src/HOL/Probability/Lebesgue_Integration.thy
 author hoelzl Mon Jan 24 22:29:50 2011 +0100 (2011-01-24) changeset 41661 baf1964bc468 parent 41545 9c869baf1c66 child 41689 3e39b0e730d6 permissions -rw-r--r--
use pre-image measure, instead of image
```     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 measure_space) simple_function_vimage:
```
```   475   assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
```
```   476     and f: "sigma_algebra.simple_function M' f"
```
```   477   shows "simple_function (\<lambda>x. f (T x))"
```
```   478 proof (intro simple_function_def[THEN iffD2] conjI ballI)
```
```   479   interpret T: sigma_algebra M' by fact
```
```   480   have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
```
```   481     using T unfolding measurable_def by auto
```
```   482   then show "finite ((\<lambda>x. f (T x)) ` space M)"
```
```   483     using f unfolding T.simple_function_def by (auto intro: finite_subset)
```
```   484   fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
```
```   485   then have "i \<in> f ` space M'"
```
```   486     using T unfolding measurable_def by auto
```
```   487   then have "f -` {i} \<inter> space M' \<in> sets M'"
```
```   488     using f unfolding T.simple_function_def by auto
```
```   489   then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
```
```   490     using T unfolding measurable_def by auto
```
```   491   also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
```
```   492     using T unfolding measurable_def by auto
```
```   493   finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
```
```   494 qed
```
```   495
```
```   496 section "Simple integral"
```
```   497
```
```   498 definition (in measure_space) simple_integral (binder "\<integral>\<^isup>S " 10) where
```
```   499   "simple_integral f = (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M))"
```
```   500
```
```   501 lemma (in measure_space) simple_integral_cong:
```
```   502   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
```
```   503   shows "simple_integral f = simple_integral g"
```
```   504 proof -
```
```   505   have "f ` space M = g ` space M"
```
```   506     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
```
```   507     using assms by (auto intro!: image_eqI)
```
```   508   thus ?thesis unfolding simple_integral_def by simp
```
```   509 qed
```
```   510
```
```   511 lemma (in measure_space) simple_integral_cong_measure:
```
```   512   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A" and "simple_function f"
```
```   513   shows "measure_space.simple_integral M \<nu> f = simple_integral f"
```
```   514 proof -
```
```   515   interpret v: measure_space M \<nu>
```
```   516     by (rule measure_space_cong) fact
```
```   517   from simple_functionD[OF `simple_function f`] assms show ?thesis
```
```   518     unfolding simple_integral_def v.simple_integral_def
```
```   519     by (auto intro!: setsum_cong)
```
```   520 qed
```
```   521
```
```   522 lemma (in measure_space) simple_integral_const[simp]:
```
```   523   "(\<integral>\<^isup>Sx. c) = c * \<mu> (space M)"
```
```   524 proof (cases "space M = {}")
```
```   525   case True thus ?thesis unfolding simple_integral_def by simp
```
```   526 next
```
```   527   case False hence "(\<lambda>x. c) ` space M = {c}" by auto
```
```   528   thus ?thesis unfolding simple_integral_def by simp
```
```   529 qed
```
```   530
```
```   531 lemma (in measure_space) simple_function_partition:
```
```   532   assumes "simple_function f" and "simple_function g"
```
```   533   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)"
```
```   534     (is "_ = setsum _ (?p ` space M)")
```
```   535 proof-
```
```   536   let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
```
```   537   let ?SIGMA = "Sigma (f`space M) ?sub"
```
```   538
```
```   539   have [intro]:
```
```   540     "finite (f ` space M)"
```
```   541     "finite (g ` space M)"
```
```   542     using assms unfolding simple_function_def by simp_all
```
```   543
```
```   544   { fix A
```
```   545     have "?p ` (A \<inter> space M) \<subseteq>
```
```   546       (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
```
```   547       by auto
```
```   548     hence "finite (?p ` (A \<inter> space M))"
```
```   549       by (rule finite_subset) auto }
```
```   550   note this[intro, simp]
```
```   551
```
```   552   { fix x assume "x \<in> space M"
```
```   553     have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
```
```   554     moreover {
```
```   555       fix x y
```
```   556       have *: "f -` {f x} \<inter> g -` {g x} \<inter> space M
```
```   557           = (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" by auto
```
```   558       assume "x \<in> space M" "y \<in> space M"
```
```   559       hence "f -` {f x} \<inter> g -` {g x} \<inter> space M \<in> sets M"
```
```   560         using assms unfolding simple_function_def * by auto }
```
```   561     ultimately
```
```   562     have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
```
```   563       by (subst measure_finitely_additive) auto }
```
```   564   hence "simple_integral f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
```
```   565     unfolding simple_integral_def
```
```   566     by (subst setsum_Sigma[symmetric],
```
```   567        auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
```
```   568   also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
```
```   569   proof -
```
```   570     have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
```
```   571     have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
```
```   572       = (\<lambda>x. (f x, ?p x)) ` space M"
```
```   573     proof safe
```
```   574       fix x assume "x \<in> space M"
```
```   575       thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
```
```   576         by (auto intro!: image_eqI[of _ _ "?p x"])
```
```   577     qed auto
```
```   578     thus ?thesis
```
```   579       apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
```
```   580       apply (rule_tac x="xa" in image_eqI)
```
```   581       by simp_all
```
```   582   qed
```
```   583   finally show ?thesis .
```
```   584 qed
```
```   585
```
```   586 lemma (in measure_space) simple_integral_add[simp]:
```
```   587   assumes "simple_function f" and "simple_function g"
```
```   588   shows "(\<integral>\<^isup>Sx. f x + g x) = simple_integral f + simple_integral g"
```
```   589 proof -
```
```   590   { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
```
```   591     assume "x \<in> space M"
```
```   592     hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
```
```   593         "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
```
```   594       by auto }
```
```   595   thus ?thesis
```
```   596     unfolding
```
```   597       simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
```
```   598       simple_function_partition[OF `simple_function f` `simple_function g`]
```
```   599       simple_function_partition[OF `simple_function g` `simple_function f`]
```
```   600     apply (subst (3) Int_commute)
```
```   601     by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
```
```   602 qed
```
```   603
```
```   604 lemma (in measure_space) simple_integral_setsum[simp]:
```
```   605   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
```
```   606   shows "(\<integral>\<^isup>Sx. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
```
```   607 proof cases
```
```   608   assume "finite P"
```
```   609   from this assms show ?thesis
```
```   610     by induct (auto simp: simple_function_setsum simple_integral_add)
```
```   611 qed auto
```
```   612
```
```   613 lemma (in measure_space) simple_integral_mult[simp]:
```
```   614   assumes "simple_function f"
```
```   615   shows "(\<integral>\<^isup>Sx. c * f x) = c * simple_integral f"
```
```   616 proof -
```
```   617   note mult = simple_function_mult[OF simple_function_const[of c] assms]
```
```   618   { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
```
```   619     assume "x \<in> space M"
```
```   620     hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
```
```   621       by auto }
```
```   622   thus ?thesis
```
```   623     unfolding simple_function_partition[OF mult assms]
```
```   624       simple_function_partition[OF assms mult]
```
```   625     by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
```
```   626 qed
```
```   627
```
```   628 lemma (in sigma_algebra) simple_function_If:
```
```   629   assumes sf: "simple_function f" "simple_function g" and A: "A \<in> sets M"
```
```   630   shows "simple_function (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function ?IF")
```
```   631 proof -
```
```   632   def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
```
```   633   show ?thesis unfolding simple_function_def
```
```   634   proof safe
```
```   635     have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
```
```   636     from finite_subset[OF this] assms
```
```   637     show "finite (?IF ` space M)" unfolding simple_function_def by auto
```
```   638   next
```
```   639     fix x assume "x \<in> space M"
```
```   640     then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
```
```   641       then ((F (f x) \<inter> A) \<union> (G (f x) - (G (f x) \<inter> A)))
```
```   642       else ((F (g x) \<inter> A) \<union> (G (g x) - (G (g x) \<inter> A))))"
```
```   643       using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
```
```   644     have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
```
```   645       unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
```
```   646     show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
```
```   647   qed
```
```   648 qed
```
```   649
```
```   650 lemma (in measure_space) simple_integral_mono_AE:
```
```   651   assumes "simple_function f" and "simple_function g"
```
```   652   and mono: "AE x. f x \<le> g x"
```
```   653   shows "simple_integral f \<le> simple_integral g"
```
```   654 proof -
```
```   655   let "?S x" = "(g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
```
```   656   have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
```
```   657     "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
```
```   658   show ?thesis
```
```   659     unfolding *
```
```   660       simple_function_partition[OF `simple_function f` `simple_function g`]
```
```   661       simple_function_partition[OF `simple_function g` `simple_function f`]
```
```   662   proof (safe intro!: setsum_mono)
```
```   663     fix x assume "x \<in> space M"
```
```   664     then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
```
```   665     show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
```
```   666     proof (cases "f x \<le> g x")
```
```   667       case True then show ?thesis using * by (auto intro!: mult_right_mono)
```
```   668     next
```
```   669       case False
```
```   670       obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
```
```   671         using mono by (auto elim!: AE_E)
```
```   672       have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
```
```   673       moreover have "?S x \<in> sets M" using assms
```
```   674         by (rule_tac Int) (auto intro!: simple_functionD)
```
```   675       ultimately have "\<mu> (?S x) \<le> \<mu> N"
```
```   676         using `N \<in> sets M` by (auto intro!: measure_mono)
```
```   677       then show ?thesis using `\<mu> N = 0` by auto
```
```   678     qed
```
```   679   qed
```
```   680 qed
```
```   681
```
```   682 lemma (in measure_space) simple_integral_mono:
```
```   683   assumes "simple_function f" and "simple_function g"
```
```   684   and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
```
```   685   shows "simple_integral f \<le> simple_integral g"
```
```   686 proof (rule simple_integral_mono_AE[OF assms(1, 2)])
```
```   687   show "AE x. f x \<le> g x"
```
```   688     using mono by (rule AE_cong) auto
```
```   689 qed
```
```   690
```
```   691 lemma (in measure_space) simple_integral_cong_AE:
```
```   692   assumes "simple_function f" "simple_function g" and "AE x. f x = g x"
```
```   693   shows "simple_integral f = simple_integral g"
```
```   694   using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
```
```   695
```
```   696 lemma (in measure_space) simple_integral_cong':
```
```   697   assumes sf: "simple_function f" "simple_function g"
```
```   698   and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
```
```   699   shows "simple_integral f = simple_integral g"
```
```   700 proof (intro simple_integral_cong_AE sf AE_I)
```
```   701   show "\<mu> {x\<in>space M. f x \<noteq> g x} = 0" by fact
```
```   702   show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
```
```   703     using sf[THEN borel_measurable_simple_function] by auto
```
```   704 qed simp
```
```   705
```
```   706 lemma (in measure_space) simple_integral_indicator:
```
```   707   assumes "A \<in> sets M"
```
```   708   assumes "simple_function f"
```
```   709   shows "(\<integral>\<^isup>Sx. f x * indicator A x) =
```
```   710     (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
```
```   711 proof cases
```
```   712   assume "A = space M"
```
```   713   moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x) = simple_integral f"
```
```   714     by (auto intro!: simple_integral_cong)
```
```   715   moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
```
```   716   ultimately show ?thesis by (simp add: simple_integral_def)
```
```   717 next
```
```   718   assume "A \<noteq> space M"
```
```   719   then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
```
```   720   have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
```
```   721   proof safe
```
```   722     fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
```
```   723   next
```
```   724     fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
```
```   725       using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
```
```   726   next
```
```   727     show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
```
```   728   qed
```
```   729   have *: "(\<integral>\<^isup>Sx. f x * indicator A x) =
```
```   730     (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
```
```   731     unfolding simple_integral_def I
```
```   732   proof (rule setsum_mono_zero_cong_left)
```
```   733     show "finite (f ` space M \<union> {0})"
```
