f x = g x"
```
```  1734   shows "integrable M f = integrable M g"
```
```  1735 proof -
```
```  1736   have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^isup>+ x. ereal (g x) \<partial>M)"
```
```  1737     "(\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) = (\<integral>\<^isup>+ x. ereal (- g x) \<partial>M)"
```
```  1738     using assms by (auto intro!: positive_integral_cong_AE)
```
```  1739   with assms show ?thesis
```
```  1740     by (auto simp: integrable_def)
```
```  1741 qed
```
```  1742
```
```  1743 lemma (in measure_space) integrable_cong:
```
```  1744   "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
```
```  1745   by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
```
```  1746
```
```  1747 lemma (in measure_space) integral_eq_positive_integral:
```
```  1748   assumes f: "\<And>x. 0 \<le> f x"
```
```  1749   shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)"
```
```  1750 proof -
```
```  1751   { fix x have "max 0 (ereal (- f x)) = 0" using f[of x] by (simp split: split_max) }
```
```  1752   then have "0 = (\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M)" by simp
```
```  1753   also have "\<dots> = (\<integral>\<^isup>+ x. ereal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
```
```  1754   finally show ?thesis
```
```  1755     unfolding lebesgue_integral_def by simp
```
```  1756 qed
```
```  1757
```
```  1758 lemma (in measure_space) integral_vimage:
```
```  1759   assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
```
```  1760   assumes f: "f \<in> borel_measurable M'"
```
```  1761   shows "integral\<^isup>L M' f = (\<integral>x. f (T x) \<partial>M)"
```
```  1762 proof -
```
```  1763   interpret T: measure_space M' by (rule measure_space_vimage[OF T])
```
```  1764   from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
```
```  1765   have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable M'" "(\<lambda>x. ereal (- f x)) \<in> borel_measurable M'"
```
```  1766     and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
```
```  1767     using f by (auto simp: comp_def)
```
```  1768   then show ?thesis
```
```  1769     using f unfolding lebesgue_integral_def integrable_def
```
```  1770     by (auto simp: borel[THEN positive_integral_vimage[OF T]])
```
```  1771 qed
```
```  1772
```
```  1773 lemma (in measure_space) integrable_vimage:
```
```  1774   assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
```
```  1775   assumes f: "integrable M' f"
```
```  1776   shows "integrable M (\<lambda>x. f (T x))"
```
```  1777 proof -
```
```  1778   interpret T: measure_space M' by (rule measure_space_vimage[OF T])
```
```  1779   from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
```
```  1780   have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable M'" "(\<lambda>x. ereal (- f x)) \<in> borel_measurable M'"
```
```  1781     and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
```
```  1782     using f by (auto simp: comp_def)
```
```  1783   then show ?thesis
```
```  1784     using f unfolding lebesgue_integral_def integrable_def
```
```  1785     by (auto simp: borel[THEN positive_integral_vimage[OF T]])
```
```  1786 qed
```
```  1787
```
```  1788 lemma (in measure_space) integral_translated_density:
```
```  1789   assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
```
```  1790     and g: "g \<in> borel_measurable M"
```
```  1791     and N: "space N = space M" "sets N = sets M"
```
```  1792     and density: "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
```
```  1793       (is "\<And>A. _ \<Longrightarrow> _ = ?d A")
```
```  1794   shows "integral\<^isup>L N g = (\<integral> x. f x * g x \<partial>M)" (is ?integral)
```
```  1795     and "integrable N g = integrable M (\<lambda>x. f x * g x)" (is ?integrable)
```
```  1796 proof -
```
```  1797   from f have ms: "measure_space (M\<lparr>measure := ?d\<rparr>)"
```
```  1798     by (intro measure_space_density[where u="\<lambda>x. ereal (f x)"]) auto
```
```  1799
```
```  1800   from ms density N have "(\<integral>\<^isup>+ x. g x \<partial>N) =  (\<integral>\<^isup>+ x. max 0 (ereal (g x)) \<partial>M\<lparr>measure := ?d\<rparr>)"
```
```  1801     unfolding positive_integral_max_0
```
```  1802     by (intro measure_space.positive_integral_cong_measure) auto
```
```  1803   also have "\<dots> = (\<integral>\<^isup>+ x. ereal (f x) * max 0 (ereal (g x)) \<partial>M)"
```
```  1804     using f g by (intro positive_integral_translated_density) auto
```
```  1805   also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (ereal (f x * g x)) \<partial>M)"
```
```  1806     using f by (intro positive_integral_cong_AE)
```
```  1807                (auto simp: ereal_max_0 zero_le_mult_iff split: split_max)
```
```  1808   finally have pos: "(\<integral>\<^isup>+ x. g x \<partial>N) = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
```
```  1809     by (simp add: positive_integral_max_0)
```
```  1810
```
```  1811   from ms density N have "(\<integral>\<^isup>+ x. - (g x) \<partial>N) =  (\<integral>\<^isup>+ x. max 0 (ereal (- g x)) \<partial>M\<lparr>measure := ?d\<rparr>)"
```
```  1812     unfolding positive_integral_max_0
```
```  1813     by (intro measure_space.positive_integral_cong_measure) auto
```
```  1814   also have "\<dots> = (\<integral>\<^isup>+ x. ereal (f x) * max 0 (ereal (- g x)) \<partial>M)"
```
```  1815     using f g by (intro positive_integral_translated_density) auto
```
```  1816   also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (ereal (- f x * g x)) \<partial>M)"
```
```  1817     using f by (intro positive_integral_cong_AE)
```
```  1818                (auto simp: ereal_max_0 mult_le_0_iff split: split_max)
```
```  1819   finally have neg: "(\<integral>\<^isup>+ x. - g x \<partial>N) = (\<integral>\<^isup>+ x. - (f x * g x) \<partial>M)"
```
```  1820     by (simp add: positive_integral_max_0)
```
```  1821
```
```  1822   have g_N: "g \<in> borel_measurable N"
```
```  1823     using g N unfolding measurable_def by simp
```
```  1824
```
```  1825   show ?integral ?integrable
```
```  1826     unfolding lebesgue_integral_def integrable_def
```
```  1827     using pos neg f g g_N by auto
```
```  1828 qed
```
```  1829
```
```  1830 lemma (in measure_space) integral_minus[intro, simp]:
```
```  1831   assumes "integrable M f"
```
```  1832   shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
```
```  1833   using assms by (auto simp: integrable_def lebesgue_integral_def)
```
```  1834
```
```  1835 lemma (in measure_space) integral_minus_iff[simp]:
```
```  1836   "integrable M (\<lambda>x. - f x) \<longleftrightarrow> integrable M f"
```
```  1837 proof
```
```  1838   assume "integrable M (\<lambda>x. - f x)"
