src/HOL/Probability/Lebesgue_Integration.thy
 author hoelzl Tue Jan 18 21:37:23 2011 +0100 (2011-01-18) changeset 41654 32fe42892983 parent 41545 9c869baf1c66 child 41661 baf1964bc468 permissions -rw-r--r--
Gauge measure removed
```     1 (* Author: Armin Heller, Johannes Hoelzl, TU Muenchen *)
```
```     2
```
```     3 header {*Lebesgue Integration*}
```
```     4
```
```     5 theory Lebesgue_Integration
```
```     6 imports Measure Borel_Space
```
```     7 begin
```
```     8
```
```     9 lemma sums_If_finite:
```
```    10   assumes finite: "finite {r. P r}"
```
```    11   shows "(\<lambda>r. if P r then f r else 0) sums (\<Sum>r\<in>{r. P r}. f r)" (is "?F sums _")
```
```    12 proof cases
```
```    13   assume "{r. P r} = {}" hence "\<And>r. \<not> P r" by auto
```
```    14   thus ?thesis by (simp add: sums_zero)
```
```    15 next
```
```    16   assume not_empty: "{r. P r} \<noteq> {}"
```
```    17   have "?F sums (\<Sum>r = 0..< Suc (Max {r. P r}). ?F r)"
```
```    18     by (rule series_zero)
```
```    19        (auto simp add: Max_less_iff[OF finite not_empty] less_eq_Suc_le[symmetric])
```
```    20   also have "(\<Sum>r = 0..< Suc (Max {r. P r}). ?F r) = (\<Sum>r\<in>{r. P r}. f r)"
```
```    21     by (subst setsum_cases)
```
```    22        (auto intro!: setsum_cong simp: Max_ge_iff[OF finite not_empty] less_Suc_eq_le)
```
```    23   finally show ?thesis .
```
```    24 qed
```
```    25
```
```    26 lemma sums_single:
```
```    27   "(\<lambda>r. if r = i then f r else 0) sums f i"
```
```    28   using sums_If_finite[of "\<lambda>r. r = i" f] by simp
```
```    29
```
```    30 section "Simple function"
```
```    31
```
```    32 text {*
```
```    33
```
```    34 Our simple functions are not restricted to positive real numbers. Instead
```
```    35 they are just functions with a finite range and are measurable when singleton
```
```    36 sets are measurable.
```
```    37
```
```    38 *}
```
```    39
```
```    40 definition (in sigma_algebra) "simple_function g \<longleftrightarrow>
```
```    41     finite (g ` space M) \<and>
```
```    42     (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
```
```    43
```
```    44 lemma (in sigma_algebra) simple_functionD:
```
```    45   assumes "simple_function g"
```
```    46   shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
```
```    47 proof -
```
```    48   show "finite (g ` space M)"
```
```    49     using assms unfolding simple_function_def by auto
```
```    50   have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
```
```    51   also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
```
```    52   finally show "g -` X \<inter> space M \<in> sets M" using assms
```
```    53     by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
```
```    54 qed
```
```    55
```
```    56 lemma (in sigma_algebra) simple_function_indicator_representation:
```
```    57   fixes f ::"'a \<Rightarrow> pextreal"
```
```    58   assumes f: "simple_function f" and x: "x \<in> space M"
```
```    59   shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
```
```    60   (is "?l = ?r")
```
```    61 proof -
```
```    62   have "?r = (\<Sum>y \<in> f ` space M.
```
```    63     (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
```
```    64     by (auto intro!: setsum_cong2)
```
```    65   also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
```
```    66     using assms by (auto dest: simple_functionD simp: setsum_delta)
```
```    67   also have "... = f x" using x by (auto simp: indicator_def)
```
```    68   finally show ?thesis by auto
```
```    69 qed
```
```    70
```
```    71 lemma (in measure_space) simple_function_notspace:
```
```    72   "simple_function (\<lambda>x. h x * indicator (- space M) x::pextreal)" (is "simple_function ?h")
```
```    73 proof -
```
```    74   have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
```
```    75   hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
```
```    76   have "?h -` {0} \<inter> space M = space M" by auto
```
```    77   thus ?thesis unfolding simple_function_def by auto
```
```    78 qed
```
```    79
```
```    80 lemma (in sigma_algebra) simple_function_cong:
```
```    81   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
```
```    82   shows "simple_function f \<longleftrightarrow> simple_function g"
```
```    83 proof -
```
```    84   have "f ` space M = g ` space M"
```
```    85     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
```
```    86     using assms by (auto intro!: image_eqI)
```
```    87   thus ?thesis unfolding simple_function_def using assms by simp
```
```    88 qed
```
```    89
```
```    90 lemma (in sigma_algebra) borel_measurable_simple_function:
```
```    91   assumes "simple_function f"
```
```    92   shows "f \<in> borel_measurable M"
```
```    93 proof (rule borel_measurableI)
```
```    94   fix S
```
```    95   let ?I = "f ` (f -` S \<inter> space M)"
```
```    96   have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
```
```    97   have "finite ?I"
```
```    98     using assms unfolding simple_function_def by (auto intro: finite_subset)
```
```    99   hence "?U \<in> sets M"
```
```   100     apply (rule finite_UN)
```
```   101     using assms unfolding simple_function_def by auto
```
```   102   thus "f -` S \<inter> space M \<in> sets M" unfolding * .
```
```   103 qed
```
```   104
```
```   105 lemma (in sigma_algebra) simple_function_borel_measurable:
```
```   106   fixes f :: "'a \<Rightarrow> 'x::t2_space"
```
```   107   assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
```
```   108   shows "simple_function f"
```
```   109   using assms unfolding simple_function_def
```
```   110   by (auto intro: borel_measurable_vimage)
```
```   111
```
```   112 lemma (in sigma_algebra) simple_function_const[intro, simp]:
```
```   113   "simple_function (\<lambda>x. c)"
```
```   114   by (auto intro: finite_subset simp: simple_function_def)
```
```   115
```
```   116 lemma (in sigma_algebra) simple_function_compose[intro, simp]:
```
```   117   assumes "simple_function f"
```
```   118   shows "simple_function (g \<circ> f)"
```
```   119   unfolding simple_function_def
```
```   120 proof safe
```
```   121   show "finite ((g \<circ> f) ` space M)"
```
```   122     using assms unfolding simple_function_def by (auto simp: image_compose)
```
```   123 next
```
```   124   fix x assume "x \<in> space M"
```
```   125   let ?G = "g -` {g (f x)} \<inter> (f`space M)"
```
```   126   have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
```
```   127     (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
```
```   128   show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
```
```   129     using assms unfolding simple_function_def *
```
```   130     by (rule_tac finite_UN) (auto intro!: finite_UN)
```
```   131 qed
```
```   132
```
```   133 lemma (in sigma_algebra) simple_function_indicator[intro, simp]:
```
```   134   assumes "A \<in> sets M"
```
```   135   shows "simple_function (indicator A)"
```
```   136 proof -
```
```   137   have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
```
```   138     by (auto simp: indicator_def)
```
```   139   hence "finite ?S" by (rule finite_subset) simp
```
```   140   moreover have "- A \<inter> space M = space M - A" by auto
```
```   141   ultimately show ?thesis unfolding simple_function_def
```
```   142     using assms by (auto simp: indicator_def_raw)
```
```   143 qed
```
```   144
```
```   145 lemma (in sigma_algebra) simple_function_Pair[intro, simp]:
```
```   146   assumes "simple_function f"
```
```   147   assumes "simple_function g"
```
```   148   shows "simple_function (\<lambda>x. (f x, g x))" (is "simple_function ?p")
```
```   149   unfolding simple_function_def
```
```   150 proof safe
```
```   151   show "finite (?p ` space M)"
```
```   152     using assms unfolding simple_function_def
```
```   153     by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
```
```   154 next
```
```   155   fix x assume "x \<in> space M"
```
```   156   have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
```
```   157       (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
```
```   158     by auto
```
```   159   with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
```
```   160     using assms unfolding simple_function_def by auto
```
```   161 qed
```
```   162
```
```   163 lemma (in sigma_algebra) simple_function_compose1:
```
```   164   assumes "simple_function f"
```
```   165   shows "simple_function (\<lambda>x. g (f x))"
```
```   166   using simple_function_compose[OF assms, of g]
```
```   167   by (simp add: comp_def)
```
```   168
```
```   169 lemma (in sigma_algebra) simple_function_compose2:
```
```   170   assumes "simple_function f" and "simple_function g"
```
```   171   shows "simple_function (\<lambda>x. h (f x) (g x))"
```
```   172 proof -
```
```   173   have "simple_function ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
```
```   174     using assms by auto
```
```   175   thus ?thesis by (simp_all add: comp_def)
```
```   176 qed
```
```   177
```
```   178 lemmas (in sigma_algebra) simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
```
```   179   and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
```
```   180   and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
```
```   181   and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
```
```   182   and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
```
```   183   and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
```
```   184
```
```   185 lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
```
```   186   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
```
```   187   shows "simple_function (\<lambda>x. \<Sum>i\<in>P. f i x)"
```
```   188 proof cases
```
```   189   assume "finite P" from this assms show ?thesis by induct auto
```
```   190 qed auto
```
```   191
```
```   192 lemma (in sigma_algebra) simple_function_le_measurable:
```
```   193   assumes "simple_function f" "simple_function g"
```
```   194   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
```
```   195 proof -
```
```   196   have *: "{x \<in> space M. f x \<le> g x} =
```
```   197     (\<Union>(F, G)\<in>f`space M \<times> g`space M.
```
```   198       if F \<le> G then (f -` {F} \<inter> space M) \<inter> (g -` {G} \<inter> space M) else {})"
```
```   199     apply (auto split: split_if_asm)
```
```   200     apply (rule_tac x=x in bexI)
```
```   201     apply (rule_tac x=x in bexI)
```
```   202     by simp_all
```
```   203   have **: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow>
```
```   204     (f -` {f x} \<inter> space M) \<inter> (g -` {g y} \<inter> space M) \<in> sets M"
```
```   205     using assms unfolding simple_function_def by auto
```
```   206   have "finite (f`space M \<times> g`space M)"
```
```   207     using assms unfolding simple_function_def by auto
```
```   208   thus ?thesis unfolding *
```
```   209     apply (rule finite_UN)
```
```   210     using assms unfolding simple_function_def
```
```   211     by (auto intro!: **)
```
```   212 qed
```
```   213
```
```   214 lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
```
```   215   fixes u :: "'a \<Rightarrow> pextreal"
```
```   216   assumes u: "u \<in> borel_measurable M"
```
```   217   shows "\<exists>f. (\<forall>i. simple_function (f i) \<and> (\<forall>x\<in>space M. f i x \<noteq> \<omega>)) \<and> f \<up> u"
```
```   218 proof -
```
```   219   have "\<exists>f. \<forall>x j. (of_nat j \<le> u x \<longrightarrow> f x j = j*2^j) \<and>
```
```   220     (u x < of_nat j \<longrightarrow> of_nat (f x j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f x j)))"
```
```   221     (is "\<exists>f. \<forall>x j. ?P x j (f x j)")
```
```   222   proof(rule choice, rule, rule choice, rule)
```
```   223     fix x j show "\<exists>n. ?P x j n"
```
```   224     proof cases
```
```   225       assume *: "u x < of_nat j"
```
```   226       then obtain r where r: "u x = Real r" "0 \<le> r" by (cases "u x") auto
```
```   227       from reals_Archimedean6a[of "r * 2^j"]
```
```   228       obtain n :: nat where "real n \<le> r * 2 ^ j" "r * 2 ^ j < real (Suc n)"
```
```   229         using `0 \<le> r` by (auto simp: zero_le_power zero_le_mult_iff)
```
```   230       thus ?thesis using r * by (auto intro!: exI[of _ n])
```
```   231     qed auto
```
```   232   qed
```
```   233   then obtain f where top: "\<And>j x. of_nat j \<le> u x \<Longrightarrow> f x j = j*2^j" and
```
```   234     upper: "\<And>j x. u x < of_nat j \<Longrightarrow> of_nat (f x j) \<le> u x * 2^j" and
```
```   235     lower: "\<And>j x. u x < of_nat j \<Longrightarrow> u x * 2^j < of_nat (Suc (f x j))" by blast
```
```   236
```
```   237   { fix j x P
```
```   238     assume 1: "of_nat j \<le> u x \<Longrightarrow> P (j * 2^j)"
```
```   239     assume 2: "\<And>k. \<lbrakk> u x < of_nat j ; of_nat k \<le> u x * 2^j ; u x * 2^j < of_nat (Suc k) \<rbrakk> \<Longrightarrow> P k"
```
```   240     have "P (f x j)"
```
```   241     proof cases
```
```   242       assume "of_nat j \<le> u x" thus "P (f x j)"
```
```   243         using top[of j x] 1 by auto
```
```   244     next
```
```   245       assume "\<not> of_nat j \<le> u x"
```
```   246       hence "u x < of_nat j" "of_nat (f x j) \<le> u x * 2^j" "u x * 2^j < of_nat (Suc (f x j))"
```
```   247         using upper lower by auto
```
```   248       from 2[OF this] show "P (f x j)" .
