src/HOL/Probability/Set_Integral.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 59867 58043346ca64 child 60615 e5fa1d5d3952 permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
```     1 (*  Title:      HOL/Probability/Set_Integral.thy
```
```     2     Author:     Jeremy Avigad, Johannes Hölzl, Luke Serafin
```
```     3
```
```     4 Notation and useful facts for working with integrals over a set.
```
```     5
```
```     6 TODO: keep all these? Need unicode translations as well.
```
```     7 *)
```
```     8
```
```     9 theory Set_Integral
```
```    10   imports Bochner_Integration Lebesgue_Measure
```
```    11 begin
```
```    12
```
```    13 (*
```
```    14     Notation
```
```    15 *)
```
```    16
```
```    17 syntax
```
```    18 "_ascii_lebesgue_integral" :: "pttrn \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
```
```    19 ("(3LINT (1_)/|(_)./ _)" [0,110,60] 60)
```
```    20
```
```    21 translations
```
```    22 "LINT x|M. f" == "CONST lebesgue_integral M (\<lambda>x. f)"
```
```    23
```
```    24 abbreviation "set_borel_measurable M A f \<equiv> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable M"
```
```    25
```
```    26 abbreviation "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
```
```    27
```
```    28 abbreviation "set_lebesgue_integral M A f \<equiv> lebesgue_integral M (\<lambda>x. indicator A x *\<^sub>R f x)"
```
```    29
```
```    30 syntax
```
```    31 "_ascii_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
```
```    32 ("(4LINT (_):(_)/|(_)./ _)" [0,60,110,61] 60)
```
```    33
```
```    34 translations
```
```    35 "LINT x:A|M. f" == "CONST set_lebesgue_integral M A (\<lambda>x. f)"
```
```    36
```
```    37 abbreviation
```
```    38   "set_almost_everywhere A M P \<equiv> AE x in M. x \<in> A \<longrightarrow> P x"
```
```    39
```
```    40 syntax
```
```    41   "_set_almost_everywhere" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool"
```
```    42 ("AE _\<in>_ in _./ _" [0,0,0,10] 10)
```
```    43
```
```    44 translations
```
```    45   "AE x\<in>A in M. P" == "CONST set_almost_everywhere A M (\<lambda>x. P)"
```
```    46
```
```    47 (*
```
```    48     Notation for integration wrt lebesgue measure on the reals:
```
```    49
```
```    50       LBINT x. f
```
```    51       LBINT x : A. f
```
```    52
```
```    53     TODO: keep all these? Need unicode.
```
```    54 *)
```
```    55
```
```    56 syntax
```
```    57 "_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real"
```
```    58 ("(2LBINT _./ _)" [0,60] 60)
```
```    59
```
```    60 translations
```
```    61 "LBINT x. f" == "CONST lebesgue_integral CONST lborel (\<lambda>x. f)"
```
```    62
```
```    63 syntax
```
```    64 "_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> real"
```
```    65 ("(3LBINT _:_./ _)" [0,60,61] 60)
```
```    66
```
```    67 translations
```
```    68 "LBINT x:A. f" == "CONST set_lebesgue_integral CONST lborel A (\<lambda>x. f)"
```
```    69
```
```    70 (*
```
```    71     Basic properties
```
```    72 *)
```
```    73
```
```    74 (*
```
```    75 lemma indicator_abs_eq: "\<And>A x. abs (indicator A x) = ((indicator A x) :: real)"
```
```    76   by (auto simp add: indicator_def)
```
```    77 *)
```
```    78
```
```    79 lemma set_lebesgue_integral_cong:
```
```    80   assumes "A \<in> sets M" and "\<forall>x. x \<in> A \<longrightarrow> f x = g x"
```
```    81   shows "(LINT x:A|M. f x) = (LINT x:A|M. g x)"
```
```    82   using assms by (auto intro!: integral_cong split: split_indicator simp add: sets.sets_into_space)
```
```    83
```
```    84 lemma set_lebesgue_integral_cong_AE:
```
```    85   assumes [measurable]: "A \<in> sets M" "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```    86   assumes "AE x \<in> A in M. f x = g x"
```
```    87   shows "LINT x:A|M. f x = LINT x:A|M. g x"
```
```    88 proof-
```
```    89   have "AE x in M. indicator A x *\<^sub>R f x = indicator A x *\<^sub>R g x"
```
```    90     using assms by auto
```
```    91   thus ?thesis by (intro integral_cong_AE) auto
```
```    92 qed
```
```    93
```
```    94 lemma set_integrable_cong_AE:
```
```    95     "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
```
```    96     AE x \<in> A in M. f x = g x \<Longrightarrow> A \<in> sets M \<Longrightarrow>
```
```    97     set_integrable M A f = set_integrable M A g"
```
```    98   by (rule integrable_cong_AE) auto
```
```    99
```
```   100 lemma set_integrable_subset:
```
```   101   fixes M A B and f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   102   assumes "set_integrable M A f" "B \<in> sets M" "B \<subseteq> A"
```
```   103   shows "set_integrable M B f"
```
```   104 proof -
```
```   105   have "set_integrable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
```
```   106     by (rule integrable_mult_indicator) fact+
```
```   107   with `B \<subseteq> A` show ?thesis
```
```   108     by (simp add: indicator_inter_arith[symmetric] Int_absorb2)
```
```   109 qed
```
```   110
```
```   111 (* TODO: integral_cmul_indicator should be named set_integral_const *)
```
```   112 (* TODO: borel_integrable_atLeastAtMost should be named something like set_integrable_Icc_isCont *)
```
```   113
```
```   114 lemma set_integral_scaleR_right [simp]: "LINT t:A|M. a *\<^sub>R f t = a *\<^sub>R (LINT t:A|M. f t)"
```
```   115   by (subst integral_scaleR_right[symmetric]) (auto intro!: integral_cong)
```
```   116
```
```   117 lemma set_integral_mult_right [simp]:
```
```   118   fixes a :: "'a::{real_normed_field, second_countable_topology}"
```
```   119   shows "LINT t:A|M. a * f t = a * (LINT t:A|M. f t)"
```
```   120   by (subst integral_mult_right_zero[symmetric]) (auto intro!: integral_cong)
```
```   121
```
```   122 lemma set_integral_mult_left [simp]:
```
```   123   fixes a :: "'a::{real_normed_field, second_countable_topology}"
```
```   124   shows "LINT t:A|M. f t * a = (LINT t:A|M. f t) * a"
```
```   125   by (subst integral_mult_left_zero[symmetric]) (auto intro!: integral_cong)
```
```   126
```
```   127 lemma set_integral_divide_zero [simp]:
```
```   128   fixes a :: "'a::{real_normed_field, field, second_countable_topology}"
```
```   129   shows "LINT t:A|M. f t / a = (LINT t:A|M. f t) / a"
```
```   130   by (subst integral_divide_zero[symmetric], intro integral_cong)
```
```   131      (auto split: split_indicator)
```
```   132
```
```   133 lemma set_integrable_scaleR_right [simp, intro]:
```
```   134   shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a *\<^sub>R f t)"
```
```   135   unfolding scaleR_left_commute by (rule integrable_scaleR_right)
```
```   136
```
```   137 lemma set_integrable_scaleR_left [simp, intro]:
```
```   138   fixes a :: "_ :: {banach, second_countable_topology}"
```
```   139   shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t *\<^sub>R a)"
```
```   140   using integrable_scaleR_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
```
```   141
```
```   142 lemma set_integrable_mult_right [simp, intro]:
```
```   143   fixes a :: "'a::{real_normed_field, second_countable_topology}"
```
```   144   shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a * f t)"
```
```   145   using integrable_mult_right[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
```
```   146
```
```   147 lemma set_integrable_mult_left [simp, intro]:
```
```   148   fixes a :: "'a::{real_normed_field, second_countable_topology}"
```
```   149   shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t * a)"
```
```   150   using integrable_mult_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
```
```   151
```
```   152 lemma set_integrable_divide [simp, intro]:
```
```   153   fixes a :: "'a::{real_normed_field, field, second_countable_topology}"
```
```   154   assumes "a \<noteq> 0 \<Longrightarrow> set_integrable M A f"
```
```   155   shows "set_integrable M A (\<lambda>t. f t / a)"
```
```   156 proof -
```
```   157   have "integrable M (\<lambda>x. indicator A x *\<^sub>R f x / a)"
```
```   158     using assms by (rule integrable_divide_zero)
```
```   159   also have "(\<lambda>x. indicator A x *\<^sub>R f x / a) = (\<lambda>x. indicator A x *\<^sub>R (f x / a))"
```
```   160     by (auto split: split_indicator)
```
```   161   finally show ?thesis .