```
```  1839   then have "integrable M (\<lambda>x. - (- f x))"
```
```  1840     by (rule integral_minus)
```
```  1841   then show "integrable M f" by simp
```
```  1842 qed (rule integral_minus)
```
```  1843
```
```  1844 lemma (in measure_space) integral_of_positive_diff:
```
```  1845   assumes integrable: "integrable M u" "integrable M v"
```
```  1846   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"
```
```  1847   shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
```
```  1848 proof -
```
```  1849   let "?f x" = "max 0 (ereal (f x))"
```
```  1850   let "?mf x" = "max 0 (ereal (- f x))"
```
```  1851   let "?u x" = "max 0 (ereal (u x))"
```
```  1852   let "?v x" = "max 0 (ereal (v x))"
```
```  1853
```
```  1854   from borel_measurable_diff[of u v] integrable
```
```  1855   have f_borel: "?f \<in> borel_measurable M" and
```
```  1856     mf_borel: "?mf \<in> borel_measurable M" and
```
```  1857     v_borel: "?v \<in> borel_measurable M" and
```
```  1858     u_borel: "?u \<in> borel_measurable M" and
```
```  1859     "f \<in> borel_measurable M"
```
```  1860     by (auto simp: f_def[symmetric] integrable_def)
```
```  1861
```
```  1862   have "(\<integral>\<^isup>+ x. ereal (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
```
```  1863     using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
```
```  1864   moreover have "(\<integral>\<^isup>+ x. ereal (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
```
```  1865     using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
```
```  1866   ultimately show f: "integrable M f"
```
```  1867     using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
```
```  1868     by (auto simp: integrable_def f_def positive_integral_max_0)
```
```  1869
```
```  1870   have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
```
```  1871     unfolding f_def using pos by (simp split: split_max)
```
```  1872   then have "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
```
```  1873   then have "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
```
```  1874       real (integral\<^isup>P M ?v + integral\<^isup>P M ?f)"
```
```  1875     using positive_integral_add[OF u_borel _ mf_borel]
```
```  1876     using positive_integral_add[OF v_borel _ f_borel]
```
```  1877     by auto
```
```  1878   then show "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
```
```  1879     unfolding positive_integral_max_0
```
```  1880     unfolding pos[THEN integral_eq_positive_integral]
```
```  1881     using integrable f by (auto elim!: integrableE)
```
```  1882 qed
```
```  1883
```
```  1884 lemma (in measure_space) integral_linear:
```
```  1885   assumes "integrable M f" "integrable M g" and "0 \<le> a"
```
```  1886   shows "integrable M (\<lambda>t. a * f t + g t)"
```
```  1887   and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g" (is ?EQ)
```
```  1888 proof -
```
```  1889   let "?f x" = "max 0 (ereal (f x))"
```
```  1890   let "?g x" = "max 0 (ereal (g x))"
```
```  1891   let "?mf x" = "max 0 (ereal (- f x))"
```
```  1892   let "?mg x" = "max 0 (ereal (- g x))"
```
```  1893   let "?p t" = "max 0 (a * f t) + max 0 (g t)"
```
```  1894   let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
```
```  1895
```
```  1896   from assms have linear:
```
```  1897     "(\<integral>\<^isup>+ x. ereal a * ?f x + ?g x \<partial>M) = ereal a * integral\<^isup>P M ?f + integral\<^isup>P M ?g"
```
```  1898     "(\<integral>\<^isup>+ x. ereal a * ?mf x + ?mg x \<partial>M) = ereal a * integral\<^isup>P M ?mf + integral\<^isup>P M ?mg"
```
```  1899     by (auto intro!: positive_integral_linear simp: integrable_def)
```
```  1900
```
```  1901   have *: "(\<integral>\<^isup>+x. ereal (- ?p x) \<partial>M) = 0" "(\<integral>\<^isup>+x. ereal (- ?n x) \<partial>M) = 0"
```
```  1902     using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
```
```  1903   have **: "\<And>x. ereal a * ?f x + ?g x = max 0 (ereal (?p x))"
```
```  1904            "\<And>x. ereal a * ?mf x + ?mg x = max 0 (ereal (?n x))"
```
```  1905     using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
```
```  1906
```
```  1907   have "integrable M ?p" "integrable M ?n"
```
```  1908       "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
```
```  1909     using linear assms unfolding integrable_def ** *
```
```  1910     by (auto simp: positive_integral_max_0)
```
```  1911   note diff = integral_of_positive_diff[OF this]
```
```  1912
```
```  1913   show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
```
```  1914   from assms linear show ?EQ
```
```  1915     unfolding diff(2) ** positive_integral_max_0
```
```  1916     unfolding lebesgue_integral_def *
```
```  1917     by (auto elim!: integrableE simp: field_simps)
```
```  1918 qed
```
```  1919
```
```  1920 lemma (in measure_space) integral_add[simp, intro]:
```
```  1921   assumes "integrable M f" "integrable M g"
```
```  1922   shows "integrable M (\<lambda>t. f t + g t)"
```
```  1923   and "(\<integral> t. f t + g t \<partial>M) = integral\<^isup>L M f + integral\<^isup>L M g"
```
```  1924   using assms integral_linear[where a=1] by auto
```
```  1925
```
```  1926 lemma (in measure_space) integral_zero[simp, intro]:
```
```  1927   shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
```
```  1928   unfolding integrable_def lebesgue_integral_def
```
```  1929   by (auto simp add: borel_measurable_const)
```
```  1930
```
```  1931 lemma (in measure_space) integral_cmult[simp, intro]:
```
```  1932   assumes "integrable M f"
```
```  1933   shows "integrable M (\<lambda>t. a * f t)" (is ?P)
```
```  1934   and "(\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f" (is ?I)
```
```  1935 proof -
```
```  1936   have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f"
```
```  1937   proof (cases rule: le_cases)
```
```  1938     assume "0 \<le> a" show ?thesis
```
```  1939       using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
```
```  1940       by (simp add: integral_zero)
```
```  1941   next
```
```  1942     assume "a \<le> 0" hence "0 \<le> - a" by auto
```
```  1943     have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
```
```  1944     show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
```
```  1945         integral_minus(1)[of "\<lambda>t. - a * f t"]
```
```  1946       unfolding * integral_zero by simp
```
```  1947   qed
```
```  1948   thus ?P ?I by auto
```
```  1949 qed
```
```  1950
```
```  1951 lemma (in measure_space) integral_multc:
```
```  1952   assumes "integrable M f"
```
```  1953   shows "(\<integral> x. f x * c \<partial>M) = integral\<^isup>L M f * c"
```
```  1954   unfolding mult_commute[of _ c] integral_cmult[OF assms] ..
