src/HOL/Probability/Set_Integral.thy
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(*  Title:      HOL/Probability/Set_Integral.thy
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    Author:     Jeremy Avigad, Johannes Hölzl, Luke Serafin
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Notation and useful facts for working with integrals over a set.
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TODO: keep all these? Need unicode translations as well.
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*)
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theory Set_Integral
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  imports Bochner_Integration Lebesgue_Measure
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begin
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(* 
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    Notation
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*)
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syntax
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"_ascii_lebesgue_integral" :: "pttrn \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
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("(3LINT (1_)/|(_)./ _)" [0,110,60] 60)
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translations
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"LINT x|M. f" == "CONST lebesgue_integral M (\<lambda>x. f)"
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abbreviation "set_borel_measurable M A f \<equiv> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable M"
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abbreviation "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
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abbreviation "set_lebesgue_integral M A f \<equiv> lebesgue_integral M (\<lambda>x. indicator A x *\<^sub>R f x)"
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syntax
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"_ascii_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
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("(4LINT (_):(_)/|(_)./ _)" [0,60,110,61] 60)
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translations
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"LINT x:A|M. f" == "CONST set_lebesgue_integral M A (\<lambda>x. f)"
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abbreviation
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  "set_almost_everywhere A M P \<equiv> AE x in M. x \<in> A \<longrightarrow> P x"
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syntax
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  "_set_almost_everywhere" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool"
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("AE _\<in>_ in _./ _" [0,0,0,10] 10)
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translations
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  "AE x\<in>A in M. P" == "CONST set_almost_everywhere A M (\<lambda>x. P)"
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(*
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    Notation for integration wrt lebesgue measure on the reals:
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      LBINT x. f
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      LBINT x : A. f
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    TODO: keep all these? Need unicode.
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*)
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syntax
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"_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real"
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("(2LBINT _./ _)" [0,60] 60)
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translations
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"LBINT x. f" == "CONST lebesgue_integral CONST lborel (\<lambda>x. f)"
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syntax
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"_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> real"
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("(3LBINT _:_./ _)" [0,60,61] 60)
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translations
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"LBINT x:A. f" == "CONST set_lebesgue_integral CONST lborel A (\<lambda>x. f)"
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(* 
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    Basic properties 
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*)
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(*
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lemma indicator_abs_eq: "\<And>A x. abs (indicator A x) = ((indicator A x) :: real)"
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  by (auto simp add: indicator_def)
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*)
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lemma set_lebesgue_integral_cong:
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  assumes "A \<in> sets M" and "\<forall>x. x \<in> A \<longrightarrow> f x = g x"
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  shows "(LINT x:A|M. f x) = (LINT x:A|M. g x)"
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  using assms by (auto intro!: integral_cong split: split_indicator simp add: sets.sets_into_space)
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lemma set_lebesgue_integral_cong_AE:
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  assumes [measurable]: "A \<in> sets M" "f \<in> borel_measurable M" "g \<in> borel_measurable M"
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  assumes "AE x \<in> A in M. f x = g x"
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  shows "LINT x:A|M. f x = LINT x:A|M. g x"
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proof-
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  have "AE x in M. indicator A x *\<^sub>R f x = indicator A x *\<^sub>R g x"
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    using assms by auto
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  thus ?thesis by (intro integral_cong_AE) auto
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qed
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lemma set_integrable_cong_AE:
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    "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
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    AE x \<in> A in M. f x = g x \<Longrightarrow> A \<in> sets M \<Longrightarrow> 
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    set_integrable M A f = set_integrable M A g"
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  by (rule integrable_cong_AE) auto
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lemma set_integrable_subset: 
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  fixes M A B and f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
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  assumes "set_integrable M A f" "B \<in> sets M" "B \<subseteq> A"  
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  shows "set_integrable M B f"
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proof -
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  have "set_integrable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
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    by (rule integrable_mult_indicator) fact+
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  with `B \<subseteq> A` show ?thesis
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    by (simp add: indicator_inter_arith[symmetric] Int_absorb2)
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qed
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(* TODO: integral_cmul_indicator should be named set_integral_const *)
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(* TODO: borel_integrable_atLeastAtMost should be named something like set_integrable_Icc_isCont *)
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lemma set_integral_scaleR_right [simp]: "LINT t:A|M. a *\<^sub>R f t = a *\<^sub>R (LINT t:A|M. f t)"
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  by (subst integral_scaleR_right[symmetric]) (auto intro!: integral_cong)