```
```   249     qed }
```
```   250   note fI = this
```
```   251
```
```   252   { fix j x
```
```   253     have "f x j = j * 2 ^ j \<longleftrightarrow> of_nat j \<le> u x"
```
```   254       by (rule fI, simp, cases "u x") (auto split: split_if_asm) }
```
```   255   note f_eq = this
```
```   256
```
```   257   { fix j x
```
```   258     have "f x j \<le> j * 2 ^ j"
```
```   259     proof (rule fI)
```
```   260       fix k assume *: "u x < of_nat j"
```
```   261       assume "of_nat k \<le> u x * 2 ^ j"
```
```   262       also have "\<dots> \<le> of_nat (j * 2^j)"
```
```   263         using * by (cases "u x") (auto simp: zero_le_mult_iff)
```
```   264       finally show "k \<le> j*2^j" by (auto simp del: real_of_nat_mult)
```
```   265     qed simp }
```
```   266   note f_upper = this
```
```   267
```
```   268   let "?g j x" = "of_nat (f x j) / 2^j :: pextreal"
```
```   269   show ?thesis unfolding simple_function_def isoton_fun_expand unfolding isoton_def
```
```   270   proof (safe intro!: exI[of _ ?g])
```
```   271     fix j
```
```   272     have *: "?g j ` space M \<subseteq> (\<lambda>x. of_nat x / 2^j) ` {..j * 2^j}"
```
```   273       using f_upper by auto
```
```   274     thus "finite (?g j ` space M)" by (rule finite_subset) auto
```
```   275   next
```
```   276     fix j t assume "t \<in> space M"
```
```   277     have **: "?g j -` {?g j t} \<inter> space M = {x \<in> space M. f t j = f x j}"
```
```   278       by (auto simp: power_le_zero_eq Real_eq_Real mult_le_0_iff)
```
```   279
```
```   280     show "?g j -` {?g j t} \<inter> space M \<in> sets M"
```
```   281     proof cases
```
```   282       assume "of_nat j \<le> u t"
```
```   283       hence "?g j -` {?g j t} \<inter> space M = {x\<in>space M. of_nat j \<le> u x}"
```
```   284         unfolding ** f_eq[symmetric] by auto
```
```   285       thus "?g j -` {?g j t} \<inter> space M \<in> sets M"
```
```   286         using u by auto
```
```   287     next
```
```   288       assume not_t: "\<not> of_nat j \<le> u t"
```
```   289       hence less: "f t j < j*2^j" using f_eq[symmetric] `f t j \<le> j*2^j` by auto
```
```   290       have split_vimage: "?g j -` {?g j t} \<inter> space M =
```
```   291           {x\<in>space M. of_nat (f t j)/2^j \<le> u x} \<inter> {x\<in>space M. u x < of_nat (Suc (f t j))/2^j}"
```
```   292         unfolding **
```
```   293       proof safe
```
```   294         fix x assume [simp]: "f t j = f x j"
```
```   295         have *: "\<not> of_nat j \<le> u x" using not_t f_eq[symmetric] by simp
```
```   296         hence "of_nat (f t j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f t j))"
```
```   297           using upper lower by auto
```
```   298         hence "of_nat (f t j) / 2^j \<le> u x \<and> u x < of_nat (Suc (f t j))/2^j" using *
```
```   299           by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
```
```   300         thus "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j" by auto
```
```   301       next
```
```   302         fix x
```
```   303         assume "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j"
```
```   304         hence "of_nat (f t j) \<le> u x * 2 ^ j \<and> u x * 2 ^ j < of_nat (Suc (f t j))"
```
```   305           by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
```
```   306         hence 1: "of_nat (f t j) \<le> u x * 2 ^ j" and 2: "u x * 2 ^ j < of_nat (Suc (f t j))" by auto
```
```   307         note 2
```
```   308         also have "\<dots> \<le> of_nat (j*2^j)"
```
```   309           using less by (auto simp: zero_le_mult_iff simp del: real_of_nat_mult)
```
```   310         finally have bound_ux: "u x < of_nat j"
```
```   311           by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq)
```
```   312         show "f t j = f x j"
```
```   313         proof (rule antisym)
```
```   314           from 1 lower[OF bound_ux]
```
```   315           show "f t j \<le> f x j" by (cases "u x") (auto split: split_if_asm)
```
```   316           from upper[OF bound_ux] 2
```
```   317           show "f x j \<le> f t j" by (cases "u x") (auto split: split_if_asm)
```
```   318         qed
```
```   319       qed
```
```   320       show ?thesis unfolding split_vimage using u by auto
```
```   321     qed
```
```   322   next
```
```   323     fix j t assume "?g j t = \<omega>" thus False by (auto simp: power_le_zero_eq)
```
```   324   next
```
```   325     fix t
```
```   326     { fix i
```
```   327       have "f t i * 2 \<le> f t (Suc i)"
```
```   328       proof (rule fI)
```
```   329         assume "of_nat (Suc i) \<le> u t"
```
```   330         hence "of_nat i \<le> u t" by (cases "u t") auto
```
```   331         thus "f t i * 2 \<le> Suc i * 2 ^ Suc i" unfolding f_eq[symmetric] by simp
```
```   332       next
```
```   333         fix k
```
```   334         assume *: "u t * 2 ^ Suc i < of_nat (Suc k)"
```
```   335         show "f t i * 2 \<le> k"
```
```   336         proof (rule fI)
```
```   337           assume "of_nat i \<le> u t"
```
```   338           hence "of_nat (i * 2^Suc i) \<le> u t * 2^Suc i"
```
```   339             by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
```
```   340           also have "\<dots> < of_nat (Suc k)" using * by auto
```
```   341           finally show "i * 2 ^ i * 2 \<le> k"
```
```   342             by (auto simp del: real_of_nat_mult)
```
```   343         next
```
```   344           fix j assume "of_nat j \<le> u t * 2 ^ i"
```
```   345           with * show "j * 2 \<le> k" by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
```
```   346         qed
```
```   347       qed
```
```   348       thus "?g i t \<le> ?g (Suc i) t"
```
```   349         by (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps) }
```
```   350     hence mono: "mono (\<lambda>i. ?g i t)" unfolding mono_iff_le_Suc by auto
```
```   351
```
```   352     show "(SUP j. of_nat (f t j) / 2 ^ j) = u t"
```
```   353     proof (rule pextreal_SUPI)
```
```   354       fix j show "of_nat (f t j) / 2 ^ j \<le> u t"
```
```   355       proof (rule fI)
```
```   356         assume "of_nat j \<le> u t" thus "of_nat (j * 2 ^ j) / 2 ^ j \<le> u t"
```
```   357           by (cases "u t") (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps)
```
```   358       next
```
```   359         fix k assume "of_nat k \<le> u t * 2 ^ j"
```
```   360         thus "of_nat k / 2 ^ j \<le> u t"
```
```   361           by (cases "u t")
```
```   362              (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps zero_le_mult_iff)
```
```   363       qed
```
```   364     next
```
```   365       fix y :: pextreal assume *: "\<And>j. j \<in> UNIV \<Longrightarrow> of_nat (f t j) / 2 ^ j \<le> y"
```
```   366       show "u t \<le> y"
```
```   367       proof (cases "u t")
```
```   368         case (preal r)
```
```   369         show ?thesis
```
```   370         proof (rule ccontr)
```
```   371           assume "\<not> u t \<le> y"
```
```   372           then obtain p where p: "y = Real p" "0 \<le> p" "p < r" using preal by (cases y) auto
```
```   373           with LIMSEQ_inverse_realpow_zero[of 2, unfolded LIMSEQ_iff, rule_format, of "r - p"]
```
```   374           obtain n where n: "\<And>N. n \<le> N \<Longrightarrow> inverse (2^N) < r - p" by auto
```
```   375           let ?N = "max n (natfloor r + 1)"
```
```   376           have "u t < of_nat ?N" "n \<le> ?N"
```
```   377             using ge_natfloor_plus_one_imp_gt[of r n] preal
```
```   378             using real_natfloor_add_one_gt
```
```   379             by (auto simp: max_def real_of_nat_Suc)
```
```   380           from lower[OF this(1)]
```
```   381           have "r < (real (f t ?N) + 1) / 2 ^ ?N" unfolding less_divide_eq
```
```   382             using preal by (auto simp add: power_le_zero_eq zero_le_mult_iff real_of_nat_Suc split: split_if_asm)
```
```   383           hence "u t < of_nat (f t ?N) / 2 ^ ?N + 1 / 2 ^ ?N"
```
```   384             using preal by (auto simp: field_simps divide_real_def[symmetric])
```
```   385           with n[OF `n \<le> ?N`] p preal *[of ?N]
```
```   386           show False
```
```   387             by (cases "f t ?N") (auto simp: power_le_zero_eq split: split_if_asm)
```
```   388         qed
```
```   389       next
```
```   390         case infinite
```
```   391         { fix j have "f t j = j*2^j" using top[of j t] infinite by simp
```
```   392           hence "of_nat j \<le> y" using *[of j]
```
```   393             by (cases y) (auto simp: power_le_zero_eq zero_le_mult_iff field_simps) }
```
```   394         note all_less_y = this
```
```   395         show ?thesis unfolding infinite
```
```   396         proof (rule ccontr)
```
```   397           assume "\<not> \<omega> \<le> y" then obtain r where r: "y = Real r" "0 \<le> r" by (cases y) auto
```
```   398           moreover obtain n ::nat where "r < real n" using ex_less_of_nat by (auto simp: real_eq_of_nat)
```
```   399           with all_less_y[of n] r show False by auto
```
```   400         qed
```
```   401       qed
```
```   402     qed
```
```   403   qed
```
```   404 qed
```
```   405
```
```   406 lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
```
```   407   fixes u :: "'a \<Rightarrow> pextreal"
```
```   408   assumes "u \<in> borel_measurable M"
```
```   409   obtains (x) f where "f \<up> u" "\<And>i. simple_function (f i)" "\<And>i. \<omega>\<notin>f i`space M"
```
```   410 proof -
```
```   411   from borel_measurable_implies_simple_function_sequence[OF assms]
```
```   412   obtain f where x: "\<And>i. simple_function (f i)" "f \<up> u"
```
```   413     and fin: "\<And>i. \<And>x. x\<in>space M \<Longrightarrow> f i x \<noteq> \<omega>" by auto
```
```   414   { fix i from fin[of _ i] have "\<omega> \<notin> f i`space M" by fastsimp }
```
```   415   with x show thesis by (auto intro!: that[of f])
```
```   416 qed
```
```   417
```
```   418 lemma (in sigma_algebra) simple_function_eq_borel_measurable:
```
```   419   fixes f :: "'a \<Rightarrow> pextreal"
```
```   420   shows "simple_function f \<longleftrightarrow>
```
```   421     finite (f`space M) \<and> f \<in> borel_measurable M"
```
```   422   using simple_function_borel_measurable[of f]
```
```   423     borel_measurable_simple_function[of f]
```
```   424   by (fastsimp simp: simple_function_def)
```
```   425
```
```   426 lemma (in measure_space) simple_function_restricted:
```
```   427   fixes f :: "'a \<Rightarrow> pextreal" assumes "A \<in> sets M"
```
```   428   shows "sigma_algebra.simple_function (restricted_space A) f \<longleftrightarrow> simple_function (\<lambda>x. f x * indicator A x)"
```
```   429     (is "sigma_algebra.simple_function ?R f \<longleftrightarrow> simple_function ?f")
```
```   430 proof -
```
```   431   interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
```
```   432   have "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
```
```   433   proof cases
```
```   434     assume "A = space M"
```
```   435     then have "f`A = ?f`space M" by (fastsimp simp: image_iff)
```
```   436     then show ?thesis by simp
```
```   437   next
```
```   438     assume "A \<noteq> space M"
```
```   439     then obtain x where x: "x \<in> space M" "x \<notin> A"
```
```   440       using sets_into_space `A \<in> sets M` by auto
```
```   441     have *: "?f`space M = f`A \<union> {0}"
```
```   442     proof (auto simp add: image_iff)
```
```   443       show "\<exists>x\<in>space M. f x = 0 \<or> indicator A x = 0"
```
```   444         using x by (auto intro!: bexI[of _ x])
```
```   445     next
```
```   446       fix x assume "x \<in> A"
```
```   447       then show "\<exists>y\<in>space M. f x = f y * indicator A y"
```
```   448         using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
```
```   449     next
```
```   450       fix x
```
```   451       assume "indicator A x \<noteq> (0::pextreal)"
```
```   452       then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
```
```   453       moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
```
```   454       ultimately show "f x = 0" by auto
```
```   455     qed
```
```   456     then show ?thesis by auto
```
```   457   qed
```
```   458   then show ?thesis
```
```   459     unfolding simple_function_eq_borel_measurable
```
```   460       R.simple_function_eq_borel_measurable
```
```   461     unfolding borel_measurable_restricted[OF `A \<in> sets M`]
```
```   462     by auto
```
```   463 qed
```
```   464
```
```   465 lemma (in sigma_algebra) simple_function_subalgebra:
```
```   466   assumes "sigma_algebra.simple_function N f"
```
```   467   and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M" "sigma_algebra N"
```
```   468   shows "simple_function f"
```
```   469   using assms
```
```   470   unfolding simple_function_def
```
```   471   unfolding sigma_algebra.simple_function_def[OF N_subalgebra(3)]
```
```   472   by auto
```
```   473
```
```   474 lemma (in sigma_algebra) simple_function_vimage:
```
```   475   fixes g :: "'a \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
```
```   476   assumes g: "simple_function g" and f: "f \<in> S \<rightarrow> space M"
```
```   477   shows "sigma_algebra.simple_function (vimage_algebra S f) (\<lambda>x. g (f x))"
```
```   478 proof -
```
```   479   have subset: "(\<lambda>x. g (f x)) ` S \<subseteq> g ` space M"
```
```   480     using f by auto
```
```   481   interpret V: sigma_algebra "vimage_algebra S f"
```
```   482     using f by (rule sigma_algebra_vimage)
```
```   483   show ?thesis using g
```
```   484     unfolding simple_function_eq_borel_measurable
```
```   485     unfolding V.simple_function_eq_borel_measurable
```
```   486     using measurable_vimage[OF _ f, of g borel]
```
```   487     using finite_subset[OF subset] by auto
```
```   488 qed
```
```   489
```
```   490 section "Simple integral"
```
```   491
```
```   492 definition (in measure_space) simple_integral (binder "\<integral>\<^isup>S " 10) where
```
```   493   "simple_integral f = (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M))"
```
```   494
```
```   495 lemma (in measure_space) simple_integral_cong:
```
```   496   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
```
```   497   shows "simple_integral f = simple_integral g"
```
```   498 proof -
```
```   499   have "f ` space M = g ` space M"
```
```   500     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
```
```   501     using assms by (auto intro!: image_eqI)
```
```   502   thus ?thesis unfolding simple_integral_def by simp
```
```   503 qed
```
```   504
```
```   505 lemma (in measure_space) simple_integral_cong_measure:
```
```   506   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A" and "simple_function f"
```
```   507   shows "measure_space.simple_integral M \<nu> f = simple_integral f"
```
```   508 proof -
```
```   509   interpret v: measure_space M \<nu>
```
```   510     by (rule measure_space_cong) fact
```
```   511   from simple_functionD[OF `simple_function f`] assms show ?thesis
```
```   512     unfolding simple_integral_def v.simple_integral_def
```
```   513     by (auto intro!: setsum_cong)
```
```   514 qed
```
```   515
```
```   516 lemma (in measure_space) simple_integral_const[simp]:
```
```   517   "(\<integral>\<^isup>Sx. c) = c * \<mu> (space M)"
```
```   518 proof (cases "space M = {}")
```
```   519   case True thus ?thesis unfolding simple_integral_def by simp
```
```   520 next
```
```   521   case False hence "(\<lambda>x. c) ` space M = {c}" by auto
```
```   522   thus ?thesis unfolding simple_integral_def by simp
```
```   523 qed
```
```   524
```
```   525 lemma (in measure_space) simple_function_partition:
```
```   526   assumes "simple_function f" and "simple_function g"
```
```   527   shows "simple_integral f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
```
```   528     (is "_ = setsum _ (?p ` space M)")
```
```   529 proof-
```
```   530   let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
```
```   531   let ?SIGMA = "Sigma (f`space M) ?sub"
```
```   532
```
```   533   have [intro]:
```
```   534     "finite (f ` space M)"
```
```   535     "finite (g ` space M)"
```
```   536     using assms unfolding simple_function_def by simp_all
```
```   537
```
```   538   { fix A
```
```   539     have "?p ` (A \<inter> space M) \<subseteq>
```
```   540       (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
```
```   541       by auto
```
```   542     hence "finite (?p ` (A \<inter> space M))"
```
```   543       by (rule finite_subset) auto }
```
```   544   note this[intro, simp]
```
```   545
```
```   546   { fix x assume "x \<in> space M"
```
```   547     have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
```
```   548     moreover {
```
```   549       fix x y
```
```   550       have *: "f -` {f x} \<inter> g -` {g x} \<inter> space M
```
```   551           = (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" by auto
```
```   552       assume "x \<in> space M" "y \<in> space M"
```
```   553       hence "f -` {f x} \<inter> g -` {g x} \<inter> space M \<in> sets M"
```
```   554         using assms unfolding simple_function_def * by auto }
```
```   555     ultimately
```
```   556     have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
```
```   557       by (subst measure_finitely_additive) auto }
```
```   558   hence "simple_integral f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
```
```   559     unfolding simple_integral_def
```
```   560     by (subst setsum_Sigma[symmetric],
```
```   561        auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
```
```   562   also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
```
```   563   proof -
```
```   564     have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
```
```   565     have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
```
```   566       = (\<lambda>x. (f x, ?p x)) ` space M"
```
```   567     proof safe
```
```   568       fix x assume "x \<in> space M"
```
```   569       thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
```
```   570         by (auto intro!: image_eqI[of _ _ "?p x"])
```
```   571     qed auto
```
```   572     thus ?thesis
```
```   573       apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
```
```   574       apply (rule_tac x="xa" in image_eqI)
```
```   575       by simp_all
```
```   576   qed
```
```   577   finally show ?thesis .