```   734       using assms(2) unfolding simple_function_def by auto
```
```   735     show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
```
```   736       using sets_into_space[OF assms(1)] by auto
```
```   737     have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
```
```   738       by (auto simp: image_iff)
```
```   739     thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
```
```   740       i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
```
```   741   next
```
```   742     fix x assume "x \<in> f`A \<union> {0}"
```
```   743     hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
```
```   744       by (auto simp: indicator_def split: split_if_asm)
```
```   745     thus "x * \<mu> (?I -` {x} \<inter> space M) =
```
```   746       x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
```
```   747   qed
```
```   748   show ?thesis unfolding *
```
```   749     using assms(2) unfolding simple_function_def
```
```   750     by (auto intro!: setsum_mono_zero_cong_right)
```
```   751 qed
```
```   752
```
```   753 lemma (in measure_space) simple_integral_indicator_only[simp]:
```
```   754   assumes "A \<in> sets M"
```
```   755   shows "simple_integral (indicator A) = \<mu> A"
```
```   756 proof cases
```
```   757   assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
```
```   758   thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
```
```   759 next
```
```   760   assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pextreal}" by auto
```
```   761   thus ?thesis
```
```   762     using simple_integral_indicator[OF assms simple_function_const[of 1]]
```
```   763     using sets_into_space[OF assms]
```
```   764     by (auto intro!: arg_cong[where f="\<mu>"])
```
```   765 qed
```
```   766
```
```   767 lemma (in measure_space) simple_integral_null_set:
```
```   768   assumes "simple_function u" "N \<in> null_sets"
```
```   769   shows "(\<integral>\<^isup>Sx. u x * indicator N x) = 0"
```
```   770 proof -
```
```   771   have "AE x. indicator N x = (0 :: pextreal)"
```
```   772     using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
```
```   773   then have "(\<integral>\<^isup>Sx. u x * indicator N x) = (\<integral>\<^isup>Sx. 0)"
```
```   774     using assms by (intro simple_integral_cong_AE) (auto intro!: AE_disjI2)
```
```   775   then show ?thesis by simp
```
```   776 qed
```
```   777
```
```   778 lemma (in measure_space) simple_integral_cong_AE_mult_indicator:
```
```   779   assumes sf: "simple_function f" and eq: "AE x. x \<in> S" and "S \<in> sets M"
```
```   780   shows "simple_integral f = (\<integral>\<^isup>Sx. f x * indicator S x)"
```
```   781 proof (rule simple_integral_cong_AE)
```
```   782   show "simple_function f" by fact
```
```   783   show "simple_function (\<lambda>x. f x * indicator S x)"
```
```   784     using sf `S \<in> sets M` by auto
```
```   785   from eq show "AE x. f x = f x * indicator S x"
```
```   786     by (rule AE_mp) simp
```
```   787 qed
```
```   788
```
```   789 lemma (in measure_space) simple_integral_restricted:
```
```   790   assumes "A \<in> sets M"
```
```   791   assumes sf: "simple_function (\<lambda>x. f x * indicator A x)"
```
```   792   shows "measure_space.simple_integral (restricted_space A) \<mu> f = (\<integral>\<^isup>Sx. f x * indicator A x)"
```
```   793     (is "_ = simple_integral ?f")
```
```   794   unfolding measure_space.simple_integral_def[OF restricted_measure_space[OF `A \<in> sets M`]]
```
```   795   unfolding simple_integral_def
```
```   796 proof (simp, safe intro!: setsum_mono_zero_cong_left)
```
```   797   from sf show "finite (?f ` space M)"
```
```   798     unfolding simple_function_def by auto
```
```   799 next
```
```   800   fix x assume "x \<in> A"
```
```   801   then show "f x \<in> ?f ` space M"
```
```   802     using sets_into_space `A \<in> sets M` by (auto intro!: image_eqI[of _ _ x])
```
```   803 next
```
```   804   fix x assume "x \<in> space M" "?f x \<notin> f`A"
```
```   805   then have "x \<notin> A" by (auto simp: image_iff)
```
```   806   then show "?f x * \<mu> (?f -` {?f x} \<inter> space M) = 0" by simp
```
```   807 next
```
```   808   fix x assume "x \<in> A"
```
```   809   then have "f x \<noteq> 0 \<Longrightarrow>
```
```   810     f -` {f x} \<inter> A = ?f -` {f x} \<inter> space M"
```
```   811     using `A \<in> sets M` sets_into_space
```
```   812     by (auto simp: indicator_def split: split_if_asm)
```
```   813   then show "f x * \<mu> (f -` {f x} \<inter> A) =
```
```   814     f x * \<mu> (?f -` {f x} \<inter> space M)"
```
```   815     unfolding pextreal_mult_cancel_left by auto
```
```   816 qed
```
```   817
```
```   818 lemma (in measure_space) simple_integral_subalgebra:
```
```   819   assumes N: "measure_space N \<mu>" and [simp]: "space N = space M"
```
```   820   shows "measure_space.simple_integral N \<mu> = simple_integral"
```
```   821   unfolding simple_integral_def_raw
```
```   822   unfolding measure_space.simple_integral_def_raw[OF N] by simp
```
```   823
```
```   824 lemma (in measure_space) simple_integral_vimage:
```
```   825   assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
```
```   826     and f: "sigma_algebra.simple_function M' f"
```
```   827   shows "measure_space.simple_integral M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f = (\<integral>\<^isup>S x. f (T x))"
```
```   828     (is "measure_space.simple_integral M' ?nu f = _")
```
```   829 proof -
```
```   830   interpret T: measure_space M' ?nu using T by (rule measure_space_vimage) auto
```
```   831   show "T.simple_integral f = (\<integral>\<^isup>S x. f (T x))"
```
```   832     unfolding simple_integral_def T.simple_integral_def
```
```   833   proof (intro setsum_mono_zero_cong_right ballI)
```
```   834     show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
```
```   835       using T unfolding measurable_def by auto
```
```   836     show "finite (f ` space M')"
```
```   837       using f unfolding T.simple_function_def by auto
```
```   838   next
```
```   839     fix i assume "i \<in> f ` space M' - (\<lambda>x. f (T x)) ` space M"
```
```   840     then have "T -` (f -` {i} \<inter> space M') \<inter> space M = {}" by (auto simp: image_iff)
```
```   841     then show "i * \<mu> (T -` (f -` {i} \<inter> space M') \<inter> space M) = 0" by simp
```
```   842   next
```
```   843     fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M"
```
```   844     then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
```
```   845       using T unfolding measurable_def by auto
```
```   846     then show "i * \<mu> (T -` (f -` {i} \<inter> space M') \<inter> space M) = i * \<mu> ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
```
```   847       by auto
```
```   848   qed
```
```   849 qed
```
```   850
```
```   851 section "Continuous posititve integration"
```
```   852
```
```   853 definition (in measure_space) positive_integral (binder "\<integral>\<^isup>+ " 10) where
```
```   854   "positive_integral f = SUPR {g. simple_function g \<and> g \<le> f} simple_integral"
```
```   855
```
```   856 lemma (in measure_space) positive_integral_alt:
```
```   857   "positive_integral f =
```
```   858     (SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M} simple_integral)" (is "_ = ?alt")
```
```   859 proof (rule antisym SUP_leI)
```
```   860   show "positive_integral f \<le> ?alt" unfolding positive_integral_def
```
```   861   proof (safe intro!: SUP_leI)
```
```   862     fix g assume g: "simple_function g" "g \<le> f"
```
```   863     let ?G = "g -` {\<omega>} \<inter> space M"
```
```   864     show "simple_integral g \<le>
```
```   865       SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g ` space M} simple_integral"
```
```   866       (is "simple_integral g \<le> SUPR ?A simple_integral")
```
```   867     proof cases
```
```   868       let ?g = "\<lambda>x. indicator (space M - ?G) x * g x"
```
```   869       have g': "simple_function ?g"
```
```   870         using g by (auto intro: simple_functionD)
```
```   871       moreover
```
```   872       assume "\<mu> ?G = 0"
```
```   873       then have "AE x. g x = ?g x" using g
```
```   874         by (intro AE_I[where N="?G"])
```
```   875            (auto intro: simple_functionD simp: indicator_def)
```
```   876       with g(1) g' have "simple_integral g = simple_integral ?g"
```
```   877         by (rule simple_integral_cong_AE)
```
```   878       moreover have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
```
```   879       from this `g \<le> f` have "?g \<le> f" by (rule order_trans)
```
```   880       moreover have "\<omega> \<notin> ?g ` space M"
```
```   881         by (auto simp: indicator_def split: split_if_asm)
```
```   882       ultimately show ?thesis by (auto intro!: le_SUPI)
```
```   883     next
```
```   884       assume "\<mu> ?G \<noteq> 0"
```
```   885       then have "?G \<noteq> {}" by auto
```
```   886       then have "\<omega> \<in> g`space M" by force
```
```   887       then have "space M \<noteq> {}" by auto
```
```   888       have "SUPR ?A simple_integral = \<omega>"
```
```   889       proof (intro SUP_\<omega>[THEN iffD2] allI impI)
```
```   890         fix x assume "x < \<omega>"
```
```   891         then guess n unfolding less_\<omega>_Ex_of_nat .. note n = this
```
```   892         then have "0 < n" by (intro neq0_conv[THEN iffD1] notI) simp
```
```   893         let ?g = "\<lambda>x. (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * indicator ?G x"
```
```   894         show "\<exists>i\<in>?A. x < simple_integral i"
```
```   895         proof (intro bexI impI CollectI conjI)
```
```   896           show "simple_function ?g" using g
```
```   897             by (auto intro!: simple_functionD simple_function_add)
```
```   898           have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
```
```   899           from this g(2) show "?g \<le> f" by (rule order_trans)
```
```   900           show "\<omega> \<notin> ?g ` space M"
```
```   901             using `\<mu> ?G \<noteq> 0` by (auto simp: indicator_def split: split_if_asm)
```
```   902           have "x < (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * \<mu> ?G"
```
```   903             using n `\<mu> ?G \<noteq> 0` `0 < n`
```
```   904             by (auto simp: pextreal_noteq_omega_Ex field_simps)
```
```   905           also have "\<dots> = simple_integral ?g" using g `space M \<noteq> {}`
```
```   906             by (subst simple_integral_indicator)
```
```   907                (auto simp: image_constant ac_simps dest: simple_functionD)
```
```   908           finally show "x < simple_integral ?g" .
```
```   909         qed
```
```   910       qed
```
```   911       then show ?thesis by simp
```
```   912     qed
```
```   913   qed
```
```   914 qed (auto intro!: SUP_subset simp: positive_integral_def)
```
```   915
```
```   916 lemma (in measure_space) positive_integral_cong_measure:
```
```   917   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A"
```
```   918   shows "measure_space.positive_integral M \<nu> f = positive_integral f"
```
```   919 proof -
```
```   920   interpret v: measure_space M \<nu>
```
```   921     by (rule measure_space_cong) fact
```
```   922   with assms show ?thesis
```
```   923     unfolding positive_integral_def v.positive_integral_def SUPR_def
```
```   924     by (auto intro!: arg_cong[where f=Sup] image_cong
```
```   925              simp: simple_integral_cong_measure[of \<nu>])
```
```   926 qed
```
```   927
```
```   928 lemma (in measure_space) positive_integral_alt1:
```
```   929   "positive_integral f =
```
```   930     (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)"
```
```   931   unfolding positive_integral_alt SUPR_def
```
```   932 proof (safe intro!: arg_cong[where f=Sup])
```
```   933   fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
```
```   934   assume "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
```
```   935   hence "?g \<le> f" "simple_function ?g" "simple_integral g = simple_integral ?g"
```
```   936     "\<omega> \<notin> g`space M"
```
```   937     unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
```
```   938   thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
```
```   939     by auto
```
```   940 next
```
```   941   fix g assume "simple_function g" "g \<le> f" "\<omega> \<notin> g`space M"
```
```   942   hence "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
```
```   943     by (auto simp add: le_fun_def image_iff)
```
```   944   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>)}"
```
```   945     by auto
```
```   946 qed
```
```   947
```
```   948 lemma (in measure_space) positive_integral_cong:
```
```   949   assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
```
```   950   shows "positive_integral f = positive_integral g"
```
```   951 proof -
```
```   952   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>)"
```
```   953     using assms by auto
```
```   954   thus ?thesis unfolding positive_integral_alt1 by auto
```
```   955 qed
```
```   956
```
```   957 lemma (in measure_space) positive_integral_eq_simple_integral:
```
```   958   assumes "simple_function f"
```
```   959   shows "positive_integral f = simple_integral f"
```
```   960   unfolding positive_integral_def
```
```   961 proof (safe intro!: pextreal_SUPI)
```
```   962   fix g assume "simple_function g" "g \<le> f"
```
```   963   with assms show "simple_integral g \<le> simple_integral f"
```
```   964     by (auto intro!: simple_integral_mono simp: le_fun_def)
```
```   965 next
```
```   966   fix y assume "\<forall>x. x\<in>{g. simple_function g \<and> g \<le> f} \<longrightarrow> simple_integral x \<le> y"
```
```   967   with assms show "simple_integral f \<le> y" by auto
```
```   968 qed
```
```   969
```
```   970 lemma (in measure_space) positive_integral_mono_AE:
```
```   971   assumes ae: "AE x. u x \<le> v x"
```
```   972   shows "positive_integral u \<le> positive_integral v"
```
```   973   unfolding positive_integral_alt1
```
```   974 proof (safe intro!: SUPR_mono)
```
```   975   fix a assume a: "simple_function a" and mono: "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
```
```   976   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"
```
```   977     by (auto elim!: AE_E)
```
```   978   have "simple_function (\<lambda>x. a x * indicator (space M - N) x)"
```
```   979     using `N \<in> sets M` a by auto
```
```   980   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>)}.