```
```   162 qed
```
```   163
```
```   164 lemma set_integral_add [simp, intro]:
```
```   165   fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   166   assumes "set_integrable M A f" "set_integrable M A g"
```
```   167   shows "set_integrable M A (\<lambda>x. f x + g x)"
```
```   168     and "LINT x:A|M. f x + g x = (LINT x:A|M. f x) + (LINT x:A|M. g x)"
```
```   169   using assms by (simp_all add: scaleR_add_right)
```
```   170
```
```   171 lemma set_integral_diff [simp, intro]:
```
```   172   assumes "set_integrable M A f" "set_integrable M A g"
```
```   173   shows "set_integrable M A (\<lambda>x. f x - g x)" and "LINT x:A|M. f x - g x =
```
```   174     (LINT x:A|M. f x) - (LINT x:A|M. g x)"
```
```   175   using assms by (simp_all add: scaleR_diff_right)
```
```   176
```
```   177 lemma set_integral_reflect:
```
```   178   fixes S and f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```   179   shows "(LBINT x : S. f x) = (LBINT x : {x. - x \<in> S}. f (- x))"
```
```   180   using assms
```
```   181   by (subst lborel_integral_real_affine[where c="-1" and t=0])
```
```   182      (auto intro!: integral_cong split: split_indicator)
```
```   183
```
```   184 (* question: why do we have this for negation, but multiplication by a constant
```
```   185    requires an integrability assumption? *)
```
```   186 lemma set_integral_uminus: "set_integrable M A f \<Longrightarrow> LINT x:A|M. - f x = - (LINT x:A|M. f x)"
```
```   187   by (subst integral_minus[symmetric]) simp_all
```
```   188
```
```   189 lemma set_integral_complex_of_real:
```
```   190   "LINT x:A|M. complex_of_real (f x) = of_real (LINT x:A|M. f x)"
```
```   191   by (subst integral_complex_of_real[symmetric])
```
```   192      (auto intro!: integral_cong split: split_indicator)
```
```   193
```
```   194 lemma set_integral_mono:
```
```   195   fixes f g :: "_ \<Rightarrow> real"
```
```   196   assumes "set_integrable M A f" "set_integrable M A g"
```
```   197     "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
```
```   198   shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)"
```
```   199 using assms by (auto intro: integral_mono split: split_indicator)
```
```   200
```
```   201 lemma set_integral_mono_AE:
```
```   202   fixes f g :: "_ \<Rightarrow> real"
```
```   203   assumes "set_integrable M A f" "set_integrable M A g"
```
```   204     "AE x \<in> A in M. f x \<le> g x"
```
```   205   shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)"
```
```   206 using assms by (auto intro: integral_mono_AE split: split_indicator)
```
```   207
```
```   208 lemma set_integrable_abs: "set_integrable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar> :: real)"
```
```   209   using integrable_abs[of M "\<lambda>x. f x * indicator A x"] by (simp add: abs_mult ac_simps)
```
```   210
```
```   211 lemma set_integrable_abs_iff:
```
```   212   fixes f :: "_ \<Rightarrow> real"
```
```   213   shows "set_borel_measurable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
```
```   214   by (subst (2) integrable_abs_iff[symmetric]) (simp_all add: abs_mult ac_simps)
```
```   215
```
```   216 lemma set_integrable_abs_iff':
```
```   217   fixes f :: "_ \<Rightarrow> real"
```
```   218   shows "f \<in> borel_measurable M \<Longrightarrow> A \<in> sets M \<Longrightarrow>
```
```   219     set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
```
```   220 by (intro set_integrable_abs_iff) auto
```
```   221
```
```   222 lemma set_integrable_discrete_difference:
```
```   223   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   224   assumes "countable X"
```
```   225   assumes diff: "(A - B) \<union> (B - A) \<subseteq> X"
```
```   226   assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
```
```   227   shows "set_integrable M A f \<longleftrightarrow> set_integrable M B f"
```
```   228 proof (rule integrable_discrete_difference[where X=X])
```
```   229   show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x"
```
```   230     using diff by (auto split: split_indicator)
```
```   231 qed fact+
```
```   232
```
```   233 lemma set_integral_discrete_difference:
```
```   234   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   235   assumes "countable X"
```
```   236   assumes diff: "(A - B) \<union> (B - A) \<subseteq> X"
```
```   237   assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
```
```   238   shows "set_lebesgue_integral M A f = set_lebesgue_integral M B f"
```
```   239 proof (rule integral_discrete_difference[where X=X])
```
```   240   show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x"