```
```   578 qed
```
```   579
```
```   580 lemma (in measure_space) simple_integral_add[simp]:
```
```   581   assumes "simple_function f" and "simple_function g"
```
```   582   shows "(\<integral>\<^isup>Sx. f x + g x) = simple_integral f + simple_integral g"
```
```   583 proof -
```
```   584   { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
```
```   585     assume "x \<in> space M"
```
```   586     hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
```
```   587         "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
```
```   588       by auto }
```
```   589   thus ?thesis
```
```   590     unfolding
```
```   591       simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
```
```   592       simple_function_partition[OF `simple_function f` `simple_function g`]
```
```   593       simple_function_partition[OF `simple_function g` `simple_function f`]
```
```   594     apply (subst (3) Int_commute)
```
```   595     by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
```
```   596 qed
```
```   597
```
```   598 lemma (in measure_space) simple_integral_setsum[simp]:
```
```   599   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
```
```   600   shows "(\<integral>\<^isup>Sx. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
```
```   601 proof cases
```
```   602   assume "finite P"
```
```   603   from this assms show ?thesis
```
```   604     by induct (auto simp: simple_function_setsum simple_integral_add)
```
```   605 qed auto
```
```   606
```
```   607 lemma (in measure_space) simple_integral_mult[simp]:
```
```   608   assumes "simple_function f"
```
```   609   shows "(\<integral>\<^isup>Sx. c * f x) = c * simple_integral f"
```
```   610 proof -
```
```   611   note mult = simple_function_mult[OF simple_function_const[of c] assms]
```
```   612   { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
```
```   613     assume "x \<in> space M"
```
```   614     hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
```
```   615       by auto }
```
```   616   thus ?thesis
```
```   617     unfolding simple_function_partition[OF mult assms]
```
```   618       simple_function_partition[OF assms mult]
```
```   619     by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
```
```   620 qed
```
```   621
```
```   622 lemma (in sigma_algebra) simple_function_If:
```
```   623   assumes sf: "simple_function f" "simple_function g" and A: "A \<in> sets M"
```
```   624   shows "simple_function (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function ?IF")
```
```   625 proof -
```
```   626   def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
```
```   627   show ?thesis unfolding simple_function_def
```
```   628   proof safe
```
```   629     have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
```
```   630     from finite_subset[OF this] assms
```
```   631     show "finite (?IF ` space M)" unfolding simple_function_def by auto
```
```   632   next
```
```   633     fix x assume "x \<in> space M"
```
```   634     then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
```
```   635       then ((F (f x) \<inter> A) \<union> (G (f x) - (G (f x) \<inter> A)))
```
```   636       else ((F (g x) \<inter> A) \<union> (G (g x) - (G (g x) \<inter> A))))"
```
```   637       using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
```
```   638     have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
```
```   639       unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
```
```   640     show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
```
```   641   qed
```
```   642 qed
```
```   643
```
```   644 lemma (in measure_space) simple_integral_mono_AE:
```
```   645   assumes "simple_function f" and "simple_function g"
```
```   646   and mono: "AE x. f x \<le> g x"
```
```   647   shows "simple_integral f \<le> simple_integral g"
```
```   648 proof -
```
```   649   let "?S x" = "(g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
```
```   650   have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
```
```   651     "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
```
```   652   show ?thesis
```
```   653     unfolding *
```
```   654       simple_function_partition[OF `simple_function f` `simple_function g`]
```
```   655       simple_function_partition[OF `simple_function g` `simple_function f`]
```
```   656   proof (safe intro!: setsum_mono)
```
```   657     fix x assume "x \<in> space M"
```
```   658     then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
```
```   659     show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
```
```   660     proof (cases "f x \<le> g x")
```
```   661       case True then show ?thesis using * by (auto intro!: mult_right_mono)
```
```   662     next
```
```   663       case False
```
```   664       obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
```
```   665         using mono by (auto elim!: AE_E)
```
```   666       have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
```
```   667       moreover have "?S x \<in> sets M" using assms
```
```   668         by (rule_tac Int) (auto intro!: simple_functionD)
```
```   669       ultimately have "\<mu> (?S x) \<le> \<mu> N"
```
```   670         using `N \<in> sets M` by (auto intro!: measure_mono)
```
```   671       then show ?thesis using `\<mu> N = 0` by auto
```
```   672     qed
```
```   673   qed
```
```   674 qed
```
```   675
```
```   676 lemma (in measure_space) simple_integral_mono:
```
```   677   assumes "simple_function f" and "simple_function g"
```
```   678   and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
```
```   679   shows "simple_integral f \<le> simple_integral g"
```
```   680 proof (rule simple_integral_mono_AE[OF assms(1, 2)])
```
```   681   show "AE x. f x \<le> g x"
```
```   682     using mono by (rule AE_cong) auto
```
```   683 qed
```
```   684
```
```   685 lemma (in measure_space) simple_integral_cong_AE:
```
```   686   assumes "simple_function f" "simple_function g" and "AE x. f x = g x"
```
```   687   shows "simple_integral f = simple_integral g"
```
```   688   using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
```
```   689
```
```   690 lemma (in measure_space) simple_integral_cong':
```
```   691   assumes sf: "simple_function f" "simple_function g"
```
```   692   and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
```
```   693   shows "simple_integral f = simple_integral g"
```
```   694 proof (intro simple_integral_cong_AE sf AE_I)
```
```   695   show "\<mu> {x\<in>space M. f x \<noteq> g x} = 0" by fact
```
```   696   show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
```
```   697     using sf[THEN borel_measurable_simple_function] by auto
```
```   698 qed simp
```
```   699
```
```   700 lemma (in measure_space) simple_integral_indicator:
```
```   701   assumes "A \<in> sets M"
```
```   702   assumes "simple_function f"
```
```   703   shows "(\<integral>\<^isup>Sx. f x * indicator A x) =
```
```   704     (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
```
```   705 proof cases
```
```   706   assume "A = space M"
```
```   707   moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x) = simple_integral f"
```
```   708     by (auto intro!: simple_integral_cong)
```
```   709   moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
```
```   710   ultimately show ?thesis by (simp add: simple_integral_def)
```
```   711 next
```
```   712   assume "A \<noteq> space M"
```
```   713   then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
```
```   714   have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
```
```   715   proof safe
```
```   716     fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
```
```   717   next
```
```   718     fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
```
```   719       using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
```
```   720   next
```
```   721     show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
```
```   722   qed
```
```   723   have *: "(\<integral>\<^isup>Sx. f x * indicator A x) =
```
```   724     (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
```
```   725     unfolding simple_integral_def I
```
```   726   proof (rule setsum_mono_zero_cong_left)
```
```   727     show "finite (f ` space M \<union> {0})"
```
```   728       using assms(2) unfolding simple_function_def by auto
```
```   729     show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
```
```   730       using sets_into_space[OF assms(1)] by auto
```
```   731     have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
```
```   732       by (auto simp: image_iff)
```
```   733     thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
```
```   734       i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
```
```   735   next
```
```   736     fix x assume "x \<in> f`A \<union> {0}"
```
```   737     hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
```
```   738       by (auto simp: indicator_def split: split_if_asm)
```
```   739     thus "x * \<mu> (?I -` {x} \<inter> space M) =
```
```   740       x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
```
```   741   qed
```
```   742   show ?thesis unfolding *
```
```   743     using assms(2) unfolding simple_function_def
```
```   744     by (auto intro!: setsum_mono_zero_cong_right)
```
```   745 qed
```
```   746
```
```   747 lemma (in measure_space) simple_integral_indicator_only[simp]:
```
```   748   assumes "A \<in> sets M"
```
```   749   shows "simple_integral (indicator A) = \<mu> A"
```
```   750 proof cases
```
```   751   assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
```
```   752   thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
```
```   753 next
```
```   754   assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pextreal}" by auto
```
```   755   thus ?thesis
```
```   756     using simple_integral_indicator[OF assms simple_function_const[of 1]]
```
```   757     using sets_into_space[OF assms]
```
```   758     by (auto intro!: arg_cong[where f="\<mu>"])
```
```   759 qed
```
```   760
```
```   761 lemma (in measure_space) simple_integral_null_set:
```
```   762   assumes "simple_function u" "N \<in> null_sets"
```
```   763   shows "(\<integral>\<^isup>Sx. u x * indicator N x) = 0"
```
```   764 proof -
```
```   765   have "AE x. indicator N x = (0 :: pextreal)"
```
```   766     using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
```
```   767   then have "(\<integral>\<^isup>Sx. u x * indicator N x) = (\<integral>\<^isup>Sx. 0)"
```
```   768     using assms by (intro simple_integral_cong_AE) (auto intro!: AE_disjI2)
```
```   769   then show ?thesis by simp
```
```   770 qed
```
```   771
```
```   772 lemma (in measure_space) simple_integral_cong_AE_mult_indicator:
```
```   773   assumes sf: "simple_function f" and eq: "AE x. x \<in> S" and "S \<in> sets M"
```
```   774   shows "simple_integral f = (\<integral>\<^isup>Sx. f x * indicator S x)"
```
```   775 proof (rule simple_integral_cong_AE)
```
```   776   show "simple_function f" by fact
```
```   777   show "simple_function (\<lambda>x. f x * indicator S x)"
```
```   778     using sf `S \<in> sets M` by auto
```
```   779   from eq show "AE x. f x = f x * indicator S x"
```
```   780     by (rule AE_mp) simp
```
```   781 qed
```
```   782
```
```   783 lemma (in measure_space) simple_integral_restricted:
```
```   784   assumes "A \<in> sets M"
```
```   785   assumes sf: "simple_function (\<lambda>x. f x * indicator A x)"
```
```   786   shows "measure_space.simple_integral (restricted_space A) \<mu> f = (\<integral>\<^isup>Sx. f x * indicator A x)"
```
```   787     (is "_ = simple_integral ?f")
```
```   788   unfolding measure_space.simple_integral_def[OF restricted_measure_space[OF `A \<in> sets M`]]
```
```   789   unfolding simple_integral_def
```
```   790 proof (simp, safe intro!: setsum_mono_zero_cong_left)
```
```   791   from sf show "finite (?f ` space M)"
```
```   792     unfolding simple_function_def by auto
```
```   793 next
```
```   794   fix x assume "x \<in> A"
```
```   795   then show "f x \<in> ?f ` space M"
```
```   796     using sets_into_space `A \<in> sets M` by (auto intro!: image_eqI[of _ _ x])
```
```   797 next
```
```   798   fix x assume "x \<in> space M" "?f x \<notin> f`A"
```
```   799   then have "x \<notin> A" by (auto simp: image_iff)
```
```   800   then show "?f x * \<mu> (?f -` {?f x} \<inter> space M) = 0" by simp
```
```   801 next
```
```   802   fix x assume "x \<in> A"
```
```   803   then have "f x \<noteq> 0 \<Longrightarrow>
```
```   804     f -` {f x} \<inter> A = ?f -` {f x} \<inter> space M"
```
```   805     using `A \<in> sets M` sets_into_space
```
```   806     by (auto simp: indicator_def split: split_if_asm)
```
```   807   then show "f x * \<mu> (f -` {f x} \<inter> A) =
```
```   808     f x * \<mu> (?f -` {f x} \<inter> space M)"
```
```   809     unfolding pextreal_mult_cancel_left by auto
```
```   810 qed
```
```   811
```
```   812 lemma (in measure_space) simple_integral_subalgebra:
```
```   813   assumes N: "measure_space N \<mu>" and [simp]: "space N = space M"
```
```   814   shows "measure_space.simple_integral N \<mu> = simple_integral"
```
```   815   unfolding simple_integral_def_raw
```
```   816   unfolding measure_space.simple_integral_def_raw[OF N] by simp
```
```   817
```
```   818 lemma (in measure_space) simple_integral_vimage:
```
```   819   fixes g :: "'a \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
```
```   820   assumes f: "bij_betw f S (space M)"
```
```   821   shows "simple_integral g =
```
```   822          measure_space.simple_integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>x. g (f x))"
```
```   823     (is "_ = measure_space.simple_integral ?T ?\<mu> _")
```
```   824 proof -
```
```   825   from f interpret T: measure_space ?T ?\<mu> by (rule measure_space_isomorphic)
```
```   826   have surj: "f`S = space M"
```
```   827     using f unfolding bij_betw_def by simp
```
```   828   have *: "(\<lambda>x. g (f x)) ` S = g ` f ` S" by auto
```
```   829   have **: "f`S = space M" using f unfolding bij_betw_def by auto
```
```   830   { fix x assume "x \<in> space M"
```
```   831     have "(f ` ((\<lambda>x. g (f x)) -` {g x} \<inter> S)) =
```
```   832       (f ` (f -` (g -` {g x}) \<inter> S))" by auto
```
```   833     also have "f -` (g -` {g x}) \<inter> S = f -` (g -` {g x} \<inter> space M) \<inter> S"
```
```   834       using f unfolding bij_betw_def by auto
```
```   835     also have "(f ` (f -` (g -` {g x} \<inter> space M) \<inter> S)) = g -` {g x} \<inter> space M"
```
```   836       using ** by (intro image_vimage_inter_eq) auto
```
```   837     finally have "(f ` ((\<lambda>x. g (f x)) -` {g x} \<inter> S)) = g -` {g x} \<inter> space M" by auto }
```
```   838   then show ?thesis using assms
```
```   839     unfolding simple_integral_def T.simple_integral_def bij_betw_def
```
```   840     by (auto simp add: * intro!: setsum_cong)
```
```   841 qed
```
```   842
```
```   843 section "Continuous posititve integration"
```
```   844
```
```   845 definition (in measure_space) positive_integral (binder "\<integral>\<^isup>+ " 10) where
```
```   846   "positive_integral f = SUPR {g. simple_function g \<and> g \<le> f} simple_integral"
```
```   847
```
```   848 lemma (in measure_space) positive_integral_alt:
```
```   849   "positive_integral f =
```
```   850     (SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M} simple_integral)" (is "_ = ?alt")
```
```   851 proof (rule antisym SUP_leI)
```
```   852   show "positive_integral f \<le> ?alt" unfolding positive_integral_def
```
```   853   proof (safe intro!: SUP_leI)
```
```   854     fix g assume g: "simple_function g" "g \<le> f"
```
```   855     let ?G = "g -` {\<omega>} \<inter> space M"
```
```   856     show "simple_integral g \<le>
```
```   857       SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g ` space M} simple_integral"
```
```   858       (is "simple_integral g \<le> SUPR ?A simple_integral")
```
```   859     proof cases
```
```   860       let ?g = "\<lambda>x. indicator (space M - ?G) x * g x"
```
```   861       have g': "simple_function ?g"
```
```   862         using g by (auto intro: simple_functionD)
```
```   863       moreover
```
```   864       assume "\<mu> ?G = 0"
```
```   865       then have "AE x. g x = ?g x" using g
```
```   866         by (intro AE_I[where N="?G"])
```
```   867            (auto intro: simple_functionD simp: indicator_def)
```
```   868       with g(1) g' have "simple_integral g = simple_integral ?g"
```
```   869         by (rule simple_integral_cong_AE)
```
```   870       moreover have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
```
```   871       from this `g \<le> f` have "?g \<le> f" by (rule order_trans)
```
```   872       moreover have "\<omega> \<notin> ?g ` space M"
```
```   873         by (auto simp: indicator_def split: split_if_asm)
```
```   874       ultimately show ?thesis by (auto intro!: le_SUPI)
```
```   875     next
```
```   876       assume "\<mu> ?G \<noteq> 0"
```
```   877       then have "?G \<noteq> {}" by auto
```
```   878       then have "\<omega> \<in> g`space M" by force
```
```   879       then have "space M \<noteq> {}" by auto
```
```   880       have "SUPR ?A simple_integral = \<omega>"
```
```   881       proof (intro SUP_\<omega>[THEN iffD2] allI impI)
```
```   882         fix x assume "x < \<omega>"
```
```   883         then guess n unfolding less_\<omega>_Ex_of_nat .. note n = this
```
```   884         then have "0 < n" by (intro neq0_conv[THEN iffD1] notI) simp
```
```   885         let ?g = "\<lambda>x. (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * indicator ?G x"
```
```   886         show "\<exists>i\<in>?A. x < simple_integral i"
```
```   887         proof (intro bexI impI CollectI conjI)
```
```   888           show "simple_function ?g" using g
```
```   889             by (auto intro!: simple_functionD simple_function_add)
```
```   890           have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
```
```   891           from this g(2) show "?g \<le> f" by (rule order_trans)
```
```   892           show "\<omega> \<notin> ?g ` space M"
```
```   893             using `\<mu> ?G \<noteq> 0` by (auto simp: indicator_def split: split_if_asm)
```
```   894           have "x < (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * \<mu> ?G"
```
```   895             using n `\<mu> ?G \<noteq> 0` `0 < n`
```
```   896             by (auto simp: pextreal_noteq_omega_Ex field_simps)
```
```   897           also have "\<dots> = simple_integral ?g" using g `space M \<noteq> {}`
```
```   898             by (subst simple_integral_indicator)
```
```   899                (auto simp: image_constant ac_simps dest: simple_functionD)
```
```   900           finally show "x < simple_integral ?g" .
```
```   901         qed
```
```   902       qed
```
```   903       then show ?thesis by simp
```
```   904     qed
```
```   905   qed
```
```   906 qed (auto intro!: SUP_subset simp: positive_integral_def)
```
```   907
```
```   908 lemma (in measure_space) positive_integral_cong_measure:
```
```   909   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A"
```
```   910   shows "measure_space.positive_integral M \<nu> f = positive_integral f"
```
```   911 proof -
```
```   912   interpret v: measure_space M \<nu>
```
```   913     by (rule measure_space_cong) fact
```
```   914   with assms show ?thesis
```
```   915     unfolding positive_integral_def v.positive_integral_def SUPR_def
```
```   916     by (auto intro!: arg_cong[where f=Sup] image_cong
```
```   917              simp: simple_integral_cong_measure[of \<nu>])
```
```   918 qed
```
```   919
```
```   920 lemma (in measure_space) positive_integral_alt1:
```
```   921   "positive_integral f =
```
```   922     (SUP g : {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}. simple_integral g)"
```
```   923   unfolding positive_integral_alt SUPR_def
```
```   924 proof (safe intro!: arg_cong[where f=Sup])
```
```   925   fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
```
```   926   assume "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
```
```   927   hence "?g \<le> f" "simple_function ?g" "simple_integral g = simple_integral ?g"
```
```   928     "\<omega> \<notin> g`space M"
```
```   929     unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
```
```   930   thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
```
```   931     by auto
```
```   932 next
```
```   933   fix g assume "simple_function g" "g \<le> f" "\<omega> \<notin> g`space M"
```
```   934   hence "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
```
```   935     by (auto simp add: le_fun_def image_iff)
```
```   936   thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}"
```
```   937     by auto
```
```   938 qed
```
```   939
```
```   940 lemma (in measure_space) positive_integral_cong:
```
```   941   assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
```
```   942   shows "positive_integral f = positive_integral g"
```
```   943 proof -
```
```   944   have "\<And>h. (\<forall>x\<in>space M. h x \<le> f x \<and> h x \<noteq> \<omega>) = (\<forall>x\<in>space M. h x \<le> g x \<and> h x \<noteq> \<omega>)"
```
```   945     using assms by auto
```
```   946   thus ?thesis unfolding positive_integral_alt1 by auto
```
```   947 qed
```
```   948
```
```   949 lemma (in measure_space) positive_integral_eq_simple_integral:
```
```   950   assumes "simple_function f"
```
```   951   shows "positive_integral f = simple_integral f"
```
```   952   unfolding positive_integral_def
```
```   953 proof (safe intro!: pextreal_SUPI)
```
```   954   fix g assume "simple_function g" "g \<le> f"
```
```   955   with assms show "simple_integral g \<le> simple_integral f"
```
```   956     by (auto intro!: simple_integral_mono simp: le_fun_def)
```
```   957 next
```
```   958   fix y assume "\<forall>x. x\<in>{g. simple_function g \<and> g \<le> f} \<longrightarrow> simple_integral x \<le> y"
```
```   959   with assms show "simple_integral f \<le> y" by auto
```
```   960 qed
```
```   961
```
```   962 lemma (in measure_space) positive_integral_mono_AE:
```
```   963   assumes ae: "AE x. u x \<le> v x"
```
```   964   shows "positive_integral u \<le> positive_integral v"
```
```   965   unfolding positive_integral_alt1
```
```   966 proof (safe intro!: SUPR_mono)
```
```   967   fix a assume a: "simple_function a" and mono: "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
```
```   968   from ae obtain N where N: "{x\<in>space M. \<not> u x \<le> v x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
```
```   969     by (auto elim!: AE_E)
```
```   970   have "simple_function (\<lambda>x. a x * indicator (space M - N) x)"
```
```   971     using `N \<in> sets M` a by auto
```
```   972   with a show "\<exists>b\<in>{g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> v x \<and> g x \<noteq> \<omega>)}.
