src/HOL/Probability/Bochner_Integration.thy
author hoelzl
Tue Jan 27 16:12:40 2015 +0100 (2015-01-27)
changeset 59452 2538b2c51769
parent 59425 c5e79df8cc21
child 59587 8ea7b22525cb
permissions -rw-r--r--
ereal: tuned proofs concerning continuity and suprema
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(*  Title:      HOL/Probability/Bochner_Integration.thy
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    Author:     Johannes Hölzl, TU München
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*)
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section {* Bochner Integration for Vector-Valued Functions *}
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theory Bochner_Integration
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  imports Finite_Product_Measure
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begin
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text {*
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In the following development of the Bochner integral we use second countable topologies instead
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of separable spaces. A second countable topology is also separable.
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*}
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lemma borel_measurable_implies_sequence_metric:
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  fixes f :: "'a \<Rightarrow> 'b :: {metric_space, second_countable_topology}"
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  assumes [measurable]: "f \<in> borel_measurable M"
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  shows "\<exists>F. (\<forall>i. simple_function M (F i)) \<and> (\<forall>x\<in>space M. (\<lambda>i. F i x) ----> f x) \<and>
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    (\<forall>i. \<forall>x\<in>space M. dist (F i x) z \<le> 2 * dist (f x) z)"
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proof -
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  obtain D :: "'b set" where "countable D" and D: "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d\<in>D. d \<in> X"
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    by (erule countable_dense_setE)
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  def e \<equiv> "from_nat_into D"
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  { fix n x
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    obtain d where "d \<in> D" and d: "d \<in> ball x (1 / Suc n)"
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      using D[of "ball x (1 / Suc n)"] by auto
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    from `d \<in> D` D[of UNIV] `countable D` obtain i where "d = e i"
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      unfolding e_def by (auto dest: from_nat_into_surj)
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    with d have "\<exists>i. dist x (e i) < 1 / Suc n"
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      by auto }
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  note e = this
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  def A \<equiv> "\<lambda>m n. {x\<in>space M. dist (f x) (e n) < 1 / (Suc m) \<and> 1 / (Suc m) \<le> dist (f x) z}"
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  def B \<equiv> "\<lambda>m. disjointed (A m)"
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  def m \<equiv> "\<lambda>N x. Max {m::nat. m \<le> N \<and> x \<in> (\<Union>n\<le>N. B m n)}"
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  def F \<equiv> "\<lambda>N::nat. \<lambda>x. if (\<exists>m\<le>N. x \<in> (\<Union>n\<le>N. B m n)) \<and> (\<exists>n\<le>N. x \<in> B (m N x) n) 
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    then e (LEAST n. x \<in> B (m N x) n) else z"
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  have B_imp_A[intro, simp]: "\<And>x m n. x \<in> B m n \<Longrightarrow> x \<in> A m n"
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    using disjointed_subset[of "A m" for m] unfolding B_def by auto
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  { fix m
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    have "\<And>n. A m n \<in> sets M"
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      by (auto simp: A_def)
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    then have "\<And>n. B m n \<in> sets M"
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      using sets.range_disjointed_sets[of "A m" M] by (auto simp: B_def) }
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  note this[measurable]
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  { fix N i x assume "\<exists>m\<le>N. x \<in> (\<Union>n\<le>N. B m n)"
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    then have "m N x \<in> {m::nat. m \<le> N \<and> x \<in> (\<Union>n\<le>N. B m n)}"
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      unfolding m_def by (intro Max_in) auto
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    then have "m N x \<le> N" "\<exists>n\<le>N. x \<in> B (m N x) n"
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      by auto }
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  note m = this
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  { fix j N i x assume "j \<le> N" "i \<le> N" "x \<in> B j i"
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    then have "j \<le> m N x"
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      unfolding m_def by (intro Max_ge) auto }
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  note m_upper = this
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  show ?thesis
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    unfolding simple_function_def
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  proof (safe intro!: exI[of _ F])
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    have [measurable]: "\<And>i. F i \<in> borel_measurable M"
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      unfolding F_def m_def by measurable
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    show "\<And>x i. F i -` {x} \<inter> space M \<in> sets M"
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      by measurable
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    { fix i
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      { fix n x assume "x \<in> B (m i x) n"
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        then have "(LEAST n. x \<in> B (m i x) n) \<le> n"
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          by (intro Least_le)
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        also assume "n \<le> i" 
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        finally have "(LEAST n. x \<in> B (m i x) n) \<le> i" . }
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      then have "F i ` space M \<subseteq> {z} \<union> e ` {.. i}"
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        by (auto simp: F_def)
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      then show "finite (F i ` space M)"
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        by (rule finite_subset) auto }
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    { fix N i n x assume "i \<le> N" "n \<le> N" "x \<in> B i n"
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      then have 1: "\<exists>m\<le>N. x \<in> (\<Union> n\<le>N. B m n)" by auto
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      from m[OF this] obtain n where n: "m N x \<le> N" "n \<le> N" "x \<in> B (m N x) n" by auto
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      moreover
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      def L \<equiv> "LEAST n. x \<in> B (m N x) n"
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      have "dist (f x) (e L) < 1 / Suc (m N x)"
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      proof -
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        have "x \<in> B (m N x) L"
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          using n(3) unfolding L_def by (rule LeastI)
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        then have "x \<in> A (m N x) L"
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          by auto
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        then show ?thesis
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          unfolding A_def by simp
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      qed
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      ultimately have "dist (f x) (F N x) < 1 / Suc (m N x)"
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        by (auto simp add: F_def L_def) }
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    note * = this
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    fix x assume "x \<in> space M"
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    show "(\<lambda>i. F i x) ----> f x"
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    proof cases
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      assume "f x = z"
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      then have "\<And>i n. x \<notin> A i n"
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        unfolding A_def by auto
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      then have "\<And>i. F i x = z"
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        by (auto simp: F_def)
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      then show ?thesis
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        using `f x = z` by auto
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    next
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      assume "f x \<noteq> z"
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      show ?thesis
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      proof (rule tendstoI)
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        fix e :: real assume "0 < e"
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        with `f x \<noteq> z` obtain n where "1 / Suc n < e" "1 / Suc n < dist (f x) z"
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          by (metis dist_nz order_less_trans neq_iff nat_approx_posE)
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        with `x\<in>space M` `f x \<noteq> z` have "x \<in> (\<Union>i. B n i)"
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          unfolding A_def B_def UN_disjointed_eq using e by auto
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        then obtain i where i: "x \<in> B n i" by auto
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        show "eventually (\<lambda>i. dist (F i x) (f x) < e) sequentially"
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          using eventually_ge_at_top[of "max n i"]
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        proof eventually_elim
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          fix j assume j: "max n i \<le> j"
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          with i have "dist (f x) (F j x) < 1 / Suc (m j x)"
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            by (intro *[OF _ _ i]) auto
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          also have "\<dots> \<le> 1 / Suc n"