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lemma set_integral_mult_right [simp]: 
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  fixes a :: "'a::{real_normed_field, second_countable_topology}"
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  shows "LINT t:A|M. a * f t = a * (LINT t:A|M. f t)"
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  by (subst integral_mult_right_zero[symmetric]) (auto intro!: integral_cong)
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hoelzl
parents:
diff changeset
   122
lemma set_integral_mult_left [simp]: 
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   123
  fixes a :: "'a::{real_normed_field, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   124
  shows "LINT t:A|M. f t * a = (LINT t:A|M. f t) * a"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   125
  by (subst integral_mult_left_zero[symmetric]) (auto intro!: integral_cong)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   126
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   127
lemma set_integral_divide_zero [simp]: 
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   128
  fixes a :: "'a::{real_normed_field, field_inverse_zero, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   129
  shows "LINT t:A|M. f t / a = (LINT t:A|M. f t) / a"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   130
  by (subst integral_divide_zero[symmetric], intro integral_cong)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   131
     (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   132
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   133
lemma set_integrable_scaleR_right [simp, intro]:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   134
  shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a *\<^sub>R f t)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   135
  unfolding scaleR_left_commute by (rule integrable_scaleR_right)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   136
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   137
lemma set_integrable_scaleR_left [simp, intro]:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   138
  fixes a :: "_ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   139
  shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t *\<^sub>R a)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   140
  using integrable_scaleR_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   141
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   142
lemma set_integrable_mult_right [simp, intro]:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   143
  fixes a :: "'a::{real_normed_field, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   144
  shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a * f t)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   145
  using integrable_mult_right[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   146
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   147
lemma set_integrable_mult_left [simp, intro]:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   148
  fixes a :: "'a::{real_normed_field, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   149
  shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t * a)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   150
  using integrable_mult_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   151
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   152
lemma set_integrable_divide [simp, intro]:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   153
  fixes a :: "'a::{real_normed_field, field_inverse_zero, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   154
  assumes "a \<noteq> 0 \<Longrightarrow> set_integrable M A f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   155
  shows "set_integrable M A (\<lambda>t. f t / a)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   156
proof -
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   157
  have "integrable M (\<lambda>x. indicator A x *\<^sub>R f x / a)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   158
    using assms by (rule integrable_divide_zero)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   159
  also have "(\<lambda>x. indicator A x *\<^sub>R f x / a) = (\<lambda>x. indicator A x *\<^sub>R (f x / a))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   160
    by (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   161
  finally show ?thesis .
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   162
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   163
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   164
lemma set_integral_add [simp, intro]:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   165
  fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   166
  assumes "set_integrable M A f" "set_integrable M A g"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   167
  shows "set_integrable M A (\<lambda>x. f x + g x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   168
    and "LINT x:A|M. f x + g x = (LINT x:A|M. f x) + (LINT x:A|M. g x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   169
  using assms by (simp_all add: scaleR_add_right)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   170
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   171
lemma set_integral_diff [simp, intro]:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   172
  assumes "set_integrable M A f" "set_integrable M A g"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   173
  shows "set_integrable M A (\<lambda>x. f x - g x)" and "LINT x:A|M. f x - g x =
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   174
    (LINT x:A|M. f x) - (LINT x:A|M. g x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   175
  using assms by (simp_all add: scaleR_diff_right)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   176
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   177
lemma set_integral_reflect:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   178
  fixes S and f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   179
  shows "(LBINT x : S. f x) = (LBINT x : {x. - x \<in> S}. f (- x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   180
  using assms
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   181
  by (subst lborel_integral_real_affine[where c="-1" and t=0])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   182
     (auto intro!: integral_cong split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   183
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   184
(* question: why do we have this for negation, but multiplication by a constant
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   185
   requires an integrability assumption? *)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   186
lemma set_integral_uminus: "set_integrable M A f \<Longrightarrow> LINT x:A|M. - f x = - (LINT x:A|M. f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   187
  by (subst integral_minus[symmetric]) simp_all
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   188
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   189
lemma set_integral_complex_of_real:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   190
  "LINT x:A|M. complex_of_real (f x) = of_real (LINT x:A|M. f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   191
  by (subst integral_complex_of_real[symmetric])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   192
     (auto intro!: integral_cong split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   193
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   194
lemma set_integral_mono:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   195
  fixes f g :: "_ \<Rightarrow> real"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   196
  assumes "set_integrable M A f" "set_integrable M A g"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   197
    "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   198
  shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   199
using assms by (auto intro: integral_mono split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   200
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   201
lemma set_integral_mono_AE: 