```
```   973     simple_integral a \<le> simple_integral b"
```
```   974   proof (safe intro!: bexI[of _ "\<lambda>x. a x * indicator (space M - N) x"]
```
```   975                       simple_integral_mono_AE)
```
```   976     show "AE x. a x \<le> a x * indicator (space M - N) x"
```
```   977     proof (rule AE_I, rule subset_refl)
```
```   978       have *: "{x \<in> space M. \<not> a x \<le> a x * indicator (space M - N) x} =
```
```   979         N \<inter> {x \<in> space M. a x \<noteq> 0}" (is "?N = _")
```
```   980         using `N \<in> sets M`[THEN sets_into_space] by (auto simp: indicator_def)
```
```   981       then show "?N \<in> sets M"
```
```   982         using `N \<in> sets M` `simple_function a`[THEN borel_measurable_simple_function]
```
```   983         by (auto intro!: measure_mono Int)
```
```   984       then have "\<mu> ?N \<le> \<mu> N"
```
```   985         unfolding * using `N \<in> sets M` by (auto intro!: measure_mono)
```
```   986       then show "\<mu> ?N = 0" using `\<mu> N = 0` by auto
```
```   987     qed
```
```   988   next
```
```   989     fix x assume "x \<in> space M"
```
```   990     show "a x * indicator (space M - N) x \<le> v x"
```
```   991     proof (cases "x \<in> N")
```
```   992       case True then show ?thesis by simp
```
```   993     next
```
```   994       case False
```
```   995       with N mono have "a x \<le> u x" "u x \<le> v x" using `x \<in> space M` by auto
```
```   996       with False `x \<in> space M` show "a x * indicator (space M - N) x \<le> v x" by auto
```
```   997     qed
```
```   998     assume "a x * indicator (space M - N) x = \<omega>"
```
```   999     with mono `x \<in> space M` show False
```
```  1000       by (simp split: split_if_asm add: indicator_def)
```
```  1001   qed
```
```  1002 qed
```
```  1003
```
```  1004 lemma (in measure_space) positive_integral_cong_AE:
```
```  1005   "AE x. u x = v x \<Longrightarrow> positive_integral u = positive_integral v"
```
```  1006   by (auto simp: eq_iff intro!: positive_integral_mono_AE)
```
```  1007
```
```  1008 lemma (in measure_space) positive_integral_mono:
```
```  1009   assumes mono: "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x"
```
```  1010   shows "positive_integral u \<le> positive_integral v"
```
```  1011   using mono by (auto intro!: AE_cong positive_integral_mono_AE)
```
```  1012
```
```  1013 lemma image_set_cong:
```
```  1014   assumes A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. f x = g y"
```
```  1015   assumes B: "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. g y = f x"
```
```  1016   shows "f ` A = g ` B"
```
```  1017   using assms by blast
```
```  1018
```
```  1019 lemma (in measure_space) positive_integral_vimage:
```
```  1020   fixes g :: "'a \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
```
```  1021   assumes f: "bij_betw f S (space M)"
```
```  1022   shows "positive_integral g =
```
```  1023          measure_space.positive_integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>x. g (f x))"
```
```  1024     (is "_ = measure_space.positive_integral ?T ?\<mu> _")
```
```  1025 proof -
```
```  1026   from f interpret T: measure_space ?T ?\<mu> by (rule measure_space_isomorphic)
```
```  1027   have f_fun: "f \<in> S \<rightarrow> space M" using assms unfolding bij_betw_def by auto
```
```  1028   from assms have inv: "bij_betw (the_inv_into S f) (space M) S"
```
```  1029     by (rule bij_betw_the_inv_into)
```
```  1030   then have inv_fun: "the_inv_into S f \<in> space M \<rightarrow> S" unfolding bij_betw_def by auto
```
```  1031   have surj: "f`S = space M"
```
```  1032     using f unfolding bij_betw_def by simp
```
```  1033   have inj: "inj_on f S"
```
```  1034     using f unfolding bij_betw_def by simp
```
```  1035   have inv_f: "\<And>x. x \<in> space M \<Longrightarrow> f (the_inv_into S f x) = x"
```
```  1036     using f_the_inv_into_f[of f S] f unfolding bij_betw_def by auto
```
```  1037   from simple_integral_vimage[OF assms, symmetric]
```
```  1038   have *: "simple_integral = T.simple_integral \<circ> (\<lambda>g. g \<circ> f)" by (simp add: comp_def)
```
```  1039   show ?thesis
```
```  1040     unfolding positive_integral_alt1 T.positive_integral_alt1 SUPR_def * image_compose
```
```  1041   proof (safe intro!: arg_cong[where f=Sup] image_set_cong, simp_all add: comp_def)
```
```  1042     fix g' :: "'a \<Rightarrow> pextreal" assume "simple_function g'" "\<forall>x\<in>space M. g' x \<le> g x \<and> g' x \<noteq> \<omega>"
```
```  1043     then show "\<exists>h. T.simple_function h \<and> (\<forall>x\<in>S. h x \<le> g (f x) \<and> h x \<noteq> \<omega>) \<and>
```
```  1044                    T.simple_integral (\<lambda>x. g' (f x)) = T.simple_integral h"
```
```  1045       using f unfolding bij_betw_def
```
```  1046       by (auto intro!: exI[of _ "\<lambda>x. g' (f x)"]
```
```  1047                simp add: le_fun_def simple_function_vimage[OF _ f_fun])
```
```  1048   next
```
```  1049     fix g' :: "'d \<Rightarrow> pextreal" assume g': "T.simple_function g'" "\<forall>x\<in>S. g' x \<le> g (f x) \<and> g' x \<noteq> \<omega>"
```
```  1050     let ?g = "\<lambda>x. g' (the_inv_into S f x)"
```
```  1051     show "\<exists>h. simple_function h \<and> (\<forall>x\<in>space M. h x \<le> g x \<and> h x \<noteq> \<omega>) \<and>
```
```  1052               T.simple_integral g' = T.simple_integral (\<lambda>x. h (f x))"
```
```  1053     proof (intro exI[of _ ?g] conjI ballI)
```
```  1054       { fix x assume x: "x \<in> space M"
```
```  1055         then have "the_inv_into S f x \<in> S" using inv_fun by auto
```
```  1056         with g' have "g' (the_inv_into S f x) \<le> g (f (the_inv_into S f x)) \<and> g' (the_inv_into S f x) \<noteq> \<omega>"
```
```  1057           by auto
```
```  1058         then show "g' (the_inv_into S f x) \<le> g x" "g' (the_inv_into S f x) \<noteq> \<omega>"
```
```  1059           using f_the_inv_into_f[of f S x] x f unfolding bij_betw_def by auto }
```
```  1060       note vimage_vimage_inv[OF f inv_f inv_fun, simp]
```
```  1061       from T.simple_function_vimage[OF g'(1), unfolded space_vimage_algebra, OF inv_fun]
```
```  1062       show "simple_function (\<lambda>x. g' (the_inv_into S f x))"
```
```  1063         unfolding simple_function_def by (simp add: simple_function_def)
```
```  1064       show "T.simple_integral g' = T.simple_integral (\<lambda>x. ?g (f x))"
```
```  1065         using the_inv_into_f_f[OF inj] by (auto intro!: T.simple_integral_cong)
```
```  1066     qed
```
```  1067   qed
```
```  1068 qed
```
```  1069
```
```  1070 lemma (in measure_space) positive_integral_vimage_inv:
```
```  1071   fixes g :: "'d \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
```
```  1072   assumes f: "bij_inv S (space M) f h"
```
```  1073   shows "measure_space.positive_integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) g =
```
```  1074       (\<integral>\<^isup>+x. g (h x))"
```
```  1075 proof -
```
```  1076   interpret v: measure_space "vimage_algebra S f" "\<lambda>A. \<mu> (f ` A)"
```
```  1077     using f by (rule measure_space_isomorphic[OF bij_inv_bij_betw(1)])
```
```  1078   show ?thesis
```
```  1079     unfolding positive_integral_vimage[OF f[THEN bij_inv_bij_betw(1)], of "\<lambda>x. g (h x)"]
```
```  1080     using f[unfolded bij_inv_def]
```
```  1081     by (auto intro!: v.positive_integral_cong)
```
```  1082 qed
```
```  1083
```
```  1084 lemma (in measure_space) positive_integral_SUP_approx:
```
```  1085   assumes "f \<up> s"
```
```  1086   and f: "\<And>i. f i \<in> borel_measurable M"
```
```  1087   and "simple_function u"
```
```  1088   and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
```
```  1089   shows "simple_integral u \<le> (SUP i. positive_integral (f i))" (is "_ \<le> ?S")
```
```  1090 proof (rule pextreal_le_mult_one_interval)
```
```  1091   fix a :: pextreal assume "0 < a" "a < 1"
```
```  1092   hence "a \<noteq> 0" by auto
```
```  1093   let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
```
```  1094   have B: "\<And>i. ?B i \<in> sets M"
```
```  1095     using f `simple_function u` by (auto simp: borel_measurable_simple_function)
```
```  1096
```
```  1097   let "?uB i x" = "u x * indicator (?B i) x"
```
```  1098
```
```  1099   { fix i have "?B i \<subseteq> ?B (Suc i)"
```
```  1100     proof safe
```
```  1101       fix i x assume "a * u x \<le> f i x"
```
```  1102       also have "\<dots> \<le> f (Suc i) x"
```
```  1103         using `f \<up> s` unfolding isoton_def le_fun_def by auto
```
```  1104       finally show "a * u x \<le> f (Suc i) x" .
```
```  1105     qed }
```
```  1106   note B_mono = this
```
```  1107
```
```  1108   have u: "\<And>x. x \<in> space M \<Longrightarrow> u -` {u x} \<inter> space M \<in> sets M"
```
```  1109     using `simple_function u` by (auto simp add: simple_function_def)
```
```  1110
```
```  1111   have "\<And>i. (\<lambda>n. (u -` {i} \<inter> space M) \<inter> ?B n) \<up> (u -` {i} \<inter> space M)" using B_mono unfolding isoton_def
```
```  1112   proof safe
```
```  1113     fix x i assume "x \<in> space M"
```
```  1114     show "x \<in> (\<Union>i. (u -` {u x} \<inter> space M) \<inter> ?B i)"
```
```  1115     proof cases
```
```  1116       assume "u x = 0" thus ?thesis using `x \<in> space M` by simp
```
```  1117     next
```
```  1118       assume "u x \<noteq> 0"
```
```  1119       with `a < 1` real `x \<in> space M`
```
```  1120       have "a * u x < 1 * u x" by (rule_tac pextreal_mult_strict_right_mono) (auto simp: image_iff)
```
```  1121       also have "\<dots> \<le> (SUP i. f i x)" using le `f \<up> s`
```
```  1122         unfolding isoton_fun_expand by (auto simp: isoton_def le_fun_def)
```
```  1123       finally obtain i where "a * u x < f i x" unfolding SUPR_def
```
```  1124         by (auto simp add: less_Sup_iff)
```
```  1125       hence "a * u x \<le> f i x" by auto
```
```  1126       thus ?thesis using `x \<in> space M` by auto
```
```  1127     qed
```
```  1128   qed auto
```
```  1129   note measure_conv = measure_up[OF Int[OF u B] this]
```
```  1130
```
```  1131   have "simple_integral u = (SUP i. simple_integral (?uB i))"
```
```  1132     unfolding simple_integral_indicator[OF B `simple_function u`]
```
```  1133   proof (subst SUPR_pextreal_setsum, safe)
```
```  1134     fix x n assume "x \<in> space M"
```
```  1135     have "\<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f n x})
```
```  1136       \<le> \<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f (Suc n) x})"
```
```  1137       using B_mono Int[OF u[OF `x \<in> space M`] B] by (auto intro!: measure_mono)
```
```  1138     thus "u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B n)
```
```  1139             \<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
```
```  1140       by (auto intro: mult_left_mono)
```
```  1141   next
```
```  1142     show "simple_integral u =
```
```  1143       (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (u -` {i} \<inter> space M \<inter> ?B n))"
```
```  1144       using measure_conv unfolding simple_integral_def isoton_def
```
```  1145       by (auto intro!: setsum_cong simp: pextreal_SUP_cmult)
```
```  1146   qed
```
```  1147   moreover
```
```  1148   have "a * (SUP i. simple_integral (?uB i)) \<le> ?S"
```
```  1149     unfolding pextreal_SUP_cmult[symmetric]
```
```  1150   proof (safe intro!: SUP_mono bexI)
```
```  1151     fix i
```
```  1152     have "a * simple_integral (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x)"
```
```  1153       using B `simple_function u`
```
```  1154       by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
```
```  1155     also have "\<dots> \<le> positive_integral (f i)"
```
```  1156     proof -
```
```  1157       have "\<And>x. a * ?uB i x \<le> f i x" unfolding indicator_def by auto
```
```  1158       hence *: "simple_function (\<lambda>x. a * ?uB i x)" using B assms(3)
```
```  1159         by (auto intro!: simple_integral_mono)
```
```  1160       show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
```
```  1161         by (auto intro!: positive_integral_mono simp: indicator_def)
```
```  1162     qed
```
```  1163     finally show "a * simple_integral (?uB i) \<le> positive_integral (f i)"
```
```  1164       by auto
```
```  1165   qed simp
```
```  1166   ultimately show "a * simple_integral u \<le> ?S" by simp
```
```  1167 qed
```
```  1168
```
```  1169 text {* Beppo-Levi monotone convergence theorem *}
```
```  1170 lemma (in measure_space) positive_integral_isoton:
```
```  1171   assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
```
```  1172   shows "(\<lambda>i. positive_integral (f i)) \<up> positive_integral u"
```
```  1173   unfolding isoton_def
```
```  1174 proof safe
```
```  1175   fix i show "positive_integral (f i) \<le> positive_integral (f (Suc i))"
```
```  1176     apply (rule positive_integral_mono)
```
```  1177     using `f \<up> u` unfolding isoton_def le_fun_def by auto
```
```  1178 next
```
```  1179   have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
```
```  1180
```
```  1181   show "(SUP i. positive_integral (f i)) = positive_integral u"
```
```  1182   proof (rule antisym)
```
```  1183     from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
```
```  1184     show "(SUP j. positive_integral (f j)) \<le> positive_integral u"
```
```  1185       by (auto intro!: SUP_leI positive_integral_mono)
```
```  1186   next
```
```  1187     show "positive_integral u \<le> (SUP i. positive_integral (f i))"
```
```  1188       unfolding positive_integral_alt[of u]
```
```  1189       by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
```
```  1190   qed
```
```  1191 qed
```
```  1192
```
```  1193 lemma (in measure_space) positive_integral_monotone_convergence_SUP:
```
```  1194   assumes "\<And>i x. x \<in> space M \<Longrightarrow> f i x \<le> f (Suc i) x"
```
```  1195   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1196   shows "(SUP i. positive_integral (f i)) = (\<integral>\<^isup>+ x. SUP i. f i x)"
```
```  1197     (is "_ = positive_integral ?u")
```
```  1198 proof -
```
```  1199   show ?thesis
```
```  1200   proof (rule antisym)
```
```  1201     show "(SUP j. positive_integral (f j)) \<le> positive_integral ?u"
```
```  1202       by (auto intro!: SUP_leI positive_integral_mono le_SUPI)
```
```  1203   next
```
```  1204     def rf \<equiv> "\<lambda>i. \<lambda>x\<in>space M. f i x" and ru \<equiv> "\<lambda>x\<in>space M. ?u x"
```
```  1205     have "\<And>i. rf i \<in> borel_measurable M" unfolding rf_def
```
```  1206       using assms by (simp cong: measurable_cong)
```
```  1207     moreover have iso: "rf \<up> ru" using assms unfolding rf_def ru_def
```
```  1208       unfolding isoton_def le_fun_def fun_eq_iff SUPR_apply
```
```  1209       using SUP_const[OF UNIV_not_empty]
```
```  1210       by (auto simp: restrict_def le_fun_def fun_eq_iff)
```
```  1211     ultimately have "positive_integral ru \<le> (SUP i. positive_integral (rf i))"
```
```  1212       unfolding positive_integral_alt[of ru]
```
```  1213       by (auto simp: le_fun_def intro!: SUP_leI positive_integral_SUP_approx)
```
```  1214     then show "positive_integral ?u \<le> (SUP i. positive_integral (f i))"
```
```  1215       unfolding ru_def rf_def by (simp cong: positive_integral_cong)
```
```  1216   qed
```
```  1217 qed
```
```  1218
```
```  1219 lemma (in measure_space) SUP_simple_integral_sequences:
```
```  1220   assumes f: "f \<up> u" "\<And>i. simple_function (f i)"
```
```  1221   and g: "g \<up> u" "\<And>i. simple_function (g i)"
```
```  1222   shows "(SUP i. simple_integral (f i)) = (SUP i. simple_integral (g i))"
```
```  1223     (is "SUPR _ ?F = SUPR _ ?G")
```
```  1224 proof -
```
```  1225   have "(SUP i. ?F i) = (SUP i. positive_integral (f i))"
```
```  1226     using assms by (simp add: positive_integral_eq_simple_integral)
```
```  1227   also have "\<dots> = positive_integral u"
```
```  1228     using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
```
```  1229     unfolding isoton_def by simp
```
```  1230   also have "\<dots> = (SUP i. positive_integral (g i))"
```
```  1231     using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
```
```  1232     unfolding isoton_def by simp
```
```  1233   also have "\<dots> = (SUP i. ?G i)"
```
```  1234     using assms by (simp add: positive_integral_eq_simple_integral)
```
```  1235   finally show ?thesis .