```
```   981     simple_integral a \<le> simple_integral b"
```
```   982   proof (safe intro!: bexI[of _ "\<lambda>x. a x * indicator (space M - N) x"]
```
```   983                       simple_integral_mono_AE)
```
```   984     show "AE x. a x \<le> a x * indicator (space M - N) x"
```
```   985     proof (rule AE_I, rule subset_refl)
```
```   986       have *: "{x \<in> space M. \<not> a x \<le> a x * indicator (space M - N) x} =
```
```   987         N \<inter> {x \<in> space M. a x \<noteq> 0}" (is "?N = _")
```
```   988         using `N \<in> sets M`[THEN sets_into_space] by (auto simp: indicator_def)
```
```   989       then show "?N \<in> sets M"
```
```   990         using `N \<in> sets M` `simple_function a`[THEN borel_measurable_simple_function]
```
```   991         by (auto intro!: measure_mono Int)
```
```   992       then have "\<mu> ?N \<le> \<mu> N"
```
```   993         unfolding * using `N \<in> sets M` by (auto intro!: measure_mono)
```
```   994       then show "\<mu> ?N = 0" using `\<mu> N = 0` by auto
```
```   995     qed
```
```   996   next
```
```   997     fix x assume "x \<in> space M"
```
```   998     show "a x * indicator (space M - N) x \<le> v x"
```
```   999     proof (cases "x \<in> N")
```
```  1000       case True then show ?thesis by simp
```
```  1001     next
```
```  1002       case False
```
```  1003       with N mono have "a x \<le> u x" "u x \<le> v x" using `x \<in> space M` by auto
```
```  1004       with False `x \<in> space M` show "a x * indicator (space M - N) x \<le> v x" by auto
```
```  1005     qed
```
```  1006     assume "a x * indicator (space M - N) x = \<omega>"
```
```  1007     with mono `x \<in> space M` show False
```
```  1008       by (simp split: split_if_asm add: indicator_def)
```
```  1009   qed
```
```  1010 qed
```
```  1011
```
```  1012 lemma (in measure_space) positive_integral_cong_AE:
```
```  1013   "AE x. u x = v x \<Longrightarrow> positive_integral u = positive_integral v"
```
```  1014   by (auto simp: eq_iff intro!: positive_integral_mono_AE)
```
```  1015
```
```  1016 lemma (in measure_space) positive_integral_mono:
```
```  1017   assumes mono: "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x"
```
```  1018   shows "positive_integral u \<le> positive_integral v"
```
```  1019   using mono by (auto intro!: AE_cong positive_integral_mono_AE)
```
```  1020
```
```  1021 lemma image_set_cong:
```
```  1022   assumes A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. f x = g y"
```
```  1023   assumes B: "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. g y = f x"
```
```  1024   shows "f ` A = g ` B"
```
```  1025   using assms by blast
```
```  1026
```
```  1027 lemma (in measure_space) positive_integral_SUP_approx:
```
```  1028   assumes "f \<up> s"
```
```  1029   and f: "\<And>i. f i \<in> borel_measurable M"
```
```  1030   and "simple_function u"
```
```  1031   and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
```
```  1032   shows "simple_integral u \<le> (SUP i. positive_integral (f i))" (is "_ \<le> ?S")
```
```  1033 proof (rule pextreal_le_mult_one_interval)
```
```  1034   fix a :: pextreal assume "0 < a" "a < 1"
```
```  1035   hence "a \<noteq> 0" by auto
```
```  1036   let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
```
```  1037   have B: "\<And>i. ?B i \<in> sets M"
```
```  1038     using f `simple_function u` by (auto simp: borel_measurable_simple_function)
```
```  1039
```
```  1040   let "?uB i x" = "u x * indicator (?B i) x"
```
```  1041
```
```  1042   { fix i have "?B i \<subseteq> ?B (Suc i)"
```
```  1043     proof safe
```
```  1044       fix i x assume "a * u x \<le> f i x"
```
```  1045       also have "\<dots> \<le> f (Suc i) x"
```
```  1046         using `f \<up> s` unfolding isoton_def le_fun_def by auto
```
```  1047       finally show "a * u x \<le> f (Suc i) x" .
```
```  1048     qed }
```
```  1049   note B_mono = this
```
```  1050
```
```  1051   have u: "\<And>x. x \<in> space M \<Longrightarrow> u -` {u x} \<inter> space M \<in> sets M"
```
```  1052     using `simple_function u` by (auto simp add: simple_function_def)
```
```  1053
```
```  1054   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
```
```  1055   proof safe
```
```  1056     fix x i assume "x \<in> space M"
```
```  1057     show "x \<in> (\<Union>i. (u -` {u x} \<inter> space M) \<inter> ?B i)"
```
```  1058     proof cases
```
```  1059       assume "u x = 0" thus ?thesis using `x \<in> space M` by simp
```
```  1060     next
```
```  1061       assume "u x \<noteq> 0"
```
```  1062       with `a < 1` real `x \<in> space M`
```
```  1063       have "a * u x < 1 * u x" by (rule_tac pextreal_mult_strict_right_mono) (auto simp: image_iff)
```
```  1064       also have "\<dots> \<le> (SUP i. f i x)" using le `f \<up> s`
```
```  1065         unfolding isoton_fun_expand by (auto simp: isoton_def le_fun_def)
```
```  1066       finally obtain i where "a * u x < f i x" unfolding SUPR_def
```
```  1067         by (auto simp add: less_Sup_iff)
```
```  1068       hence "a * u x \<le> f i x" by auto
```
```  1069       thus ?thesis using `x \<in> space M` by auto
```
```  1070     qed
```
```  1071   qed auto
```
```  1072   note measure_conv = measure_up[OF Int[OF u B] this]
```
```  1073
```
```  1074   have "simple_integral u = (SUP i. simple_integral (?uB i))"
```
```  1075     unfolding simple_integral_indicator[OF B `simple_function u`]
```
```  1076   proof (subst SUPR_pextreal_setsum, safe)
```
```  1077     fix x n assume "x \<in> space M"
```
```  1078     have "\<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f n x})
```
```  1079       \<le> \<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f (Suc n) x})"
```
```  1080       using B_mono Int[OF u[OF `x \<in> space M`] B] by (auto intro!: measure_mono)
```
```  1081     thus "u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B n)
```
```  1082             \<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
```
```  1083       by (auto intro: mult_left_mono)
```
```  1084   next
```
```  1085     show "simple_integral u =
```
```  1086       (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (u -` {i} \<inter> space M \<inter> ?B n))"
```
```  1087       using measure_conv unfolding simple_integral_def isoton_def
```
```  1088       by (auto intro!: setsum_cong simp: pextreal_SUP_cmult)
```
```  1089   qed
```
```  1090   moreover
```
```  1091   have "a * (SUP i. simple_integral (?uB i)) \<le> ?S"
```
```  1092     unfolding pextreal_SUP_cmult[symmetric]
```
```  1093   proof (safe intro!: SUP_mono bexI)
```
```  1094     fix i
```
```  1095     have "a * simple_integral (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x)"
```
```  1096       using B `simple_function u`
```
```  1097       by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
```
```  1098     also have "\<dots> \<le> positive_integral (f i)"
```
```  1099     proof -
```
```  1100       have "\<And>x. a * ?uB i x \<le> f i x" unfolding indicator_def by auto
```
```  1101       hence *: "simple_function (\<lambda>x. a * ?uB i x)" using B assms(3)
```
```  1102         by (auto intro!: simple_integral_mono)
```
```  1103       show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
```
```  1104         by (auto intro!: positive_integral_mono simp: indicator_def)
```
```  1105     qed
```
```  1106     finally show "a * simple_integral (?uB i) \<le> positive_integral (f i)"
```
```  1107       by auto
```
```  1108   qed simp
```
```  1109   ultimately show "a * simple_integral u \<le> ?S" by simp
```
```  1110 qed
```
```  1111
```
```  1112 text {* Beppo-Levi monotone convergence theorem *}
```
```  1113 lemma (in measure_space) positive_integral_isoton:
```
```  1114   assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
```
```  1115   shows "(\<lambda>i. positive_integral (f i)) \<up> positive_integral u"
```
```  1116   unfolding isoton_def
```
```  1117 proof safe
```
```  1118   fix i show "positive_integral (f i) \<le> positive_integral (f (Suc i))"
```
```  1119     apply (rule positive_integral_mono)
```
```  1120     using `f \<up> u` unfolding isoton_def le_fun_def by auto
```
```  1121 next
```
```  1122   have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
```
```  1123
```
```  1124   show "(SUP i. positive_integral (f i)) = positive_integral u"
```
```  1125   proof (rule antisym)
```
```  1126     from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
```
```  1127     show "(SUP j. positive_integral (f j)) \<le> positive_integral u"
```
```  1128       by (auto intro!: SUP_leI positive_integral_mono)
```
```  1129   next
```
```  1130     show "positive_integral u \<le> (SUP i. positive_integral (f i))"
```
```  1131       unfolding positive_integral_alt[of u]
```
```  1132       by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
```
```  1133   qed
```
```  1134 qed
```
```  1135
```
```  1136 lemma (in measure_space) positive_integral_monotone_convergence_SUP:
```
```  1137   assumes "\<And>i x. x \<in> space M \<Longrightarrow> f i x \<le> f (Suc i) x"
```
```  1138   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1139   shows "(SUP i. positive_integral (f i)) = (\<integral>\<^isup>+ x. SUP i. f i x)"
```
```  1140     (is "_ = positive_integral ?u")
```
```  1141 proof -
```
```  1142   show ?thesis
```
```  1143   proof (rule antisym)
```
```  1144     show "(SUP j. positive_integral (f j)) \<le> positive_integral ?u"
```
```  1145       by (auto intro!: SUP_leI positive_integral_mono le_SUPI)
```
```  1146   next
```
```  1147     def rf \<equiv> "\<lambda>i. \<lambda>x\<in>space M. f i x" and ru \<equiv> "\<lambda>x\<in>space M. ?u x"
```
```  1148     have "\<And>i. rf i \<in> borel_measurable M" unfolding rf_def
```
```  1149       using assms by (simp cong: measurable_cong)
```
```  1150     moreover have iso: "rf \<up> ru" using assms unfolding rf_def ru_def
```
```  1151       unfolding isoton_def le_fun_def fun_eq_iff SUPR_apply
```
```  1152       using SUP_const[OF UNIV_not_empty]
```
```  1153       by (auto simp: restrict_def le_fun_def fun_eq_iff)
```
```  1154     ultimately have "positive_integral ru \<le> (SUP i. positive_integral (rf i))"
```
```  1155       unfolding positive_integral_alt[of ru]
```
```  1156       by (auto simp: le_fun_def intro!: SUP_leI positive_integral_SUP_approx)
```
```  1157     then show "positive_integral ?u \<le> (SUP i. positive_integral (f i))"
```
```  1158       unfolding ru_def rf_def by (simp cong: positive_integral_cong)
```
```  1159   qed
```
```  1160 qed
```
```  1161
```
```  1162 lemma (in measure_space) SUP_simple_integral_sequences:
```
```  1163   assumes f: "f \<up> u" "\<And>i. simple_function (f i)"
```
```  1164   and g: "g \<up> u" "\<And>i. simple_function (g i)"
```
```  1165   shows "(SUP i. simple_integral (f i)) = (SUP i. simple_integral (g i))"
```
```  1166     (is "SUPR _ ?F = SUPR _ ?G")
```
```  1167 proof -
```
```  1168   have "(SUP i. ?F i) = (SUP i. positive_integral (f i))"
```
```  1169     using assms by (simp add: positive_integral_eq_simple_integral)
```
```  1170   also have "\<dots> = positive_integral u"
```
```  1171     using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
```
```  1172     unfolding isoton_def by simp
```
```  1173   also have "\<dots> = (SUP i. positive_integral (g i))"
```
```  1174     using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
```
```  1175     unfolding isoton_def by simp
```
```  1176   also have "\<dots> = (SUP i. ?G i)"
```
```  1177     using assms by (simp add: positive_integral_eq_simple_integral)
```
```  1178   finally show ?thesis .
```
```  1179 qed
```
```  1180
```
```  1181 lemma (in measure_space) positive_integral_const[simp]:
```
```  1182   "(\<integral>\<^isup>+ x. c) = c * \<mu> (space M)"
```
```  1183   by (subst positive_integral_eq_simple_integral) auto
```
```  1184
```
```  1185 lemma (in measure_space) positive_integral_isoton_simple:
```
```  1186   assumes "f \<up> u" and e: "\<And>i. simple_function (f i)"
```
```  1187   shows "(\<lambda>i. simple_integral (f i)) \<up> positive_integral u"
```
```  1188   using positive_integral_isoton[OF `f \<up> u` e(1)[THEN borel_measurable_simple_function]]
```
```  1189   unfolding positive_integral_eq_simple_integral[OF e] .
```
```  1190
```
```  1191 lemma (in measure_space) positive_integral_vimage:
```
```  1192   assumes T: "sigma_algebra M'" "T \<in> measurable M M'" and f: "f \<in> borel_measurable M'"
```
```  1193   shows "measure_space.positive_integral M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f = (\<integral>\<^isup>+ x. f (T x))"
```
```  1194     (is "measure_space.positive_integral M' ?nu f = _")
```
```  1195 proof -
```
```  1196   interpret T: measure_space M' ?nu using T by (rule measure_space_vimage) auto
```
```  1197   obtain f' where f': "f' \<up> f" "\<And>i. T.simple_function (f' i)"
```
```  1198     using T.borel_measurable_implies_simple_function_sequence[OF f] by blast
```
```  1199   then have f: "(\<lambda>i x. f' i (T x)) \<up> (\<lambda>x. f (T x))" "\<And>i. simple_function (\<lambda>x. f' i (T x))"
```
```  1200     using simple_function_vimage[OF T] unfolding isoton_fun_expand by auto
```
```  1201   show "T.positive_integral f = (\<integral>\<^isup>+ x. f (T x))"
```
```  1202     using positive_integral_isoton_simple[OF f]
```
```  1203     using T.positive_integral_isoton_simple[OF f']
```
```  1204     unfolding simple_integral_vimage[OF T f'(2)] isoton_def
```
```  1205     by simp
```
```  1206 qed
```
```  1207
```
```  1208 lemma (in measure_space) positive_integral_linear:
```
```  1209   assumes f: "f \<in> borel_measurable M"
```
```  1210   and g: "g \<in> borel_measurable M"
```
```  1211   shows "(\<integral>\<^isup>+ x. a * f x + g x) =
```
```  1212       a * positive_integral f + positive_integral g"
```
```  1213     (is "positive_integral ?L = _")
```
```  1214 proof -
```
```  1215   from borel_measurable_implies_simple_function_sequence'[OF f] guess u .