```
```  1955
```
```  1956 lemma (in measure_space) integral_mono_AE:
```
```  1957   assumes fg: "integrable M f" "integrable M g"
```
```  1958   and mono: "AE t. f t \<le> g t"
```
```  1959   shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
```
```  1960 proof -
```
```  1961   have "AE x. ereal (f x) \<le> ereal (g x)"
```
```  1962     using mono by auto
```
```  1963   moreover have "AE x. ereal (- g x) \<le> ereal (- f x)"
```
```  1964     using mono by auto
```
```  1965   ultimately show ?thesis using fg
```
```  1966     by (auto intro!: add_mono positive_integral_mono_AE real_of_ereal_positive_mono
```
```  1967              simp: positive_integral_positive lebesgue_integral_def diff_minus)
```
```  1968 qed
```
```  1969
```
```  1970 lemma (in measure_space) integral_mono:
```
```  1971   assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
```
```  1972   shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
```
```  1973   using assms by (auto intro: integral_mono_AE)
```
```  1974
```
```  1975 lemma (in measure_space) integral_diff[simp, intro]:
```
```  1976   assumes f: "integrable M f" and g: "integrable M g"
```
```  1977   shows "integrable M (\<lambda>t. f t - g t)"
```
```  1978   and "(\<integral> t. f t - g t \<partial>M) = integral\<^isup>L M f - integral\<^isup>L M g"
```
```  1979   using integral_add[OF f integral_minus(1)[OF g]]
```
```  1980   unfolding diff_minus integral_minus(2)[OF g]
```
```  1981   by auto
```
```  1982
```
```  1983 lemma (in measure_space) integral_indicator[simp, intro]:
```
```  1984   assumes "A \<in> sets M" and "\<mu> A \<noteq> \<infinity>"
```
```  1985   shows "integral\<^isup>L M (indicator A) = real (\<mu> A)" (is ?int)
```
```  1986   and "integrable M (indicator A)" (is ?able)
```
```  1987 proof -
```
```  1988   from `A \<in> sets M` have *:
```
```  1989     "\<And>x. ereal (indicator A x) = indicator A x"
```
```  1990     "(\<integral>\<^isup>+x. ereal (- indicator A x) \<partial>M) = 0"
```
```  1991     by (auto split: split_indicator simp: positive_integral_0_iff_AE one_ereal_def)
```
```  1992   show ?int ?able
```
```  1993     using assms unfolding lebesgue_integral_def integrable_def
```
```  1994     by (auto simp: * positive_integral_indicator borel_measurable_indicator)
```
```  1995 qed
```
```  1996
```
```  1997 lemma (in measure_space) integral_cmul_indicator:
```
```  1998   assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<infinity>"
```
```  1999   shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
```
```  2000   and "(\<integral>x. c * indicator A x \<partial>M) = c * real (\<mu> A)" (is ?I)
```
```  2001 proof -
```
```  2002   show ?P
```
```  2003   proof (cases "c = 0")
```
```  2004     case False with assms show ?thesis by simp
```
```  2005   qed simp
```
```  2006
```
```  2007   show ?I
```
```  2008   proof (cases "c = 0")
```
```  2009     case False with assms show ?thesis by simp
```
```  2010   qed simp
```
```  2011 qed
```
```  2012
```
```  2013 lemma (in measure_space) integral_setsum[simp, intro]:
```
```  2014   assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f n)"
```
```  2015   shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^isup>L M (f i))" (is "?int S")
```
```  2016     and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
```
```  2017 proof -
```
```  2018   have "?int S \<and> ?I S"
```
```  2019   proof (cases "finite S")
```
```  2020     assume "finite S"
```
```  2021     from this assms show ?thesis by (induct S) simp_all
```
```  2022   qed simp
```
```  2023   thus "?int S" and "?I S" by auto
```
```  2024 qed
```
```  2025
```
```  2026 lemma (in measure_space) integrable_abs:
```
```  2027   assumes "integrable M f"
```
```  2028   shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
```
```  2029 proof -
```
```  2030   from assms have *: "(\<integral>\<^isup>+x. ereal (- \<bar>f x\<bar>)\<partial>M) = 0"
```
```  2031     "\<And>x. ereal \<bar>f x\<bar> = max 0 (ereal (f x)) + max 0 (ereal (- f x))"
```
```  2032     by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
```
```  2033   with assms show ?thesis
```
```  2034     by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
```
```  2035 qed
```
```  2036
```
```  2037 lemma (in measure_space) integral_subalgebra:
```
```  2038   assumes borel: "f \<in> borel_measurable N"
```
```  2039   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A" and sa: "sigma_algebra N"
```
```  2040   shows "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
```
```  2041     and "integral\<^isup>L N f = integral\<^isup>L M f" (is ?I)
```
```  2042 proof -
```
```  2043   interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
```
```  2044   have "(\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>M)"
```
```  2045        "(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M)"
```
```  2046     using borel by (auto intro!: positive_integral_subalgebra N sa)
```
```  2047   moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
```
```  2048     using assms unfolding measurable_def by auto
```
```  2049   ultimately show ?P ?I
```
```  2050     by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
```
```  2051 qed
```
```  2052
```
```  2053 lemma (in measure_space) integrable_bound:
```
```  2054   assumes "integrable M f"
```
```  2055   and f: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  2056     "\<And>x. x \<in> space M \<Longrightarrow> \<bar>g x\<bar> \<le> f x"
```
```  2057   assumes borel: "g \<in> borel_measurable M"
```
```  2058   shows "integrable M g"
```
```  2059 proof -
```
```  2060   have "(\<integral>\<^isup>+ x. ereal (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. ereal \<bar>g x\<bar> \<partial>M)"
```
```  2061     by (auto intro!: positive_integral_mono)
```
```  2062   also have "\<dots> \<le> (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)"
```
```  2063     using f by (auto intro!: positive_integral_mono)
```
```  2064   also have "\<dots> < \<infinity>"
```
```  2065     using `integrable M f` unfolding integrable_def by auto
```
```  2066   finally have pos: "(\<integral>\<^isup>+ x. ereal (g x) \<partial>M) < \<infinity>" .
```
```  2067
```
```  2068   have "(\<integral>\<^isup>+ x. ereal (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. ereal (\<bar>g x\<bar>) \<partial>M)"
```
```  2069     by (auto intro!: positive_integral_mono)
```
```  2070   also have "\<dots> \<le> (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)"
```
```  2071     using f by (auto intro!: positive_integral_mono)
```
```  2072   also have "\<dots> < \<infinity>"
```
```  2073     using `integrable M f` unfolding integrable_def by auto
```
```  2074   finally have neg: "(\<integral>\<^isup>+ x. ereal (- g x) \<partial>M) < \<infinity>" .