```
```  1236 qed
```
```  1237
```
```  1238 lemma (in measure_space) positive_integral_const[simp]:
```
```  1239   "(\<integral>\<^isup>+ x. c) = c * \<mu> (space M)"
```
```  1240   by (subst positive_integral_eq_simple_integral) auto
```
```  1241
```
```  1242 lemma (in measure_space) positive_integral_isoton_simple:
```
```  1243   assumes "f \<up> u" and e: "\<And>i. simple_function (f i)"
```
```  1244   shows "(\<lambda>i. simple_integral (f i)) \<up> positive_integral u"
```
```  1245   using positive_integral_isoton[OF `f \<up> u` e(1)[THEN borel_measurable_simple_function]]
```
```  1246   unfolding positive_integral_eq_simple_integral[OF e] .
```
```  1247
```
```  1248 lemma (in measure_space) positive_integral_linear:
```
```  1249   assumes f: "f \<in> borel_measurable M"
```
```  1250   and g: "g \<in> borel_measurable M"
```
```  1251   shows "(\<integral>\<^isup>+ x. a * f x + g x) =
```
```  1252       a * positive_integral f + positive_integral g"
```
```  1253     (is "positive_integral ?L = _")
```
```  1254 proof -
```
```  1255   from borel_measurable_implies_simple_function_sequence'[OF f] guess u .
```
```  1256   note u = this positive_integral_isoton_simple[OF this(1-2)]
```
```  1257   from borel_measurable_implies_simple_function_sequence'[OF g] guess v .
```
```  1258   note v = this positive_integral_isoton_simple[OF this(1-2)]
```
```  1259   let "?L' i x" = "a * u i x + v i x"
```
```  1260
```
```  1261   have "?L \<in> borel_measurable M"
```
```  1262     using assms by simp
```
```  1263   from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
```
```  1264   note positive_integral_isoton_simple[OF this(1-2)] and l = this
```
```  1265   moreover have
```
```  1266       "(SUP i. simple_integral (l i)) = (SUP i. simple_integral (?L' i))"
```
```  1267   proof (rule SUP_simple_integral_sequences[OF l(1-2)])
```
```  1268     show "?L' \<up> ?L" "\<And>i. simple_function (?L' i)"
```
```  1269       using u v by (auto simp: isoton_fun_expand isoton_add isoton_cmult_right)
```
```  1270   qed
```
```  1271   moreover from u v have L'_isoton:
```
```  1272       "(\<lambda>i. simple_integral (?L' i)) \<up> a * positive_integral f + positive_integral g"
```
```  1273     by (simp add: isoton_add isoton_cmult_right)
```
```  1274   ultimately show ?thesis by (simp add: isoton_def)
```
```  1275 qed
```
```  1276
```
```  1277 lemma (in measure_space) positive_integral_cmult:
```
```  1278   assumes "f \<in> borel_measurable M"
```
```  1279   shows "(\<integral>\<^isup>+ x. c * f x) = c * positive_integral f"
```
```  1280   using positive_integral_linear[OF assms, of "\<lambda>x. 0" c] by auto
```
```  1281
```
```  1282 lemma (in measure_space) positive_integral_multc:
```
```  1283   assumes "f \<in> borel_measurable M"
```
```  1284   shows "(\<integral>\<^isup>+ x. f x * c) = positive_integral f * c"
```
```  1285   unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
```
```  1286
```
```  1287 lemma (in measure_space) positive_integral_indicator[simp]:
```
```  1288   "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x) = \<mu> A"
```
```  1289   by (subst positive_integral_eq_simple_integral)
```
```  1290      (auto simp: simple_function_indicator simple_integral_indicator)
```
```  1291
```
```  1292 lemma (in measure_space) positive_integral_cmult_indicator:
```
```  1293   "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x) = c * \<mu> A"
```
```  1294   by (subst positive_integral_eq_simple_integral)
```
```  1295      (auto simp: simple_function_indicator simple_integral_indicator)
```
```  1296
```
```  1297 lemma (in measure_space) positive_integral_add:
```
```  1298   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1299   shows "(\<integral>\<^isup>+ x. f x + g x) = positive_integral f + positive_integral g"
```
```  1300   using positive_integral_linear[OF assms, of 1] by simp
```
```  1301
```
```  1302 lemma (in measure_space) positive_integral_setsum:
```
```  1303   assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1304   shows "(\<integral>\<^isup>+ x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. positive_integral (f i))"
```
```  1305 proof cases
```
```  1306   assume "finite P"
```
```  1307   from this assms show ?thesis
```
```  1308   proof induct
```
```  1309     case (insert i P)
```
```  1310     have "f i \<in> borel_measurable M"
```
```  1311       "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M"
```
```  1312       using insert by (auto intro!: borel_measurable_pextreal_setsum)
```
```  1313     from positive_integral_add[OF this]
```
```  1314     show ?case using insert by auto
```
```  1315   qed simp
```
```  1316 qed simp
```
```  1317
```
```  1318 lemma (in measure_space) positive_integral_diff:
```
```  1319   assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
```
```  1320   and fin: "positive_integral g \<noteq> \<omega>"
```
```  1321   and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
```
```  1322   shows "(\<integral>\<^isup>+ x. f x - g x) = positive_integral f - positive_integral g"
```
```  1323 proof -
```
```  1324   have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1325     using f g by (rule borel_measurable_pextreal_diff)
```
```  1326   have "(\<integral>\<^isup>+x. f x - g x) + positive_integral g =
```
```  1327     positive_integral f"
```
```  1328     unfolding positive_integral_add[OF borel g, symmetric]
```
```  1329   proof (rule positive_integral_cong)
```
```  1330     fix x assume "x \<in> space M"
```
```  1331     from mono[OF this] show "f x - g x + g x = f x"
```
```  1332       by (cases "f x", cases "g x", simp, simp, cases "g x", auto)
```
```  1333   qed
```
```  1334   with mono show ?thesis
```
```  1335     by (subst minus_pextreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
```
```  1336 qed
```
```  1337
```
```  1338 lemma (in measure_space) positive_integral_psuminf:
```
```  1339   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1340   shows "(\<integral>\<^isup>+ x. \<Sum>\<^isub>\<infinity> i. f i x) = (\<Sum>\<^isub>\<infinity> i. positive_integral (f i))"
```
```  1341 proof -
```
```  1342   have "(\<lambda>i. (\<integral>\<^isup>+x. \<Sum>i<i. f i x)) \<up> (\<integral>\<^isup>+x. \<Sum>\<^isub>\<infinity>i. f i x)"
```
```  1343     by (rule positive_integral_isoton)
```
```  1344        (auto intro!: borel_measurable_pextreal_setsum assms positive_integral_mono
```
```  1345                      arg_cong[where f=Sup]
```
```  1346              simp: isoton_def le_fun_def psuminf_def fun_eq_iff SUPR_def Sup_fun_def)
```
```  1347   thus ?thesis
```
```  1348     by (auto simp: isoton_def psuminf_def positive_integral_setsum[OF assms])
```
```  1349 qed
```
```  1350
```
```  1351 text {* Fatou's lemma: convergence theorem on limes inferior *}
```
```  1352 lemma (in measure_space) positive_integral_lim_INF:
```
```  1353   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1354   assumes "\<And>i. u i \<in> borel_measurable M"
```
```  1355   shows "(\<integral>\<^isup>+ x. SUP n. INF m. u (m + n) x) \<le>
```
```  1356     (SUP n. INF m. positive_integral (u (m + n)))"
```
```  1357 proof -
```
```  1358   have "(\<integral>\<^isup>+x. SUP n. INF m. u (m + n) x)
```
```  1359       = (SUP n. (\<integral>\<^isup>+x. INF m. u (m + n) x))"
```
```  1360     using assms
```
```  1361     by (intro positive_integral_monotone_convergence_SUP[symmetric] INF_mono)
```
```  1362        (auto simp del: add_Suc simp add: add_Suc[symmetric])
```
```  1363   also have "\<dots> \<le> (SUP n. INF m. positive_integral (u (m + n)))"
```
```  1364     by (auto intro!: SUP_mono bexI le_INFI positive_integral_mono INF_leI)
```
```  1365   finally show ?thesis .
```
```  1366 qed
```
```  1367
```
```  1368 lemma (in measure_space) measure_space_density:
```
```  1369   assumes borel: "u \<in> borel_measurable M"
```
```  1370   shows "measure_space M (\<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x))" (is "measure_space M ?v")
```
```  1371 proof
```
```  1372   show "?v {} = 0" by simp
```
```  1373   show "countably_additive M ?v"
```
```  1374     unfolding countably_additive_def
```
```  1375   proof safe
```
```  1376     fix A :: "nat \<Rightarrow> 'a set"
```
```  1377     assume "range A \<subseteq> sets M"
```
```  1378     hence "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
```
```  1379       using borel by (auto intro: borel_measurable_indicator)
```
```  1380     moreover assume "disjoint_family A"
```
```  1381     note psuminf_indicator[OF this]
```
```  1382     ultimately show "(\<Sum>\<^isub>\<infinity>n. ?v (A n)) = ?v (\<Union>x. A x)"
```
```  1383       by (simp add: positive_integral_psuminf[symmetric])
```
```  1384   qed
```
```  1385 qed
```
```  1386
```
```  1387 lemma (in measure_space) positive_integral_translated_density:
```
```  1388   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1389   shows "measure_space.positive_integral M (\<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x)) g =
```
```  1390          (\<integral>\<^isup>+ x. f x * g x)" (is "measure_space.positive_integral M ?T _ = _")
```
```  1391 proof -
```
```  1392   from measure_space_density[OF assms(1)]
```
```  1393   interpret T: measure_space M ?T .
```
```  1394   from borel_measurable_implies_simple_function_sequence[OF assms(2)]
```
```  1395   obtain G where G: "\<And>i. simple_function (G i)" "G \<up> g" by blast
```
```  1396   note G_borel = borel_measurable_simple_function[OF this(1)]
```
```  1397   from T.positive_integral_isoton[OF `G \<up> g` G_borel]
```
```  1398   have *: "(\<lambda>i. T.positive_integral (G i)) \<up> T.positive_integral g" .
```
```  1399   { fix i
```
```  1400     have [simp]: "finite (G i ` space M)"
```
```  1401       using G(1) unfolding simple_function_def by auto
```
```  1402     have "T.positive_integral (G i) = T.simple_integral (G i)"
```
```  1403       using G T.positive_integral_eq_simple_integral by simp
```
```  1404     also have "\<dots> = (\<integral>\<^isup>+x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x))"
```
```  1405       apply (simp add: T.simple_integral_def)
```
```  1406       apply (subst positive_integral_cmult[symmetric])
```
```  1407       using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
```
```  1408       apply (subst positive_integral_setsum[symmetric])
```
```  1409       using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
```
```  1410       by (simp add: setsum_right_distrib field_simps)
```
```  1411     also have "\<dots> = (\<integral>\<^isup>+x. f x * G i x)"
```
```  1412       by (auto intro!: positive_integral_cong
```
```  1413                simp: indicator_def if_distrib setsum_cases)
```
```  1414     finally have "T.positive_integral (G i) = (\<integral>\<^isup>+x. f x * G i x)" . }
```
```  1415   with * have eq_Tg: "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x)) \<up> T.positive_integral g" by simp
```
```  1416   from G(2) have "(\<lambda>i x. f x * G i x) \<up> (\<lambda>x. f x * g x)"
```
```  1417     unfolding isoton_fun_expand by (auto intro!: isoton_cmult_right)
```
```  1418   then have "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x)) \<up> (\<integral>\<^isup>+x. f x * g x)"
```
```  1419     using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pextreal_times)
```
```  1420   with eq_Tg show "T.positive_integral g = (\<integral>\<^isup>+x. f x * g x)"
```
```  1421     unfolding isoton_def by simp
```
```  1422 qed
```
```  1423
```
```  1424 lemma (in measure_space) positive_integral_null_set:
```
```  1425   assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x) = 0"
```
```  1426 proof -
```
```  1427   have "(\<integral>\<^isup>+ x. u x * indicator N x) = (\<integral>\<^isup>+ x. 0)"
```
```  1428   proof (intro positive_integral_cong_AE AE_I)
```
```  1429     show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
```
```  1430       by (auto simp: indicator_def)
```
```  1431     show "\<mu> N = 0" "N \<in> sets M"
```
```  1432       using assms by auto
```
```  1433   qed
```
```  1434   then show ?thesis by simp
```
```  1435 qed
```
```  1436
```
```  1437 lemma (in measure_space) positive_integral_Markov_inequality:
```
```  1438   assumes borel: "u \<in> borel_measurable M" and "A \<in> sets M" and c: "c \<noteq> \<omega>"
```
```  1439   shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x)"
```
```  1440     (is "\<mu> ?A \<le> _ * ?PI")
```
```  1441 proof -
```
```  1442   have "?A \<in> sets M"
```
```  1443     using `A \<in> sets M` borel by auto
```
```  1444   hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x)"
```
```  1445     using positive_integral_indicator by simp
```
```  1446   also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x))"
```
```  1447   proof (rule positive_integral_mono)
```
```  1448     fix x assume "x \<in> space M"
```
```  1449     show "indicator ?A x \<le> c * (u x * indicator A x)"
```
```  1450       by (cases "x \<in> ?A") auto
```
```  1451   qed
```
```  1452   also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x)"
```
```  1453     using assms
```
```  1454     by (auto intro!: positive_integral_cmult borel_measurable_indicator)
```
```  1455   finally show ?thesis .
```
```  1456 qed
```
```  1457
```
```  1458 lemma (in measure_space) positive_integral_0_iff:
```
```  1459   assumes borel: "u \<in> borel_measurable M"
```
```  1460   shows "positive_integral u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
```
```  1461     (is "_ \<longleftrightarrow> \<mu> ?A = 0")
```
```  1462 proof -
```
```  1463   have A: "?A \<in> sets M" using borel by auto
```
```  1464   have u: "(\<integral>\<^isup>+ x. u x * indicator ?A x) = positive_integral u"
```
```  1465     by (auto intro!: positive_integral_cong simp: indicator_def)
```
```  1466
```
```  1467   show ?thesis
```
```  1468   proof
```
```  1469     assume "\<mu> ?A = 0"
```
```  1470     hence "?A \<in> null_sets" using `?A \<in> sets M` by auto
```
```  1471     from positive_integral_null_set[OF this]
```
```  1472     have "0 = (\<integral>\<^isup>+ x. u x * indicator ?A x)" by simp
```
```  1473     thus "positive_integral u = 0" unfolding u by simp
```
```  1474   next
```
```  1475     assume *: "positive_integral u = 0"
```
```  1476     let "?M n" = "{x \<in> space M. 1 \<le> of_nat n * u x}"
```
```  1477     have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
```
```  1478     proof -
```
```  1479       { fix n
```
```  1480         from positive_integral_Markov_inequality[OF borel `?A \<in> sets M`, of "of_nat n"]
```
```  1481         have "\<mu> (?M n \<inter> ?A) = 0" unfolding * u by simp }
```
```  1482       thus ?thesis by simp
```
```  1483     qed
```
```  1484     also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
```
```  1485     proof (safe intro!: continuity_from_below)
```
```  1486       fix n show "?M n \<inter> ?A \<in> sets M"
```
```  1487         using borel by (auto intro!: Int)
```
```  1488     next
```
```  1489       fix n x assume "1 \<le> of_nat n * u x"
```
```  1490       also have "\<dots> \<le> of_nat (Suc n) * u x"
```
```  1491         by (subst (1 2) mult_commute) (auto intro!: pextreal_mult_cancel)
```
```  1492       finally show "1 \<le> of_nat (Suc n) * u x" .
```
```  1493     qed
```
```  1494     also have "\<dots> = \<mu> ?A"
```
```  1495     proof (safe intro!: arg_cong[where f="\<mu>"])
```
```  1496       fix x assume "u x \<noteq> 0" and [simp, intro]: "x \<in> space M"
```
```  1497       show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
```
```  1498       proof (cases "u x")
```
```  1499         case (preal r)
```
```  1500         obtain j where "1 / r \<le> of_nat j" using ex_le_of_nat ..