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            using j m_upper[OF _ _ i]
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            by (auto simp: field_simps)
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          also note `1 / Suc n < e`
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          finally show "dist (F j x) (f x) < e"
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            by (simp add: less_imp_le dist_commute)
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        qed
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      qed
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    qed
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    fix i 
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    { fix n m assume "x \<in> A n m"
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      then have "dist (e m) (f x) + dist (f x) z \<le> 2 * dist (f x) z"
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        unfolding A_def by (auto simp: dist_commute)
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      also have "dist (e m) z \<le> dist (e m) (f x) + dist (f x) z"
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        by (rule dist_triangle)
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      finally (xtrans) have "dist (e m) z \<le> 2 * dist (f x) z" . }
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    then show "dist (F i x) z \<le> 2 * dist (f x) z"
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      unfolding F_def
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      apply auto
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      apply (rule LeastI2)
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      apply auto
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      done
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  qed
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qed
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lemma real_indicator: "real (indicator A x :: ereal) = indicator A x"
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  unfolding indicator_def by auto
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lemma split_indicator_asm:
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  "P (indicator S x) \<longleftrightarrow> \<not> ((x \<in> S \<and> \<not> P 1) \<or> (x \<notin> S \<and> \<not> P 0))"
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  unfolding indicator_def by auto
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lemma
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  fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A"
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  shows setsum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator (B x) (g x)) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
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  and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator (B x) (g x) * f x) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
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  unfolding indicator_def
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  using assms by (auto intro!: setsum.mono_neutral_cong_right split: split_if_asm)
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lemma borel_measurable_induct_real[consumes 2, case_names set mult add seq]:
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  fixes P :: "('a \<Rightarrow> real) \<Rightarrow> bool"
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  assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
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  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
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  assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
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  assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
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  assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow> (\<And>i x. 0 \<le> U i x) \<Longrightarrow> (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. U i x) ----> u x) \<Longrightarrow> P u"
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  shows "P u"
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proof -
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  have "(\<lambda>x. ereal (u x)) \<in> borel_measurable M" using u by auto
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  from borel_measurable_implies_simple_function_sequence'[OF this]
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  obtain U where U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and
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    sup: "\<And>x. (SUP i. U i x) = max 0 (ereal (u x))" and nn: "\<And>i x. 0 \<le> U i x"
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    by blast
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  def U' \<equiv> "\<lambda>i x. indicator (space M) x * real (U i x)"
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  then have U'_sf[measurable]: "\<And>i. simple_function M (U' i)"
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    using U by (auto intro!: simple_function_compose1[where g=real])
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  show "P u"
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  proof (rule seq)
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    fix i show "U' i \<in> borel_measurable M" "\<And>x. 0 \<le> U' i x"
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      using U nn by (auto
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          intro: borel_measurable_simple_function 
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          intro!: borel_measurable_real_of_ereal real_of_ereal_pos borel_measurable_times
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          simp: U'_def zero_le_mult_iff)
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    show "incseq U'"
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      using U(2,3) nn
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      by (auto simp: incseq_def le_fun_def image_iff eq_commute U'_def indicator_def
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               intro!: real_of_ereal_positive_mono)
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  next
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    fix x assume x: "x \<in> space M"
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    have "(\<lambda>i. U i x) ----> (SUP i. U i x)"
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      using U(2) by (intro LIMSEQ_SUP) (auto simp: incseq_def le_fun_def)
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    moreover have "(\<lambda>i. U i x) = (\<lambda>i. ereal (U' i x))"
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      using x nn U(3) by (auto simp: fun_eq_iff U'_def ereal_real image_iff eq_commute)
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    moreover have "(SUP i. U i x) = ereal (u x)"
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      using sup u(2) by (simp add: max_def)
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    ultimately show "(\<lambda>i. U' i x) ----> u x" 
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      by simp
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  next
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    fix i
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    have "U' i ` space M \<subseteq> real ` (U i ` space M)" "finite (U i ` space M)"
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      unfolding U'_def using U(1) by (auto dest: simple_functionD)
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    then have fin: "finite (U' i ` space M)"
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      by (metis finite_subset finite_imageI)
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    moreover have "\<And>z. {y. U' i z = y \<and> y \<in> U' i ` space M \<and> z \<in> space M} = (if z \<in> space M then {U' i z} else {})"
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      by auto
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    ultimately have U': "(\<lambda>z. \<Sum>y\<in>U' i`space M. y * indicator {x\<in>space M. U' i x = y} z) = U' i"
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      by (simp add: U'_def fun_eq_iff)
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    have "\<And>x. x \<in> U' i ` space M \<Longrightarrow> 0 \<le> x"
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      using nn by (auto simp: U'_def real_of_ereal_pos)
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    with fin have "P (\<lambda>z. \<Sum>y\<in>U' i`space M. y * indicator {x\<in>space M. U' i x = y} z)"
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    proof induct
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      case empty from set[of "{}"] show ?case
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        by (simp add: indicator_def[abs_def])
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    next
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      case (insert x F)
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      then show ?case
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        by (auto intro!: add mult set setsum_nonneg split: split_indicator split_indicator_asm
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                 simp del: setsum_mult_indicator simp: setsum_nonneg_eq_0_iff )
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    qed
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    with U' show "P (U' i)" by simp
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  qed
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qed
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lemma scaleR_cong_right:
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  fixes x :: "'a :: real_vector"
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  shows "(x \<noteq> 0 \<Longrightarrow> r = p) \<Longrightarrow> r *\<^sub>R x = p *\<^sub>R x"
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  by (cases "x = 0") auto
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inductive simple_bochner_integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" for M f where
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  "simple_function M f \<Longrightarrow> emeasure M {y\<in>space M. f y \<noteq> 0} \<noteq> \<infinity> \<Longrightarrow>
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    simple_bochner_integrable M f"
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lemma simple_bochner_integrable_compose2:
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  assumes p_0: "p 0 0 = 0"
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  shows "simple_bochner_integrable M f \<Longrightarrow> simple_bochner_integrable M g \<Longrightarrow>
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    simple_bochner_integrable M (\<lambda>x. p (f x) (g x))"
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proof (safe intro!: simple_bochner_integrable.intros elim!: simple_bochner_integrable.cases del: notI)
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  assume sf: "simple_function M f" "simple_function M g"