```
```   241     using diff by (auto split: split_indicator)
```
```   242 qed fact+
```
```   243
```
```   244 lemma set_integrable_Un:
```
```   245   fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   246   assumes f_A: "set_integrable M A f" and f_B:  "set_integrable M B f"
```
```   247     and [measurable]: "A \<in> sets M" "B \<in> sets M"
```
```   248   shows "set_integrable M (A \<union> B) f"
```
```   249 proof -
```
```   250   have "set_integrable M (A - B) f"
```
```   251     using f_A by (rule set_integrable_subset) auto
```
```   252   from integrable_add[OF this f_B] show ?thesis
```
```   253     by (rule integrable_cong[THEN iffD1, rotated 2]) (auto split: split_indicator)
```
```   254 qed
```
```   255
```
```   256 lemma set_integrable_UN:
```
```   257   fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   258   assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> set_integrable M (A i) f"
```
```   259     "\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M"
```
```   260   shows "set_integrable M (\<Union>i\<in>I. A i) f"
```
```   261 using assms by (induct I) (auto intro!: set_integrable_Un)
```
```   262
```
```   263 lemma set_integral_Un:
```
```   264   fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   265   assumes "A \<inter> B = {}"
```
```   266   and "set_integrable M A f"
```
```   267   and "set_integrable M B f"
```
```   268   shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
```
```   269 by (auto simp add: indicator_union_arith indicator_inter_arith[symmetric]
```
```   270       scaleR_add_left assms)
```
```   271
```
```   272 lemma set_integral_cong_set:
```
```   273   fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   274   assumes [measurable]: "set_borel_measurable M A f" "set_borel_measurable M B f"
```
```   275     and ae: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
```
```   276   shows "LINT x:B|M. f x = LINT x:A|M. f x"
```
```   277 proof (rule integral_cong_AE)
```
```   278   show "AE x in M. indicator B x *\<^sub>R f x = indicator A x *\<^sub>R f x"
```
```   279     using ae by (auto simp: subset_eq split: split_indicator)
```
```   280 qed fact+
```
```   281
```
```   282 lemma set_borel_measurable_subset:
```
```   283   fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   284   assumes [measurable]: "set_borel_measurable M A f" "B \<in> sets M" and "B \<subseteq> A"
```
```   285   shows "set_borel_measurable M B f"
```
```   286 proof -
```
```   287   have "set_borel_measurable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
```
```   288     by measurable
```
```   289   also have "(\<lambda>x. indicator B x *\<^sub>R indicator A x *\<^sub>R f x) = (\<lambda>x. indicator B x *\<^sub>R f x)"
```
```   290     using `B \<subseteq> A` by (auto simp: fun_eq_iff split: split_indicator)
```
```   291   finally show ?thesis .
```
```   292 qed
```
```   293
```
```   294 lemma set_integral_Un_AE:
```
```   295   fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   296   assumes ae: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)" and [measurable]: "A \<in> sets M" "B \<in> sets M"
```
```   297   and "set_integrable M A f"
```
```   298   and "set_integrable M B f"
```
```   299   shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
```
```   300 proof -
```
```   301   have f: "set_integrable M (A \<union> B) f"
```
```   302     by (intro set_integrable_Un assms)
```
```   303   then have f': "set_borel_measurable M (A \<union> B) f"
```
```   304     by (rule borel_measurable_integrable)
```
```   305   have "LINT x:A\<union>B|M. f x = LINT x:(A - A \<inter> B) \<union> (B - A \<inter> B)|M. f x"
```
```   306   proof (rule set_integral_cong_set)
```
```   307     show "AE x in M. (x \<in> A - A \<inter> B \<union> (B - A \<inter> B)) = (x \<in> A \<union> B)"
```
```   308       using ae by auto
```
```   309     show "set_borel_measurable M (A - A \<inter> B \<union> (B - A \<inter> B)) f"
```
```   310       using f' by (rule set_borel_measurable_subset) auto
```
```   311   qed fact
```
```   312   also have "\<dots> = (LINT x:(A - A \<inter> B)|M. f x) + (LINT x:(B - A \<inter> B)|M. f x)"
```
```   313     by (auto intro!: set_integral_Un set_integrable_subset[OF f])
```
```   314   also have "\<dots> = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
```
```   315     using ae
```
```   316     by (intro arg_cong2[where f="op+"] set_integral_cong_set)
```
```   317        (auto intro!: set_borel_measurable_subset[OF f'])
```
```   318   finally show ?thesis .