```
```  1216   note u = this positive_integral_isoton_simple[OF this(1-2)]
```
```  1217   from borel_measurable_implies_simple_function_sequence'[OF g] guess v .
```
```  1218   note v = this positive_integral_isoton_simple[OF this(1-2)]
```
```  1219   let "?L' i x" = "a * u i x + v i x"
```
```  1220
```
```  1221   have "?L \<in> borel_measurable M"
```
```  1222     using assms by simp
```
```  1223   from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
```
```  1224   note positive_integral_isoton_simple[OF this(1-2)] and l = this
```
```  1225   moreover have
```
```  1226       "(SUP i. simple_integral (l i)) = (SUP i. simple_integral (?L' i))"
```
```  1227   proof (rule SUP_simple_integral_sequences[OF l(1-2)])
```
```  1228     show "?L' \<up> ?L" "\<And>i. simple_function (?L' i)"
```
```  1229       using u v by (auto simp: isoton_fun_expand isoton_add isoton_cmult_right)
```
```  1230   qed
```
```  1231   moreover from u v have L'_isoton:
```
```  1232       "(\<lambda>i. simple_integral (?L' i)) \<up> a * positive_integral f + positive_integral g"
```
```  1233     by (simp add: isoton_add isoton_cmult_right)
```
```  1234   ultimately show ?thesis by (simp add: isoton_def)
```
```  1235 qed
```
```  1236
```
```  1237 lemma (in measure_space) positive_integral_cmult:
```
```  1238   assumes "f \<in> borel_measurable M"
```
```  1239   shows "(\<integral>\<^isup>+ x. c * f x) = c * positive_integral f"
```
```  1240   using positive_integral_linear[OF assms, of "\<lambda>x. 0" c] by auto
```
```  1241
```
```  1242 lemma (in measure_space) positive_integral_multc:
```
```  1243   assumes "f \<in> borel_measurable M"
```
```  1244   shows "(\<integral>\<^isup>+ x. f x * c) = positive_integral f * c"
```
```  1245   unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
```
```  1246
```
```  1247 lemma (in measure_space) positive_integral_indicator[simp]:
```
```  1248   "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x) = \<mu> A"
```
```  1249   by (subst positive_integral_eq_simple_integral)
```
```  1250      (auto simp: simple_function_indicator simple_integral_indicator)
```
```  1251
```
```  1252 lemma (in measure_space) positive_integral_cmult_indicator:
```
```  1253   "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x) = c * \<mu> A"
```
```  1254   by (subst positive_integral_eq_simple_integral)
```
```  1255      (auto simp: simple_function_indicator simple_integral_indicator)
```
```  1256
```
```  1257 lemma (in measure_space) positive_integral_add:
```
```  1258   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1259   shows "(\<integral>\<^isup>+ x. f x + g x) = positive_integral f + positive_integral g"
```
```  1260   using positive_integral_linear[OF assms, of 1] by simp
```
```  1261
```
```  1262 lemma (in measure_space) positive_integral_setsum:
```
```  1263   assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M"
```
```  1264   shows "(\<integral>\<^isup>+ x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. positive_integral (f i))"
```
```  1265 proof cases
```
```  1266   assume "finite P"
```
```  1267   from this assms show ?thesis
```
```  1268   proof induct
```
```  1269     case (insert i P)
```
```  1270     have "f i \<in> borel_measurable M"
```
```  1271       "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M"
```
```  1272       using insert by (auto intro!: borel_measurable_pextreal_setsum)
```
```  1273     from positive_integral_add[OF this]
```
```  1274     show ?case using insert by auto
```
```  1275   qed simp
```
```  1276 qed simp
```
```  1277
```
```  1278 lemma (in measure_space) positive_integral_diff:
```
```  1279   assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
```
```  1280   and fin: "positive_integral g \<noteq> \<omega>"
```
```  1281   and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
```
```  1282   shows "(\<integral>\<^isup>+ x. f x - g x) = positive_integral f - positive_integral g"
```
```  1283 proof -
```
```  1284   have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
```
```  1285     using f g by (rule borel_measurable_pextreal_diff)
```
```  1286   have "(\<integral>\<^isup>+x. f x - g x) + positive_integral g =
```
```  1287     positive_integral f"
```
```  1288     unfolding positive_integral_add[OF borel g, symmetric]
```
```  1289   proof (rule positive_integral_cong)
```
```  1290     fix x assume "x \<in> space M"
```
```  1291     from mono[OF this] show "f x - g x + g x = f x"
```
```  1292       by (cases "f x", cases "g x", simp, simp, cases "g x", auto)
```
```  1293   qed
```
```  1294   with mono show ?thesis
```
```  1295     by (subst minus_pextreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
```
```  1296 qed
```
```  1297
```
```  1298 lemma (in measure_space) positive_integral_psuminf:
```
```  1299   assumes "\<And>i. f i \<in> borel_measurable M"
```
```  1300   shows "(\<integral>\<^isup>+ x. \<Sum>\<^isub>\<infinity> i. f i x) = (\<Sum>\<^isub>\<infinity> i. positive_integral (f i))"
```
```  1301 proof -
```
```  1302   have "(\<lambda>i. (\<integral>\<^isup>+x. \<Sum>i<i. f i x)) \<up> (\<integral>\<^isup>+x. \<Sum>\<^isub>\<infinity>i. f i x)"
```
```  1303     by (rule positive_integral_isoton)
```
```  1304        (auto intro!: borel_measurable_pextreal_setsum assms positive_integral_mono
```
```  1305                      arg_cong[where f=Sup]
```
```  1306              simp: isoton_def le_fun_def psuminf_def fun_eq_iff SUPR_def Sup_fun_def)
```
```  1307   thus ?thesis
```
```  1308     by (auto simp: isoton_def psuminf_def positive_integral_setsum[OF assms])
```
```  1309 qed
```
```  1310
```
```  1311 text {* Fatou's lemma: convergence theorem on limes inferior *}
```
```  1312 lemma (in measure_space) positive_integral_lim_INF:
```
```  1313   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```  1314   assumes "\<And>i. u i \<in> borel_measurable M"
```
```  1315   shows "(\<integral>\<^isup>+ x. SUP n. INF m. u (m + n) x) \<le>
```
```  1316     (SUP n. INF m. positive_integral (u (m + n)))"
```
```  1317 proof -
```
```  1318   have "(\<integral>\<^isup>+x. SUP n. INF m. u (m + n) x)
```
```  1319       = (SUP n. (\<integral>\<^isup>+x. INF m. u (m + n) x))"
```
```  1320     using assms
```
```  1321     by (intro positive_integral_monotone_convergence_SUP[symmetric] INF_mono)
```
```  1322        (auto simp del: add_Suc simp add: add_Suc[symmetric])
```
```  1323   also have "\<dots> \<le> (SUP n. INF m. positive_integral (u (m + n)))"
```
```  1324     by (auto intro!: SUP_mono bexI le_INFI positive_integral_mono INF_leI)
```
```  1325   finally show ?thesis .
```
```  1326 qed
```
```  1327
```
```  1328 lemma (in measure_space) measure_space_density:
```
```  1329   assumes borel: "u \<in> borel_measurable M"
```
```  1330   shows "measure_space M (\<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x))" (is "measure_space M ?v")
```
```  1331 proof
```
```  1332   show "?v {} = 0" by simp
```
```  1333   show "countably_additive M ?v"
```
```  1334     unfolding countably_additive_def
```
```  1335   proof safe
```
```  1336     fix A :: "nat \<Rightarrow> 'a set"
```
```  1337     assume "range A \<subseteq> sets M"
```
```  1338     hence "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
```
```  1339       using borel by (auto intro: borel_measurable_indicator)
```
```  1340     moreover assume "disjoint_family A"
```
```  1341     note psuminf_indicator[OF this]
```
```  1342     ultimately show "(\<Sum>\<^isub>\<infinity>n. ?v (A n)) = ?v (\<Union>x. A x)"
```
```  1343       by (simp add: positive_integral_psuminf[symmetric])
```
```  1344   qed
```
```  1345 qed
```
```  1346
```
```  1347 lemma (in measure_space) positive_integral_translated_density:
```
```  1348   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```  1349   shows "measure_space.positive_integral M (\<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x)) g =
```
```  1350          (\<integral>\<^isup>+ x. f x * g x)" (is "measure_space.positive_integral M ?T _ = _")
```
```  1351 proof -
```
```  1352   from measure_space_density[OF assms(1)]
```
```  1353   interpret T: measure_space M ?T .
```
```  1354   from borel_measurable_implies_simple_function_sequence[OF assms(2)]
```
```  1355   obtain G where G: "\<And>i. simple_function (G i)" "G \<up> g" by blast
```
```  1356   note G_borel = borel_measurable_simple_function[OF this(1)]
```
```  1357   from T.positive_integral_isoton[OF `G \<up> g` G_borel]
```
```  1358   have *: "(\<lambda>i. T.positive_integral (G i)) \<up> T.positive_integral g" .
```
```  1359   { fix i
```
```  1360     have [simp]: "finite (G i ` space M)"
```
```  1361       using G(1) unfolding simple_function_def by auto
```
```  1362     have "T.positive_integral (G i) = T.simple_integral (G i)"
```
```  1363       using G T.positive_integral_eq_simple_integral by simp
```
```  1364     also have "\<dots> = (\<integral>\<^isup>+x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x))"
```
```  1365       apply (simp add: T.simple_integral_def)
```
```  1366       apply (subst positive_integral_cmult[symmetric])
```
```  1367       using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
```
```  1368       apply (subst positive_integral_setsum[symmetric])
```
```  1369       using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
```
```  1370       by (simp add: setsum_right_distrib field_simps)
```
```  1371     also have "\<dots> = (\<integral>\<^isup>+x. f x * G i x)"
```
```  1372       by (auto intro!: positive_integral_cong
```
```  1373                simp: indicator_def if_distrib setsum_cases)
```
```  1374     finally have "T.positive_integral (G i) = (\<integral>\<^isup>+x. f x * G i x)" . }
```
```  1375   with * have eq_Tg: "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x)) \<up> T.positive_integral g" by simp
```
```  1376   from G(2) have "(\<lambda>i x. f x * G i x) \<up> (\<lambda>x. f x * g x)"
```
```  1377     unfolding isoton_fun_expand by (auto intro!: isoton_cmult_right)
```
```  1378   then have "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x)) \<up> (\<integral>\<^isup>+x. f x * g x)"
```
```  1379     using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pextreal_times)
```
```  1380   with eq_Tg show "T.positive_integral g = (\<integral>\<^isup>+x. f x * g x)"
```
```  1381     unfolding isoton_def by simp
```
```  1382 qed
```
```  1383
```
```  1384 lemma (in measure_space) positive_integral_null_set:
```
```  1385   assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x) = 0"
```
```  1386 proof -
```
```  1387   have "(\<integral>\<^isup>+ x. u x * indicator N x) = (\<integral>\<^isup>+ x. 0)"
```
```  1388   proof (intro positive_integral_cong_AE AE_I)
```
```  1389     show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
```
```  1390       by (auto simp: indicator_def)
```
```  1391     show "\<mu> N = 0" "N \<in> sets M"
```
```  1392       using assms by auto
```
```  1393   qed
```
```  1394   then show ?thesis by simp
```
```  1395 qed
```
```  1396
```
```  1397 lemma (in measure_space) positive_integral_Markov_inequality:
```
```  1398   assumes borel: "u \<in> borel_measurable M" and "A \<in> sets M" and c: "c \<noteq> \<omega>"
```
```  1399   shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x)"
```
```  1400     (is "\<mu> ?A \<le> _ * ?PI")
```
```  1401 proof -
```
```  1402   have "?A \<in> sets M"
```
```  1403     using `A \<in> sets M` borel by auto
```
```  1404   hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x)"
```
```  1405     using positive_integral_indicator by simp
```
```  1406   also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x))"
```
```  1407   proof (rule positive_integral_mono)
```
```  1408     fix x assume "x \<in> space M"
```
```  1409     show "indicator ?A x \<le> c * (u x * indicator A x)"
```
```  1410       by (cases "x \<in> ?A") auto
```
```  1411   qed
```
```  1412   also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x)"
```
```  1413     using assms
```
```  1414     by (auto intro!: positive_integral_cmult borel_measurable_indicator)
```
```  1415   finally show ?thesis .
```
```  1416 qed
```
```  1417
```
```  1418 lemma (in measure_space) positive_integral_0_iff:
```
```  1419   assumes borel: "u \<in> borel_measurable M"
```
```  1420   shows "positive_integral u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
```
```  1421     (is "_ \<longleftrightarrow> \<mu> ?A = 0")
```
```  1422 proof -
```
```  1423   have A: "?A \<in> sets M" using borel by auto
```
```  1424   have u: "(\<integral>\<^isup>+ x. u x * indicator ?A x) = positive_integral u"
```
```  1425     by (auto intro!: positive_integral_cong simp: indicator_def)
```
```  1426
```
```  1427   show ?thesis
```
```  1428   proof
```
```  1429     assume "\<mu> ?A = 0"
```
```  1430     hence "?A \<in> null_sets" using `?A \<in> sets M` by auto
```
```  1431     from positive_integral_null_set[OF this]
```
```  1432     have "0 = (\<integral>\<^isup>+ x. u x * indicator ?A x)" by simp
```
```  1433     thus "positive_integral u = 0" unfolding u by simp
```
```  1434   next
```
```  1435     assume *: "positive_integral u = 0"
```
```  1436     let "?M n" = "{x \<in> space M. 1 \<le> of_nat n * u x}"
```
```  1437     have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
```
```  1438     proof -
```
```  1439       { fix n
```
```  1440         from positive_integral_Markov_inequality[OF borel `?A \<in> sets M`, of "of_nat n"]
```
```  1441         have "\<mu> (?M n \<inter> ?A) = 0" unfolding * u by simp }
```
```  1442       thus ?thesis by simp
```
```  1443     qed
```
```  1444     also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
```
```  1445     proof (safe intro!: continuity_from_below)
```
```  1446       fix n show "?M n \<inter> ?A \<in> sets M"
```
```  1447         using borel by (auto intro!: Int)
```
```  1448     next
```
```  1449       fix n x assume "1 \<le> of_nat n * u x"
```
```  1450       also have "\<dots> \<le> of_nat (Suc n) * u x"
```
```  1451         by (subst (1 2) mult_commute) (auto intro!: pextreal_mult_cancel)
```
```  1452       finally show "1 \<le> of_nat (Suc n) * u x" .