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  then show "simple_function M (\<lambda>x. p (f x) (g x))"
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    by (rule simple_function_compose2)
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  from sf have [measurable]:
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      "f \<in> measurable M (count_space UNIV)"
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   256
      "g \<in> measurable M (count_space UNIV)"
hoelzl@56993
   257
    by (auto intro: measurable_simple_function)
hoelzl@56993
   258
hoelzl@56993
   259
  assume fin: "emeasure M {y \<in> space M. f y \<noteq> 0} \<noteq> \<infinity>" "emeasure M {y \<in> space M. g y \<noteq> 0} \<noteq> \<infinity>"
hoelzl@56993
   260
   
hoelzl@56993
   261
  have "emeasure M {x\<in>space M. p (f x) (g x) \<noteq> 0} \<le>
hoelzl@56993
   262
      emeasure M ({x\<in>space M. f x \<noteq> 0} \<union> {x\<in>space M. g x \<noteq> 0})"
hoelzl@56993
   263
    by (intro emeasure_mono) (auto simp: p_0)
hoelzl@56993
   264
  also have "\<dots> \<le> emeasure M {x\<in>space M. f x \<noteq> 0} + emeasure M {x\<in>space M. g x \<noteq> 0}"
hoelzl@56993
   265
    by (intro emeasure_subadditive) auto
hoelzl@56993
   266
  finally show "emeasure M {y \<in> space M. p (f y) (g y) \<noteq> 0} \<noteq> \<infinity>"
hoelzl@56993
   267
    using fin by auto
hoelzl@56993
   268
qed
hoelzl@56993
   269
hoelzl@56993
   270
lemma simple_function_finite_support:
hoelzl@56993
   271
  assumes f: "simple_function M f" and fin: "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>" and nn: "\<And>x. 0 \<le> f x"
hoelzl@56993
   272
  shows "emeasure M {x\<in>space M. f x \<noteq> 0} \<noteq> \<infinity>"
hoelzl@56993
   273
proof cases
hoelzl@56993
   274
  from f have meas[measurable]: "f \<in> borel_measurable M"
hoelzl@56993
   275
    by (rule borel_measurable_simple_function)
hoelzl@56993
   276
hoelzl@56993
   277
  assume non_empty: "\<exists>x\<in>space M. f x \<noteq> 0"
hoelzl@56993
   278
hoelzl@56993
   279
  def m \<equiv> "Min (f`space M - {0})"
hoelzl@56993
   280
  have "m \<in> f`space M - {0}"
hoelzl@56993
   281
    unfolding m_def using f non_empty by (intro Min_in) (auto simp: simple_function_def)
hoelzl@56993
   282
  then have m: "0 < m"
hoelzl@56993
   283
    using nn by (auto simp: less_le)
hoelzl@56993
   284
hoelzl@56993
   285
  from m have "m * emeasure M {x\<in>space M. 0 \<noteq> f x} = 
hoelzl@56993
   286
    (\<integral>\<^sup>+x. m * indicator {x\<in>space M. 0 \<noteq> f x} x \<partial>M)"
hoelzl@56996
   287
    using f by (intro nn_integral_cmult_indicator[symmetric]) auto
hoelzl@56993
   288
  also have "\<dots> \<le> (\<integral>\<^sup>+x. f x \<partial>M)"
hoelzl@56993
   289
    using AE_space
hoelzl@56996
   290
  proof (intro nn_integral_mono_AE, eventually_elim)
hoelzl@56993
   291
    fix x assume "x \<in> space M"
hoelzl@56993
   292
    with nn show "m * indicator {x \<in> space M. 0 \<noteq> f x} x \<le> f x"
hoelzl@56993
   293
      using f by (auto split: split_indicator simp: simple_function_def m_def)
hoelzl@56993
   294
  qed
hoelzl@56993
   295
  also note `\<dots> < \<infinity>`
hoelzl@56993
   296
  finally show ?thesis
hoelzl@56993
   297
    using m by auto 
hoelzl@56993
   298
next
hoelzl@56993
   299
  assume "\<not> (\<exists>x\<in>space M. f x \<noteq> 0)"
hoelzl@56993
   300
  with nn have *: "{x\<in>space M. f x \<noteq> 0} = {}"
hoelzl@56993
   301
    by auto
hoelzl@56993
   302
  show ?thesis unfolding * by simp
hoelzl@56993
   303
qed
hoelzl@56993
   304
hoelzl@56993
   305
lemma simple_bochner_integrableI_bounded:
hoelzl@56993
   306
  assumes f: "simple_function M f" and fin: "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
hoelzl@56993
   307
  shows "simple_bochner_integrable M f"
hoelzl@56993
   308
proof
hoelzl@56993
   309
  have "emeasure M {y \<in> space M. ereal (norm (f y)) \<noteq> 0} \<noteq> \<infinity>"
hoelzl@56993
   310
  proof (rule simple_function_finite_support)
hoelzl@56993
   311
    show "simple_function M (\<lambda>x. ereal (norm (f x)))"
hoelzl@56993
   312
      using f by (rule simple_function_compose1)
hoelzl@56993
   313
    show "(\<integral>\<^sup>+ y. ereal (norm (f y)) \<partial>M) < \<infinity>" by fact
hoelzl@56993
   314
  qed simp
hoelzl@56993
   315
  then show "emeasure M {y \<in> space M. f y \<noteq> 0} \<noteq> \<infinity>" by simp
hoelzl@56993
   316
qed fact
hoelzl@56993
   317
hoelzl@56993
   318
definition simple_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> 'b" where
hoelzl@56993
   319
  "simple_bochner_integral M f = (\<Sum>y\<in>f`space M. measure M {x\<in>space M. f x = y} *\<^sub>R y)"
hoelzl@56993
   320
hoelzl@56993
   321
lemma simple_bochner_integral_partition:
hoelzl@56993
   322
  assumes f: "simple_bochner_integrable M f" and g: "simple_function M g"
hoelzl@56993
   323
  assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y"
hoelzl@56993
   324
  assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)"
hoelzl@56993
   325
  shows "simple_bochner_integral M f = (\<Sum>y\<in>g ` space M. measure M {x\<in>space M. g x = y} *\<^sub>R v y)"
hoelzl@56993
   326
    (is "_ = ?r")
hoelzl@56993
   327
proof -
hoelzl@56993
   328
  from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
hoelzl@56993
   329
    by (auto simp: simple_function_def elim: simple_bochner_integrable.cases)
hoelzl@56993
   330
hoelzl@56993
   331
  from f have [measurable]: "f \<in> measurable M (count_space UNIV)"
hoelzl@56993
   332
    by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)
hoelzl@56993
   333
hoelzl@56993
   334
  from g have [measurable]: "g \<in> measurable M (count_space UNIV)"
hoelzl@56993
   335
    by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)
hoelzl@56993
   336
hoelzl@56993
   337
  { fix y assume "y \<in> space M"
hoelzl@56993
   338
    then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}"
hoelzl@56993
   339
      by (auto cong: sub simp: v[symmetric]) }
hoelzl@56993
   340
  note eq = this
hoelzl@56993
   341
hoelzl@56993
   342
  have "simple_bochner_integral M f =
hoelzl@56993
   343
    (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. 
hoelzl@56993
   344
      if \<exists>x\<in>space M. y = f x \<and> z = g x then measure M {x\<in>space M. g x = z} else 0) *\<^sub>R y)"
hoelzl@56993
   345
    unfolding simple_bochner_integral_def
haftmann@57418
   346
  proof (safe intro!: setsum.cong scaleR_cong_right)
hoelzl@56993
   347
    fix y assume y: "y \<in> space M" "f y \<noteq> 0"
hoelzl@56993
   348
    have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} = 
hoelzl@56993
   349
        {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
hoelzl@56993
   350
      by auto
hoelzl@56993
   351
    have eq:"{x \<in> space M. f x = f y} =
hoelzl@56993
   352
        (\<Union>i\<in>{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}. {x \<in> space M. g x = i})"
hoelzl@56993
   353
      by (auto simp: eq_commute cong: sub rev_conj_cong)
hoelzl@56993
   354
    have "finite (g`space M)" by simp
hoelzl@56993
   355
    then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
hoelzl@56993
   356
      by (rule rev_finite_subset) auto
hoelzl@56993
   357
    moreover
hoelzl@56993
   358
    { fix x assume "x \<in> space M" "f x = f y"
hoelzl@56993
   359
      then have "x \<in> space M" "f x \<noteq> 0"
hoelzl@56993
   360
        using y by auto
hoelzl@56993
   361
      then have "emeasure M {y \<in> space M. g y = g x} \<le> emeasure M {y \<in> space M. f y \<noteq> 0}"
hoelzl@56993
   362
        by (auto intro!: emeasure_mono cong: sub)
hoelzl@56993
   363
      then have "emeasure M {xa \<in> space M. g xa = g x} < \<infinity>"
hoelzl@56993
   364
        using f by (auto simp: simple_bochner_integrable.simps) }
hoelzl@56993
   365
    ultimately
hoelzl@56993
   366
    show "measure M {x \<in> space M. f x = f y} =
hoelzl@56993
   367
      (\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then measure M {x \<in> space M. g x = z} else 0)"
haftmann@57418
   368
      apply (simp add: setsum.If_cases eq)
hoelzl@56993
   369
      apply (subst measure_finite_Union[symmetric])
hoelzl@56993
   370
      apply (auto simp: disjoint_family_on_def)
hoelzl@56993
   371
      done
hoelzl@56993
   372
  qed
hoelzl@56993
   373
  also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. 
hoelzl@56993
   374
      if \<exists>x\<in>space M. y = f x \<and> z = g x then measure M {x\<in>space M. g x = z} *\<^sub>R y else 0))"
haftmann@57418
   375
    by (auto intro!: setsum.cong simp: scaleR_setsum_left)
hoelzl@56993
   376
  also have "\<dots> = ?r"
haftmann@57418
   377
    by (subst setsum.commute)
haftmann@57418
   378
       (auto intro!: setsum.cong simp: setsum.If_cases scaleR_setsum_right[symmetric] eq)
hoelzl@56993
   379
  finally show "simple_bochner_integral M f = ?r" .
hoelzl@56993
   380
qed
hoelzl@56993
   381
hoelzl@56993
   382
lemma simple_bochner_integral_add:
hoelzl@56993
   383
  assumes f: "simple_bochner_integrable M f" and g: "simple_bochner_integrable M g"
hoelzl@56993
   384
  shows "simple_bochner_integral M (\<lambda>x. f x + g x) =
hoelzl@56993
   385
    simple_bochner_integral M f + simple_bochner_integral M g"
hoelzl@56993
   386
proof -
hoelzl@56993
   387
  from f g have "simple_bochner_integral M (\<lambda>x. f x + g x) =
hoelzl@56993
   388
    (\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R (fst y + snd y))"
hoelzl@56993
   389
    by (intro simple_bochner_integral_partition)
hoelzl@56993
   390
       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
hoelzl@56993
   391
  moreover from f g have "simple_bochner_integral M f =
hoelzl@56993
   392
    (\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R fst y)"
hoelzl@56993
   393
    by (intro simple_bochner_integral_partition)
hoelzl@56993
   394
       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
hoelzl@56993
   395
  moreover from f g have "simple_bochner_integral M g =
hoelzl@56993
   396
    (\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R snd y)"
hoelzl@56993
   397
    by (intro simple_bochner_integral_partition)
hoelzl@56993
   398
       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
hoelzl@56993
   399
  ultimately show ?thesis
haftmann@57418
   400
    by (simp add: setsum.distrib[symmetric] scaleR_add_right)
hoelzl@56993
   401
qed
hoelzl@56993
   402
hoelzl@56993
   403
lemma (in linear) simple_bochner_integral_linear:
hoelzl@56993
   404
  assumes g: "simple_bochner_integrable M g"
hoelzl@56993
   405
  shows "simple_bochner_integral M (\<lambda>x. f (g x)) = f (simple_bochner_integral M g)"
hoelzl@56993
   406
proof -
hoelzl@56993
   407
  from g have "simple_bochner_integral M (\<lambda>x. f (g x)) =
hoelzl@56993
   408
    (\<Sum>y\<in>g ` space M. measure M {x \<in> space M. g x = y} *\<^sub>R f y)"
hoelzl@56993
   409
    by (intro simple_bochner_integral_partition)
hoelzl@56993
   410
       (auto simp: simple_bochner_integrable_compose2[where p="\<lambda>x y. f x"] zero
hoelzl@56993
   411
             elim: simple_bochner_integrable.cases)
hoelzl@56993
   412
  also have "\<dots> = f (simple_bochner_integral M g)"
hoelzl@56993
   413
    by (simp add: simple_bochner_integral_def setsum scaleR)
hoelzl@56993
   414
  finally show ?thesis .