```
```   319 qed
```
```   320
```
```   321 lemma set_integral_finite_Union:
```
```   322   fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
```
```   323   assumes "finite I" "disjoint_family_on A I"
```
```   324     and "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
```
```   325   shows "(LINT x:(\<Union>i\<in>I. A i)|M. f x) = (\<Sum>i\<in>I. LINT x:A i|M. f x)"
```
```   326   using assms
```
```   327   apply induct
```
```   328   apply (auto intro!: set_integral_Un set_integrable_Un set_integrable_UN simp: disjoint_family_on_def)
```
```   329 by (subst set_integral_Un, auto intro: set_integrable_UN)
```
```   330
```
```   331 (* TODO: find a better name? *)
```
```   332 lemma pos_integrable_to_top:
```
```   333   fixes l::real
```
```   334   assumes "\<And>i. A i \<in> sets M" "mono A"
```
```   335   assumes nneg: "\<And>x i. x \<in> A i \<Longrightarrow> 0 \<le> f x"
```
```   336   and intgbl: "\<And>i::nat. set_integrable M (A i) f"
```
```   337   and lim: "(\<lambda>i::nat. LINT x:A i|M. f x) ----> l"
```
```   338   shows "set_integrable M (\<Union>i. A i) f"
```
```   339   apply (rule integrable_monotone_convergence[where f = "\<lambda>i::nat. \<lambda>x. indicator (A i) x *\<^sub>R f x" and x = l])
```
```   340   apply (rule intgbl)
```
```   341   prefer 3 apply (rule lim)
```
```   342   apply (rule AE_I2)
```
```   343   using `mono A` apply (auto simp: mono_def nneg split: split_indicator) []
```
```   344 proof (rule AE_I2)
```
```   345   { fix x assume "x \<in> space M"
```
```   346     show "(\<lambda>i. indicator (A i) x *\<^sub>R f x) ----> indicator (\<Union>i. A i) x *\<^sub>R f x"
```
```   347     proof cases
```
```   348       assume "\<exists>i. x \<in> A i"
```
```   349       then guess i ..
```
```   350       then have *: "eventually (\<lambda>i. x \<in> A i) sequentially"
```
```   351         using `x \<in> A i` `mono A` by (auto simp: eventually_sequentially mono_def)
```
```   352       show ?thesis
```
```   353         apply (intro Lim_eventually)
```
```   354         using *
```
```   355         apply eventually_elim
```
```   356         apply (auto split: split_indicator)
```
```   357         done
```
```   358     qed auto }
```
```   359   then show "(\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R f x) \<in> borel_measurable M"
```
```   360     apply (rule borel_measurable_LIMSEQ)
```
```   361     apply assumption
```
```   362     apply (intro borel_measurable_integrable intgbl)
```
```   363     done
```
```   364 qed
```
```   365
```
```   366 (* Proof from Royden Real Analysis, p. 91. *)
```
```   367 lemma lebesgue_integral_countable_add:
```
```   368   fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```   369   assumes meas[intro]: "\<And>i::nat. A i \<in> sets M"
```
```   370     and disj: "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
```
```   371     and intgbl: "set_integrable M (\<Union>i. A i) f"
```
```   372   shows "LINT x:(\<Union>i. A i)|M. f x = (\<Sum>i. (LINT x:(A i)|M. f x))"
```
```   373 proof (subst integral_suminf[symmetric])
```
```   374   show int_A: "\<And>i. set_integrable M (A i) f"
```
```   375     using intgbl by (rule set_integrable_subset) auto
```
```   376   { fix x assume "x \<in> space M"
```
```   377     have "(\<lambda>i. indicator (A i) x *\<^sub>R f x) sums (indicator (\<Union>i. A i) x *\<^sub>R f x)"
```
```   378       by (intro sums_scaleR_left indicator_sums) fact }
```
```   379   note sums = this
```
```   380
```
```   381   have norm_f: "\<And>i. set_integrable M (A i) (\<lambda>x. norm (f x))"
```
```   382     using int_A[THEN integrable_norm] by auto
```
```   383
```
```   384   show "AE x in M. summable (\<lambda>i. norm (indicator (A i) x *\<^sub>R f x))"
```
```   385     using disj by (intro AE_I2) (auto intro!: summable_mult2 sums_summable[OF indicator_sums])
```
```   386
```
```   387   show "summable (\<lambda>i. LINT x|M. norm (indicator (A i) x *\<^sub>R f x))"
```
```   388   proof (rule summableI_nonneg_bounded)