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   202
  fixes f g :: "_ \<Rightarrow> real"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   203
  assumes "set_integrable M A f" "set_integrable M A g"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   204
    "AE x \<in> A in M. f x \<le> g x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   205
  shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   206
using assms by (auto intro: integral_mono_AE split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   207
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   208
lemma set_integrable_abs: "set_integrable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar> :: real)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   209
  using integrable_abs[of M "\<lambda>x. f x * indicator A x"] by (simp add: abs_mult ac_simps)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   210
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   211
lemma set_integrable_abs_iff:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   212
  fixes f :: "_ \<Rightarrow> real"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   213
  shows "set_borel_measurable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f" 
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   214
  by (subst (2) integrable_abs_iff[symmetric]) (simp_all add: abs_mult ac_simps)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   215
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   216
lemma set_integrable_abs_iff':
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   217
  fixes f :: "_ \<Rightarrow> real"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   218
  shows "f \<in> borel_measurable M \<Longrightarrow> A \<in> sets M \<Longrightarrow> 
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   219
    set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   220
by (intro set_integrable_abs_iff) auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   221
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   222
lemma set_integrable_discrete_difference:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   223
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   224
  assumes "countable X"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   225
  assumes diff: "(A - B) \<union> (B - A) \<subseteq> X"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   226
  assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   227
  shows "set_integrable M A f \<longleftrightarrow> set_integrable M B f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   228
proof (rule integrable_discrete_difference[where X=X])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   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"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   230
    using diff by (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   231
qed fact+
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   232
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   233
lemma set_integral_discrete_difference:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   234
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   235
  assumes "countable X"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   236
  assumes diff: "(A - B) \<union> (B - A) \<subseteq> X"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   237
  assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   238
  shows "set_lebesgue_integral M A f = set_lebesgue_integral M B f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   239
proof (rule integral_discrete_difference[where X=X])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   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"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   241
    using diff by (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   242
qed fact+
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   243
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   244
lemma set_integrable_Un:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   245
  fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   246
  assumes f_A: "set_integrable M A f" and f_B:  "set_integrable M B f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   247
    and [measurable]: "A \<in> sets M" "B \<in> sets M"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   248
  shows "set_integrable M (A \<union> B) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   249
proof -
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   250
  have "set_integrable M (A - B) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   251
    using f_A by (rule set_integrable_subset) auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   252
  from integrable_add[OF this f_B] show ?thesis
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   253
    by (rule integrable_cong[THEN iffD1, rotated 2]) (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   254
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   255
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   256
lemma set_integrable_UN:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   257
  fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   258
  assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> set_integrable M (A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   259
    "\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   260
  shows "set_integrable M (\<Union>i\<in>I. A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   261
using assms by (induct I) (auto intro!: set_integrable_Un)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   262
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   263
lemma set_integral_Un:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   264
  fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   265
  assumes "A \<inter> B = {}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   266
  and "set_integrable M A f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   267
  and "set_integrable M B f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   268
  shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   269
by (auto simp add: indicator_union_arith indicator_inter_arith[symmetric]
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   270
      scaleR_add_left assms)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   271
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   272
lemma set_integral_cong_set:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   273
  fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   274
  assumes [measurable]: "set_borel_measurable M A f" "set_borel_measurable M B f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   275
    and ae: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   276
  shows "LINT x:B|M. f x = LINT x:A|M. f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   277
proof (rule integral_cong_AE)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   278
  show "AE x in M. indicator B x *\<^sub>R f x = indicator A x *\<^sub>R f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   279
    using ae by (auto simp: subset_eq split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   280
qed fact+
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   281
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   282
lemma set_borel_measurable_subset:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   283
  fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   284
  assumes [measurable]: "set_borel_measurable M A f" "B \<in> sets M" and "B \<subseteq> A"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   285
  shows "set_borel_measurable M B f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   286
proof -
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   287
  have "set_borel_measurable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   288
    by measurable
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   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)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   290
    using `B \<subseteq> A` by (auto simp: fun_eq_iff split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   291
  finally show ?thesis .