hoelzl@56993
   415
qed
hoelzl@56993
   416
hoelzl@56993
   417
lemma simple_bochner_integral_minus:
hoelzl@56993
   418
  assumes f: "simple_bochner_integrable M f"
hoelzl@56993
   419
  shows "simple_bochner_integral M (\<lambda>x. - f x) = - simple_bochner_integral M f"
hoelzl@56993
   420
proof -
hoelzl@56993
   421
  interpret linear uminus by unfold_locales auto
hoelzl@56993
   422
  from f show ?thesis
hoelzl@56993
   423
    by (rule simple_bochner_integral_linear)
hoelzl@56993
   424
qed
hoelzl@56993
   425
hoelzl@56993
   426
lemma simple_bochner_integral_diff:
hoelzl@56993
   427
  assumes f: "simple_bochner_integrable M f" and g: "simple_bochner_integrable M g"
hoelzl@56993
   428
  shows "simple_bochner_integral M (\<lambda>x. f x - g x) =
hoelzl@56993
   429
    simple_bochner_integral M f - simple_bochner_integral M g"
hoelzl@56993
   430
  unfolding diff_conv_add_uminus using f g
hoelzl@56993
   431
  by (subst simple_bochner_integral_add)
hoelzl@56993
   432
     (auto simp: simple_bochner_integral_minus simple_bochner_integrable_compose2[where p="\<lambda>x y. - y"])
hoelzl@56993
   433
hoelzl@56993
   434
lemma simple_bochner_integral_norm_bound:
hoelzl@56993
   435
  assumes f: "simple_bochner_integrable M f"
hoelzl@56993
   436
  shows "norm (simple_bochner_integral M f) \<le> simple_bochner_integral M (\<lambda>x. norm (f x))"
hoelzl@56993
   437
proof -
hoelzl@56993
   438
  have "norm (simple_bochner_integral M f) \<le> 
hoelzl@56993
   439
    (\<Sum>y\<in>f ` space M. norm (measure M {x \<in> space M. f x = y} *\<^sub>R y))"
hoelzl@56993
   440
    unfolding simple_bochner_integral_def by (rule norm_setsum)
hoelzl@56993
   441
  also have "\<dots> = (\<Sum>y\<in>f ` space M. measure M {x \<in> space M. f x = y} *\<^sub>R norm y)"
hoelzl@56993
   442
    by (simp add: measure_nonneg)
hoelzl@56993
   443
  also have "\<dots> = simple_bochner_integral M (\<lambda>x. norm (f x))"
hoelzl@56993
   444
    using f
hoelzl@56993
   445
    by (intro simple_bochner_integral_partition[symmetric])
hoelzl@56993
   446
       (auto intro: f simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
hoelzl@56993
   447
  finally show ?thesis .
hoelzl@56993
   448
qed
hoelzl@56993
   449
hoelzl@56996
   450
lemma simple_bochner_integral_eq_nn_integral:
hoelzl@56993
   451
  assumes f: "simple_bochner_integrable M f" "\<And>x. 0 \<le> f x"
hoelzl@56993
   452
  shows "simple_bochner_integral M f = (\<integral>\<^sup>+x. f x \<partial>M)"
hoelzl@56993
   453
proof -
hoelzl@56993
   454
  { fix x y z have "(x \<noteq> 0 \<Longrightarrow> y = z) \<Longrightarrow> ereal x * y = ereal x * z"
hoelzl@56993
   455
      by (cases "x = 0") (auto simp: zero_ereal_def[symmetric]) }
hoelzl@56993
   456
  note ereal_cong_mult = this
hoelzl@56993
   457
hoelzl@56993
   458
  have [measurable]: "f \<in> borel_measurable M"
hoelzl@56993
   459
    using f(1) by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
hoelzl@56993
   460
hoelzl@56993
   461
  { fix y assume y: "y \<in> space M" "f y \<noteq> 0"
hoelzl@56993
   462
    have "ereal (measure M {x \<in> space M. f x = f y}) = emeasure M {x \<in> space M. f x = f y}"
hoelzl@56993
   463
    proof (rule emeasure_eq_ereal_measure[symmetric])
hoelzl@56993
   464
      have "emeasure M {x \<in> space M. f x = f y} \<le> emeasure M {x \<in> space M. f x \<noteq> 0}"
hoelzl@56993
   465
        using y by (intro emeasure_mono) auto
hoelzl@56993
   466
      with f show "emeasure M {x \<in> space M. f x = f y} \<noteq> \<infinity>"
hoelzl@56993
   467
        by (auto simp: simple_bochner_integrable.simps)
hoelzl@56993
   468
    qed
hoelzl@56993
   469
    moreover have "{x \<in> space M. f x = f y} = (\<lambda>x. ereal (f x)) -` {ereal (f y)} \<inter> space M"
hoelzl@56993
   470
      by auto
hoelzl@56993
   471
    ultimately have "ereal (measure M {x \<in> space M. f x = f y}) =
hoelzl@56993
   472
          emeasure M ((\<lambda>x. ereal (f x)) -` {ereal (f y)} \<inter> space M)" by simp }
hoelzl@56993
   473
  with f have "simple_bochner_integral M f = (\<integral>\<^sup>Sx. f x \<partial>M)"
hoelzl@56993
   474
    unfolding simple_integral_def
hoelzl@56993
   475
    by (subst simple_bochner_integral_partition[OF f(1), where g="\<lambda>x. ereal (f x)" and v=real])
hoelzl@56993
   476
       (auto intro: f simple_function_compose1 elim: simple_bochner_integrable.cases
haftmann@57418
   477
             intro!: setsum.cong ereal_cong_mult
haftmann@57514
   478
             simp: setsum_ereal[symmetric] times_ereal.simps(1)[symmetric] ac_simps
hoelzl@56993
   479
             simp del: setsum_ereal times_ereal.simps(1))
hoelzl@56993
   480
  also have "\<dots> = (\<integral>\<^sup>+x. f x \<partial>M)"
hoelzl@56993
   481
    using f
hoelzl@56996
   482
    by (intro nn_integral_eq_simple_integral[symmetric])
hoelzl@56993
   483
       (auto simp: simple_function_compose1 simple_bochner_integrable.simps)
hoelzl@56993
   484
  finally show ?thesis .
hoelzl@56993
   485
qed
hoelzl@56993
   486
hoelzl@56993
   487
lemma simple_bochner_integral_bounded:
hoelzl@56993
   488
  fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector, second_countable_topology}"
hoelzl@56993
   489
  assumes f[measurable]: "f \<in> borel_measurable M"
hoelzl@56993
   490
  assumes s: "simple_bochner_integrable M s" and t: "simple_bochner_integrable M t"
hoelzl@56993
   491
  shows "ereal (norm (simple_bochner_integral M s - simple_bochner_integral M t)) \<le>
hoelzl@56993
   492
    (\<integral>\<^sup>+ x. norm (f x - s x) \<partial>M) + (\<integral>\<^sup>+ x. norm (f x - t x) \<partial>M)"
hoelzl@56993
   493
    (is "ereal (norm (?s - ?t)) \<le> ?S + ?T")
hoelzl@56993
   494
proof -
hoelzl@56993
   495
  have [measurable]: "s \<in> borel_measurable M" "t \<in> borel_measurable M"
hoelzl@56993
   496
    using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
hoelzl@56993
   497
hoelzl@56993
   498
  have "ereal (norm (?s - ?t)) = norm (simple_bochner_integral M (\<lambda>x. s x - t x))"
hoelzl@56993
   499
    using s t by (subst simple_bochner_integral_diff) auto
hoelzl@56993
   500
  also have "\<dots> \<le> simple_bochner_integral M (\<lambda>x. norm (s x - t x))"
hoelzl@56993
   501
    using simple_bochner_integrable_compose2[of "op -" M "s" "t"] s t
hoelzl@56993
   502
    by (auto intro!: simple_bochner_integral_norm_bound)
hoelzl@56993
   503
  also have "\<dots> = (\<integral>\<^sup>+x. norm (s x - t x) \<partial>M)"
hoelzl@56993
   504
    using simple_bochner_integrable_compose2[of "\<lambda>x y. norm (x - y)" M "s" "t"] s t
hoelzl@56996
   505
    by (auto intro!: simple_bochner_integral_eq_nn_integral)
hoelzl@56993
   506
  also have "\<dots> \<le> (\<integral>\<^sup>+x. ereal (norm (f x - s x)) + ereal (norm (f x - t x)) \<partial>M)"
hoelzl@56996
   507
    by (auto intro!: nn_integral_mono)
hoelzl@56993
   508
       (metis (erased, hide_lams) add_diff_cancel_left add_diff_eq diff_add_eq order_trans
hoelzl@56993
   509
              norm_minus_commute norm_triangle_ineq4 order_refl)
hoelzl@56993
   510
  also have "\<dots> = ?S + ?T"
hoelzl@56996
   511
   by (rule nn_integral_add) auto
hoelzl@56993
   512
  finally show ?thesis .