```
```  1501         hence "1 / r * r \<le> of_nat j * r" using preal unfolding mult_le_cancel_right by auto
```
```  1502         hence "1 \<le> of_nat j * r" using preal `u x \<noteq> 0` by auto
```
```  1503         thus ?thesis using `u x \<noteq> 0` preal by (auto simp: real_of_nat_def[symmetric])
```
```  1504       qed auto
```
```  1505     qed
```
```  1506     finally show "\<mu> ?A = 0" by simp
```
```  1507   qed
```
```  1508 qed
```
```  1509
```
```  1510 lemma (in measure_space) positive_integral_restricted:
```
```  1511   assumes "A \<in> sets M"
```
```  1512   shows "measure_space.positive_integral (restricted_space A) \<mu> f = (\<integral>\<^isup>+ x. f x * indicator A x)"
```
```  1513     (is "measure_space.positive_integral ?R \<mu> f = positive_integral ?f")
```
```  1514 proof -
```
```  1515   have msR: "measure_space ?R \<mu>" by (rule restricted_measure_space[OF `A \<in> sets M`])
```
```  1516   then interpret R: measure_space ?R \<mu> .
```
```  1517   have saR: "sigma_algebra ?R" by fact
```
```  1518   have *: "R.positive_integral f = R.positive_integral ?f"
```
```  1519     by (intro R.positive_integral_cong) auto
```
```  1520   show ?thesis
```
```  1521     unfolding * R.positive_integral_def positive_integral_def
```
```  1522     unfolding simple_function_restricted[OF `A \<in> sets M`]
```
```  1523     apply (simp add: SUPR_def)
```
```  1524     apply (rule arg_cong[where f=Sup])
```
```  1525   proof (auto simp add: image_iff simple_integral_restricted[OF `A \<in> sets M`])
```
```  1526     fix g assume "simple_function (\<lambda>x. g x * indicator A x)"
```
```  1527       "g \<le> f"
```
```  1528     then show "\<exists>x. simple_function x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and>
```
```  1529       (\<integral>\<^isup>Sx. g x * indicator A x) = simple_integral x"
```
```  1530       apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
```
```  1531       by (auto simp: indicator_def le_fun_def)
```
```  1532   next
```
```  1533     fix g assume g: "simple_function g" "g \<le> (\<lambda>x. f x * indicator A x)"
```
```  1534     then have *: "(\<lambda>x. g x * indicator A x) = g"
```
```  1535       "\<And>x. g x * indicator A x = g x"
```
```  1536       "\<And>x. g x \<le> f x"
```
```  1537       by (auto simp: le_fun_def fun_eq_iff indicator_def split: split_if_asm)
```
```  1538     from g show "\<exists>x. simple_function (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and>
```
```  1539       simple_integral g = simple_integral (\<lambda>xa. x xa * indicator A xa)"
```
```  1540       using `A \<in> sets M`[THEN sets_into_space]
```
```  1541       apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
```
```  1542       by (fastsimp simp: le_fun_def *)
```
```  1543   qed
```
```  1544 qed
```
```  1545
```
```  1546 lemma (in measure_space) positive_integral_subalgebra:
```
```  1547   assumes borel: "f \<in> borel_measurable N"
```
```  1548   and N: "sets N \<subseteq> sets M" "space N = space M" and sa: "sigma_algebra N"
```
```  1549   shows "measure_space.positive_integral N \<mu> f = positive_integral f"
```
```  1550 proof -
```
```  1551   interpret N: measure_space N \<mu> using measure_space_subalgebra[OF sa N] .
```
```  1552   from N.borel_measurable_implies_simple_function_sequence[OF borel]
```
```  1553   obtain fs where Nsf: "\<And>i. N.simple_function (fs i)" and "fs \<up> f" by blast
```
```  1554   then have sf: "\<And>i. simple_function (fs i)"
```
```  1555     using simple_function_subalgebra[OF _ N sa] by blast
```
```  1556   from positive_integral_isoton_simple[OF `fs \<up> f` sf] N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
```
```  1557   show ?thesis unfolding isoton_def simple_integral_def N.simple_integral_def `space N = space M` by simp
```
```  1558 qed
```
```  1559
```
```  1560 section "Lebesgue Integral"
```
```  1561
```
```  1562 definition (in measure_space) integrable where
```
```  1563   "integrable f \<longleftrightarrow> f \<in> borel_measurable M \<and>
```
```  1564     (\<integral>\<^isup>+ x. Real (f x)) \<noteq> \<omega> \<and>
```
```  1565     (\<integral>\<^isup>+ x. Real (- f x)) \<noteq> \<omega>"
```
```  1566
```
```  1567 lemma (in measure_space) integrableD[dest]:
```
```  1568   assumes "integrable f"
```
```  1569   shows "f \<in> borel_measurable M"
```
```  1570   "(\<integral>\<^isup>+ x. Real (f x)) \<noteq> \<omega>"
```
```  1571   "(\<integral>\<^isup>+ x. Real (- f x)) \<noteq> \<omega>"
```
```  1572   using assms unfolding integrable_def by auto
```
```  1573
```
```  1574 definition (in measure_space) integral (binder "\<integral> " 10) where
```
```  1575   "integral f = real ((\<integral>\<^isup>+ x. Real (f x))) - real ((\<integral>\<^isup>+ x. Real (- f x)))"
```
```  1576
```
```  1577 lemma (in measure_space) integral_cong:
```
```  1578   assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
```
```  1579   shows "integral f = integral g"
```
```  1580   using assms by (simp cong: positive_integral_cong add: integral_def)
```
```  1581
```
```  1582 lemma (in measure_space) integral_cong_measure:
```
```  1583   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A"
```
```  1584   shows "measure_space.integral M \<nu> f = integral f"
```
```  1585 proof -
```
```  1586   interpret v: measure_space M \<nu>
```
```  1587     by (rule measure_space_cong) fact
```
```  1588   show ?thesis
```
```  1589     unfolding integral_def v.integral_def
```
```  1590     by (simp add: positive_integral_cong_measure[OF assms])
```
```  1591 qed
```
```  1592
```
```  1593 lemma (in measure_space) integral_cong_AE:
```
```  1594   assumes cong: "AE x. f x = g x"
```
```  1595   shows "integral f = integral g"
```
```  1596 proof -
```
```  1597   have "AE x. Real (f x) = Real (g x)"
```
```  1598     using cong by (rule AE_mp) simp
```
```  1599   moreover have "AE x. Real (- f x) = Real (- g x)"
```
```  1600     using cong by (rule AE_mp) simp
```
```  1601   ultimately show ?thesis
```
```  1602     by (simp cong: positive_integral_cong_AE add: integral_def)
```
```  1603 qed
```
```  1604
```
```  1605 lemma (in measure_space) integrable_cong:
```
```  1606   "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable f \<longleftrightarrow> integrable g"
```
```  1607   by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
```
```  1608
```
```  1609 lemma (in measure_space) integral_eq_positive_integral:
```
```  1610   assumes "\<And>x. 0 \<le> f x"
```
```  1611   shows "integral f = real ((\<integral>\<^isup>+ x. Real (f x)))"
```
```  1612 proof -
```
```  1613   have "\<And>x. Real (- f x) = 0" using assms by simp
```
```  1614   thus ?thesis by (simp del: Real_eq_0 add: integral_def)
```
```  1615 qed
```
```  1616
```
```  1617 lemma (in measure_space) integral_vimage_inv:
```
```  1618   assumes f: "bij_betw f S (space M)"
```
```  1619   shows "measure_space.integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>x. g x) = (\<integral>x. g (the_inv_into S f x))"
```
```  1620 proof -
```
```  1621   interpret v: measure_space "vimage_algebra S f" "\<lambda>A. \<mu> (f ` A)"
```
```  1622     using f by (rule measure_space_isomorphic)
```
```  1623   have "\<And>x. x \<in> space (vimage_algebra S f) \<Longrightarrow> the_inv_into S f (f x) = x"
```
```  1624     using f[unfolded bij_betw_def] by (simp add: the_inv_into_f_f)
```
```  1625   then have *: "v.positive_integral (\<lambda>x. Real (g (the_inv_into S f (f x)))) = v.positive_integral (\<lambda>x. Real (g x))"
```
```  1626      "v.positive_integral (\<lambda>x. Real (- g (the_inv_into S f (f x)))) = v.positive_integral (\<lambda>x. Real (- g x))"
```
```  1627     by (auto intro!: v.positive_integral_cong)
```
```  1628   show ?thesis
```
```  1629     unfolding integral_def v.integral_def
```
```  1630     unfolding positive_integral_vimage[OF f]
```
```  1631     by (simp add: *)
```
```  1632 qed
```
```  1633
```
```  1634 lemma (in measure_space) integral_minus[intro, simp]:
```
```  1635   assumes "integrable f"
```
```  1636   shows "integrable (\<lambda>x. - f x)" "(\<integral>x. - f x) = - integral f"
```
```  1637   using assms by (auto simp: integrable_def integral_def)
```
```  1638
```
```  1639 lemma (in measure_space) integral_of_positive_diff:
```
```  1640   assumes integrable: "integrable u" "integrable v"
```
```  1641   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"
```
```  1642   shows "integrable f" and "integral f = integral u - integral v"
```
```  1643 proof -
```
```  1644   let ?PI = positive_integral
```
```  1645   let "?f x" = "Real (f x)"
```
```  1646   let "?mf x" = "Real (- f x)"
```
```  1647   let "?u x" = "Real (u x)"
```
```  1648   let "?v x" = "Real (v x)"
```
```  1649
```
```  1650   from borel_measurable_diff[of u v] integrable
```
```  1651   have f_borel: "?f \<in> borel_measurable M" and
```
```  1652     mf_borel: "?mf \<in> borel_measurable M" and
```
```  1653     v_borel: "?v \<in> borel_measurable M" and
```
```  1654     u_borel: "?u \<in> borel_measurable M" and
```
```  1655     "f \<in> borel_measurable M"
```
```  1656     by (auto simp: f_def[symmetric] integrable_def)
```
```  1657
```
```  1658   have "?PI (\<lambda>x. Real (u x - v x)) \<le> ?PI ?u"
```
```  1659     using pos by (auto intro!: positive_integral_mono)
```
```  1660   moreover have "?PI (\<lambda>x. Real (v x - u x)) \<le> ?PI ?v"
```
```  1661     using pos by (auto intro!: positive_integral_mono)
```
```  1662   ultimately show f: "integrable f"
```
```  1663     using `integrable u` `integrable v` `f \<in> borel_measurable M`
```
```  1664     by (auto simp: integrable_def f_def)
```
```  1665   hence mf: "integrable (\<lambda>x. - f x)" ..
```
```  1666
```
```  1667   have *: "\<And>x. Real (- v x) = 0" "\<And>x. Real (- u x) = 0"
```
```  1668     using pos by auto
```
```  1669
```
```  1670   have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
```
```  1671     unfolding f_def using pos by simp
```
```  1672   hence "?PI (\<lambda>x. ?u x + ?mf x) = ?PI (\<lambda>x. ?v x + ?f x)" by simp
```
```  1673   hence "real (?PI ?u + ?PI ?mf) = real (?PI ?v + ?PI ?f)"
```
```  1674     using positive_integral_add[OF u_borel mf_borel]
```
```  1675     using positive_integral_add[OF v_borel f_borel]
```
```  1676     by auto
```
```  1677   then show "integral f = integral u - integral v"
```
```  1678     using f mf `integrable u` `integrable v`
```
```  1679     unfolding integral_def integrable_def *
```
```  1680     by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?v", cases "?PI ?u")
```
```  1681        (auto simp add: field_simps)
```
```  1682 qed
```
```  1683
```
```  1684 lemma (in measure_space) integral_linear:
```
```  1685   assumes "integrable f" "integrable g" and "0 \<le> a"
```
```  1686   shows "integrable (\<lambda>t. a * f t + g t)"
```
```  1687   and "(\<integral> t. a * f t + g t) = a * integral f + integral g"
```
```  1688 proof -
```
```  1689   let ?PI = positive_integral
```
```  1690   let "?f x" = "Real (f x)"
```
```  1691   let "?g x" = "Real (g x)"
```
```  1692   let "?mf x" = "Real (- f x)"
```
```  1693   let "?mg x" = "Real (- g x)"
```
```  1694   let "?p t" = "max 0 (a * f t) + max 0 (g t)"
```
```  1695   let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
```
```  1696
```
```  1697   have pos: "?f \<in> borel_measurable M" "?g \<in> borel_measurable M"
```
```  1698     and neg: "?mf \<in> borel_measurable M" "?mg \<in> borel_measurable M"
```
```  1699     and p: "?p \<in> borel_measurable M"
```
```  1700     and n: "?n \<in> borel_measurable M"
```
```  1701     using assms by (simp_all add: integrable_def)
```
```  1702
```
```  1703   have *: "\<And>x. Real (?p x) = Real a * ?f x + ?g x"
```
```  1704           "\<And>x. Real (?n x) = Real a * ?mf x + ?mg x"
```
```  1705           "\<And>x. Real (- ?p x) = 0"
```
```  1706           "\<And>x. Real (- ?n x) = 0"
```
```  1707     using `0 \<le> a` by (auto simp: max_def min_def zero_le_mult_iff mult_le_0_iff add_nonpos_nonpos)
```
```  1708
```
```  1709   note linear =
```
```  1710     positive_integral_linear[OF pos]
```
```  1711     positive_integral_linear[OF neg]
```
```  1712
```
```  1713   have "integrable ?p" "integrable ?n"
```
```  1714       "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
```
```  1715     using assms p n unfolding integrable_def * linear by auto
```
```  1716   note diff = integral_of_positive_diff[OF this]
```
```  1717
```
```  1718   show "integrable (\<lambda>t. a * f t + g t)" by (rule diff)
```
```  1719
```
```  1720   from assms show "(\<integral> t. a * f t + g t) = a * integral f + integral g"
```
```  1721     unfolding diff(2) unfolding integral_def * linear integrable_def
```
```  1722     by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?g", cases "?PI ?mg")
```
```  1723        (auto simp add: field_simps zero_le_mult_iff)
```
```  1724 qed
```
```  1725
```
```  1726 lemma (in measure_space) integral_add[simp, intro]:
```
```  1727   assumes "integrable f" "integrable g"
```
```  1728   shows "integrable (\<lambda>t. f t + g t)"
```
```  1729   and "(\<integral> t. f t + g t) = integral f + integral g"
```
```  1730   using assms integral_linear[where a=1] by auto
```
```  1731
```
```  1732 lemma (in measure_space) integral_zero[simp, intro]:
```
```  1733   shows "integrable (\<lambda>x. 0)"
```
```  1734   and "(\<integral> x.0) = 0"
```
```  1735   unfolding integrable_def integral_def
```
```  1736   by (auto simp add: borel_measurable_const)
```
```  1737
```
```  1738 lemma (in measure_space) integral_cmult[simp, intro]:
```
```  1739   assumes "integrable f"
```
```  1740   shows "integrable (\<lambda>t. a * f t)" (is ?P)
```
```  1741   and "(\<integral> t. a * f t) = a * integral f" (is ?I)
```
```  1742 proof -
```
```  1743   have "integrable (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t) = a * integral f"
```
```  1744   proof (cases rule: le_cases)
```
```  1745     assume "0 \<le> a" show ?thesis
```
```  1746       using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
```
```  1747       by (simp add: integral_zero)
```
```  1748   next
```
```  1749     assume "a \<le> 0" hence "0 \<le> - a" by auto
```
```  1750     have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
```
```  1751     show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
```
```  1752         integral_minus(1)[of "\<lambda>t. - a * f t"]
```
```  1753       unfolding * integral_zero by simp
```
```  1754   qed
```
```  1755   thus ?P ?I by auto
```
```  1756 qed
```
```  1757
```
```  1758 lemma (in measure_space) integral_multc:
```
```  1759   assumes "integrable f"
```
```  1760   shows "(\<integral> x. f x * c) = integral f * c"
```
```  1761   unfolding mult_commute[of _ c] integral_cmult[OF assms] ..