```
```   389     fix n
```
```   390     show "0 \<le> LINT x|M. norm (indicator (A n) x *\<^sub>R f x)"
```
```   391       using norm_f by (auto intro!: integral_nonneg_AE)
```
```   392
```
```   393     have "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) =
```
```   394       (\<Sum>i<n. set_lebesgue_integral M (A i) (\<lambda>x. norm (f x)))"
```
```   395       by (simp add: abs_mult)
```
```   396     also have "\<dots> = set_lebesgue_integral M (\<Union>i<n. A i) (\<lambda>x. norm (f x))"
```
```   397       using norm_f
```
```   398       by (subst set_integral_finite_Union) (auto simp: disjoint_family_on_def disj)
```
```   399     also have "\<dots> \<le> set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))"
```
```   400       using intgbl[THEN integrable_norm]
```
```   401       by (intro integral_mono set_integrable_UN[of "{..<n}"] norm_f)
```
```   402          (auto split: split_indicator)
```
```   403     finally show "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) \<le>
```
```   404       set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))"
```
```   405       by simp
```
```   406   qed
```
```   407   show "set_lebesgue_integral M (UNION UNIV A) f = LINT x|M. (\<Sum>i. indicator (A i) x *\<^sub>R f x)"
```
```   408     apply (rule integral_cong[OF refl])
```
```   409     apply (subst suminf_scaleR_left[OF sums_summable[OF indicator_sums, OF disj], symmetric])
```
```   410     using sums_unique[OF indicator_sums[OF disj]]
```
```   411     apply auto
```
```   412     done
```
```   413 qed
```
```   414
```
```   415 lemma set_integral_cont_up:
```
```   416   fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```   417   assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "incseq A"
```
```   418   and intgbl: "set_integrable M (\<Union>i. A i) f"
```
```   419   shows "(\<lambda>i. LINT x:(A i)|M. f x) ----> LINT x:(\<Union>i. A i)|M. f x"
```
```   420 proof (intro integral_dominated_convergence[where w="\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R norm (f x)"])
```
```   421   have int_A: "\<And>i. set_integrable M (A i) f"
```
```   422     using intgbl by (rule set_integrable_subset) auto
```
```   423   then show "\<And>i. set_borel_measurable M (A i) f" "set_borel_measurable M (\<Union>i. A i) f"
```
```   424     "set_integrable M (\<Union>i. A i) (\<lambda>x. norm (f x))"
```
```   425     using intgbl integrable_norm[OF intgbl] by auto
```
```   426
```
```   427   { fix x i assume "x \<in> A i"
```
```   428     with A have "(\<lambda>xa. indicator (A xa) x::real) ----> 1 \<longleftrightarrow> (\<lambda>xa. 1::real) ----> 1"
```
```   429       by (intro filterlim_cong refl)
```
```   430          (fastforce simp: eventually_sequentially incseq_def subset_eq intro!: exI[of _ i]) }
```
```   431   then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) ----> indicator (\<Union>i. A i) x *\<^sub>R f x"
```
```   432     by (intro AE_I2 tendsto_intros) (auto split: split_indicator)
```
```   433 qed (auto split: split_indicator)
```
```   434
```
```   435 (* Can the int0 hypothesis be dropped? *)
```
```   436 lemma set_integral_cont_down:
```
```   437   fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```   438   assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "decseq A"
```
```   439   and int0: "set_integrable M (A 0) f"
```
```   440   shows "(\<lambda>i::nat. LINT x:(A i)|M. f x) ----> LINT x:(\<Inter>i. A i)|M. f x"
```
```   441 proof (rule integral_dominated_convergence)
```
```   442   have int_A: "\<And>i. set_integrable M (A i) f"
```
```   443     using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
```
```   444   show "set_integrable M (A 0) (\<lambda>x. norm (f x))"
```
```   445     using int0[THEN integrable_norm] by simp
```
```   446   have "set_integrable M (\<Inter>i. A i) f"
```
```   447     using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
```
```   448   with int_A show "set_borel_measurable M (\<Inter>i. A i) f" "\<And>i. set_borel_measurable M (A i) f"