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   292
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   293
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   294
lemma set_integral_Un_AE:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   295
  fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   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"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   297
  and "set_integrable M A f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   298
  and "set_integrable M B f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   299
  shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   300
proof -
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   301
  have f: "set_integrable M (A \<union> B) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   302
    by (intro set_integrable_Un assms)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   303
  then have f': "set_borel_measurable M (A \<union> B) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   304
    by (rule borel_measurable_integrable)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   305
  have "LINT x:A\<union>B|M. f x = LINT x:(A - A \<inter> B) \<union> (B - A \<inter> B)|M. f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   306
  proof (rule set_integral_cong_set)  
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   307
    show "AE x in M. (x \<in> A - A \<inter> B \<union> (B - A \<inter> B)) = (x \<in> A \<union> B)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   308
      using ae by auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   309
    show "set_borel_measurable M (A - A \<inter> B \<union> (B - A \<inter> B)) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   310
      using f' by (rule set_borel_measurable_subset) auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   311
  qed fact
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   312
  also have "\<dots> = (LINT x:(A - A \<inter> B)|M. f x) + (LINT x:(B - A \<inter> B)|M. f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   313
    by (auto intro!: set_integral_Un set_integrable_subset[OF f])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   314
  also have "\<dots> = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   315
    using ae
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   316
    by (intro arg_cong2[where f="op+"] set_integral_cong_set)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   317
       (auto intro!: set_borel_measurable_subset[OF f'])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   318
  finally show ?thesis .
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   319
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   320
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   321
lemma set_integral_finite_Union:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   322
  fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   323
  assumes "finite I" "disjoint_family_on A I"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   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"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   325
  shows "(LINT x:(\<Union>i\<in>I. A i)|M. f x) = (\<Sum>i\<in>I. LINT x:A i|M. f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   326
  using assms
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   327
  apply induct
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   328
  apply (auto intro!: set_integral_Un set_integrable_Un set_integrable_UN simp: disjoint_family_on_def)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   329
by (subst set_integral_Un, auto intro: set_integrable_UN)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   330
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   331
(* TODO: find a better name? *)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   332
lemma pos_integrable_to_top:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   333
  fixes l::real
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   334
  assumes "\<And>i. A i \<in> sets M" "mono A"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   335
  assumes nneg: "\<And>x i. x \<in> A i \<Longrightarrow> 0 \<le> f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   336
  and intgbl: "\<And>i::nat. set_integrable M (A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   337
  and lim: "(\<lambda>i::nat. LINT x:A i|M. f x) ----> l"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   338
  shows "set_integrable M (\<Union>i. A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   339
  apply (rule integrable_monotone_convergence[where f = "\<lambda>i::nat. \<lambda>x. indicator (A i) x *\<^sub>R f x" and x = l])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   340
  apply (rule intgbl)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   341
  prefer 3 apply (rule lim)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   342
  apply (rule AE_I2)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   343
  using `mono A` apply (auto simp: mono_def nneg split: split_indicator) []
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   344
proof (rule AE_I2)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   345
  { fix x assume "x \<in> space M"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   346
    show "(\<lambda>i. indicator (A i) x *\<^sub>R f x) ----> indicator (\<Union>i. A i) x *\<^sub>R f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   347
    proof cases
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   348
      assume "\<exists>i. x \<in> A i"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   349
      then guess i ..