```
```  1453     qed
```
```  1454     also have "\<dots> = \<mu> ?A"
```
```  1455     proof (safe intro!: arg_cong[where f="\<mu>"])
```
```  1456       fix x assume "u x \<noteq> 0" and [simp, intro]: "x \<in> space M"
```
```  1457       show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
```
```  1458       proof (cases "u x")
```
```  1459         case (preal r)
```
```  1460         obtain j where "1 / r \<le> of_nat j" using ex_le_of_nat ..
```
```  1461         hence "1 / r * r \<le> of_nat j * r" using preal unfolding mult_le_cancel_right by auto
```
```  1462         hence "1 \<le> of_nat j * r" using preal `u x \<noteq> 0` by auto
```
```  1463         thus ?thesis using `u x \<noteq> 0` preal by (auto simp: real_of_nat_def[symmetric])
```
```  1464       qed auto
```
```  1465     qed
```
```  1466     finally show "\<mu> ?A = 0" by simp
```
```  1467   qed
```
```  1468 qed
```
```  1469
```
```  1470 lemma (in measure_space) positive_integral_restricted:
```
```  1471   assumes "A \<in> sets M"
```
```  1472   shows "measure_space.positive_integral (restricted_space A) \<mu> f = (\<integral>\<^isup>+ x. f x * indicator A x)"
```
```  1473     (is "measure_space.positive_integral ?R \<mu> f = positive_integral ?f")
```
```  1474 proof -
```
```  1475   have msR: "measure_space ?R \<mu>" by (rule restricted_measure_space[OF `A \<in> sets M`])
```
```  1476   then interpret R: measure_space ?R \<mu> .
```
```  1477   have saR: "sigma_algebra ?R" by fact
```
```  1478   have *: "R.positive_integral f = R.positive_integral ?f"
```
```  1479     by (intro R.positive_integral_cong) auto
```
```  1480   show ?thesis
```
```  1481     unfolding * R.positive_integral_def positive_integral_def
```
```  1482     unfolding simple_function_restricted[OF `A \<in> sets M`]
```
```  1483     apply (simp add: SUPR_def)
```
```  1484     apply (rule arg_cong[where f=Sup])
```
```  1485   proof (auto simp add: image_iff simple_integral_restricted[OF `A \<in> sets M`])
```
```  1486     fix g assume "simple_function (\<lambda>x. g x * indicator A x)"
```
```  1487       "g \<le> f"
```
```  1488     then show "\<exists>x. simple_function x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and>
```
```  1489       (\<integral>\<^isup>Sx. g x * indicator A x) = simple_integral x"
```
```  1490       apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
```
```  1491       by (auto simp: indicator_def le_fun_def)
```
```  1492   next
```
```  1493     fix g assume g: "simple_function g" "g \<le> (\<lambda>x. f x * indicator A x)"
```
```  1494     then have *: "(\<lambda>x. g x * indicator A x) = g"
```
```  1495       "\<And>x. g x * indicator A x = g x"
```
```  1496       "\<And>x. g x \<le> f x"
```
```  1497       by (auto simp: le_fun_def fun_eq_iff indicator_def split: split_if_asm)
```
```  1498     from g show "\<exists>x. simple_function (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and>
```
```  1499       simple_integral g = simple_integral (\<lambda>xa. x xa * indicator A xa)"
```
```  1500       using `A \<in> sets M`[THEN sets_into_space]
```
```  1501       apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
```
```  1502       by (fastsimp simp: le_fun_def *)
```
```  1503   qed
```
```  1504 qed
```
```  1505
```
```  1506 lemma (in measure_space) positive_integral_subalgebra:
```
```  1507   assumes borel: "f \<in> borel_measurable N"
```
```  1508   and N: "sets N \<subseteq> sets M" "space N = space M" and sa: "sigma_algebra N"
```
```  1509   shows "measure_space.positive_integral N \<mu> f = positive_integral f"
```
```  1510 proof -
```
```  1511   interpret N: measure_space N \<mu> using measure_space_subalgebra[OF sa N] .
```
```  1512   from N.borel_measurable_implies_simple_function_sequence[OF borel]
```
```  1513   obtain fs where Nsf: "\<And>i. N.simple_function (fs i)" and "fs \<up> f" by blast
```
```  1514   then have sf: "\<And>i. simple_function (fs i)"
```
```  1515     using simple_function_subalgebra[OF _ N sa] by blast
```
```  1516   from positive_integral_isoton_simple[OF `fs \<up> f` sf] N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
```
```  1517   show ?thesis unfolding isoton_def simple_integral_def N.simple_integral_def `space N = space M` by simp
```
```  1518 qed
```
```  1519
```
```  1520 section "Lebesgue Integral"
```
```  1521
```
```  1522 definition (in measure_space) integrable where
```
```  1523   "integrable f \<longleftrightarrow> f \<in> borel_measurable M \<and>
```
```  1524     (\<integral>\<^isup>+ x. Real (f x)) \<noteq> \<omega> \<and>
```
```  1525     (\<integral>\<^isup>+ x. Real (- f x)) \<noteq> \<omega>"
```
```  1526
```
```  1527 lemma (in measure_space) integrableD[dest]:
```
```  1528   assumes "integrable f"
```
```  1529   shows "f \<in> borel_measurable M"
```
```  1530   "(\<integral>\<^isup>+ x. Real (f x)) \<noteq> \<omega>"
```
```  1531   "(\<integral>\<^isup>+ x. Real (- f x)) \<noteq> \<omega>"
```
```  1532   using assms unfolding integrable_def by auto
```
```  1533
```
```  1534 definition (in measure_space) integral (binder "\<integral> " 10) where
```
```  1535   "integral f = real ((\<integral>\<^isup>+ x. Real (f x))) - real ((\<integral>\<^isup>+ x. Real (- f x)))"
```
```  1536
```
```  1537 lemma (in measure_space) integral_cong:
```
```  1538   assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
```
```  1539   shows "integral f = integral g"
```
```  1540   using assms by (simp cong: positive_integral_cong add: integral_def)
```
```  1541
```
```  1542 lemma (in measure_space) integral_cong_measure:
```
```  1543   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A"
```
```  1544   shows "measure_space.integral M \<nu> f = integral f"
```
```  1545 proof -
```
```  1546   interpret v: measure_space M \<nu>
```
```  1547     by (rule measure_space_cong) fact
```
```  1548   show ?thesis
```
```  1549     unfolding integral_def v.integral_def
```
```  1550     by (simp add: positive_integral_cong_measure[OF assms])
```
```  1551 qed
```
```  1552
```
```  1553 lemma (in measure_space) integral_cong_AE:
```
```  1554   assumes cong: "AE x. f x = g x"
```
```  1555   shows "integral f = integral g"
```
```  1556 proof -
```
```  1557   have "AE x. Real (f x) = Real (g x)"
```
```  1558     using cong by (rule AE_mp) simp
```
```  1559   moreover have "AE x. Real (- f x) = Real (- g x)"
```
```  1560     using cong by (rule AE_mp) simp
```
```  1561   ultimately show ?thesis
```
```  1562     by (simp cong: positive_integral_cong_AE add: integral_def)
```
```  1563 qed
```
```  1564
```
```  1565 lemma (in measure_space) integrable_cong:
```
```  1566   "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable f \<longleftrightarrow> integrable g"
```
```  1567   by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
```
```  1568
```
```  1569 lemma (in measure_space) integral_eq_positive_integral:
```
```  1570   assumes "\<And>x. 0 \<le> f x"
```
```  1571   shows "integral f = real ((\<integral>\<^isup>+ x. Real (f x)))"
```
```  1572 proof -
```
```  1573   have "\<And>x. Real (- f x) = 0" using assms by simp
```
```  1574   thus ?thesis by (simp del: Real_eq_0 add: integral_def)
```
```  1575 qed
```
```  1576
```
```  1577 lemma (in measure_space) integral_vimage:
```
```  1578   assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
```
```  1579   assumes f: "measure_space.integrable M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f"
```
```  1580   shows "integrable (\<lambda>x. f (T x))" (is ?P)
```
```  1581     and "measure_space.integral M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f = (\<integral>x. f (T x))" (is ?I)
```
```  1582 proof -
```
```  1583   interpret T: measure_space M' "\<lambda>A. \<mu> (T -` A \<inter> space M)"
```
```  1584     using T by (rule measure_space_vimage) auto
```
```  1585   from measurable_comp[OF T(2), of f borel]
```
```  1586   have borel: "(\<lambda>x. Real (f x)) \<in> borel_measurable M'" "(\<lambda>x. Real (- f x)) \<in> borel_measurable M'"
```
```  1587     and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
```
```  1588     using f unfolding T.integrable_def by (auto simp: comp_def)
```
```  1589   then show ?P ?I
```
```  1590     using f unfolding T.integral_def integral_def T.integrable_def integrable_def
```
```  1591     unfolding borel[THEN positive_integral_vimage[OF T]] by auto
```
```  1592 qed
```
```  1593
```
```  1594 lemma (in measure_space) integral_minus[intro, simp]:
```
```  1595   assumes "integrable f"
```
```  1596   shows "integrable (\<lambda>x. - f x)" "(\<integral>x. - f x) = - integral f"
```
```  1597   using assms by (auto simp: integrable_def integral_def)
```
```  1598
```
```  1599 lemma (in measure_space) integral_of_positive_diff:
```
```  1600   assumes integrable: "integrable u" "integrable v"
```
```  1601   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"
```
```  1602   shows "integrable f" and "integral f = integral u - integral v"
```
```  1603 proof -
```
```  1604   let ?PI = positive_integral
```
```  1605   let "?f x" = "Real (f x)"
```
```  1606   let "?mf x" = "Real (- f x)"
```
```  1607   let "?u x" = "Real (u x)"
```
```  1608   let "?v x" = "Real (v x)"
```
```  1609
```
```  1610   from borel_measurable_diff[of u v] integrable
```
```  1611   have f_borel: "?f \<in> borel_measurable M" and
```
```  1612     mf_borel: "?mf \<in> borel_measurable M" and
```
```  1613     v_borel: "?v \<in> borel_measurable M" and
```
```  1614     u_borel: "?u \<in> borel_measurable M" and
```
```  1615     "f \<in> borel_measurable M"
```
```  1616     by (auto simp: f_def[symmetric] integrable_def)
```
```  1617
```
```  1618   have "?PI (\<lambda>x. Real (u x - v x)) \<le> ?PI ?u"
```
```  1619     using pos by (auto intro!: positive_integral_mono)
```
```  1620   moreover have "?PI (\<lambda>x. Real (v x - u x)) \<le> ?PI ?v"
```
```  1621     using pos by (auto intro!: positive_integral_mono)
```
```  1622   ultimately show f: "integrable f"
```
```  1623     using `integrable u` `integrable v` `f \<in> borel_measurable M`
```
```  1624     by (auto simp: integrable_def f_def)
```
```  1625   hence mf: "integrable (\<lambda>x. - f x)" ..