```
```   449     by auto
```
```   450   show "\<And>i. AE x in M. norm (indicator (A i) x *\<^sub>R f x) \<le> indicator (A 0) x *\<^sub>R norm (f x)"
```
```   451     using A by (auto split: split_indicator simp: decseq_def)
```
```   452   { fix x i assume "x \<in> space M" "x \<notin> A i"
```
```   453     with A have "(\<lambda>i. indicator (A i) x::real) ----> 0 \<longleftrightarrow> (\<lambda>i. 0::real) ----> 0"
```
```   454       by (intro filterlim_cong refl)
```
```   455          (auto split: split_indicator simp: eventually_sequentially decseq_def intro!: exI[of _ i]) }
```
```   456   then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) ----> indicator (\<Inter>i. A i) x *\<^sub>R f x"
```
```   457     by (intro AE_I2 tendsto_intros) (auto split: split_indicator)
```
```   458 qed
```
```   459
```
```   460 lemma set_integral_at_point:
```
```   461   fixes a :: real
```
```   462   assumes "set_integrable M {a} f"
```
```   463   and [simp]: "{a} \<in> sets M" and "(emeasure M) {a} \<noteq> \<infinity>"
```
```   464   shows "(LINT x:{a} | M. f x) = f a * measure M {a}"
```
```   465 proof-
```
```   466   have "set_lebesgue_integral M {a} f = set_lebesgue_integral M {a} (%x. f a)"
```
```   467     by (intro set_lebesgue_integral_cong) simp_all
```
```   468   then show ?thesis using assms by simp
```
```   469 qed
```
```   470
```
```   471
```
```   472 abbreviation complex_integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> bool" where
```
```   473   "complex_integrable M f \<equiv> integrable M f"
```
```   474
```
```   475 abbreviation complex_lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> complex" ("integral\<^sup>C") where
```
```   476   "integral\<^sup>C M f == integral\<^sup>L M f"
```
```   477
```
```   478 syntax
```
```   479   "_complex_lebesgue_integral" :: "pttrn \<Rightarrow> complex \<Rightarrow> 'a measure \<Rightarrow> complex"
```
```   480  ("\<integral>\<^sup>C _. _ \<partial>_" [60,61] 110)
```
```   481
```
```   482 translations
```
```   483   "\<integral>\<^sup>Cx. f \<partial>M" == "CONST complex_lebesgue_integral M (\<lambda>x. f)"
```
```   484
```
```   485 syntax
```
```   486   "_ascii_complex_lebesgue_integral" :: "pttrn \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
```
```   487   ("(3CLINT _|_. _)" [0,110,60] 60)
```
```   488
```
```   489 translations
```
```   490   "CLINT x|M. f" == "CONST complex_lebesgue_integral M (\<lambda>x. f)"
```
```   491
```
```   492 lemma complex_integrable_cnj [simp]:
```
```   493   "complex_integrable M (\<lambda>x. cnj (f x)) \<longleftrightarrow> complex_integrable M f"
```
```   494 proof
```
```   495   assume "complex_integrable M (\<lambda>x. cnj (f x))"
```
```   496   then have "complex_integrable M (\<lambda>x. cnj (cnj (f x)))"
```
```   497     by (rule integrable_cnj)
```
```   498   then show "complex_integrable M f"
```
```   499     by simp
```
```   500 qed simp
```
```   501
```
```   502 lemma complex_of_real_integrable_eq:
```
```   503   "complex_integrable M (\<lambda>x. complex_of_real (f x)) \<longleftrightarrow> integrable M f"
```
```   504 proof
```
```   505   assume "complex_integrable M (\<lambda>x. complex_of_real (f x))"
```
```   506   then have "integrable M (\<lambda>x. Re (complex_of_real (f x)))"
```
```   507     by (rule integrable_Re)
```
```   508   then show "integrable M f"
```
```   509     by simp
```
```   510 qed simp
```
```   511
```
```   512
```
```   513 abbreviation complex_set_integrable :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> bool" where
```
```   514   "complex_set_integrable M A f \<equiv> set_integrable M A f"
```
```   515
```
```   516 abbreviation complex_set_lebesgue_integral :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> complex" where
```
```   517   "complex_set_lebesgue_integral M A f \<equiv> set_lebesgue_integral M A f"
```
```   518
```
```   519 syntax
```
```   520 "_ascii_complex_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
```