```
```  2075
```
```  2076   from neg pos borel show ?thesis
```
```  2077     unfolding integrable_def by auto
```
```  2078 qed
```
```  2079
```
```  2080 lemma (in measure_space) integrable_abs_iff:
```
```  2081   "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
```
```  2082   by (auto intro!: integrable_bound[where g=f] integrable_abs)
```
```  2083
```
```  2084 lemma (in measure_space) integrable_max:
```
```  2085   assumes int: "integrable M f" "integrable M g"
```
```  2086   shows "integrable M (\<lambda> x. max (f x) (g x))"
```
```  2087 proof (rule integrable_bound)
```
```  2088   show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
```
```  2089     using int by (simp add: integrable_abs)
```
```  2090   show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
```
```  2091     using int unfolding integrable_def by auto
```
```  2092 next
```
```  2093   fix x assume "x \<in> space M"
```
```  2094   show "0 \<le> \<bar>f x\<bar> + \<bar>g x\<bar>" "\<bar>max (f x) (g x)\<bar> \<le> \<bar>f x\<bar> + \<bar>g x\<bar>"
```
```  2095     by auto
```
```  2096 qed
```
```  2097
```
```  2098 lemma (in measure_space) integrable_min:
```
```  2099   assumes int: "integrable M f" "integrable M g"
```
```  2100   shows "integrable M (\<lambda> x. min (f x) (g x))"
```
```  2101 proof (rule integrable_bound)
```
```  2102   show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
```
```  2103     using int by (simp add: integrable_abs)
```
```  2104   show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
```
```  2105     using int unfolding integrable_def by auto
```
```  2106 next
```
```  2107   fix x assume "x \<in> space M"
```
```  2108   show "0 \<le> \<bar>f x\<bar> + \<bar>g x\<bar>" "\<bar>min (f x) (g x)\<bar> \<le> \<bar>f x\<bar> + \<bar>g x\<bar>"
```
```  2109     by auto
```
```  2110 qed
```
```  2111
```
```  2112 lemma (in measure_space) integral_triangle_inequality:
```
```  2113   assumes "integrable M f"
```
```  2114   shows "\<bar>integral\<^isup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
```
```  2115 proof -
```
```  2116   have "\<bar>integral\<^isup>L M f\<bar> = max (integral\<^isup>L M f) (- integral\<^isup>L M f)" by auto
```
```  2117   also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
```
```  2118       using assms integral_minus(2)[of f, symmetric]
```
```  2119       by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
```
```  2120   finally show ?thesis .
```
```  2121 qed
```
```  2122
```
```  2123 lemma (in measure_space) integral_positive:
```
```  2124   assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  2125   shows "0 \<le> integral\<^isup>L M f"
```
```  2126 proof -
```
```  2127   have "0 = (\<integral>x. 0 \<partial>M)" by (auto simp: integral_zero)
```
```  2128   also have "\<dots> \<le> integral\<^isup>L M f"
```
```  2129     using assms by (rule integral_mono[OF integral_zero(1)])
```
```  2130   finally show ?thesis .
```
```  2131 qed
```
```  2132
```
```  2133 lemma (in measure_space) integral_monotone_convergence_pos:
```
```  2134   assumes i: "\<And>i. integrable M (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
```
```  2135   and pos: "\<And>x i. 0 \<le> f i x"
```
```  2136   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
```
```  2137   and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
```
```  2138   shows "integrable M u"
```
```  2139   and "integral\<^isup>L M u = x"
```
```  2140 proof -
```
```  2141   { fix x have "0 \<le> u x"
```
```  2142       using mono pos[of 0 x] incseq_le[OF _ lim, of x 0]
```
```  2143       by (simp add: mono_def incseq_def) }
```
```  2144   note pos_u = this
```
```  2145
```
```  2146   have SUP_F: "\<And>x. (SUP n. ereal (f n x)) = ereal (u x)"
```
```  2147     unfolding SUP_eq_LIMSEQ[OF mono] by (rule lim)
```
```  2148
```
```  2149   have borel_f: "\<And>i. (\<lambda>x. ereal (f i x)) \<in> borel_measurable M"
```
```  2150     using i unfolding integrable_def by auto
```
```  2151   hence "(\<lambda>x. SUP i. ereal (f i x)) \<in> borel_measurable M"
```
```  2152     by auto
```
```  2153   hence borel_u: "u \<in> borel_measurable M"
```
```  2154     by (auto simp: borel_measurable_ereal_iff SUP_F)
```
```  2155
```
```  2156   hence [simp]: "\<And>i. (\<integral>\<^isup>+x. ereal (- f i x) \<partial>M) = 0" "(\<integral>\<^isup>+x. ereal (- u x) \<partial>M) = 0"
```
```  2157     using i borel_u pos pos_u by (auto simp: positive_integral_0_iff_AE integrable_def)
```
```  2158
```
```  2159   have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. ereal (f n x) \<partial>M) = ereal (integral\<^isup>L M (f n))"
```
```  2160     using i positive_integral_positive by (auto simp: ereal_real lebesgue_integral_def integrable_def)
```
```  2161
```
```  2162   have pos_integral: "\<And>n. 0 \<le> integral\<^isup>L M (f n)"
```
```  2163     using pos i by (auto simp: integral_positive)
```
```  2164   hence "0 \<le> x"
```
```  2165     using LIMSEQ_le_const[OF ilim, of 0] by auto
```
```  2166
```
```  2167   from mono pos i have pI: "(\<integral>\<^isup>+ x. ereal (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. ereal (f n x) \<partial>M))"
```
```  2168     by (auto intro!: positive_integral_monotone_convergence_SUP
```
```  2169       simp: integrable_def incseq_mono incseq_Suc_iff le_fun_def SUP_F[symmetric])
```
```  2170   also have "\<dots> = ereal x" unfolding integral_eq
```
```  2171   proof (rule SUP_eq_LIMSEQ[THEN iffD2])
```
```  2172     show "mono (\<lambda>n. integral\<^isup>L M (f n))"
```
```  2173       using mono i by (auto simp: mono_def intro!: integral_mono)
```
```  2174     show "(\<lambda>n. integral\<^isup>L M (f n)) ----> x" using ilim .