hoelzl@56993
   513
qed
hoelzl@56993
   514
hoelzl@56993
   515
inductive has_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b::{real_normed_vector, second_countable_topology} \<Rightarrow> bool"
hoelzl@56993
   516
  for M f x where
hoelzl@56993
   517
  "f \<in> borel_measurable M \<Longrightarrow>
hoelzl@56993
   518
    (\<And>i. simple_bochner_integrable M (s i)) \<Longrightarrow>
hoelzl@56993
   519
    (\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) ----> 0 \<Longrightarrow>
hoelzl@56993
   520
    (\<lambda>i. simple_bochner_integral M (s i)) ----> x \<Longrightarrow>
hoelzl@56993
   521
    has_bochner_integral M f x"
hoelzl@56993
   522
hoelzl@56993
   523
lemma has_bochner_integral_cong:
hoelzl@56993
   524
  assumes "M = N" "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" "x = y"
hoelzl@56993
   525
  shows "has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral N g y"
hoelzl@56993
   526
  unfolding has_bochner_integral.simps assms(1,3)
hoelzl@56996
   527
  using assms(2) by (simp cong: measurable_cong_strong nn_integral_cong_strong)
hoelzl@56993
   528
hoelzl@56993
   529
lemma has_bochner_integral_cong_AE:
hoelzl@56993
   530
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow>
hoelzl@56993
   531
    has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral M g x"
hoelzl@56993
   532
  unfolding has_bochner_integral.simps
hoelzl@56993
   533
  by (intro arg_cong[where f=Ex] ext conj_cong rev_conj_cong refl arg_cong[where f="\<lambda>x. x ----> 0"]
hoelzl@56996
   534
            nn_integral_cong_AE)
hoelzl@56993
   535
     auto
hoelzl@56993
   536
hoelzl@59353
   537
lemma borel_measurable_has_bochner_integral:
hoelzl@56993
   538
  "has_bochner_integral M f x \<Longrightarrow> f \<in> borel_measurable M"
hoelzl@59353
   539
  by (rule has_bochner_integral.cases)
hoelzl@59353
   540
hoelzl@59353
   541
lemma borel_measurable_has_bochner_integral'[measurable_dest]:
hoelzl@59353
   542
  "has_bochner_integral M f x \<Longrightarrow> g \<in> measurable N M \<Longrightarrow> (\<lambda>x. f (g x)) \<in> borel_measurable N"
hoelzl@59353
   543
  using borel_measurable_has_bochner_integral[measurable] by measurable
hoelzl@56993
   544
hoelzl@56993
   545
lemma has_bochner_integral_simple_bochner_integrable:
hoelzl@56993
   546
  "simple_bochner_integrable M f \<Longrightarrow> has_bochner_integral M f (simple_bochner_integral M f)"
hoelzl@56993
   547
  by (rule has_bochner_integral.intros[where s="\<lambda>_. f"])
hoelzl@56993
   548
     (auto intro: borel_measurable_simple_function 
hoelzl@56993
   549
           elim: simple_bochner_integrable.cases
hoelzl@56993
   550
           simp: zero_ereal_def[symmetric])
hoelzl@56993
   551
hoelzl@56993
   552
lemma has_bochner_integral_real_indicator:
hoelzl@56993
   553
  assumes [measurable]: "A \<in> sets M" and A: "emeasure M A < \<infinity>"
hoelzl@56993
   554
  shows "has_bochner_integral M (indicator A) (measure M A)"
hoelzl@56993
   555
proof -
hoelzl@56993
   556
  have sbi: "simple_bochner_integrable M (indicator A::'a \<Rightarrow> real)"
hoelzl@56993
   557
  proof
hoelzl@56993
   558
    have "{y \<in> space M. (indicator A y::real) \<noteq> 0} = A"
hoelzl@56993
   559
      using sets.sets_into_space[OF `A\<in>sets M`] by (auto split: split_indicator)
hoelzl@56993
   560
    then show "emeasure M {y \<in> space M. (indicator A y::real) \<noteq> 0} \<noteq> \<infinity>"
hoelzl@56993
   561
      using A by auto
hoelzl@56993
   562
  qed (rule simple_function_indicator assms)+
hoelzl@56993
   563
  moreover have "simple_bochner_integral M (indicator A) = measure M A"
hoelzl@56996
   564
    using simple_bochner_integral_eq_nn_integral[OF sbi] A
hoelzl@56993
   565
    by (simp add: ereal_indicator emeasure_eq_ereal_measure)
hoelzl@56993
   566
  ultimately show ?thesis
hoelzl@56993
   567
    by (metis has_bochner_integral_simple_bochner_integrable)
hoelzl@56993
   568
qed
hoelzl@56993
   569
hoelzl@57036
   570
lemma has_bochner_integral_add[intro]:
hoelzl@56993
   571
  "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M g y \<Longrightarrow>
hoelzl@56993
   572
    has_bochner_integral M (\<lambda>x. f x + g x) (x + y)"
hoelzl@56993
   573
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
hoelzl@56993
   574
  fix sf sg
hoelzl@56993
   575
  assume f_sf: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - sf i x) \<partial>M) ----> 0"
hoelzl@56993
   576
  assume g_sg: "(\<lambda>i. \<integral>\<^sup>+ x. norm (g x - sg i x) \<partial>M) ----> 0"
hoelzl@56993
   577
hoelzl@56993
   578
  assume sf: "\<forall>i. simple_bochner_integrable M (sf i)"
hoelzl@56993
   579
    and sg: "\<forall>i. simple_bochner_integrable M (sg i)"
hoelzl@56993
   580
  then have [measurable]: "\<And>i. sf i \<in> borel_measurable M" "\<And>i. sg i \<in> borel_measurable M"
hoelzl@56993
   581
    by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
hoelzl@56993
   582
  assume [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@56993
   583
hoelzl@56993
   584
  show "\<And>i. simple_bochner_integrable M (\<lambda>x. sf i x + sg i x)"
hoelzl@56993
   585
    using sf sg by (simp add: simple_bochner_integrable_compose2)
hoelzl@56993
   586
hoelzl@56993
   587
  show "(\<lambda>i. \<integral>\<^sup>+ x. (norm (f x + g x - (sf i x + sg i x))) \<partial>M) ----> 0"
hoelzl@56993
   588
    (is "?f ----> 0")
hoelzl@56993
   589
  proof (rule tendsto_sandwich)
hoelzl@56993
   590
    show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) ----> 0"
hoelzl@56996
   591
      by (auto simp: nn_integral_nonneg)
hoelzl@56993
   592
    show "eventually (\<lambda>i. ?f i \<le> (\<integral>\<^sup>+ x. (norm (f x - sf i x)) \<partial>M) + \<integral>\<^sup>+ x. (norm (g x - sg i x)) \<partial>M) sequentially"
hoelzl@56993
   593
      (is "eventually (\<lambda>i. ?f i \<le> ?g i) sequentially")
hoelzl@56993
   594
    proof (intro always_eventually allI)
hoelzl@56993
   595
      fix i have "?f i \<le> (\<integral>\<^sup>+ x. (norm (f x - sf i x)) + ereal (norm (g x - sg i x)) \<partial>M)"
hoelzl@56996
   596
        by (auto intro!: nn_integral_mono norm_diff_triangle_ineq)
hoelzl@56993
   597
      also have "\<dots> = ?g i"
hoelzl@56996
   598
        by (intro nn_integral_add) auto
hoelzl@56993
   599
      finally show "?f i \<le> ?g i" .