```
```  1762
```
```  1763 lemma (in measure_space) integral_mono_AE:
```
```  1764   assumes fg: "integrable f" "integrable g"
```
```  1765   and mono: "AE t. f t \<le> g t"
```
```  1766   shows "integral f \<le> integral g"
```
```  1767 proof -
```
```  1768   have "AE x. Real (f x) \<le> Real (g x)"
```
```  1769     using mono by (rule AE_mp) (auto intro!: AE_cong)
```
```  1770   moreover have "AE x. Real (- g x) \<le> Real (- f x)"
```
```  1771     using mono by (rule AE_mp) (auto intro!: AE_cong)
```
```  1772   ultimately show ?thesis using fg
```
```  1773     by (auto simp: integral_def integrable_def diff_minus
```
```  1774              intro!: add_mono real_of_pextreal_mono positive_integral_mono_AE)
```
```  1775 qed
```
```  1776
```
```  1777 lemma (in measure_space) integral_mono:
```
```  1778   assumes fg: "integrable f" "integrable g"
```
```  1779   and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
```
```  1780   shows "integral f \<le> integral g"
```
```  1781   apply (rule integral_mono_AE[OF fg])
```
```  1782   using mono by (rule AE_cong) auto
```
```  1783
```
```  1784 lemma (in measure_space) integral_diff[simp, intro]:
```
```  1785   assumes f: "integrable f" and g: "integrable g"
```
```  1786   shows "integrable (\<lambda>t. f t - g t)"
```
```  1787   and "(\<integral> t. f t - g t) = integral f - integral g"
```
```  1788   using integral_add[OF f integral_minus(1)[OF g]]
```
```  1789   unfolding diff_minus integral_minus(2)[OF g]
```
```  1790   by auto
```
```  1791
```
```  1792 lemma (in measure_space) integral_indicator[simp, intro]:
```
```  1793   assumes "a \<in> sets M" and "\<mu> a \<noteq> \<omega>"
```
```  1794   shows "integral (indicator a) = real (\<mu> a)" (is ?int)
```
```  1795   and "integrable (indicator a)" (is ?able)
```
```  1796 proof -
```
```  1797   have *:
```
```  1798     "\<And>A x. Real (indicator A x) = indicator A x"
```
```  1799     "\<And>A x. Real (- indicator A x) = 0" unfolding indicator_def by auto
```
```  1800   show ?int ?able
```
```  1801     using assms unfolding integral_def integrable_def
```
```  1802     by (auto simp: * positive_integral_indicator borel_measurable_indicator)
```
```  1803 qed
```
```  1804
```
```  1805 lemma (in measure_space) integral_cmul_indicator:
```
```  1806   assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<omega>"
```
```  1807   shows "integrable (\<lambda>x. c * indicator A x)" (is ?P)
```
```  1808   and "(\<integral>x. c * indicator A x) = c * real (\<mu> A)" (is ?I)
```
```  1809 proof -
```
```  1810   show ?P
```
```  1811   proof (cases "c = 0")
```
```  1812     case False with assms show ?thesis by simp
```
```  1813   qed simp
```
```  1814
```
```  1815   show ?I
```
```  1816   proof (cases "c = 0")
```
```  1817     case False with assms show ?thesis by simp
```
```  1818   qed simp
```
```  1819 qed
```
```  1820
```
```  1821 lemma (in measure_space) integral_setsum[simp, intro]:
```
```  1822   assumes "\<And>n. n \<in> S \<Longrightarrow> integrable (f n)"
```
```  1823   shows "(\<integral>x. \<Sum> i \<in> S. f i x) = (\<Sum> i \<in> S. integral (f i))" (is "?int S")
```
```  1824     and "integrable (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
```
```  1825 proof -
```
```  1826   have "?int S \<and> ?I S"
```
```  1827   proof (cases "finite S")
```
```  1828     assume "finite S"
```
```  1829     from this assms show ?thesis by (induct S) simp_all
```
```  1830   qed simp
```
```  1831   thus "?int S" and "?I S" by auto
```
```  1832 qed
```
```  1833
```
```  1834 lemma (in measure_space) integrable_abs:
```
```  1835   assumes "integrable f"
```
```  1836   shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
```
```  1837 proof -
```
```  1838   have *:
```
```  1839     "\<And>x. Real \<bar>f x\<bar> = Real (f x) + Real (- f x)"
```
```  1840     "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
```
```  1841   have abs: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" and
```
```  1842     f: "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
```
```  1843         "(\<lambda>x. Real (- f x)) \<in> borel_measurable M"
```
```  1844     using assms unfolding integrable_def by auto
```
```  1845   from abs assms show ?thesis unfolding integrable_def *
```
```  1846     using positive_integral_linear[OF f, of 1] by simp
```
```  1847 qed
```
```  1848
```
```  1849 lemma (in measure_space) integral_subalgebra:
```
```  1850   assumes borel: "f \<in> borel_measurable N"
```
```  1851   and N: "sets N \<subseteq> sets M" "space N = space M" and sa: "sigma_algebra N"
```
```  1852   shows "measure_space.integrable N \<mu> f \<longleftrightarrow> integrable f" (is ?P)
```
```  1853     and "measure_space.integral N \<mu> f = integral f" (is ?I)
```
```  1854 proof -
```
```  1855   interpret N: measure_space N \<mu> using measure_space_subalgebra[OF sa N] .
```
```  1856   have "(\<lambda>x. Real (f x)) \<in> borel_measurable N" "(\<lambda>x. Real (- f x)) \<in> borel_measurable N"
```
```  1857     using borel by auto
```
```  1858   note * = this[THEN positive_integral_subalgebra[OF _ N sa]]
```
```  1859   have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
```
```  1860     using assms unfolding measurable_def by auto
```
```  1861   then show ?P ?I unfolding integrable_def N.integrable_def integral_def N.integral_def
```
```  1862     unfolding * by auto
```
```  1863 qed
```
```  1864
```
```  1865 lemma (in measure_space) integrable_bound:
```
```  1866   assumes "integrable f"
```
```  1867   and f: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1868     "\<And>x. x \<in> space M \<Longrightarrow> \<bar>g x\<bar> \<le> f x"
```
```  1869   assumes borel: "g \<in> borel_measurable M"
```
```  1870   shows "integrable g"
```
```  1871 proof -
```
```  1872   have "(\<integral>\<^isup>+ x. Real (g x)) \<le> (\<integral>\<^isup>+ x. Real \<bar>g x\<bar>)"
```
```  1873     by (auto intro!: positive_integral_mono)
```
```  1874   also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x))"
```
```  1875     using f by (auto intro!: positive_integral_mono)
```
```  1876   also have "\<dots> < \<omega>"
```
```  1877     using `integrable f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
```
```  1878   finally have pos: "(\<integral>\<^isup>+ x. Real (g x)) < \<omega>" .
```
```  1879
```
```  1880   have "(\<integral>\<^isup>+ x. Real (- g x)) \<le> (\<integral>\<^isup>+ x. Real (\<bar>g x\<bar>))"
```
```  1881     by (auto intro!: positive_integral_mono)
```
```  1882   also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x))"
```
```  1883     using f by (auto intro!: positive_integral_mono)
```
```  1884   also have "\<dots> < \<omega>"
```
```  1885     using `integrable f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
```
```  1886   finally have neg: "(\<integral>\<^isup>+ x. Real (- g x)) < \<omega>" .
```
```  1887
```
```  1888   from neg pos borel show ?thesis
```
```  1889     unfolding integrable_def by auto
```
```  1890 qed
```
```  1891
```
```  1892 lemma (in measure_space) integrable_abs_iff:
```
```  1893   "f \<in> borel_measurable M \<Longrightarrow> integrable (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable f"
```
```  1894   by (auto intro!: integrable_bound[where g=f] integrable_abs)
```
```  1895
```
```  1896 lemma (in measure_space) integrable_max:
```
```  1897   assumes int: "integrable f" "integrable g"
```
```  1898   shows "integrable (\<lambda> x. max (f x) (g x))"
```
```  1899 proof (rule integrable_bound)
```
```  1900   show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
```
```  1901     using int by (simp add: integrable_abs)
```
```  1902   show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
```
```  1903     using int unfolding integrable_def by auto
```
```  1904 next
```
```  1905   fix x assume "x \<in> space M"
```
```  1906   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>"
```
```  1907     by auto
```
```  1908 qed
```
```  1909
```
```  1910 lemma (in measure_space) integrable_min:
```
```  1911   assumes int: "integrable f" "integrable g"
```
```  1912   shows "integrable (\<lambda> x. min (f x) (g x))"
```
```  1913 proof (rule integrable_bound)
```
```  1914   show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
```
```  1915     using int by (simp add: integrable_abs)
```
```  1916   show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
```
```  1917     using int unfolding integrable_def by auto
```
```  1918 next
```
```  1919   fix x assume "x \<in> space M"
```
```  1920   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>"
```
```  1921     by auto
```
```  1922 qed
```
```  1923
```
```  1924 lemma (in measure_space) integral_triangle_inequality:
```
```  1925   assumes "integrable f"
```
```  1926   shows "\<bar>integral f\<bar> \<le> (\<integral>x. \<bar>f x\<bar>)"
```
```  1927 proof -
```
```  1928   have "\<bar>integral f\<bar> = max (integral f) (- integral f)" by auto
```
```  1929   also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar>)"
```
```  1930       using assms integral_minus(2)[of f, symmetric]
```
```  1931       by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
```
```  1932   finally show ?thesis .
```
```  1933 qed
```
```  1934
```
```  1935 lemma (in measure_space) integral_positive:
```
```  1936   assumes "integrable f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1937   shows "0 \<le> integral f"
```
```  1938 proof -
```
```  1939   have "0 = (\<integral>x. 0)" by (auto simp: integral_zero)
```
```  1940   also have "\<dots> \<le> integral f"
```
```  1941     using assms by (rule integral_mono[OF integral_zero(1)])
```
```  1942   finally show ?thesis .
```
```  1943 qed
```
```  1944
```
```  1945 lemma (in measure_space) integral_monotone_convergence_pos:
```
```  1946   assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
```
```  1947   and pos: "\<And>x i. 0 \<le> f i x"
```
```  1948   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
```
```  1949   and ilim: "(\<lambda>i. integral (f i)) ----> x"
```
```  1950   shows "integrable u"
```
```  1951   and "integral u = x"
```
```  1952 proof -
```
```  1953   { fix x have "0 \<le> u x"
```
```  1954       using mono pos[of 0 x] incseq_le[OF _ lim, of x 0]
```
```  1955       by (simp add: mono_def incseq_def) }
```
```  1956   note pos_u = this
```
```  1957
```
```  1958   hence [simp]: "\<And>i x. Real (- f i x) = 0" "\<And>x. Real (- u x) = 0"
```
```  1959     using pos by auto
```
```  1960
```
```  1961   have SUP_F: "\<And>x. (SUP n. Real (f n x)) = Real (u x)"
```
```  1962     using mono pos pos_u lim by (rule SUP_eq_LIMSEQ[THEN iffD2])
```
```  1963
```
```  1964   have borel_f: "\<And>i. (\<lambda>x. Real (f i x)) \<in> borel_measurable M"
```
```  1965     using i unfolding integrable_def by auto
```
```  1966   hence "(\<lambda>x. SUP i. Real (f i x)) \<in> borel_measurable M"
```
```  1967     by auto
```
```  1968   hence borel_u: "u \<in> borel_measurable M"
```
```  1969     using pos_u by (auto simp: borel_measurable_Real_eq SUP_F)
```
```  1970
```
```  1971   have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. Real (f n x)) = Real (integral (f n))"
```
```  1972     using i unfolding integral_def integrable_def by (auto simp: Real_real)
```
```  1973
```
```  1974   have pos_integral: "\<And>n. 0 \<le> integral (f n)"
```
```  1975     using pos i by (auto simp: integral_positive)
```
```  1976   hence "0 \<le> x"
```
```  1977     using LIMSEQ_le_const[OF ilim, of 0] by auto
```
```  1978
```
```  1979   have "(\<lambda>i. (\<integral>\<^isup>+ x. Real (f i x))) \<up> (\<integral>\<^isup>+ x. Real (u x))"
```
```  1980   proof (rule positive_integral_isoton)
```
```  1981     from SUP_F mono pos
```
```  1982     show "(\<lambda>i x. Real (f i x)) \<up> (\<lambda>x. Real (u x))"
```
```  1983       unfolding isoton_fun_expand by (auto simp: isoton_def mono_def)
```
```  1984   qed (rule borel_f)
```
```  1985   hence pI: "(\<integral>\<^isup>+ x. Real (u x)) =
```
```  1986       (SUP n. (\<integral>\<^isup>+ x. Real (f n x)))"
```
```  1987     unfolding isoton_def by simp
```
```  1988   also have "\<dots> = Real x" unfolding integral_eq
```
```  1989   proof (rule SUP_eq_LIMSEQ[THEN iffD2])
```
```  1990     show "mono (\<lambda>n. integral (f n))"
```
```  1991       using mono i by (auto simp: mono_def intro!: integral_mono)
```
```  1992     show "\<And>n. 0 \<le> integral (f n)" using pos_integral .
```
```  1993     show "0 \<le> x" using `0 \<le> x` .
```
```  1994     show "(\<lambda>n. integral (f n)) ----> x" using ilim .
```
```  1995   qed
```
```  1996   finally show  "integrable u" "integral u = x" using borel_u `0 \<le> x`
```
```  1997     unfolding integrable_def integral_def by auto
```
```  1998 qed
```
```  1999
```
```  2000 lemma (in measure_space) integral_monotone_convergence:
```
```  2001   assumes f: "\<And>i. integrable (f i)" and "mono f"
```
```  2002   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
```
```  2003   and ilim: "(\<lambda>i. integral (f i)) ----> x"
```
```  2004   shows "integrable u"
```
```  2005   and "integral u = x"
```
```  2006 proof -
```
```  2007   have 1: "\<And>i. integrable (\<lambda>x. f i x - f 0 x)"
```
```  2008       using f by (auto intro!: integral_diff)
```
```  2009   have 2: "\<And>x. mono (\<lambda>n. f n x - f 0 x)" using `mono f`
```
```  2010       unfolding mono_def le_fun_def by auto
```
```  2011   have 3: "\<And>x n. 0 \<le> f n x - f 0 x" using `mono f`
```
```  2012       unfolding mono_def le_fun_def by (auto simp: field_simps)
```
```  2013   have 4: "\<And>x. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
```
```  2014     using lim by (auto intro!: LIMSEQ_diff)
```
```  2015   have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x)) ----> x - integral (f 0)"
```
```  2016     using f ilim by (auto intro!: LIMSEQ_diff simp: integral_diff)
```
```  2017   note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5]
```
```  2018   have "integrable (\<lambda>x. (u x - f 0 x) + f 0 x)"
```
```  2019     using diff(1) f by (rule integral_add(1))
```
```  2020   with diff(2) f show "integrable u" "integral u = x"
```
```  2021     by (auto simp: integral_diff)
```
```  2022 qed
```
```  2023
```
```  2024 lemma (in measure_space) integral_0_iff:
```
```  2025   assumes "integrable f"
```
```  2026   shows "(\<integral>x. \<bar>f x\<bar>) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
```
```  2027 proof -
```
```  2028   have *: "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
```
```  2029   have "integrable (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
```
```  2030   hence "(\<lambda>x. Real (\<bar>f x\<bar>)) \<in> borel_measurable M"
```
```  2031     "(\<integral>\<^isup>+ x. Real \<bar>f x\<bar>) \<noteq> \<omega>" unfolding integrable_def by auto
```
```  2032   from positive_integral_0_iff[OF this(1)] this(2)
```
```  2033   show ?thesis unfolding integral_def *
```
```  2034     by (simp add: real_of_pextreal_eq_0)
```
```  2035 qed
```
```  2036
```
```  2037 lemma (in measure_space) positive_integral_omega:
```
```  2038   assumes "f \<in> borel_measurable M"
```
```  2039   and "positive_integral f \<noteq> \<omega>"
```
```  2040   shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
```
```  2041 proof -
```
```  2042   have "\<omega> * \<mu> (f -` {\<omega>} \<inter> space M) = (\<integral>\<^isup>+ x. \<omega> * indicator (f -` {\<omega>} \<inter> space M) x)"
```
```  2043     using positive_integral_cmult_indicator[OF borel_measurable_vimage, OF assms(1), of \<omega> \<omega>] by simp
```
```  2044   also have "\<dots> \<le> positive_integral f"
```
```  2045     by (auto intro!: positive_integral_mono simp: indicator_def)
```
```  2046   finally show ?thesis
```
```  2047     using assms(2) by (cases ?thesis) auto
```
```  2048 qed
```
```  2049
```
```  2050 lemma (in measure_space) positive_integral_omega_AE:
```
```  2051   assumes "f \<in> borel_measurable M" "positive_integral f \<noteq> \<omega>" shows "AE x. f x \<noteq> \<omega>"
```
```  2052 proof (rule AE_I)
```
```  2053   show "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
```
```  2054     by (rule positive_integral_omega[OF assms])
```
```  2055   show "f -` {\<omega>} \<inter> space M \<in> sets M"
```
```  2056     using assms by (auto intro: borel_measurable_vimage)
```
```  2057 qed auto
```
```  2058
```
```  2059 lemma (in measure_space) simple_integral_omega:
```
```  2060   assumes "simple_function f"
```
```  2061   and "simple_integral f \<noteq> \<omega>"
```
```  2062   shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
```
```  2063 proof (rule positive_integral_omega)
```
```  2064   show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
```
```  2065   show "positive_integral f \<noteq> \<omega>"
```
```  2066     using assms by (simp add: positive_integral_eq_simple_integral)
```
```  2067 qed
```
```  2068
```
```  2069 lemma (in measure_space) integral_real:
```
```  2070   fixes f :: "'a \<Rightarrow> pextreal"
```
```  2071   assumes "AE x. f x \<noteq> \<omega>"
```
```  2072   shows "(\<integral>x. real (f x)) = real (positive_integral f)" (is ?plus)
```
```  2073     and "(\<integral>x. - real (f x)) = - real (positive_integral f)" (is ?minus)
```
```  2074 proof -
```
```  2075   have "(\<integral>\<^isup>+ x. Real (real (f x))) = positive_integral f"
```
```  2076     apply (rule positive_integral_cong_AE)
```
```  2077     apply (rule AE_mp[OF assms(1)])
```
```  2078     by (auto intro!: AE_cong simp: Real_real)
```
```  2079   moreover
```
```  2080   have "(\<integral>\<^isup>+ x. Real (- real (f x))) = (\<integral>\<^isup>+ x. 0)"
```
```  2081     by (intro positive_integral_cong) auto
```
```  2082   ultimately show ?plus ?minus
```
```  2083     by (auto simp: integral_def integrable_def)
```
```  2084 qed
```
```  2085
```
```  2086 lemma (in measure_space) integral_dominated_convergence:
```
```  2087   assumes u: "\<And>i. integrable (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
```
```  2088   and w: "integrable w" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> w x"
```
```  2089   and u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  2090   shows "integrable u'"
```
```  2091   and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar>)) ----> 0" (is "?lim_diff")
```
```  2092   and "(\<lambda>i. integral (u i)) ----> integral u'" (is ?lim)
```
```  2093 proof -
```
```  2094   { fix x j assume x: "x \<in> space M"
```
```  2095     from u'[OF x] have "(\<lambda>i. \<bar>u i x\<bar>) ----> \<bar>u' x\<bar>" by (rule LIMSEQ_imp_rabs)
```
```  2096     from LIMSEQ_le_const2[OF this]
```
```  2097     have "\<bar>u' x\<bar> \<le> w x" using bound[OF x] by auto }
```
```  2098   note u'_bound = this
```
```  2099
```
```  2100   from u[unfolded integrable_def]
```
```  2101   have u'_borel: "u' \<in> borel_measurable M"
```
```  2102     using u' by (blast intro: borel_measurable_LIMSEQ[of u])
```
```  2103
```
```  2104   show "integrable u'"
```
```  2105   proof (rule integrable_bound)
```
```  2106     show "integrable w" by fact
```
```  2107     show "u' \<in> borel_measurable M" by fact
```
```  2108   next
```
```  2109     fix x assume x: "x \<in> space M"
```
```  2110     thus "0 \<le> w x" by fact
```
```  2111     show "\<bar>u' x\<bar> \<le> w x" using u'_bound[OF x] .