```
```  1626
```
```  1627   have *: "\<And>x. Real (- v x) = 0" "\<And>x. Real (- u x) = 0"
```
```  1628     using pos by auto
```
```  1629
```
```  1630   have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
```
```  1631     unfolding f_def using pos by simp
```
```  1632   hence "?PI (\<lambda>x. ?u x + ?mf x) = ?PI (\<lambda>x. ?v x + ?f x)" by simp
```
```  1633   hence "real (?PI ?u + ?PI ?mf) = real (?PI ?v + ?PI ?f)"
```
```  1634     using positive_integral_add[OF u_borel mf_borel]
```
```  1635     using positive_integral_add[OF v_borel f_borel]
```
```  1636     by auto
```
```  1637   then show "integral f = integral u - integral v"
```
```  1638     using f mf `integrable u` `integrable v`
```
```  1639     unfolding integral_def integrable_def *
```
```  1640     by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?v", cases "?PI ?u")
```
```  1641        (auto simp add: field_simps)
```
```  1642 qed
```
```  1643
```
```  1644 lemma (in measure_space) integral_linear:
```
```  1645   assumes "integrable f" "integrable g" and "0 \<le> a"
```
```  1646   shows "integrable (\<lambda>t. a * f t + g t)"
```
```  1647   and "(\<integral> t. a * f t + g t) = a * integral f + integral g"
```
```  1648 proof -
```
```  1649   let ?PI = positive_integral
```
```  1650   let "?f x" = "Real (f x)"
```
```  1651   let "?g x" = "Real (g x)"
```
```  1652   let "?mf x" = "Real (- f x)"
```
```  1653   let "?mg x" = "Real (- g x)"
```
```  1654   let "?p t" = "max 0 (a * f t) + max 0 (g t)"
```
```  1655   let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
```
```  1656
```
```  1657   have pos: "?f \<in> borel_measurable M" "?g \<in> borel_measurable M"
```
```  1658     and neg: "?mf \<in> borel_measurable M" "?mg \<in> borel_measurable M"
```
```  1659     and p: "?p \<in> borel_measurable M"
```
```  1660     and n: "?n \<in> borel_measurable M"
```
```  1661     using assms by (simp_all add: integrable_def)
```
```  1662
```
```  1663   have *: "\<And>x. Real (?p x) = Real a * ?f x + ?g x"
```
```  1664           "\<And>x. Real (?n x) = Real a * ?mf x + ?mg x"
```
```  1665           "\<And>x. Real (- ?p x) = 0"
```
```  1666           "\<And>x. Real (- ?n x) = 0"
```
```  1667     using `0 \<le> a` by (auto simp: max_def min_def zero_le_mult_iff mult_le_0_iff add_nonpos_nonpos)
```
```  1668
```
```  1669   note linear =
```
```  1670     positive_integral_linear[OF pos]
```
```  1671     positive_integral_linear[OF neg]
```
```  1672
```
```  1673   have "integrable ?p" "integrable ?n"
```
```  1674       "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
```
```  1675     using assms p n unfolding integrable_def * linear by auto
```
```  1676   note diff = integral_of_positive_diff[OF this]
```
```  1677
```
```  1678   show "integrable (\<lambda>t. a * f t + g t)" by (rule diff)
```
```  1679
```
```  1680   from assms show "(\<integral> t. a * f t + g t) = a * integral f + integral g"
```
```  1681     unfolding diff(2) unfolding integral_def * linear integrable_def
```
```  1682     by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?g", cases "?PI ?mg")
```
```  1683        (auto simp add: field_simps zero_le_mult_iff)
```
```  1684 qed
```
```  1685
```
```  1686 lemma (in measure_space) integral_add[simp, intro]:
```
```  1687   assumes "integrable f" "integrable g"
```
```  1688   shows "integrable (\<lambda>t. f t + g t)"
```
```  1689   and "(\<integral> t. f t + g t) = integral f + integral g"
```
```  1690   using assms integral_linear[where a=1] by auto
```
```  1691
```
```  1692 lemma (in measure_space) integral_zero[simp, intro]:
```
```  1693   shows "integrable (\<lambda>x. 0)"
```
```  1694   and "(\<integral> x.0) = 0"
```
```  1695   unfolding integrable_def integral_def
```
```  1696   by (auto simp add: borel_measurable_const)
```
```  1697
```
```  1698 lemma (in measure_space) integral_cmult[simp, intro]:
```
```  1699   assumes "integrable f"
```
```  1700   shows "integrable (\<lambda>t. a * f t)" (is ?P)
```
```  1701   and "(\<integral> t. a * f t) = a * integral f" (is ?I)
```
```  1702 proof -
```
```  1703   have "integrable (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t) = a * integral f"
```
```  1704   proof (cases rule: le_cases)
```
```  1705     assume "0 \<le> a" show ?thesis
```
```  1706       using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
```
```  1707       by (simp add: integral_zero)
```
```  1708   next
```
```  1709     assume "a \<le> 0" hence "0 \<le> - a" by auto
```
```  1710     have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
```
```  1711     show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
```
```  1712         integral_minus(1)[of "\<lambda>t. - a * f t"]
```
```  1713       unfolding * integral_zero by simp
```
```  1714   qed
```
```  1715   thus ?P ?I by auto
```
```  1716 qed
```
```  1717
```
```  1718 lemma (in measure_space) integral_multc:
```
```  1719   assumes "integrable f"
```
```  1720   shows "(\<integral> x. f x * c) = integral f * c"
```
```  1721   unfolding mult_commute[of _ c] integral_cmult[OF assms] ..
```
```  1722
```
```  1723 lemma (in measure_space) integral_mono_AE:
```
```  1724   assumes fg: "integrable f" "integrable g"
```
```  1725   and mono: "AE t. f t \<le> g t"
```
```  1726   shows "integral f \<le> integral g"
```
```  1727 proof -
```
```  1728   have "AE x. Real (f x) \<le> Real (g x)"
```
```  1729     using mono by (rule AE_mp) (auto intro!: AE_cong)
```
```  1730   moreover have "AE x. Real (- g x) \<le> Real (- f x)"
```
```  1731     using mono by (rule AE_mp) (auto intro!: AE_cong)
```
```  1732   ultimately show ?thesis using fg
```
```  1733     by (auto simp: integral_def integrable_def diff_minus
```
```  1734              intro!: add_mono real_of_pextreal_mono positive_integral_mono_AE)
```
```  1735 qed
```
```  1736
```
```  1737 lemma (in measure_space) integral_mono:
```
```  1738   assumes fg: "integrable f" "integrable g"
```
```  1739   and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
```
```  1740   shows "integral f \<le> integral g"
```
```  1741   apply (rule integral_mono_AE[OF fg])
```
```  1742   using mono by (rule AE_cong) auto
```
```  1743
```
```  1744 lemma (in measure_space) integral_diff[simp, intro]:
```
```  1745   assumes f: "integrable f" and g: "integrable g"
```
```  1746   shows "integrable (\<lambda>t. f t - g t)"
```
```  1747   and "(\<integral> t. f t - g t) = integral f - integral g"
```
```  1748   using integral_add[OF f integral_minus(1)[OF g]]
```
```  1749   unfolding diff_minus integral_minus(2)[OF g]
```
```  1750   by auto
```
```  1751
```
```  1752 lemma (in measure_space) integral_indicator[simp, intro]:
```
```  1753   assumes "a \<in> sets M" and "\<mu> a \<noteq> \<omega>"
```
```  1754   shows "integral (indicator a) = real (\<mu> a)" (is ?int)
```
```  1755   and "integrable (indicator a)" (is ?able)
```
```  1756 proof -
```
```  1757   have *:
```
```  1758     "\<And>A x. Real (indicator A x) = indicator A x"
```
```  1759     "\<And>A x. Real (- indicator A x) = 0" unfolding indicator_def by auto
```
```  1760   show ?int ?able
```
```  1761     using assms unfolding integral_def integrable_def
```
```  1762     by (auto simp: * positive_integral_indicator borel_measurable_indicator)
```
```  1763 qed
```
```  1764
```
```  1765 lemma (in measure_space) integral_cmul_indicator:
```
```  1766   assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<omega>"
```
```  1767   shows "integrable (\<lambda>x. c * indicator A x)" (is ?P)
```
```  1768   and "(\<integral>x. c * indicator A x) = c * real (\<mu> A)" (is ?I)
```
```  1769 proof -
```
```  1770   show ?P
```
```  1771   proof (cases "c = 0")
```
```  1772     case False with assms show ?thesis by simp
```
```  1773   qed simp
```
```  1774
```
```  1775   show ?I
```
```  1776   proof (cases "c = 0")
```
```  1777     case False with assms show ?thesis by simp
```
```  1778   qed simp
```
```  1779 qed
```
```  1780
```
```  1781 lemma (in measure_space) integral_setsum[simp, intro]:
```
```  1782   assumes "\<And>n. n \<in> S \<Longrightarrow> integrable (f n)"
```
```  1783   shows "(\<integral>x. \<Sum> i \<in> S. f i x) = (\<Sum> i \<in> S. integral (f i))" (is "?int S")
```
```  1784     and "integrable (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
```
```  1785 proof -
```
```  1786   have "?int S \<and> ?I S"
```
```  1787   proof (cases "finite S")
```
```  1788     assume "finite S"
```
```  1789     from this assms show ?thesis by (induct S) simp_all
```
```  1790   qed simp
```
```  1791   thus "?int S" and "?I S" by auto
```
```  1792 qed
```
```  1793
```
```  1794 lemma (in measure_space) integrable_abs:
```
```  1795   assumes "integrable f"
```
```  1796   shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
```
```  1797 proof -
```
```  1798   have *:
```
```  1799     "\<And>x. Real \<bar>f x\<bar> = Real (f x) + Real (- f x)"
```
```  1800     "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
```
```  1801   have abs: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" and
```
```  1802     f: "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
```
```  1803         "(\<lambda>x. Real (- f x)) \<in> borel_measurable M"
```
```  1804     using assms unfolding integrable_def by auto
```
```  1805   from abs assms show ?thesis unfolding integrable_def *
```
```  1806     using positive_integral_linear[OF f, of 1] by simp
```
```  1807 qed
```
```  1808
```
```  1809 lemma (in measure_space) integral_subalgebra:
```
```  1810   assumes borel: "f \<in> borel_measurable N"
```
```  1811   and N: "sets N \<subseteq> sets M" "space N = space M" and sa: "sigma_algebra N"
```
```  1812   shows "measure_space.integrable N \<mu> f \<longleftrightarrow> integrable f" (is ?P)
```
```  1813     and "measure_space.integral N \<mu> f = integral f" (is ?I)
```
```  1814 proof -
```
```  1815   interpret N: measure_space N \<mu> using measure_space_subalgebra[OF sa N] .
```
```  1816   have "(\<lambda>x. Real (f x)) \<in> borel_measurable N" "(\<lambda>x. Real (- f x)) \<in> borel_measurable N"
```
```  1817     using borel by auto
```
```  1818   note * = this[THEN positive_integral_subalgebra[OF _ N sa]]
```
```  1819   have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
```
```  1820     using assms unfolding measurable_def by auto
```
```  1821   then show ?P ?I unfolding integrable_def N.integrable_def integral_def N.integral_def
```
```  1822     unfolding * by auto
```
```  1823 qed
```
```  1824
```
```  1825 lemma (in measure_space) integrable_bound:
```
```  1826   assumes "integrable f"
```
```  1827   and f: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1828     "\<And>x. x \<in> space M \<Longrightarrow> \<bar>g x\<bar> \<le> f x"
```
```  1829   assumes borel: "g \<in> borel_measurable M"
```
```  1830   shows "integrable g"
```
```  1831 proof -
```
```  1832   have "(\<integral>\<^isup>+ x. Real (g x)) \<le> (\<integral>\<^isup>+ x. Real \<bar>g x\<bar>)"
```
```  1833     by (auto intro!: positive_integral_mono)
```
```  1834   also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x))"
```
```  1835     using f by (auto intro!: positive_integral_mono)
```
```  1836   also have "\<dots> < \<omega>"
```
```  1837     using `integrable f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
```
```  1838   finally have pos: "(\<integral>\<^isup>+ x. Real (g x)) < \<omega>" .
```
```  1839
```
```  1840   have "(\<integral>\<^isup>+ x. Real (- g x)) \<le> (\<integral>\<^isup>+ x. Real (\<bar>g x\<bar>))"
```
```  1841     by (auto intro!: positive_integral_mono)
```
```  1842   also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x))"
```
```  1843     using f by (auto intro!: positive_integral_mono)
```
```  1844   also have "\<dots> < \<omega>"
```
```  1845     using `integrable f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
```
```  1846   finally have neg: "(\<integral>\<^isup>+ x. Real (- g x)) < \<omega>" .
```
```  1847
```
```  1848   from neg pos borel show ?thesis
```
```  1849     unfolding integrable_def by auto
```
```  1850 qed
```
```  1851
```
```  1852 lemma (in measure_space) integrable_abs_iff:
```
```  1853   "f \<in> borel_measurable M \<Longrightarrow> integrable (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable f"
```
```  1854   by (auto intro!: integrable_bound[where g=f] integrable_abs)
```
```  1855
```
```  1856 lemma (in measure_space) integrable_max:
```
```  1857   assumes int: "integrable f" "integrable g"
```
```  1858   shows "integrable (\<lambda> x. max (f x) (g x))"
```
```  1859 proof (rule integrable_bound)
```
```  1860   show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
```
```  1861     using int by (simp add: integrable_abs)
```
```  1862   show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
```
```  1863     using int unfolding integrable_def by auto
```
```  1864 next
```
```  1865   fix x assume "x \<in> space M"
```
```  1866   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>"
```
```  1867     by auto
```
```  1868 qed
```
```  1869
```
```  1870 lemma (in measure_space) integrable_min:
```
```  1871   assumes int: "integrable f" "integrable g"
```
```  1872   shows "integrable (\<lambda> x. min (f x) (g x))"
```
```  1873 proof (rule integrable_bound)
```
```  1874   show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
```
```  1875     using int by (simp add: integrable_abs)
```
```  1876   show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
```
```  1877     using int unfolding integrable_def by auto
```
```  1878 next
```
```  1879   fix x assume "x \<in> space M"
```
```  1880   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>"
```
```  1881     by auto
```
```  1882 qed
```
```  1883
```
```  1884 lemma (in measure_space) integral_triangle_inequality:
```
```  1885   assumes "integrable f"
```
```  1886   shows "\<bar>integral f\<bar> \<le> (\<integral>x. \<bar>f x\<bar>)"
```
```  1887 proof -
```
```  1888   have "\<bar>integral f\<bar> = max (integral f) (- integral f)" by auto
```
```  1889   also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar>)"
```
```  1890       using assms integral_minus(2)[of f, symmetric]
```
```  1891       by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
```
```  1892   finally show ?thesis .