```
```  2175   qed
```
```  2176   finally show  "integrable M u" "integral\<^isup>L M u = x" using borel_u `0 \<le> x`
```
```  2177     unfolding integrable_def lebesgue_integral_def by auto
```
```  2178 qed
```
```  2179
```
```  2180 lemma (in measure_space) integral_monotone_convergence:
```
```  2181   assumes f: "\<And>i. integrable M (f i)" and "mono f"
```
```  2182   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
```
```  2183   and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
```
```  2184   shows "integrable M u"
```
```  2185   and "integral\<^isup>L M u = x"
```
```  2186 proof -
```
```  2187   have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
```
```  2188       using f by (auto intro!: integral_diff)
```
```  2189   have 2: "\<And>x. mono (\<lambda>n. f n x - f 0 x)" using `mono f`
```
```  2190       unfolding mono_def le_fun_def by auto
```
```  2191   have 3: "\<And>x n. 0 \<le> f n x - f 0 x" using `mono f`
```
```  2192       unfolding mono_def le_fun_def by (auto simp: field_simps)
```
```  2193   have 4: "\<And>x. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
```
```  2194     using lim by (auto intro!: LIMSEQ_diff)
```
```  2195   have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^isup>L M (f 0)"
```
```  2196     using f ilim by (auto intro!: LIMSEQ_diff simp: integral_diff)
```
```  2197   note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5]
```
```  2198   have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
```
```  2199     using diff(1) f by (rule integral_add(1))
```
```  2200   with diff(2) f show "integrable M u" "integral\<^isup>L M u = x"
```
```  2201     by (auto simp: integral_diff)
```
```  2202 qed
```
```  2203
```
```  2204 lemma (in measure_space) integral_0_iff:
```
```  2205   assumes "integrable M f"
```
```  2206   shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
```
```  2207 proof -
```
```  2208   have *: "(\<integral>\<^isup>+x. ereal (- \<bar>f x\<bar>) \<partial>M) = 0"
```
```  2209     using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
```
```  2210   have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
```
```  2211   hence "(\<lambda>x. ereal (\<bar>f x\<bar>)) \<in> borel_measurable M"
```
```  2212     "(\<integral>\<^isup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
```
```  2213   from positive_integral_0_iff[OF this(1)] this(2)
```
```  2214   show ?thesis unfolding lebesgue_integral_def *
```
```  2215     using positive_integral_positive[of "\<lambda>x. ereal \<bar>f x\<bar>"]
```
```  2216     by (auto simp add: real_of_ereal_eq_0)
```
```  2217 qed
```
```  2218
```
```  2219 lemma (in measure_space) positive_integral_PInf:
```
```  2220   assumes f: "f \<in> borel_measurable M"
```
```  2221   and not_Inf: "integral\<^isup>P M f \<noteq> \<infinity>"
```
```  2222   shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
```
```  2223 proof -
```
```  2224   have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^isup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
```
```  2225     using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
```
```  2226   also have "\<dots> \<le> integral\<^isup>P M (\<lambda>x. max 0 (f x))"
```
```  2227     by (auto intro!: positive_integral_mono simp: indicator_def max_def)
```
```  2228   finally have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) \<le> integral\<^isup>P M f"
```
```  2229     by (simp add: positive_integral_max_0)
```
```  2230   moreover have "0 \<le> \<mu> (f -` {\<infinity>} \<inter> space M)"
```
```  2231     using f by (simp add: measurable_sets)
```
```  2232   ultimately show ?thesis
```
```  2233     using assms by (auto split: split_if_asm)
```
```  2234 qed
```
```  2235
```
```  2236 lemma (in measure_space) positive_integral_PInf_AE:
```
```  2237   assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<infinity>" shows "AE x. f x \<noteq> \<infinity>"
```
```  2238 proof (rule AE_I)
```
```  2239   show "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
```
```  2240     by (rule positive_integral_PInf[OF assms])
```
```  2241   show "f -` {\<infinity>} \<inter> space M \<in> sets M"
```
```  2242     using assms by (auto intro: borel_measurable_vimage)
```
```  2243 qed auto
```
```  2244
```
```  2245 lemma (in measure_space) simple_integral_PInf:
```
```  2246   assumes "simple_function M f" "\<And>x. 0 \<le> f x"
```
```  2247   and "integral\<^isup>S M f \<noteq> \<infinity>"
```
```  2248   shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
```
```  2249 proof (rule positive_integral_PInf)
```
```  2250   show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
```
```  2251   show "integral\<^isup>P M f \<noteq> \<infinity>"
```
```  2252     using assms by (simp add: positive_integral_eq_simple_integral)
```
```  2253 qed
```
```  2254
```
```  2255 lemma (in measure_space) integral_real:
```
```  2256   "AE x. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f) - real (integral\<^isup>P M (\<lambda>x. - f x))"
```
```  2257   using assms unfolding lebesgue_integral_def
```
```  2258   by (subst (1 2) positive_integral_cong_AE) (auto simp add: ereal_real)
```
```  2259
```
```  2260 lemma (in finite_measure) lebesgue_integral_const[simp]:
```
```  2261   shows "integrable M (\<lambda>x. a)"
```
```  2262   and  "(\<integral>x. a \<partial>M) = a * \<mu>' (space M)"
```
```  2263 proof -
```
```  2264   { fix a :: real assume "0 \<le> a"
```
```  2265     then have "(\<integral>\<^isup>+ x. ereal a \<partial>M) = ereal a * \<mu> (space M)"
```
```  2266       by (subst positive_integral_const) auto
```
```  2267     moreover
```
```  2268     from `0 \<le> a` have "(\<integral>\<^isup>+ x. ereal (-a) \<partial>M) = 0"
```
```  2269       by (subst positive_integral_0_iff_AE) auto
```
```  2270     ultimately have "integrable M (\<lambda>x. a)" by (auto simp: integrable_def) }
```
```  2271   note * = this
```
```  2272   show "integrable M (\<lambda>x. a)"
```
```  2273   proof cases
```
```  2274     assume "0 \<le> a" with * show ?thesis .