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   350
      then have *: "eventually (\<lambda>i. x \<in> A i) sequentially"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   351
        using `x \<in> A i` `mono A` by (auto simp: eventually_sequentially mono_def)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   352
      show ?thesis
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   353
        apply (intro Lim_eventually)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   354
        using *
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   355
        apply eventually_elim
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   356
        apply (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   357
        done
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   358
    qed auto }
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   359
  then show "(\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R f x) \<in> borel_measurable M"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   360
    apply (rule borel_measurable_LIMSEQ)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   361
    apply assumption
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   362
    apply (intro borel_measurable_integrable intgbl)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   363
    done
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   364
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   365
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   366
(* Proof from Royden Real Analysis, p. 91. *)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   367
lemma lebesgue_integral_countable_add:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   368
  fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   369
  assumes meas[intro]: "\<And>i::nat. A i \<in> sets M"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   370
    and disj: "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   371
    and intgbl: "set_integrable M (\<Union>i. A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   372
  shows "LINT x:(\<Union>i. A i)|M. f x = (\<Sum>i. (LINT x:(A i)|M. f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   373
proof (subst integral_suminf[symmetric])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   374
  show int_A: "\<And>i. set_integrable M (A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   375
    using intgbl by (rule set_integrable_subset) auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   376
  { fix x assume "x \<in> space M"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   377
    have "(\<lambda>i. indicator (A i) x *\<^sub>R f x) sums (indicator (\<Union>i. A i) x *\<^sub>R f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   378
      by (intro sums_scaleR_left indicator_sums) fact }
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   379
  note sums = this
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   380
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   381
  have norm_f: "\<And>i. set_integrable M (A i) (\<lambda>x. norm (f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   382
    using int_A[THEN integrable_norm] by auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   383
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   384
  show "AE x in M. summable (\<lambda>i. norm (indicator (A i) x *\<^sub>R f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   385
    using disj by (intro AE_I2) (auto intro!: summable_mult2 sums_summable[OF indicator_sums])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   386
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   387
  show "summable (\<lambda>i. LINT x|M. norm (indicator (A i) x *\<^sub>R f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   388
  proof (rule summableI_nonneg_bounded)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   389
    fix n
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   390
    show "0 \<le> LINT x|M. norm (indicator (A n) x *\<^sub>R f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   391
      using norm_f by (auto intro!: integral_nonneg_AE)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   392
    
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   393
    have "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) =
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   394
      (\<Sum>i<n. set_lebesgue_integral M (A i) (\<lambda>x. norm (f x)))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   395
      by (simp add: abs_mult)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   396
    also have "\<dots> = set_lebesgue_integral M (\<Union>i<n. A i) (\<lambda>x. norm (f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   397
      using norm_f
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   398
      by (subst set_integral_finite_Union) (auto simp: disjoint_family_on_def disj)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   399
    also have "\<dots> \<le> set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   400
      using intgbl[THEN integrable_norm]
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   401
      by (intro integral_mono set_integrable_UN[of "{..<n}"] norm_f)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   402
         (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   403
    finally show "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) \<le>
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   404
      set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   405
      by simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   406
  qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   407
  show "set_lebesgue_integral M (UNION UNIV A) f = LINT x|M. (\<Sum>i. indicator (A i) x *\<^sub>R f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   408
    apply (rule integral_cong[OF refl])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   409
    apply (subst suminf_scaleR_left[OF sums_summable[OF indicator_sums, OF disj], symmetric])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   410
    using sums_unique[OF indicator_sums[OF disj]]
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   411
    apply auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   412
    done
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   413
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   414
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   415
lemma set_integral_cont_up:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   416
  fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   417
  assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "incseq A"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   418
  and intgbl: "set_integrable M (\<Union>i. A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   419
  shows "(\<lambda>i. LINT x:(A i)|M. f x) ----> LINT x:(\<Union>i. A i)|M. f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   420
proof (intro integral_dominated_convergence[where w="\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R norm (f x)"])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   421
  have int_A: "\<And>i. set_integrable M (A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   422
    using intgbl by (rule set_integrable_subset) auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   423
  then show "\<And>i. set_borel_measurable M (A i) f" "set_borel_measurable M (\<Union>i. A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   424
    "set_integrable M (\<Union>i. A i) (\<lambda>x. norm (f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   425
    using intgbl integrable_norm[OF intgbl] by auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   426
  
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   427
  { fix x i assume "x \<in> A i"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   428
    with A have "(\<lambda>xa. indicator (A xa) x::real) ----> 1 \<longleftrightarrow> (\<lambda>xa. 1::real) ----> 1"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   429