```
```  1893 qed
```
```  1894
```
```  1895 lemma (in measure_space) integral_positive:
```
```  1896   assumes "integrable f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
```
```  1897   shows "0 \<le> integral f"
```
```  1898 proof -
```
```  1899   have "0 = (\<integral>x. 0)" by (auto simp: integral_zero)
```
```  1900   also have "\<dots> \<le> integral f"
```
```  1901     using assms by (rule integral_mono[OF integral_zero(1)])
```
```  1902   finally show ?thesis .
```
```  1903 qed
```
```  1904
```
```  1905 lemma (in measure_space) integral_monotone_convergence_pos:
```
```  1906   assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
```
```  1907   and pos: "\<And>x i. 0 \<le> f i x"
```
```  1908   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
```
```  1909   and ilim: "(\<lambda>i. integral (f i)) ----> x"
```
```  1910   shows "integrable u"
```
```  1911   and "integral u = x"
```
```  1912 proof -
```
```  1913   { fix x have "0 \<le> u x"
```
```  1914       using mono pos[of 0 x] incseq_le[OF _ lim, of x 0]
```
```  1915       by (simp add: mono_def incseq_def) }
```
```  1916   note pos_u = this
```
```  1917
```
```  1918   hence [simp]: "\<And>i x. Real (- f i x) = 0" "\<And>x. Real (- u x) = 0"
```
```  1919     using pos by auto
```
```  1920
```
```  1921   have SUP_F: "\<And>x. (SUP n. Real (f n x)) = Real (u x)"
```
```  1922     using mono pos pos_u lim by (rule SUP_eq_LIMSEQ[THEN iffD2])
```
```  1923
```
```  1924   have borel_f: "\<And>i. (\<lambda>x. Real (f i x)) \<in> borel_measurable M"
```
```  1925     using i unfolding integrable_def by auto
```
```  1926   hence "(\<lambda>x. SUP i. Real (f i x)) \<in> borel_measurable M"
```
```  1927     by auto
```
```  1928   hence borel_u: "u \<in> borel_measurable M"
```
```  1929     using pos_u by (auto simp: borel_measurable_Real_eq SUP_F)
```
```  1930
```
```  1931   have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. Real (f n x)) = Real (integral (f n))"
```
```  1932     using i unfolding integral_def integrable_def by (auto simp: Real_real)
```
```  1933
```
```  1934   have pos_integral: "\<And>n. 0 \<le> integral (f n)"
```
```  1935     using pos i by (auto simp: integral_positive)
```
```  1936   hence "0 \<le> x"
```
```  1937     using LIMSEQ_le_const[OF ilim, of 0] by auto
```
```  1938
```
```  1939   have "(\<lambda>i. (\<integral>\<^isup>+ x. Real (f i x))) \<up> (\<integral>\<^isup>+ x. Real (u x))"
```
```  1940   proof (rule positive_integral_isoton)
```
```  1941     from SUP_F mono pos
```
```  1942     show "(\<lambda>i x. Real (f i x)) \<up> (\<lambda>x. Real (u x))"
```
```  1943       unfolding isoton_fun_expand by (auto simp: isoton_def mono_def)
```
```  1944   qed (rule borel_f)
```
```  1945   hence pI: "(\<integral>\<^isup>+ x. Real (u x)) =
```
```  1946       (SUP n. (\<integral>\<^isup>+ x. Real (f n x)))"
```
```  1947     unfolding isoton_def by simp
```
```  1948   also have "\<dots> = Real x" unfolding integral_eq
```
```  1949   proof (rule SUP_eq_LIMSEQ[THEN iffD2])
```
```  1950     show "mono (\<lambda>n. integral (f n))"
```
```  1951       using mono i by (auto simp: mono_def intro!: integral_mono)
```
```  1952     show "\<And>n. 0 \<le> integral (f n)" using pos_integral .
```
```  1953     show "0 \<le> x" using `0 \<le> x` .
```
```  1954     show "(\<lambda>n. integral (f n)) ----> x" using ilim .
```
```  1955   qed
```
```  1956   finally show  "integrable u" "integral u = x" using borel_u `0 \<le> x`
```
```  1957     unfolding integrable_def integral_def by auto
```
```  1958 qed
```
```  1959
```
```  1960 lemma (in measure_space) integral_monotone_convergence:
```
```  1961   assumes f: "\<And>i. integrable (f i)" and "mono f"
```
```  1962   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
```
```  1963   and ilim: "(\<lambda>i. integral (f i)) ----> x"
```
```  1964   shows "integrable u"
```
```  1965   and "integral u = x"
```
```  1966 proof -
```
```  1967   have 1: "\<And>i. integrable (\<lambda>x. f i x - f 0 x)"
```
```  1968       using f by (auto intro!: integral_diff)
```
```  1969   have 2: "\<And>x. mono (\<lambda>n. f n x - f 0 x)" using `mono f`
```
```  1970       unfolding mono_def le_fun_def by auto
```
```  1971   have 3: "\<And>x n. 0 \<le> f n x - f 0 x" using `mono f`
```
```  1972       unfolding mono_def le_fun_def by (auto simp: field_simps)
```
```  1973   have 4: "\<And>x. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
```
```  1974     using lim by (auto intro!: LIMSEQ_diff)
```
```  1975   have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x)) ----> x - integral (f 0)"
```
```  1976     using f ilim by (auto intro!: LIMSEQ_diff simp: integral_diff)
```
```  1977   note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5]
```
```  1978   have "integrable (\<lambda>x. (u x - f 0 x) + f 0 x)"
```
```  1979     using diff(1) f by (rule integral_add(1))
```
```  1980   with diff(2) f show "integrable u" "integral u = x"
```
```  1981     by (auto simp: integral_diff)
```
```  1982 qed
```
```  1983
```
```  1984 lemma (in measure_space) integral_0_iff:
```
```  1985   assumes "integrable f"
```
```  1986   shows "(\<integral>x. \<bar>f x\<bar>) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
```
```  1987 proof -
```
```  1988   have *: "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
```
```  1989   have "integrable (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
```
```  1990   hence "(\<lambda>x. Real (\<bar>f x\<bar>)) \<in> borel_measurable M"
```
```  1991     "(\<integral>\<^isup>+ x. Real \<bar>f x\<bar>) \<noteq> \<omega>" unfolding integrable_def by auto
```
```  1992   from positive_integral_0_iff[OF this(1)] this(2)
```
```  1993   show ?thesis unfolding integral_def *
```
```  1994     by (simp add: real_of_pextreal_eq_0)
```
```  1995 qed
```
```  1996
```
```  1997 lemma (in measure_space) positive_integral_omega:
```
```  1998   assumes "f \<in> borel_measurable M"
```
```  1999   and "positive_integral f \<noteq> \<omega>"
```
```  2000   shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
```
```  2001 proof -
```
```  2002   have "\<omega> * \<mu> (f -` {\<omega>} \<inter> space M) = (\<integral>\<^isup>+ x. \<omega> * indicator (f -` {\<omega>} \<inter> space M) x)"
```
```  2003     using positive_integral_cmult_indicator[OF borel_measurable_vimage, OF assms(1), of \<omega> \<omega>] by simp
```
```  2004   also have "\<dots> \<le> positive_integral f"
```
```  2005     by (auto intro!: positive_integral_mono simp: indicator_def)
```
```  2006   finally show ?thesis
```
```  2007     using assms(2) by (cases ?thesis) auto
```
```  2008 qed
```
```  2009
```
```  2010 lemma (in measure_space) positive_integral_omega_AE:
```
```  2011   assumes "f \<in> borel_measurable M" "positive_integral f \<noteq> \<omega>" shows "AE x. f x \<noteq> \<omega>"
```
```  2012 proof (rule AE_I)
```
```  2013   show "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
```
```  2014     by (rule positive_integral_omega[OF assms])
```
```  2015   show "f -` {\<omega>} \<inter> space M \<in> sets M"
```
```  2016     using assms by (auto intro: borel_measurable_vimage)
```
```  2017 qed auto
```
```  2018
```
```  2019 lemma (in measure_space) simple_integral_omega:
```
```  2020   assumes "simple_function f"
```
```  2021   and "simple_integral f \<noteq> \<omega>"
```
```  2022   shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
```
```  2023 proof (rule positive_integral_omega)
```
```  2024   show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
```
```  2025   show "positive_integral f \<noteq> \<omega>"
```
```  2026     using assms by (simp add: positive_integral_eq_simple_integral)
```
```  2027 qed
```
```  2028
```
```  2029 lemma (in measure_space) integral_real:
```
```  2030   fixes f :: "'a \<Rightarrow> pextreal"
```
```  2031   assumes "AE x. f x \<noteq> \<omega>"
```
```  2032   shows "(\<integral>x. real (f x)) = real (positive_integral f)" (is ?plus)
```
```  2033     and "(\<integral>x. - real (f x)) = - real (positive_integral f)" (is ?minus)
```
```  2034 proof -
```
```  2035   have "(\<integral>\<^isup>+ x. Real (real (f x))) = positive_integral f"
```
```  2036     apply (rule positive_integral_cong_AE)
```
```  2037     apply (rule AE_mp[OF assms(1)])
```
```  2038     by (auto intro!: AE_cong simp: Real_real)
```
```  2039   moreover
```
```  2040   have "(\<integral>\<^isup>+ x. Real (- real (f x))) = (\<integral>\<^isup>+ x. 0)"
```
```  2041     by (intro positive_integral_cong) auto
```
```  2042   ultimately show ?plus ?minus
```
```  2043     by (auto simp: integral_def integrable_def)
```
```  2044 qed
```
```  2045
```
```  2046 lemma (in measure_space) integral_dominated_convergence:
```
```  2047   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"
```
```  2048   and w: "integrable w" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> w x"
```
```  2049   and u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
```
```  2050   shows "integrable u'"
```
```  2051   and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar>)) ----> 0" (is "?lim_diff")
```
```  2052   and "(\<lambda>i. integral (u i)) ----> integral u'" (is ?lim)
```
```  2053 proof -
```
```  2054   { fix x j assume x: "x \<in> space M"
```
```  2055     from u'[OF x] have "(\<lambda>i. \<bar>u i x\<bar>) ----> \<bar>u' x\<bar>" by (rule LIMSEQ_imp_rabs)
```
```  2056     from LIMSEQ_le_const2[OF this]
```
```  2057     have "\<bar>u' x\<bar> \<le> w x" using bound[OF x] by auto }
```
```  2058   note u'_bound = this
```
```  2059
```
```  2060   from u[unfolded integrable_def]
```
```  2061   have u'_borel: "u' \<in> borel_measurable M"
```
```  2062     using u' by (blast intro: borel_measurable_LIMSEQ[of u])
```
```  2063
```
```  2064   show "integrable u'"
```
```  2065   proof (rule integrable_bound)
```
```  2066     show "integrable w" by fact
```
```  2067     show "u' \<in> borel_measurable M" by fact
```
```  2068   next
```
```  2069     fix x assume x: "x \<in> space M"
```
```  2070     thus "0 \<le> w x" by fact
```
```  2071     show "\<bar>u' x\<bar> \<le> w x" using u'_bound[OF x] .
```
```  2072   qed
```
```  2073
```
```  2074   let "?diff n x" = "2 * w x - \<bar>u n x - u' x\<bar>"
```
```  2075   have diff: "\<And>n. integrable (\<lambda>x. \<bar>u n x - u' x\<bar>)"
```
```  2076     using w u `integrable u'`
```
```  2077     by (auto intro!: integral_add integral_diff integral_cmult integrable_abs)
```
```  2078
```
```  2079   { fix j x assume x: "x \<in> space M"
```
```  2080     have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
```
```  2081     also have "\<dots> \<le> w x + w x"
```
```  2082       by (rule add_mono[OF bound[OF x] u'_bound[OF x]])
```
```  2083     finally have "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp }
```
```  2084   note diff_less_2w = this
```
```  2085
```
```  2086   have PI_diff: "\<And>m n. (\<integral>\<^isup>+ x. Real (?diff (m + n) x)) =
```
```  2087     (\<integral>\<^isup>+ x. Real (2 * w x)) - (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)"
```
```  2088     using diff w diff_less_2w
```
```  2089     by (subst positive_integral_diff[symmetric])
```
```  2090        (auto simp: integrable_def intro!: positive_integral_cong)
```
```  2091
```
```  2092   have "integrable (\<lambda>x. 2 * w x)"
```
```  2093     using w by (auto intro: integral_cmult)
```
```  2094   hence I2w_fin: "(\<integral>\<^isup>+ x. Real (2 * w x)) \<noteq> \<omega>" and
```
```  2095     borel_2w: "(\<lambda>x. Real (2 * w x)) \<in> borel_measurable M"
```
```  2096     unfolding integrable_def by auto
```
```  2097
```
```  2098   have "(INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)) = 0" (is "?lim_SUP = 0")
```
```  2099   proof cases
```
```  2100     assume eq_0: "(\<integral>\<^isup>+ x. Real (2 * w x)) = 0"
```
```  2101     have "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar>) \<le> (\<integral>\<^isup>+ x. Real (2 * w x))"
```
```  2102     proof (rule positive_integral_mono)
```
```  2103       fix i x assume "x \<in> space M" from diff_less_2w[OF this, of i]
```
```  2104       show "Real \<bar>u i x - u' x\<bar> \<le> Real (2 * w x)" by auto
```
```  2105     qed
```
```  2106     hence "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar>) = 0" using eq_0 by auto
```
```  2107     thus ?thesis by simp
```
```  2108   next
```
```  2109     assume neq_0: "(\<integral>\<^isup>+ x. Real (2 * w x)) \<noteq> 0"
```
```  2110     have "(\<integral>\<^isup>+ x. Real (2 * w x)) = (\<integral>\<^isup>+ x. SUP n. INF m. Real (?diff (m + n) x))"
```
```  2111     proof (rule positive_integral_cong, subst add_commute)
```
```  2112       fix x assume x: "x \<in> space M"
```
```  2113       show "Real (2 * w x) = (SUP n. INF m. Real (?diff (n + m) x))"
```
```  2114       proof (rule LIMSEQ_imp_lim_INF[symmetric])
```
```  2115         fix j show "0 \<le> ?diff j x" using diff_less_2w[OF x, of j] by simp
```
```  2116       next
```
```  2117         have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
```
```  2118           using u'[OF x] by (safe intro!: LIMSEQ_diff LIMSEQ_const LIMSEQ_imp_rabs)
```
```  2119         thus "(\<lambda>i. ?diff i x) ----> 2 * w x" by simp
```
```  2120       qed
```
```  2121     qed
```
```  2122     also have "\<dots> \<le> (SUP n. INF m. (\<integral>\<^isup>+ x. Real (?diff (m + n) x)))"
```
```  2123       using u'_borel w u unfolding integrable_def
```
```  2124       by (auto intro!: positive_integral_lim_INF)
```
```  2125     also have "\<dots> = (\<integral>\<^isup>+ x. Real (2 * w x)) -
```
```  2126         (INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>))"
```
```  2127       unfolding PI_diff pextreal_INF_minus[OF I2w_fin] pextreal_SUP_minus ..