```
```  2275   next
```
```  2276     assume "\<not> 0 \<le> a"
```
```  2277     then have "0 \<le> -a" by auto
```
```  2278     from *[OF this] show ?thesis by simp
```
```  2279   qed
```
```  2280   show "(\<integral>x. a \<partial>M) = a * \<mu>' (space M)"
```
```  2281     by (simp add: \<mu>'_def lebesgue_integral_def positive_integral_const_If)
```
```  2282 qed
```
```  2283
```
```  2284 lemma indicator_less[simp]:
```
```  2285   "indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
```
```  2286   by (simp add: indicator_def not_le)
```
```  2287
```
```  2288 lemma (in finite_measure) integral_less_AE:
```
```  2289   assumes int: "integrable M X" "integrable M Y"
```
```  2290   assumes A: "\<mu> A \<noteq> 0" "A \<in> sets M" "AE x. x \<in> A \<longrightarrow> X x \<noteq> Y x"
```
```  2291   assumes gt: "AE x. X x \<le> Y x"
```
```  2292   shows "integral\<^isup>L M X < integral\<^isup>L M Y"
```
```  2293 proof -
```
```  2294   have "integral\<^isup>L M X \<le> integral\<^isup>L M Y"
```
```  2295     using gt int by (intro integral_mono_AE) auto
```
```  2296   moreover
```
```  2297   have "integral\<^isup>L M X \<noteq> integral\<^isup>L M Y"
```
```  2298   proof
```
```  2299     assume eq: "integral\<^isup>L M X = integral\<^isup>L M Y"
```
```  2300     have "integral\<^isup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^isup>L M (\<lambda>x. Y x - X x)"
```
```  2301       using gt by (intro integral_cong_AE) auto
```
```  2302     also have "\<dots> = 0"
```
```  2303       using eq int by simp
```
```  2304     finally have "\<mu> {x \<in> space M. Y x - X x \<noteq> 0} = 0"
```
```  2305       using int by (simp add: integral_0_iff)
```
```  2306     moreover
```
```  2307     have "(\<integral>\<^isup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^isup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
```
```  2308       using A by (intro positive_integral_mono_AE) auto
```
```  2309     then have "\<mu> A \<le> \<mu> {x \<in> space M. Y x - X x \<noteq> 0}"
```
```  2310       using int A by (simp add: integrable_def)
```
```  2311     moreover note `\<mu> A \<noteq> 0` positive_measure[OF `A \<in> sets M`]
```
```  2312     ultimately show False by auto
```
```  2313   qed
```
```  2314   ultimately show ?thesis by auto
```
```  2315 qed
```
```  2316
```
```  2317 lemma (in finite_measure) integral_less_AE_space:
```
```  2318   assumes int: "integrable M X" "integrable M Y"
```
```  2319   assumes gt: "AE x. X x < Y x" "\<mu> (space M) \<noteq> 0"
```
```  2320   shows "integral\<^isup>L M X < integral\<^isup>L M Y"
```
```  2321   using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
```
```  2322
```
```  2323 lemma (in measure_space) integral_dominated_convergence:
```
```  2324   assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
```
```  2325   and w: "integrable M w"
```
```  2326   and u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  2327   shows "integrable M u'"
```
```  2328   and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar> \<partial>M)) ----> 0" (is "?lim_diff")
```
```  2329   and "(\<lambda>i. integral\<^isup>L M (u i)) ----> integral\<^isup>L M u'" (is ?lim)
```
```  2330 proof -
```
```  2331   { fix x j assume x: "x \<in> space M"
```
```  2332     from u'[OF x] have "(\<lambda>i. \<bar>u i x\<bar>) ----> \<bar>u' x\<bar>" by (rule LIMSEQ_imp_rabs)
```
```  2333     from LIMSEQ_le_const2[OF this]
```
```  2334     have "\<bar>u' x\<bar> \<le> w x" using bound[OF x] by auto }
```
```  2335   note u'_bound = this
```
```  2336
```
```  2337   from u[unfolded integrable_def]
```
```  2338   have u'_borel: "u' \<in> borel_measurable M"
```
```  2339     using u' by (blast intro: borel_measurable_LIMSEQ[of u])
```
```  2340
```
```  2341   { fix x assume x: "x \<in> space M"
```
```  2342     then have "0 \<le> \<bar>u 0 x\<bar>" by auto
```
```  2343     also have "\<dots> \<le> w x" using bound[OF x] by auto
```
```  2344     finally have "0 \<le> w x" . }
```
```  2345   note w_pos = this
```
```  2346
```
```  2347   show "integrable M u'"
```
```  2348   proof (rule integrable_bound)
```
```  2349     show "integrable M w" by fact
```
```  2350     show "u' \<in> borel_measurable M" by fact
```
```  2351   next
```
```  2352     fix x assume x: "x \<in> space M" then show "0 \<le> w x" by fact
```
```  2353     show "\<bar>u' x\<bar> \<le> w x" using u'_bound[OF x] .
```
```  2354   qed
```
```  2355
```
```  2356   let "?diff n x" = "2 * w x - \<bar>u n x - u' x\<bar>"
```
```  2357   have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)"
```
```  2358     using w u `integrable M u'`
```
```  2359     by (auto intro!: integral_add integral_diff integral_cmult integrable_abs)
```
```  2360
```
```  2361   { fix j x assume x: "x \<in> space M"
```
```  2362     have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
```
```  2363     also have "\<dots> \<le> w x + w x"
```
```  2364       by (rule add_mono[OF bound[OF x] u'_bound[OF x]])
```
```  2365     finally have "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp }
```
```  2366   note diff_less_2w = this
```
```  2367
```
```  2368   have PI_diff: "\<And>n. (\<integral>\<^isup>+ x. ereal (?diff n x) \<partial>M) =
```
```  2369     (\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
```
```  2370     using diff w diff_less_2w w_pos
```
```  2371     by (subst positive_integral_diff[symmetric])
```
```  2372        (auto simp: integrable_def intro!: positive_integral_cong)
```
```  2373
```
```  2374   have "integrable M (\<lambda>x. 2 * w x)"
```
```  2375     using w by (auto intro: integral_cmult)
```
```  2376   hence I2w_fin: "(\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
```
```  2377     borel_2w: "(\<lambda>x. ereal (2 * w x)) \<in> borel_measurable M"
```
```  2378     unfolding integrable_def by auto
```
```  2379
```
```  2380   have "limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
```
```  2381   proof cases
```
```  2382     assume eq_0: "(\<integral>\<^isup>+ x. max 0 (ereal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
```
```  2383     { fix n
```
```  2384       have "?f n \<le> ?wx" (is "integral\<^isup>P M ?f' \<le> _")
```
```  2385         using diff_less_2w[of _ n] unfolding positive_integral_max_0
```
```  2386         by (intro positive_integral_mono) auto
```
```  2387       then have "?f n = 0"
```
```  2388         using positive_integral_positive[of ?f'] eq_0 by auto }
```
```  2389     then show ?thesis by (simp add: Limsup_const)
```
```  2390   next
```
```  2391     assume neq_0: "(\<integral>\<^isup>+ x. max 0 (ereal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
```
```  2392     have "0 = limsup (\<lambda>n. 0 :: ereal)" by (simp add: Limsup_const)
```
```  2393     also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
```
```  2394       by (intro limsup_mono positive_integral_positive)
```
```  2395     finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" .