hoelzl@56993
   600
    qed
hoelzl@56993
   601
    show "?g ----> 0"
hoelzl@56993
   602
      using tendsto_add_ereal[OF _ _ f_sf g_sg] by simp
hoelzl@56993
   603
  qed
hoelzl@56993
   604
qed (auto simp: simple_bochner_integral_add tendsto_add)
hoelzl@56993
   605
hoelzl@56993
   606
lemma has_bochner_integral_bounded_linear:
hoelzl@56993
   607
  assumes "bounded_linear T"
hoelzl@56993
   608
  shows "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M (\<lambda>x. T (f x)) (T x)"
hoelzl@56993
   609
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
hoelzl@56993
   610
  interpret T: bounded_linear T by fact
hoelzl@56993
   611
  have [measurable]: "T \<in> borel_measurable borel"
hoelzl@56993
   612
    by (intro borel_measurable_continuous_on1 T.continuous_on continuous_on_id)
hoelzl@56993
   613
  assume [measurable]: "f \<in> borel_measurable M"
hoelzl@56993
   614
  then show "(\<lambda>x. T (f x)) \<in> borel_measurable M"
hoelzl@56993
   615
    by auto
hoelzl@56993
   616
hoelzl@56993
   617
  fix s assume f_s: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) ----> 0"
hoelzl@56993
   618
  assume s: "\<forall>i. simple_bochner_integrable M (s i)"
hoelzl@56993
   619
  then show "\<And>i. simple_bochner_integrable M (\<lambda>x. T (s i x))"
hoelzl@56993
   620
    by (auto intro: simple_bochner_integrable_compose2 T.zero)
hoelzl@56993
   621
hoelzl@56993
   622
  have [measurable]: "\<And>i. s i \<in> borel_measurable M"
hoelzl@56993
   623
    using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
hoelzl@56993
   624
hoelzl@56993
   625
  obtain K where K: "K > 0" "\<And>x i. norm (T (f x) - T (s i x)) \<le> norm (f x - s i x) * K"
hoelzl@56993
   626
    using T.pos_bounded by (auto simp: T.diff[symmetric])
hoelzl@56993
   627
hoelzl@56993
   628
  show "(\<lambda>i. \<integral>\<^sup>+ x. norm (T (f x) - T (s i x)) \<partial>M) ----> 0"
hoelzl@56993
   629
    (is "?f ----> 0")
hoelzl@56993
   630
  proof (rule tendsto_sandwich)
hoelzl@56993
   631
    show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) ----> 0"
hoelzl@56996
   632
      by (auto simp: nn_integral_nonneg)
hoelzl@56993
   633
hoelzl@56993
   634
    show "eventually (\<lambda>i. ?f i \<le> K * (\<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M)) sequentially"
hoelzl@56993
   635
      (is "eventually (\<lambda>i. ?f i \<le> ?g i) sequentially")
hoelzl@56993
   636
    proof (intro always_eventually allI)
hoelzl@56993
   637
      fix i have "?f i \<le> (\<integral>\<^sup>+ x. ereal K * norm (f x - s i x) \<partial>M)"
haftmann@57514
   638
        using K by (intro nn_integral_mono) (auto simp: ac_simps)
hoelzl@56993
   639
      also have "\<dots> = ?g i"
hoelzl@56996
   640
        using K by (intro nn_integral_cmult) auto
hoelzl@56993
   641
      finally show "?f i \<le> ?g i" .
hoelzl@56993
   642
    qed
hoelzl@56993
   643
    show "?g ----> 0"
hoelzl@59452
   644
      using tendsto_cmult_ereal[OF _ f_s, of "ereal K"] by simp
hoelzl@56993
   645
  qed
hoelzl@56993
   646
hoelzl@56993
   647
  assume "(\<lambda>i. simple_bochner_integral M (s i)) ----> x"
hoelzl@56993
   648
  with s show "(\<lambda>i. simple_bochner_integral M (\<lambda>x. T (s i x))) ----> T x"
hoelzl@56993
   649
    by (auto intro!: T.tendsto simp: T.simple_bochner_integral_linear)
hoelzl@56993
   650
qed
hoelzl@56993
   651
hoelzl@56993
   652
lemma has_bochner_integral_zero[intro]: "has_bochner_integral M (\<lambda>x. 0) 0"
hoelzl@56993
   653
  by (auto intro!: has_bochner_integral.intros[where s="\<lambda>_ _. 0"]
hoelzl@56993
   654
           simp: zero_ereal_def[symmetric] simple_bochner_integrable.simps
hoelzl@56993
   655
                 simple_bochner_integral_def image_constant_conv)
hoelzl@56993
   656
hoelzl@56993
   657
lemma has_bochner_integral_scaleR_left[intro]:
hoelzl@56993
   658
  "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x *\<^sub>R c) (x *\<^sub>R c)"
hoelzl@56993
   659
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_left])
hoelzl@56993
   660
hoelzl@56993
   661
lemma has_bochner_integral_scaleR_right[intro]:
hoelzl@56993
   662
  "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c *\<^sub>R f x) (c *\<^sub>R x)"
hoelzl@56993
   663
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_right])
hoelzl@56993
   664
hoelzl@56993
   665
lemma has_bochner_integral_mult_left[intro]:
hoelzl@56993
   666
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
hoelzl@56993
   667
  shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x * c) (x * c)"
hoelzl@56993
   668
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_left])
hoelzl@56993
   669
hoelzl@56993
   670
lemma has_bochner_integral_mult_right[intro]:
hoelzl@56993
   671
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
hoelzl@56993
   672
  shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c * f x) (c * x)"
hoelzl@56993
   673
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_right])
hoelzl@56993
   674
hoelzl@56993
   675
lemmas has_bochner_integral_divide = 
hoelzl@56993
   676
  has_bochner_integral_bounded_linear[OF bounded_linear_divide]
hoelzl@56993
   677
hoelzl@56993
   678
lemma has_bochner_integral_divide_zero[intro]:
hoelzl@56993
   679
  fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
hoelzl@56993
   680
  shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x / c) (x / c)"
hoelzl@56993
   681
  using has_bochner_integral_divide by (cases "c = 0") auto
hoelzl@56993
   682
hoelzl@56993
   683
lemma has_bochner_integral_inner_left[intro]:
hoelzl@56993
   684
  "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x \<bullet> c) (x \<bullet> c)"
hoelzl@56993
   685
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_left])
hoelzl@56993
   686
hoelzl@56993
   687
lemma has_bochner_integral_inner_right[intro]:
hoelzl@56993
   688
  "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c \<bullet> f x) (c \<bullet> x)"
hoelzl@56993
   689
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_right])
hoelzl@56993
   690
hoelzl@56993
   691
lemmas has_bochner_integral_minus =
hoelzl@56993
   692
  has_bochner_integral_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
hoelzl@56993
   693
lemmas has_bochner_integral_Re =
hoelzl@56993
   694
  has_bochner_integral_bounded_linear[OF bounded_linear_Re]
hoelzl@56993
   695
lemmas has_bochner_integral_Im =
hoelzl@56993
   696
  has_bochner_integral_bounded_linear[OF bounded_linear_Im]
hoelzl@56993
   697
lemmas has_bochner_integral_cnj =
hoelzl@56993
   698
  has_bochner_integral_bounded_linear[OF bounded_linear_cnj]
hoelzl@56993
   699
lemmas has_bochner_integral_of_real =
hoelzl@56993
   700
  has_bochner_integral_bounded_linear[OF bounded_linear_of_real]