```
```  2128     finally show ?thesis using neq_0 I2w_fin by (rule pextreal_le_minus_imp_0)
```
```  2129   qed
```
```  2130
```
```  2131   have [simp]: "\<And>n m x. Real (- \<bar>u (m + n) x - u' x\<bar>) = 0" by auto
```
```  2132
```
```  2133   have [simp]: "\<And>n m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>) =
```
```  2134     Real ((\<integral>x. \<bar>u (n + m) x - u' x\<bar>))"
```
```  2135     using diff by (subst add_commute) (simp add: integral_def integrable_def Real_real)
```
```  2136
```
```  2137   have "(SUP n. INF m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)) \<le> ?lim_SUP"
```
```  2138     (is "?lim_INF \<le> _") by (subst (1 2) add_commute) (rule lim_INF_le_lim_SUP)
```
```  2139   hence "?lim_INF = Real 0" "?lim_SUP = Real 0" using `?lim_SUP = 0` by auto
```
```  2140   thus ?lim_diff using diff by (auto intro!: integral_positive lim_INF_eq_lim_SUP)
```
```  2141
```
```  2142   show ?lim
```
```  2143   proof (rule LIMSEQ_I)
```
```  2144     fix r :: real assume "0 < r"
```
```  2145     from LIMSEQ_D[OF `?lim_diff` this]
```
```  2146     obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar>) < r"
```
```  2147       using diff by (auto simp: integral_positive)
```
```  2148
```
```  2149     show "\<exists>N. \<forall>n\<ge>N. norm (integral (u n) - integral u') < r"
```
```  2150     proof (safe intro!: exI[of _ N])
```
```  2151       fix n assume "N \<le> n"
```
```  2152       have "\<bar>integral (u n) - integral u'\<bar> = \<bar>(\<integral>x. u n x - u' x)\<bar>"
```
```  2153         using u `integrable u'` by (auto simp: integral_diff)
```
```  2154       also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar>)" using u `integrable u'`
```
```  2155         by (rule_tac integral_triangle_inequality) (auto intro!: integral_diff)
```
```  2156       also note N[OF `N \<le> n`]
```
```  2157       finally show "norm (integral (u n) - integral u') < r" by simp
```
```  2158     qed
```
```  2159   qed
```
```  2160 qed
```
```  2161
```
```  2162 lemma (in measure_space) integral_sums:
```
```  2163   assumes borel: "\<And>i. integrable (f i)"
```
```  2164   and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
```
```  2165   and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar>))"
```
```  2166   shows "integrable (\<lambda>x. (\<Sum>i. f i x))" (is "integrable ?S")
```
```  2167   and "(\<lambda>i. integral (f i)) sums (\<integral>x. (\<Sum>i. f i x))" (is ?integral)
```
```  2168 proof -
```
```  2169   have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
```
```  2170     using summable unfolding summable_def by auto
```
```  2171   from bchoice[OF this]
```
```  2172   obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
```
```  2173
```
```  2174   let "?w y" = "if y \<in> space M then w y else 0"
```
```  2175
```
```  2176   obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar>)) sums x"
```
```  2177     using sums unfolding summable_def ..
```
```  2178
```
```  2179   have 1: "\<And>n. integrable (\<lambda>x. \<Sum>i = 0..<n. f i x)"
```
```  2180     using borel by (auto intro!: integral_setsum)
```
```  2181
```
```  2182   { fix j x assume [simp]: "x \<in> space M"
```
```  2183     have "\<bar>\<Sum>i = 0..< j. f i x\<bar> \<le> (\<Sum>i = 0..< j. \<bar>f i x\<bar>)" by (rule setsum_abs)
```
```  2184     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
```
```  2185     finally have "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp }
```
```  2186   note 2 = this
```
```  2187
```
```  2188   have 3: "integrable ?w"
```
```  2189   proof (rule integral_monotone_convergence(1))
```
```  2190     let "?F n y" = "(\<Sum>i = 0..<n. \<bar>f i y\<bar>)"
```
```  2191     let "?w' n y" = "if y \<in> space M then ?F n y else 0"
```
```  2192     have "\<And>n. integrable (?F n)"
```
```  2193       using borel by (auto intro!: integral_setsum integrable_abs)
```
```  2194     thus "\<And>n. integrable (?w' n)" by (simp cong: integrable_cong)
```
```  2195     show "mono ?w'"
```
```  2196       by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
```
```  2197     { fix x show "(\<lambda>n. ?w' n x) ----> ?w x"
```
```  2198         using w by (cases "x \<in> space M") (simp_all add: LIMSEQ_const sums_def) }
```
```  2199     have *: "\<And>n. integral (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar>))"
```
```  2200       using borel by (simp add: integral_setsum integrable_abs cong: integral_cong)
```
```  2201     from abs_sum
```
```  2202     show "(\<lambda>i. integral (?w' i)) ----> x" unfolding * sums_def .
```
```  2203   qed
```
```  2204
```
```  2205   have 4: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> ?w x" using 2[of _ 0] by simp
```
```  2206
```
```  2207   from summable[THEN summable_rabs_cancel]
```
```  2208   have 5: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>n. \<Sum>i = 0..<n. f i x) ----> (\<Sum>i. f i x)"
```
```  2209     by (auto intro: summable_sumr_LIMSEQ_suminf)
```
```  2210
```
```  2211   note int = integral_dominated_convergence(1,3)[OF 1 2 3 4 5]
```
```  2212
```
```  2213   from int show "integrable ?S" by simp
```
```  2214
```
```  2215   show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF borel]
```
```  2216     using int(2) by simp
```
```  2217 qed
```
```  2218
```
```  2219 section "Lebesgue integration on countable spaces"
```
```  2220
```
```  2221 lemma (in measure_space) integral_on_countable:
```
```  2222   assumes f: "f \<in> borel_measurable M"
```
```  2223   and bij: "bij_betw enum S (f ` space M)"
```
```  2224   and enum_zero: "enum ` (-S) \<subseteq> {0}"
```
```  2225   and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
```
```  2226   and abs_summable: "summable (\<lambda>r. \<bar>enum r * real (\<mu> (f -` {enum r} \<inter> space M))\<bar>)"
```
```  2227   shows "integrable f"
```
```  2228   and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral f" (is ?sums)
```
```  2229 proof -
```
```  2230   let "?A r" = "f -` {enum r} \<inter> space M"
```
```  2231   let "?F r x" = "enum r * indicator (?A r) x"
```
```  2232   have enum_eq: "\<And>r. enum r * real (\<mu> (?A r)) = integral (?F r)"
```
```  2233     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2234
```
```  2235   { fix x assume "x \<in> space M"
```
```  2236     hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
```
```  2237     then obtain i where "i\<in>S" "enum i = f x" by auto
```
```  2238     have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
```
```  2239     proof cases
```
```  2240       fix j assume "j = i"
```
```  2241       thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
```
```  2242     next
```
```  2243       fix j assume "j \<noteq> i"
```
```  2244       show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
```
```  2245         by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
```
```  2246     qed
```
```  2247     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
```
```  2248     have "(\<lambda>i. ?F i x) sums f x"
```
```  2249          "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
```
```  2250       by (auto intro!: sums_single simp: F F_abs) }
```
```  2251   note F_sums_f = this(1) and F_abs_sums_f = this(2)
```
```  2252
```
```  2253   have int_f: "integral f = (\<integral>x. \<Sum>r. ?F r x)" "integrable f = integrable (\<lambda>x. \<Sum>r. ?F r x)"
```
```  2254     using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
```
```  2255
```
```  2256   { fix r
```
```  2257     have "(\<integral>x. \<bar>?F r x\<bar>) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x)"
```
```  2258       by (auto simp: indicator_def intro!: integral_cong)
```
```  2259     also have "\<dots> = \<bar>enum r\<bar> * real (\<mu> (?A r))"
```
```  2260       using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2261     finally have "(\<integral>x. \<bar>?F r x\<bar>) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
```
```  2262       by (simp add: abs_mult_pos real_pextreal_pos) }
```
```  2263   note int_abs_F = this
```
```  2264
```
```  2265   have 1: "\<And>i. integrable (\<lambda>x. ?F i x)"
```
```  2266     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
```
```  2267
```
```  2268   have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
```
```  2269     using F_abs_sums_f unfolding sums_iff by auto
```
```  2270
```
```  2271   from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
```
```  2272   show ?sums unfolding enum_eq int_f by simp
```
```  2273
```
```  2274   from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
```
```  2275   show "integrable f" unfolding int_f by simp
```
```  2276 qed
```
```  2277
```
```  2278 section "Lebesgue integration on finite space"
```
```  2279
```
```  2280 lemma (in measure_space) integral_on_finite:
```
```  2281   assumes f: "f \<in> borel_measurable M" and finite: "finite (f`space M)"
```
```  2282   and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
```
```  2283   shows "integrable f"
```
```  2284   and "(\<integral>x. f x) =
```
```  2285     (\<Sum> r \<in> f`space M. r * real (\<mu> (f -` {r} \<inter> space M)))" (is "?integral")
```
```  2286 proof -
```
```  2287   let "?A r" = "f -` {r} \<inter> space M"
```
```  2288   let "?S x" = "\<Sum>r\<in>f`space M. r * indicator (?A r) x"
```
```  2289
```
```  2290   { fix x assume "x \<in> space M"
```
```  2291     have "f x = (\<Sum>r\<in>f`space M. if x \<in> ?A r then r else 0)"
```
```  2292       using finite `x \<in> space M` by (simp add: setsum_cases)
```
```  2293     also have "\<dots> = ?S x"
```
```  2294       by (auto intro!: setsum_cong)
```
```  2295     finally have "f x = ?S x" . }
```
```  2296   note f_eq = this
```
```  2297
```
```  2298   have f_eq_S: "integrable f \<longleftrightarrow> integrable ?S" "integral f = integral ?S"
```
```  2299     by (auto intro!: integrable_cong integral_cong simp only: f_eq)
```
```  2300
```
```  2301   show "integrable f" ?integral using fin f f_eq_S
```
```  2302     by (simp_all add: integral_cmul_indicator borel_measurable_vimage)
```
```  2303 qed
```
```  2304
```
```  2305 lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function f"
```
```  2306   unfolding simple_function_def using finite_space by auto
```
```  2307
```
```  2308 lemma (in finite_measure_space) borel_measurable_finite[intro, simp]: "f \<in> borel_measurable M"
```
```  2309   by (auto intro: borel_measurable_simple_function)
```
```  2310
```
```  2311 lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
```
```  2312   "positive_integral f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
```
```  2313 proof -
```
```  2314   have *: "positive_integral f = (\<integral>\<^isup>+ x. \<Sum>y\<in>space M. f y * indicator {y} x)"
```
```  2315     by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
```
```  2316   show ?thesis unfolding * using borel_measurable_finite[of f]
```
```  2317     by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
```
```  2318 qed
```
```  2319
```
```  2320 lemma (in finite_measure_space) integral_finite_singleton:
```
```  2321   shows "integrable f"
```
```  2322   and "integral f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
```
```  2323 proof -
```
```  2324   have [simp]:
```
```  2325     "(\<integral>\<^isup>+ x. Real (f x)) = (\<Sum>x \<in> space M. Real (f x) * \<mu> {x})"
```
```  2326     "(\<integral>\<^isup>+ x. Real (- f x)) = (\<Sum>x \<in> space M. Real (- f x) * \<mu> {x})"
```
```  2327     unfolding positive_integral_finite_eq_setsum by auto
```
```  2328   show "integrable f" using finite_space finite_measure
```
```  2329     by (simp add: setsum_\<omega> integrable_def)
```
```  2330   show ?I using finite_measure
```
```  2331     apply (simp add: integral_def real_of_pextreal_setsum[symmetric]
```
```  2332       real_of_pextreal_mult[symmetric] setsum_subtractf[symmetric])
```
```  2333     by (rule setsum_cong) (simp_all split: split_if)
```
```  2334 qed
```
```  2335
```
```  2336 end
```