```
```  2396     have "?wx = (\<integral>\<^isup>+ x. liminf (\<lambda>n. max 0 (ereal (?diff n x))) \<partial>M)"
```
```  2397     proof (rule positive_integral_cong)
```
```  2398       fix x assume x: "x \<in> space M"
```
```  2399       show "max 0 (ereal (2 * w x)) = liminf (\<lambda>n. max 0 (ereal (?diff n x)))"
```
```  2400         unfolding ereal_max_0
```
```  2401       proof (rule lim_imp_Liminf[symmetric], unfold lim_ereal)
```
```  2402         have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
```
```  2403           using u'[OF x] by (safe intro!: LIMSEQ_diff LIMSEQ_const LIMSEQ_imp_rabs)
```
```  2404         then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
```
```  2405           by (auto intro!: tendsto_real_max simp add: lim_ereal)
```
```  2406       qed (rule trivial_limit_sequentially)
```
```  2407     qed
```
```  2408     also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^isup>+ x. max 0 (ereal (?diff n x)) \<partial>M)"
```
```  2409       using u'_borel w u unfolding integrable_def
```
```  2410       by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
```
```  2411     also have "\<dots> = (\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) -
```
```  2412         limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
```
```  2413       unfolding PI_diff positive_integral_max_0
```
```  2414       using positive_integral_positive[of "\<lambda>x. ereal (2 * w x)"]
```
```  2415       by (subst liminf_ereal_cminus) auto
```
```  2416     finally show ?thesis
```
```  2417       using neq_0 I2w_fin positive_integral_positive[of "\<lambda>x. ereal (2 * w x)"] pos
```
```  2418       unfolding positive_integral_max_0
```
```  2419       by (cases rule: ereal2_cases[of "\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"])
```
```  2420          auto
```
```  2421   qed
```
```  2422
```
```  2423   have "liminf ?f \<le> limsup ?f"
```
```  2424     by (intro ereal_Liminf_le_Limsup trivial_limit_sequentially)
```
```  2425   moreover
```
```  2426   { have "0 = liminf (\<lambda>n. 0 :: ereal)" by (simp add: Liminf_const)
```
```  2427     also have "\<dots> \<le> liminf ?f"
```
```  2428       by (intro liminf_mono positive_integral_positive)
```
```  2429     finally have "0 \<le> liminf ?f" . }
```
```  2430   ultimately have liminf_limsup_eq: "liminf ?f = ereal 0" "limsup ?f = ereal 0"
```
```  2431     using `limsup ?f = 0` by auto
```
```  2432   have "\<And>n. (\<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = ereal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
```
```  2433     using diff positive_integral_positive
```
```  2434     by (subst integral_eq_positive_integral) (auto simp: ereal_real integrable_def)
```
```  2435   then show ?lim_diff
```
```  2436     using ereal_Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
```
```  2437     by (simp add: lim_ereal)
```
```  2438
```
```  2439   show ?lim
```
```  2440   proof (rule LIMSEQ_I)
```
```  2441     fix r :: real assume "0 < r"
```
```  2442     from LIMSEQ_D[OF `?lim_diff` this]
```
```  2443     obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r"
```
```  2444       using diff by (auto simp: integral_positive)
```
```  2445
```
```  2446     show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r"
```
```  2447     proof (safe intro!: exI[of _ N])
```
```  2448       fix n assume "N \<le> n"
```
```  2449       have "\<bar>integral\<^isup>L M (u n) - integral\<^isup>L M u'\<bar> = \<bar>(\<integral>x. u n x - u' x \<partial>M)\<bar>"
```
```  2450         using u `integrable M u'` by (auto simp: integral_diff)
```
```  2451       also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'`
```
```  2452         by (rule_tac integral_triangle_inequality) (auto intro!: integral_diff)
```
```  2453       also note N[OF `N \<le> n`]
```
```  2454       finally show "norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r" by simp
```
```  2455     qed
```
```  2456   qed
```
```  2457 qed
```
```  2458
```
```  2459 lemma (in measure_space) integral_sums:
```
```  2460   assumes borel: "\<And>i. integrable M (f i)"
```
```  2461   and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
```
```  2462   and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
```
```  2463   shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
```
```  2464   and "(\<lambda>i. integral\<^isup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
```
```  2465 proof -
```
```  2466   have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
```
```  2467     using summable unfolding summable_def by auto
```
```  2468   from bchoice[OF this]
```
```  2469   obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
```
```  2470
```
```  2471   let "?w y" = "if y \<in> space M then w y else 0"
```
```  2472
```
```  2473   obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
```
```  2474     using sums unfolding summable_def ..
```
```  2475
```
```  2476   have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i = 0..<n. f i x)"
```
```  2477     using borel by (auto intro!: integral_setsum)
```
```  2478
```
```  2479   { fix j x assume [simp]: "x \<in> space M"
```
```  2480     have "\<bar>\<Sum>i = 0..< j. f i x\<bar> \<le> (\<Sum>i = 0..< j. \<bar>f i x\<bar>)" by (rule setsum_abs)
```
```  2481     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
```
```  2482     finally have "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp }
```
```  2483   note 2 = this
```
```  2484
```
```  2485   have 3: "integrable M ?w"
```
```  2486   proof (rule integral_monotone_convergence(1))
```
```  2487     let "?F n y" = "(\<Sum>i = 0..<n. \<bar>f i y\<bar>)"
```
```  2488     let "?w' n y" = "if y \<in> space M then ?F n y else 0"
```
```  2489     have "\<And>n. integrable M (?F n)"
```
```  2490       using borel by (auto intro!: integral_setsum integrable_abs)
```
```  2491     thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
```
```  2492     show "mono ?w'"
```
```  2493       by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
```
```  2494     { fix x show "(\<lambda>n. ?w' n x) ----> ?w x"
```
```  2495         using w by (cases "x \<in> space M") (simp_all add: LIMSEQ_const sums_def) }
```
```  2496     have *: "\<And>n. integral\<^isup>L M (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
```
```  2497       using borel by (simp add: integral_setsum integrable_abs cong: integral_cong)
```
```  2498     from abs_sum
```
```  2499     show "(\<lambda>i. integral\<^isup>L M (?w' i)) ----> x" unfolding * sums_def .