```   521 ("(4CLINT _:_|_. _)" [0,60,110,61] 60)
```
```   522
```
```   523 translations
```
```   524 "CLINT x:A|M. f" == "CONST complex_set_lebesgue_integral M A (\<lambda>x. f)"
```
```   525
```
```   526 (*
```
```   527 lemma cmod_mult: "cmod ((a :: real) * (x :: complex)) = abs a * cmod x"
```
```   528   apply (simp add: norm_mult)
```
```   529   by (subst norm_mult, auto)
```
```   530 *)
```
```   531
```
```   532 lemma borel_integrable_atLeastAtMost':
```
```   533   fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
```
```   534   assumes f: "continuous_on {a..b} f"
```
```   535   shows "set_integrable lborel {a..b} f" (is "integrable _ ?f")
```
```   536   by (intro borel_integrable_compact compact_Icc f)
```
```   537
```
```   538 lemma integral_FTC_atLeastAtMost:
```
```   539   fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
```
```   540   assumes "a \<le> b"
```
```   541     and F: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
```
```   542     and f: "continuous_on {a .. b} f"
```
```   543   shows "integral\<^sup>L lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x) = F b - F a"
```
```   544 proof -
```
```   545   let ?f = "\<lambda>x. indicator {a .. b} x *\<^sub>R f x"
```
```   546   have "(?f has_integral (\<integral>x. ?f x \<partial>lborel)) UNIV"
```
```   547     using borel_integrable_atLeastAtMost'[OF f] by (rule has_integral_integral_lborel)
```
```   548   moreover
```
```   549   have "(f has_integral F b - F a) {a .. b}"
```
```   550     by (intro fundamental_theorem_of_calculus ballI assms) auto
```
```   551   then have "(?f has_integral F b - F a) {a .. b}"
```
```   552     by (subst has_integral_eq_eq[where g=f]) auto
```
```   553   then have "(?f has_integral F b - F a) UNIV"
```
```   554     by (intro has_integral_on_superset[where t=UNIV and s="{a..b}"]) auto
```
```   555   ultimately show "integral\<^sup>L lborel ?f = F b - F a"
```
```   556     by (rule has_integral_unique)
```
```   557 qed
```
```   558
```
```   559 lemma set_borel_integral_eq_integral:
```
```   560   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   561   assumes "set_integrable lborel S f"
```
```   562   shows "f integrable_on S" "LINT x : S | lborel. f x = integral S f"
```
```   563 proof -
```
```   564   let ?f = "\<lambda>x. indicator S x *\<^sub>R f x"
```
```   565   have "(?f has_integral LINT x : S | lborel. f x) UNIV"
```
```   566     by (rule has_integral_integral_lborel) fact
```
```   567   hence 1: "(f has_integral (set_lebesgue_integral lborel S f)) S"
```
```   568     apply (subst has_integral_restrict_univ [symmetric])
```
```   569     apply (rule has_integral_eq)
```
```   570     by auto
```
```   571   thus "f integrable_on S"
```
```   572     by (auto simp add: integrable_on_def)
```
```   573   with 1 have "(f has_integral (integral S f)) S"
```
```   574     by (intro integrable_integral, auto simp add: integrable_on_def)
```
```   575   thus "LINT x : S | lborel. f x = integral S f"
```
```   576     by (intro has_integral_unique [OF 1])
```
```   577 qed
```
```   578
```
```   579 lemma set_borel_measurable_continuous:
```
```   580   fixes f :: "_ \<Rightarrow> _::real_normed_vector"
```
```   581   assumes "S \<in> sets borel" "continuous_on S f"
```
```   582   shows "set_borel_measurable borel S f"
```
```   583 proof -
```
```   584   have "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable borel"
```
```   585     by (intro assms borel_measurable_continuous_on_if continuous_on_const)
```
```   586   also have "(\<lambda>x. if x \<in> S then f x else 0) = (\<lambda>x. indicator S x *\<^sub>R f x)"
```
```   587     by auto
```
```   588   finally show ?thesis .
```
```   589 qed
```
```   590
```
```   591 lemma set_measurable_continuous_on_ivl:
```
```   592   assumes "continuous_on {a..b} (f :: real \<Rightarrow> real)"
```
```   593   shows "set_borel_measurable borel {a..b} f"
```
```   594   by (rule set_borel_measurable_continuous[OF _ assms]) simp
```
```   595
```
```   596 end
```
```   597
```