hoelzl@56993
   701
lemmas has_bochner_integral_fst =
hoelzl@56993
   702
  has_bochner_integral_bounded_linear[OF bounded_linear_fst]
hoelzl@56993
   703
lemmas has_bochner_integral_snd =
hoelzl@56993
   704
  has_bochner_integral_bounded_linear[OF bounded_linear_snd]
hoelzl@56993
   705
hoelzl@56993
   706
lemma has_bochner_integral_indicator:
hoelzl@56993
   707
  "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
hoelzl@56993
   708
    has_bochner_integral M (\<lambda>x. indicator A x *\<^sub>R c) (measure M A *\<^sub>R c)"
hoelzl@56993
   709
  by (intro has_bochner_integral_scaleR_left has_bochner_integral_real_indicator)
hoelzl@56993
   710
hoelzl@56993
   711
lemma has_bochner_integral_diff:
hoelzl@56993
   712
  "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M g y \<Longrightarrow>
hoelzl@56993
   713
    has_bochner_integral M (\<lambda>x. f x - g x) (x - y)"
hoelzl@56993
   714
  unfolding diff_conv_add_uminus
hoelzl@56993
   715
  by (intro has_bochner_integral_add has_bochner_integral_minus)
hoelzl@56993
   716
hoelzl@56993
   717
lemma has_bochner_integral_setsum:
hoelzl@56993
   718
  "(\<And>i. i \<in> I \<Longrightarrow> has_bochner_integral M (f i) (x i)) \<Longrightarrow>
hoelzl@56993
   719
    has_bochner_integral M (\<lambda>x. \<Sum>i\<in>I. f i x) (\<Sum>i\<in>I. x i)"
hoelzl@57036
   720
  by (induct I rule: infinite_finite_induct) auto
hoelzl@56993
   721
hoelzl@56993
   722
lemma has_bochner_integral_implies_finite_norm:
hoelzl@56993
   723
  "has_bochner_integral M f x \<Longrightarrow> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
hoelzl@56993
   724
proof (elim has_bochner_integral.cases)
hoelzl@56993
   725
  fix s v
hoelzl@56993
   726
  assume [measurable]: "f \<in> borel_measurable M" and s: "\<And>i. simple_bochner_integrable M (s i)" and
hoelzl@56993
   727
    lim_0: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
hoelzl@56993
   728
  from order_tendstoD[OF lim_0, of "\<infinity>"]
hoelzl@56993
   729
  obtain i where f_s_fin: "(\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) < \<infinity>"
hoelzl@56993
   730
    by (metis (mono_tags, lifting) eventually_False_sequentially eventually_elim1
hoelzl@56993
   731
              less_ereal.simps(4) zero_ereal_def)
hoelzl@56993
   732
hoelzl@56993
   733
  have [measurable]: "\<And>i. s i \<in> borel_measurable M"
hoelzl@56993
   734
    using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
hoelzl@56993
   735
hoelzl@56993
   736
  def m \<equiv> "if space M = {} then 0 else Max ((\<lambda>x. norm (s i x))`space M)"
hoelzl@56993
   737
  have "finite (s i ` space M)"
hoelzl@56993
   738
    using s by (auto simp: simple_function_def simple_bochner_integrable.simps)
hoelzl@56993
   739
  then have "finite (norm ` s i ` space M)"
hoelzl@56993
   740
    by (rule finite_imageI)
hoelzl@56993
   741
  then have "\<And>x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> m" "0 \<le> m"
hoelzl@56993
   742
    by (auto simp: m_def image_comp comp_def Max_ge_iff)
hoelzl@56993
   743
  then have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) \<le> (\<integral>\<^sup>+x. ereal m * indicator {x\<in>space M. s i x \<noteq> 0} x \<partial>M)"
hoelzl@56996
   744
    by (auto split: split_indicator intro!: Max_ge nn_integral_mono simp:)
hoelzl@56993
   745
  also have "\<dots> < \<infinity>"
hoelzl@56996
   746
    using s by (subst nn_integral_cmult_indicator) (auto simp: `0 \<le> m` simple_bochner_integrable.simps)
hoelzl@56993
   747
  finally have s_fin: "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>" .
hoelzl@56993
   748
hoelzl@56993
   749
  have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) + ereal (norm (s i x)) \<partial>M)"
haftmann@57512
   750
    by (auto intro!: nn_integral_mono) (metis add.commute norm_triangle_sub)
hoelzl@56993
   751
  also have "\<dots> = (\<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) + (\<integral>\<^sup>+x. norm (s i x) \<partial>M)"
hoelzl@56996
   752
    by (rule nn_integral_add) auto
hoelzl@56993
   753
  also have "\<dots> < \<infinity>"
hoelzl@56993
   754
    using s_fin f_s_fin by auto
hoelzl@56993
   755
  finally show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" .
hoelzl@56993
   756
qed
hoelzl@56993
   757
hoelzl@56993
   758
lemma has_bochner_integral_norm_bound:
hoelzl@56993
   759
  assumes i: "has_bochner_integral M f x"
hoelzl@56993
   760
  shows "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
hoelzl@56993
   761
using assms proof
hoelzl@56993
   762
  fix s assume
hoelzl@56993
   763
    x: "(\<lambda>i. simple_bochner_integral M (s i)) ----> x" (is "?s ----> x") and
hoelzl@56993
   764
    s[simp]: "\<And>i. simple_bochner_integrable M (s i)" and
hoelzl@56993
   765
    lim: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0" and
hoelzl@56993
   766
    f[measurable]: "f \<in> borel_measurable M"
hoelzl@56993
   767
hoelzl@56993
   768
  have [measurable]: "\<And>i. s i \<in> borel_measurable M"
hoelzl@56993
   769
    using s by (auto simp: simple_bochner_integrable.simps intro: borel_measurable_simple_function)
hoelzl@56993
   770
hoelzl@56993
   771
  show "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
hoelzl@56993
   772
  proof (rule LIMSEQ_le)
hoelzl@56993
   773
    show "(\<lambda>i. ereal (norm (?s i))) ----> norm x"
hoelzl@56993
   774
      using x by (intro tendsto_intros lim_ereal[THEN iffD2])
hoelzl@56993
   775
    show "\<exists>N. \<forall>n\<ge>N. norm (?s n) \<le> (\<integral>\<^sup>+x. norm (f x - s n x) \<partial>M) + (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
hoelzl@56993
   776
      (is "\<exists>N. \<forall>n\<ge>N. _ \<le> ?t n")
hoelzl@56993
   777
    proof (intro exI allI impI)
hoelzl@56993
   778
      fix n
hoelzl@56993
   779
      have "ereal (norm (?s n)) \<le> simple_bochner_integral M (\<lambda>x. norm (s n x))"
hoelzl@56993
   780
        by (auto intro!: simple_bochner_integral_norm_bound)
hoelzl@56993
   781
      also have "\<dots> = (\<integral>\<^sup>+x. norm (s n x) \<partial>M)"
hoelzl@56996
   782
        by (intro simple_bochner_integral_eq_nn_integral)
hoelzl@56993
   783
           (auto intro: s simple_bochner_integrable_compose2)
hoelzl@56993
   784
      also have "\<dots> \<le> (\<integral>\<^sup>+x. ereal (norm (f x - s n x)) + norm (f x) \<partial>M)"
hoelzl@56996
   785
        by (auto intro!: nn_integral_mono)
haftmann@57512
   786
           (metis add.commute norm_minus_commute norm_triangle_sub)
hoelzl@56993
   787
      also have "\<dots> = ?t n" 
hoelzl@56996
   788
        by (rule nn_integral_add) auto
hoelzl@56993
   789
      finally show "norm (?s n) \<le> ?t n" .