      by (intro filterlim_cong refl)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   430
         (fastforce simp: eventually_sequentially incseq_def subset_eq intro!: exI[of _ i]) }
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   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"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   432
    by (intro AE_I2 tendsto_intros) (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   433
qed (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   434
        
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   435
(* Can the int0 hypothesis be dropped? *)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   436
lemma set_integral_cont_down:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   437
  fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   438
  assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "decseq A"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   439
  and int0: "set_integrable M (A 0) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   440
  shows "(\<lambda>i::nat. LINT x:(A i)|M. f x) ----> LINT x:(\<Inter>i. A i)|M. f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   441
proof (rule integral_dominated_convergence)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   442
  have int_A: "\<And>i. set_integrable M (A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   443
    using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   444
  show "set_integrable M (A 0) (\<lambda>x. norm (f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   445
    using int0[THEN integrable_norm] by simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   446
  have "set_integrable M (\<Inter>i. A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   447
    using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   448
  with int_A show "set_borel_measurable M (\<Inter>i. A i) f" "\<And>i. set_borel_measurable M (A i) f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   449
    by auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   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)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   451
    using A by (auto split: split_indicator simp: decseq_def)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   452
  { fix x i assume "x \<in> space M" "x \<notin> A i"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   453
    with A have "(\<lambda>i. indicator (A i) x::real) ----> 0 \<longleftrightarrow> (\<lambda>i. 0::real) ----> 0"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   454
      by (intro filterlim_cong refl)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   455
         (auto split: split_indicator simp: eventually_sequentially decseq_def intro!: exI[of _ i]) }
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   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"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   457
    by (intro AE_I2 tendsto_intros) (auto split: split_indicator)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   458
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   459
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   460
lemma set_integral_at_point:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   461
  fixes a :: real
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   462
  assumes "set_integrable M {a} f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   463
  and [simp]: "{a} \<in> sets M" and "(emeasure M) {a} \<noteq> \<infinity>"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   464
  shows "(LINT x:{a} | M. f x) = f a * measure M {a}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   465
proof-
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   466
  have "set_lebesgue_integral M {a} f = set_lebesgue_integral M {a} (%x. f a)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   467
    by (intro set_lebesgue_integral_cong) simp_all
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   468
  then show ?thesis using assms by simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   469
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   470
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   471
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   472
abbreviation complex_integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> bool" where
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   473
  "complex_integrable M f \<equiv> integrable M f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   474
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   475
abbreviation complex_lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> complex" ("integral\<^sup>C") where
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   476
  "integral\<^sup>C M f == integral\<^sup>L M f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   477
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   478
syntax
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   479
  "_complex_lebesgue_integral" :: "pttrn \<Rightarrow> complex \<Rightarrow> 'a measure \<Rightarrow> complex"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   480
 ("\<integral>\<^sup>C _. _ \<partial>_" [60,61] 110)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   481
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   482
translations
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   483
  "\<integral>\<^sup>Cx. f \<partial>M" == "CONST complex_lebesgue_integral M (\<lambda>x. f)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   484
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   485
syntax
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   486
  "_ascii_complex_lebesgue_integral" :: "pttrn \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   487
  ("(3CLINT _|_. _)" [0,110,60] 60)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   488
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   489
translations
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   490
  "CLINT x|M. f" == "CONST complex_lebesgue_integral M (\<lambda>x. f)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   491
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   492
lemma complex_integrable_cnj [simp]:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   493
  "complex_integrable M (\<lambda>x. cnj (f x)) \<longleftrightarrow> complex_integrable M f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   494
proof
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   495
  assume "complex_integrable M (\<lambda>x. cnj (f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   496
  then have "complex_integrable M (\<lambda>x. cnj (cnj (f x)))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   497
    by (rule integrable_cnj)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   498
  then show "complex_integrable M f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   499
    by simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   500
qed simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   501
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   502
lemma complex_of_real_integrable_eq:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   503
  "complex_integrable M (\<lambda>x. complex_of_real (f x)) \<longleftrightarrow> integrable M f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   504
proof
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   505
  assume "complex_integrable M (\<lambda>x. complex_of_real (f x))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   506
  then have "integrable M (\<lambda>x. Re (complex_of_real (f x)))"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   507
    by (rule integrable_Re)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   508
  then show "integrable M f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   509
    by simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   510
qed simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   511
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   512
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   513