```
```  2500   qed
```
```  2501
```
```  2502   from summable[THEN summable_rabs_cancel]
```
```  2503   have 4: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>n. \<Sum>i = 0..<n. f i x) ----> (\<Sum>i. f i x)"
```
```  2504     by (auto intro: summable_sumr_LIMSEQ_suminf)
```
```  2505
```
```  2506   note int = integral_dominated_convergence(1,3)[OF 1 2 3 4]
```
```  2507
```
```  2508   from int show "integrable M ?S" by simp
```
```  2509
```
```  2510   show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF borel]
```
```  2511     using int(2) by simp
```
```  2512 qed
```
```  2513
```
```  2514 section "Lebesgue integration on countable spaces"
```
```  2515
```
```  2516 lemma (in measure_space) integral_on_countable:
```
```  2517   assumes f: "f \<in> borel_measurable M"
```
```  2518   and bij: "bij_betw enum S (f ` space M)"
```
```  2519   and enum_zero: "enum ` (-S) \<subseteq> {0}"
```
```  2520   and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
```
```  2521   and abs_summable: "summable (\<lambda>r. \<bar>enum r * real (\<mu> (f -` {enum r} \<inter> space M))\<bar>)"
```
```  2522   shows "integrable M f"
```
```  2523   and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums)
```
```  2524 proof -
```
```  2525   let "?A r" = "f -` {enum r} \<inter> space M"
```
```  2526   let "?F r x" = "enum r * indicator (?A r) x"
```
```  2527   have enum_eq: "\<And>r. enum r * real (\<mu> (?A r)) = integral\<^isup>L M (?F r)"
```
```  2528     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2529
```
```  2530   { fix x assume "x \<in> space M"
```
```  2531     hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
```
```  2532     then obtain i where "i\<in>S" "enum i = f x" by auto
```
```  2533     have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
```
```  2534     proof cases
```
```  2535       fix j assume "j = i"
```
```  2536       thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
```
```  2537     next
```
```  2538       fix j assume "j \<noteq> i"
```
```  2539       show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
```
```  2540         by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
```
```  2541     qed
```
```  2542     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
```
```  2543     have "(\<lambda>i. ?F i x) sums f x"
```
```  2544          "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
```
```  2545       by (auto intro!: sums_single simp: F F_abs) }
```
```  2546   note F_sums_f = this(1) and F_abs_sums_f = this(2)
```
```  2547
```
```  2548   have int_f: "integral\<^isup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
```
```  2549     using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
```
```  2550
```
```  2551   { fix r
```
```  2552     have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
```
```  2553       by (auto simp: indicator_def intro!: integral_cong)
```
```  2554     also have "\<dots> = \<bar>enum r\<bar> * real (\<mu> (?A r))"
```
```  2555       using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2556     finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
```
```  2557       using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) }
```
```  2558   note int_abs_F = this
```
```  2559
```
```  2560   have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
```
```  2561     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2562
```
```  2563   have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
```
```  2564     using F_abs_sums_f unfolding sums_iff by auto
```
```  2565
```
```  2566   from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
```
```  2567   show ?sums unfolding enum_eq int_f by simp
```
```  2568
```
```  2569   from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
```
```  2570   show "integrable M f" unfolding int_f by simp
```
```  2571 qed
```
```  2572
```
```  2573 section "Lebesgue integration on finite space"
```
```  2574
```
```  2575 lemma (in measure_space) integral_on_finite:
```
```  2576   assumes f: "f \<in> borel_measurable M" and finite: "finite (f`space M)"
```
```  2577   and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
```
```  2578   shows "integrable M f"
```
```  2579   and "(\<integral>x. f x \<partial>M) =
```
```  2580     (\<Sum> r \<in> f`space M. r * real (\<mu> (f -` {r} \<inter> space M)))" (is "?integral")
```
```  2581 proof -
```
```  2582   let "?A r" = "f -` {r} \<inter> space M"
```
```  2583   let "?S x" = "\<Sum>r\<in>f`space M. r * indicator (?A r) x"
```
```  2584
```
```  2585   { fix x assume "x \<in> space M"
```
```  2586     have "f x = (\<Sum>r\<in>f`space M. if x \<in> ?A r then r else 0)"
```
```  2587       using finite `x \<in> space M` by (simp add: setsum_cases)
```
```  2588     also have "\<dots> = ?S x"
```
```  2589       by (auto intro!: setsum_cong)
```
```  2590     finally have "f x = ?S x" . }
```
```  2591   note f_eq = this
```
```  2592
```
```  2593   have f_eq_S: "integrable M f \<longleftrightarrow> integrable M ?S" "integral\<^isup>L M f = integral\<^isup>L M ?S"
```
```  2594     by (auto intro!: integrable_cong integral_cong simp only: f_eq)
```
```  2595
```
```  2596   show "integrable M f" ?integral using fin f f_eq_S
```
```  2597     by (simp_all add: integral_cmul_indicator borel_measurable_vimage)
```
```  2598 qed
```
```  2599
```
```  2600 lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function M f"
```
```  2601   unfolding simple_function_def using finite_space by auto
```
```  2602
```
```  2603 lemma (in finite_measure_space) borel_measurable_finite[intro, simp]: "f \<in> borel_measurable M"
```
```  2604   by (auto intro: borel_measurable_simple_function)
```
```  2605
```
```  2606 lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
```
```  2607   assumes pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  2608   shows "integral\<^isup>P M f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
```
```  2609 proof -
```
```  2610   have *: "integral\<^isup>P M f = (\<integral>\<^isup>+ x. (\<Sum>y\<in>space M. f y * indicator {y} x) \<partial>M)"
```
```  2611     by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
```
```  2612   show ?thesis unfolding * using borel_measurable_finite[of f] pos
```
```  2613     by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
```
```  2614 qed
```
```  2615
```
```  2616 lemma (in finite_measure_space) integral_finite_singleton:
```
```  2617   shows "integrable M f"
```
```  2618   and "integral\<^isup>L M f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
```
```  2619 proof -
```
```  2620   have *:
```
```  2621     "(\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (ereal (f x)) * \<mu> {x})"
```
```  2622     "(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (ereal (- f x)) * \<mu> {x})"
```
```  2623     by (simp_all add: positive_integral_finite_eq_setsum)
```
```  2624   then show "integrable M f" using finite_space finite_measure
```
```  2625     by (simp add: setsum_Pinfty integrable_def positive_integral_max_0
```
```  2626              split: split_max)
```
```  2627   show ?I using finite_measure *
```
```  2628     apply (simp add: positive_integral_max_0 lebesgue_integral_def)
```
```  2629     apply (subst (1 2) setsum_real_of_ereal[symmetric])
```
```  2630     apply (simp_all split: split_max add: setsum_subtractf[symmetric])
```
```  2631     apply (intro setsum_cong[OF refl])
```
```  2632     apply (simp split: split_max)
```
```  2633     done
```
```  2634 qed
```
```  2635
```
```  2636 end
```