```
```  2112   qed
```
```  2113
```
```  2114   let "?diff n x" = "2 * w x - \<bar>u n x - u' x\<bar>"
```
```  2115   have diff: "\<And>n. integrable (\<lambda>x. \<bar>u n x - u' x\<bar>)"
```
```  2116     using w u `integrable u'`
```
```  2117     by (auto intro!: integral_add integral_diff integral_cmult integrable_abs)
```
```  2118
```
```  2119   { fix j x assume x: "x \<in> space M"
```
```  2120     have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
```
```  2121     also have "\<dots> \<le> w x + w x"
```
```  2122       by (rule add_mono[OF bound[OF x] u'_bound[OF x]])
```
```  2123     finally have "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp }
```
```  2124   note diff_less_2w = this
```
```  2125
```
```  2126   have PI_diff: "\<And>m n. (\<integral>\<^isup>+ x. Real (?diff (m + n) x)) =
```
```  2127     (\<integral>\<^isup>+ x. Real (2 * w x)) - (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)"
```
```  2128     using diff w diff_less_2w
```
```  2129     by (subst positive_integral_diff[symmetric])
```
```  2130        (auto simp: integrable_def intro!: positive_integral_cong)
```
```  2131
```
```  2132   have "integrable (\<lambda>x. 2 * w x)"
```
```  2133     using w by (auto intro: integral_cmult)
```
```  2134   hence I2w_fin: "(\<integral>\<^isup>+ x. Real (2 * w x)) \<noteq> \<omega>" and
```
```  2135     borel_2w: "(\<lambda>x. Real (2 * w x)) \<in> borel_measurable M"
```
```  2136     unfolding integrable_def by auto
```
```  2137
```
```  2138   have "(INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)) = 0" (is "?lim_SUP = 0")
```
```  2139   proof cases
```
```  2140     assume eq_0: "(\<integral>\<^isup>+ x. Real (2 * w x)) = 0"
```
```  2141     have "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar>) \<le> (\<integral>\<^isup>+ x. Real (2 * w x))"
```
```  2142     proof (rule positive_integral_mono)
```
```  2143       fix i x assume "x \<in> space M" from diff_less_2w[OF this, of i]
```
```  2144       show "Real \<bar>u i x - u' x\<bar> \<le> Real (2 * w x)" by auto
```
```  2145     qed
```
```  2146     hence "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar>) = 0" using eq_0 by auto
```
```  2147     thus ?thesis by simp
```
```  2148   next
```
```  2149     assume neq_0: "(\<integral>\<^isup>+ x. Real (2 * w x)) \<noteq> 0"
```
```  2150     have "(\<integral>\<^isup>+ x. Real (2 * w x)) = (\<integral>\<^isup>+ x. SUP n. INF m. Real (?diff (m + n) x))"
```
```  2151     proof (rule positive_integral_cong, subst add_commute)
```
```  2152       fix x assume x: "x \<in> space M"
```
```  2153       show "Real (2 * w x) = (SUP n. INF m. Real (?diff (n + m) x))"
```
```  2154       proof (rule LIMSEQ_imp_lim_INF[symmetric])
```
```  2155         fix j show "0 \<le> ?diff j x" using diff_less_2w[OF x, of j] by simp
```
```  2156       next
```
```  2157         have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
```
```  2158           using u'[OF x] by (safe intro!: LIMSEQ_diff LIMSEQ_const LIMSEQ_imp_rabs)
```
```  2159         thus "(\<lambda>i. ?diff i x) ----> 2 * w x" by simp
```
```  2160       qed
```
```  2161     qed
```
```  2162     also have "\<dots> \<le> (SUP n. INF m. (\<integral>\<^isup>+ x. Real (?diff (m + n) x)))"
```
```  2163       using u'_borel w u unfolding integrable_def
```
```  2164       by (auto intro!: positive_integral_lim_INF)
```
```  2165     also have "\<dots> = (\<integral>\<^isup>+ x. Real (2 * w x)) -
```
```  2166         (INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>))"
```
```  2167       unfolding PI_diff pextreal_INF_minus[OF I2w_fin] pextreal_SUP_minus ..
```
```  2168     finally show ?thesis using neq_0 I2w_fin by (rule pextreal_le_minus_imp_0)
```
```  2169   qed
```
```  2170
```
```  2171   have [simp]: "\<And>n m x. Real (- \<bar>u (m + n) x - u' x\<bar>) = 0" by auto
```
```  2172
```
```  2173   have [simp]: "\<And>n m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>) =
```
```  2174     Real ((\<integral>x. \<bar>u (n + m) x - u' x\<bar>))"
```
```  2175     using diff by (subst add_commute) (simp add: integral_def integrable_def Real_real)
```
```  2176
```
```  2177   have "(SUP n. INF m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)) \<le> ?lim_SUP"
```
```  2178     (is "?lim_INF \<le> _") by (subst (1 2) add_commute) (rule lim_INF_le_lim_SUP)
```
```  2179   hence "?lim_INF = Real 0" "?lim_SUP = Real 0" using `?lim_SUP = 0` by auto
```
```  2180   thus ?lim_diff using diff by (auto intro!: integral_positive lim_INF_eq_lim_SUP)
```
```  2181
```
```  2182   show ?lim
```
```  2183   proof (rule LIMSEQ_I)
```
```  2184     fix r :: real assume "0 < r"
```
```  2185     from LIMSEQ_D[OF `?lim_diff` this]
```
```  2186     obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar>) < r"
```
```  2187       using diff by (auto simp: integral_positive)
```
```  2188
```
```  2189     show "\<exists>N. \<forall>n\<ge>N. norm (integral (u n) - integral u') < r"
```
```  2190     proof (safe intro!: exI[of _ N])
```
```  2191       fix n assume "N \<le> n"
```
```  2192       have "\<bar>integral (u n) - integral u'\<bar> = \<bar>(\<integral>x. u n x - u' x)\<bar>"
```
```  2193         using u `integrable u'` by (auto simp: integral_diff)
```
```  2194       also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar>)" using u `integrable u'`
```
```  2195         by (rule_tac integral_triangle_inequality) (auto intro!: integral_diff)
```
```  2196       also note N[OF `N \<le> n`]
```
```  2197       finally show "norm (integral (u n) - integral u') < r" by simp
```
```  2198     qed
```
```  2199   qed
```
```  2200 qed
```
```  2201
```
```  2202 lemma (in measure_space) integral_sums:
```
```  2203   assumes borel: "\<And>i. integrable (f i)"
```
```  2204   and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
```
```  2205   and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar>))"
```
```  2206   shows "integrable (\<lambda>x. (\<Sum>i. f i x))" (is "integrable ?S")
```
```  2207   and "(\<lambda>i. integral (f i)) sums (\<integral>x. (\<Sum>i. f i x))" (is ?integral)
```
```  2208 proof -
```
```  2209   have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
```
```  2210     using summable unfolding summable_def by auto
```
```  2211   from bchoice[OF this]
```
```  2212   obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
```
```  2213
```
```  2214   let "?w y" = "if y \<in> space M then w y else 0"
```
```  2215
```
```  2216   obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar>)) sums x"
```
```  2217     using sums unfolding summable_def ..
```
```  2218
```
```  2219   have 1: "\<And>n. integrable (\<lambda>x. \<Sum>i = 0..<n. f i x)"
```
```  2220     using borel by (auto intro!: integral_setsum)
```
```  2221
```
```  2222   { fix j x assume [simp]: "x \<in> space M"
```
```  2223     have "\<bar>\<Sum>i = 0..< j. f i x\<bar> \<le> (\<Sum>i = 0..< j. \<bar>f i x\<bar>)" by (rule setsum_abs)
```
```  2224     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
```
```  2225     finally have "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp }
```
```  2226   note 2 = this
```
```  2227
```
```  2228   have 3: "integrable ?w"
```
```  2229   proof (rule integral_monotone_convergence(1))
```
```  2230     let "?F n y" = "(\<Sum>i = 0..<n. \<bar>f i y\<bar>)"
```
```  2231     let "?w' n y" = "if y \<in> space M then ?F n y else 0"
```
```  2232     have "\<And>n. integrable (?F n)"
```
```  2233       using borel by (auto intro!: integral_setsum integrable_abs)
```
```  2234     thus "\<And>n. integrable (?w' n)" by (simp cong: integrable_cong)
```
```  2235     show "mono ?w'"
```
```  2236       by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
```
```  2237     { fix x show "(\<lambda>n. ?w' n x) ----> ?w x"
```
```  2238         using w by (cases "x \<in> space M") (simp_all add: LIMSEQ_const sums_def) }
```
```  2239     have *: "\<And>n. integral (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar>))"
```
```  2240       using borel by (simp add: integral_setsum integrable_abs cong: integral_cong)
```
```  2241     from abs_sum
```
```  2242     show "(\<lambda>i. integral (?w' i)) ----> x" unfolding * sums_def .
```
```  2243   qed
```
```  2244
```
```  2245   have 4: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> ?w x" using 2[of _ 0] by simp
```
```  2246
```
```  2247   from summable[THEN summable_rabs_cancel]
```
```  2248   have 5: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>n. \<Sum>i = 0..<n. f i x) ----> (\<Sum>i. f i x)"
```
```  2249     by (auto intro: summable_sumr_LIMSEQ_suminf)
```
```  2250
```
```  2251   note int = integral_dominated_convergence(1,3)[OF 1 2 3 4 5]
```
```  2252
```
```  2253   from int show "integrable ?S" by simp
```
```  2254
```
```  2255   show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF borel]
```
```  2256     using int(2) by simp
```
```  2257 qed
```
```  2258
```
```  2259 section "Lebesgue integration on countable spaces"
```
```  2260
```
```  2261 lemma (in measure_space) integral_on_countable:
```
```  2262   assumes f: "f \<in> borel_measurable M"
```
```  2263   and bij: "bij_betw enum S (f ` space M)"
```
```  2264   and enum_zero: "enum ` (-S) \<subseteq> {0}"
```
```  2265   and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
```
```  2266   and abs_summable: "summable (\<lambda>r. \<bar>enum r * real (\<mu> (f -` {enum r} \<inter> space M))\<bar>)"
```
```  2267   shows "integrable f"
```
```  2268   and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral f" (is ?sums)
```
```  2269 proof -
```
```  2270   let "?A r" = "f -` {enum r} \<inter> space M"
```
```  2271   let "?F r x" = "enum r * indicator (?A r) x"
```
```  2272   have enum_eq: "\<And>r. enum r * real (\<mu> (?A r)) = integral (?F r)"
```
```  2273     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2274
```
```  2275   { fix x assume "x \<in> space M"
```
```  2276     hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
```
```  2277     then obtain i where "i\<in>S" "enum i = f x" by auto
```
```  2278     have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
```
```  2279     proof cases
```
```  2280       fix j assume "j = i"
```
```  2281       thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
```
```  2282     next
```
```  2283       fix j assume "j \<noteq> i"
```
```  2284       show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
```
```  2285         by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
```
```  2286     qed
```
```  2287     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
```
```  2288     have "(\<lambda>i. ?F i x) sums f x"
```
```  2289          "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
```
```  2290       by (auto intro!: sums_single simp: F F_abs) }
```
```  2291   note F_sums_f = this(1) and F_abs_sums_f = this(2)
```
```  2292
```
```  2293   have int_f: "integral f = (\<integral>x. \<Sum>r. ?F r x)" "integrable f = integrable (\<lambda>x. \<Sum>r. ?F r x)"
```
```  2294     using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
```
```  2295
```
```  2296   { fix r
```
```  2297     have "(\<integral>x. \<bar>?F r x\<bar>) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x)"
```
```  2298       by (auto simp: indicator_def intro!: integral_cong)
```
```  2299     also have "\<dots> = \<bar>enum r\<bar> * real (\<mu> (?A r))"
```
```  2300       using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2301     finally have "(\<integral>x. \<bar>?F r x\<bar>) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
```
```  2302       by (simp add: abs_mult_pos real_pextreal_pos) }
```
```  2303   note int_abs_F = this
```
```  2304
```
```  2305   have 1: "\<And>i. integrable (\<lambda>x. ?F i x)"
```
```  2306     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2307
```
```  2308   have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
```
```  2309     using F_abs_sums_f unfolding sums_iff by auto
```
```  2310
```
```  2311   from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
```
```  2312   show ?sums unfolding enum_eq int_f by simp
```
```  2313
```
```  2314   from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
```
```  2315   show "integrable f" unfolding int_f by simp
```
```  2316 qed
```
```  2317
```
```  2318 section "Lebesgue integration on finite space"
```
```  2319
```
```  2320 lemma (in measure_space) integral_on_finite:
```
```  2321   assumes f: "f \<in> borel_measurable M" and finite: "finite (f`space M)"
```
```  2322   and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
```
```  2323   shows "integrable f"
```
```  2324   and "(\<integral>x. f x) =
```
```  2325     (\<Sum> r \<in> f`space M. r * real (\<mu> (f -` {r} \<inter> space M)))" (is "?integral")
```
```  2326 proof -
```
```  2327   let "?A r" = "f -` {r} \<inter> space M"
```
```  2328   let "?S x" = "\<Sum>r\<in>f`space M. r * indicator (?A r) x"
```
```  2329
```
```  2330   { fix x assume "x \<in> space M"
```
```  2331     have "f x = (\<Sum>r\<in>f`space M. if x \<in> ?A r then r else 0)"
```
```  2332       using finite `x \<in> space M` by (simp add: setsum_cases)
```
```  2333     also have "\<dots> = ?S x"
```
```  2334       by (auto intro!: setsum_cong)
```
```  2335     finally have "f x = ?S x" . }
```
```  2336   note f_eq = this
```
```  2337
```
```  2338   have f_eq_S: "integrable f \<longleftrightarrow> integrable ?S" "integral f = integral ?S"
```
```  2339     by (auto intro!: integrable_cong integral_cong simp only: f_eq)
```
```  2340
```
```  2341   show "integrable f" ?integral using fin f f_eq_S
```
```  2342     by (simp_all add: integral_cmul_indicator borel_measurable_vimage)
```
```  2343 qed
```
```  2344
```
```  2345 lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function f"
```
```  2346   unfolding simple_function_def using finite_space by auto
```
```  2347
```
```  2348 lemma (in finite_measure_space) borel_measurable_finite[intro, simp]: "f \<in> borel_measurable M"
```
```  2349   by (auto intro: borel_measurable_simple_function)
```
```  2350
```
```  2351 lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
```
```  2352   "positive_integral f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
```
```  2353 proof -
```
```  2354   have *: "positive_integral f = (\<integral>\<^isup>+ x. \<Sum>y\<in>space M. f y * indicator {y} x)"
```
```  2355     by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
```
```  2356   show ?thesis unfolding * using borel_measurable_finite[of f]
```
```  2357     by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
```
```  2358 qed
```
```  2359
```
```  2360 lemma (in finite_measure_space) integral_finite_singleton:
```
```  2361   shows "integrable f"
```
```  2362   and "integral f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
```
```  2363 proof -
```
```  2364   have [simp]:
```
```  2365     "(\<integral>\<^isup>+ x. Real (f x)) = (\<Sum>x \<in> space M. Real (f x) * \<mu> {x})"
```
```  2366     "(\<integral>\<^isup>+ x. Real (- f x)) = (\<Sum>x \<in> space M. Real (- f x) * \<mu> {x})"
```
```  2367     unfolding positive_integral_finite_eq_setsum by auto
```
```  2368   show "integrable f" using finite_space finite_measure
```
```  2369     by (simp add: setsum_\<omega> integrable_def)
```
```  2370   show ?I using finite_measure
```
```  2371     apply (simp add: integral_def real_of_pextreal_setsum[symmetric]
```
```  2372       real_of_pextreal_mult[symmetric] setsum_subtractf[symmetric])
```
```  2373     by (rule setsum_cong) (simp_all split: split_if)
```
```  2374 qed
```
```  2375
```
```  2376 end
```