abbreviation complex_set_integrable :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> bool" where
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   514
  "complex_set_integrable M A f \<equiv> set_integrable M A f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   515
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   516
abbreviation complex_set_lebesgue_integral :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> complex" where
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   517
  "complex_set_lebesgue_integral M A f \<equiv> set_lebesgue_integral M A f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   518
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   519
syntax
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   520
"_ascii_complex_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   521
("(4CLINT _:_|_. _)" [0,60,110,61] 60)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   522
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   523
translations
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   524
"CLINT x:A|M. f" == "CONST complex_set_lebesgue_integral M A (\<lambda>x. f)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   525
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   526
(*
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   527
lemma cmod_mult: "cmod ((a :: real) * (x :: complex)) = abs a * cmod x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   528
  apply (simp add: norm_mult)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   529
  by (subst norm_mult, auto)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   530
*)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   531
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   532
lemma borel_integrable_atLeastAtMost':
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   533
  fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   534
  assumes f: "continuous_on {a..b} f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   535
  shows "set_integrable lborel {a..b} f" (is "integrable _ ?f")
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   536
  by (intro borel_integrable_compact compact_Icc f)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   537
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   538
lemma integral_FTC_atLeastAtMost:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   539
  fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   540
  assumes "a \<le> b"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   541
    and F: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   542
    and f: "continuous_on {a .. b} f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   543
  shows "integral\<^sup>L lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x) = F b - F a"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   544
proof -
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   545
  let ?f = "\<lambda>x. indicator {a .. b} x *\<^sub>R f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   546
  have "(?f has_integral (\<integral>x. ?f x \<partial>lborel)) UNIV"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   547
    using borel_integrable_atLeastAtMost'[OF f] by (rule has_integral_integral_lborel)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   548
  moreover
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   549
  have "(f has_integral F b - F a) {a .. b}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   550
    by (intro fundamental_theorem_of_calculus ballI assms) auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   551
  then have "(?f has_integral F b - F a) {a .. b}"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   552
    by (subst has_integral_eq_eq[where g=f]) auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   553
  then have "(?f has_integral F b - F a) UNIV"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   554
    by (intro has_integral_on_superset[where t=UNIV and s="{a..b}"]) auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   555
  ultimately show "integral\<^sup>L lborel ?f = F b - F a"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   556
    by (rule has_integral_unique)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   557
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   558
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   559
lemma set_borel_integral_eq_integral:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   560
  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   561
  assumes "set_integrable lborel S f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   562
  shows "f integrable_on S" "LINT x : S | lborel. f x = integral S f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   563
proof -
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   564
  let ?f = "\<lambda>x. indicator S x *\<^sub>R f x"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   565
  have "(?f has_integral LINT x : S | lborel. f x) UNIV"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   566
    by (rule has_integral_integral_lborel) fact
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   567
  hence 1: "(f has_integral (set_lebesgue_integral lborel S f)) S"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   568
    apply (subst has_integral_restrict_univ [symmetric])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   569
    apply (rule has_integral_eq)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   570
    by auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   571
  thus "f integrable_on S"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   572
    by (auto simp add: integrable_on_def)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   573
  with 1 have "(f has_integral (integral S f)) S"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   574
    by (intro integrable_integral, auto simp add: integrable_on_def)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   575
  thus "LINT x : S | lborel. f x = integral S f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   576
    by (intro has_integral_unique [OF 1])
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   577
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   578
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   579
lemma set_borel_measurable_continuous:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   580
  fixes f :: "_ \<Rightarrow> _::real_normed_vector"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   581
  assumes "S \<in> sets borel" "continuous_on S f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   582
  shows "set_borel_measurable borel S f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   583
proof -
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   584
  have "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable borel"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   585
    by (intro assms borel_measurable_continuous_on_if continuous_on_const)
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   586
  also have "(\<lambda>x. if x \<in> S then f x else 0) = (\<lambda>x. indicator S x *\<^sub>R f x)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   587
    by auto
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   588
  finally show ?thesis .
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   589
qed
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   590
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   591
lemma set_measurable_continuous_on_ivl:
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   592
  assumes "continuous_on {a..b} (f :: real \<Rightarrow> real)"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   593
  shows "set_borel_measurable borel {a..b} f"
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   594
  by (rule set_borel_measurable_continuous[OF _ assms]) simp
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   595
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   596
end
d469103c0737 add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff changeset
   597