hoelzl@56993
   790
    qed
hoelzl@56993
   791
    have "?t ----> 0 + (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
hoelzl@56993
   792
      using has_bochner_integral_implies_finite_norm[OF i]
hoelzl@56993
   793
      by (intro tendsto_add_ereal tendsto_const lim) auto
hoelzl@56993
   794
    then show "?t ----> \<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M"
hoelzl@56993
   795
      by simp
hoelzl@56993
   796
  qed
hoelzl@56993
   797
qed
hoelzl@56993
   798
hoelzl@56993
   799
lemma has_bochner_integral_eq:
hoelzl@56993
   800
  "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M f y \<Longrightarrow> x = y"
hoelzl@56993
   801
proof (elim has_bochner_integral.cases)
hoelzl@56993
   802
  assume f[measurable]: "f \<in> borel_measurable M"
hoelzl@56993
   803
hoelzl@56993
   804
  fix s t
hoelzl@56993
   805
  assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) ----> 0" (is "?S ----> 0")
hoelzl@56993
   806
  assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - t i x) \<partial>M) ----> 0" (is "?T ----> 0")
hoelzl@56993
   807
  assume s: "\<And>i. simple_bochner_integrable M (s i)"
hoelzl@56993
   808
  assume t: "\<And>i. simple_bochner_integrable M (t i)"
hoelzl@56993
   809
hoelzl@56993
   810
  have [measurable]: "\<And>i. s i \<in> borel_measurable M" "\<And>i. t i \<in> borel_measurable M"
hoelzl@56993
   811
    using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
hoelzl@56993
   812
hoelzl@56993
   813
  let ?s = "\<lambda>i. simple_bochner_integral M (s i)"
hoelzl@56993
   814
  let ?t = "\<lambda>i. simple_bochner_integral M (t i)"
hoelzl@56993
   815
  assume "?s ----> x" "?t ----> y"
hoelzl@56993
   816
  then have "(\<lambda>i. norm (?s i - ?t i)) ----> norm (x - y)"
hoelzl@56993
   817
    by (intro tendsto_intros)
hoelzl@56993
   818
  moreover
hoelzl@56993
   819
  have "(\<lambda>i. ereal (norm (?s i - ?t i))) ----> ereal 0"
hoelzl@56993
   820
  proof (rule tendsto_sandwich)
hoelzl@56993
   821
    show "eventually (\<lambda>i. 0 \<le> ereal (norm (?s i - ?t i))) sequentially" "(\<lambda>_. 0) ----> ereal 0"
hoelzl@56996
   822
      by (auto simp: nn_integral_nonneg zero_ereal_def[symmetric])
hoelzl@56993
   823
hoelzl@56993
   824
    show "eventually (\<lambda>i. norm (?s i - ?t i) \<le> ?S i + ?T i) sequentially"
hoelzl@56993
   825
      by (intro always_eventually allI simple_bochner_integral_bounded s t f)
hoelzl@56993
   826
    show "(\<lambda>i. ?S i + ?T i) ----> ereal 0"
hoelzl@56993
   827
      using tendsto_add_ereal[OF _ _ `?S ----> 0` `?T ----> 0`]
hoelzl@56993
   828
      by (simp add: zero_ereal_def[symmetric])
hoelzl@56993
   829
  qed
hoelzl@56993
   830
  then have "(\<lambda>i. norm (?s i - ?t i)) ----> 0"
hoelzl@56993
   831
    by simp
hoelzl@56993
   832
  ultimately have "norm (x - y) = 0"
hoelzl@56993
   833
    by (rule LIMSEQ_unique)
hoelzl@56993
   834
  then show "x = y" by simp
hoelzl@56993
   835
qed
hoelzl@56993
   836
hoelzl@56993
   837
lemma has_bochner_integralI_AE:
hoelzl@56993
   838
  assumes f: "has_bochner_integral M f x"
hoelzl@56993
   839
    and g: "g \<in> borel_measurable M"
hoelzl@56993
   840
    and ae: "AE x in M. f x = g x"
hoelzl@56993
   841
  shows "has_bochner_integral M g x"
hoelzl@56993
   842
  using f
hoelzl@56993
   843
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
hoelzl@56993
   844
  fix s assume "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
hoelzl@56993
   845
  also have "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) = (\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M)"
hoelzl@56993
   846
    using ae
hoelzl@56996
   847
    by (intro ext nn_integral_cong_AE, eventually_elim) simp
hoelzl@56993
   848
  finally show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M) ----> 0" .
hoelzl@56993
   849
qed (auto intro: g)
hoelzl@56993
   850
hoelzl@56993
   851
lemma has_bochner_integral_eq_AE:
hoelzl@56993
   852
  assumes f: "has_bochner_integral M f x"
hoelzl@56993
   853
    and g: "has_bochner_integral M g y"
hoelzl@56993
   854
    and ae: "AE x in M. f x = g x"
hoelzl@56993
   855
  shows "x = y"
hoelzl@56993
   856
proof -
hoelzl@56993
   857
  from assms have "has_bochner_integral M g x"
hoelzl@56993
   858
    by (auto intro: has_bochner_integralI_AE)
hoelzl@56993
   859
  from this g show "x = y"
hoelzl@56993
   860
    by (rule has_bochner_integral_eq)
hoelzl@56993
   861
qed
hoelzl@56993
   862
hoelzl@57137
   863
lemma simple_bochner_integrable_restrict_space:
hoelzl@57137
   864
  fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@57137
   865
  assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
hoelzl@57137
   866
  shows "simple_bochner_integrable (restrict_space M \<Omega>) f \<longleftrightarrow>
hoelzl@57137
   867
    simple_bochner_integrable M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
hoelzl@57137
   868
  by (simp add: simple_bochner_integrable.simps space_restrict_space
hoelzl@57137
   869
    simple_function_restrict_space[OF \<Omega>] emeasure_restrict_space[OF \<Omega>] Collect_restrict
hoelzl@57137
   870
    indicator_eq_0_iff conj_ac)
hoelzl@57137
   871
hoelzl@57137
   872
lemma simple_bochner_integral_restrict_space:
hoelzl@57137
   873
  fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@57137
   874
  assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
hoelzl@57137
   875
  assumes f: "simple_bochner_integrable (restrict_space M \<Omega>) f"
hoelzl@57137
   876
  shows "simple_bochner_integral (restrict_space M \<Omega>) f =
hoelzl@57137
   877
    simple_bochner_integral M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
hoelzl@57137
   878
proof -
hoelzl@57137
   879
  have "finite ((\<lambda>x. indicator \<Omega> x *\<^sub>R f x)`space M)"
hoelzl@57137
   880
    using f simple_bochner_integrable_restrict_space[OF \<Omega>, of f]
hoelzl@57137
   881
    by (simp add: simple_bochner_integrable.simps simple_function_def)
hoelzl@57137
   882
  then show ?thesis
hoelzl@57137
   883
    by (auto simp: space_restrict_space measure_restrict_space[OF \<Omega>(1)] le_infI2
hoelzl@57137
   884
                   simple_bochner_integral_def Collect_restrict
hoelzl@57137
   885
             split: split_indicator split_indicator_asm
haftmann@57418
   886
             intro!: setsum.mono_neutral_cong_left arg_cong2[where f=measure])
hoelzl@57137
   887
qed
hoelzl@57137
   888
hoelzl@56993
   889
inductive integrable for M f where
hoelzl@56993
   890
  "has_bochner_integral M f x \<Longrightarrow> integrable M f"
hoelzl@56993
   891
hoelzl@56993
   892
definition lebesgue_integral ("integral\<^sup>L") where
hoelzl@57166
   893
  "integral\<^sup>L M f = (if \<exists>x. has_bochner_integral M f x then THE x. has_bochner_integral M f x else 0)"
hoelzl@56993
   894
hoelzl@56993
   895
syntax
Andreas@59357
   896
  "_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral>((2 _./ _)/ \<partial>_)" [60,61] 110)
hoelzl@56993
   897
hoelzl@56993
   898
translations
hoelzl@56993
   899
  "\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (\<lambda>x. f)"
hoelzl@56993
   900
hoelzl@56993
   901
lemma has_bochner_integral_integral_eq: "has_bochner_integral M f x \<Longrightarrow> integral\<^sup>L M f = x"
hoelzl@56993
   902
  by (metis the_equality has_bochner_integral_eq lebesgue_integral_def)
hoelzl@56993
   903
hoelzl@56993
   904
lemma has_bochner_integral_integrable:
hoelzl@56993
   905
  "integrable M f \<Longrightarrow> has_bochner_integral M f (integral\<^sup>L M f)"
hoelzl@56993
   906
  by (auto simp: has_bochner_integral_integral_eq integrable.simps)
hoelzl@56993
   907
hoelzl@56993
   908
lemma has_bochner_integral_iff:
hoelzl@56993
   909
  "has_bochner_integral M f x \<longleftrightarrow> integrable M f \<and> integral\<^sup>L M f = x"
hoelzl@56993
   910
  by (metis has_bochner_integral_integrable has_bochner_integral_integral_eq integrable.intros)
hoelzl@56993
   911
hoelzl@56993
   912
lemma simple_bochner_integrable_eq_integral:
hoelzl@56993
   913
  "simple_bochner_integrable M f \<Longrightarrow> simple_bochner_integral M f = integral\<^sup>L M f"
hoelzl@56993
   914
  using has_bochner_integral_simple_bochner_integrable[of M f]
hoelzl@56993
   915
  by (simp add: has_bochner_integral_integral_eq)
hoelzl@56993
   916
hoelzl@57166
   917
lemma not_integrable_integral_eq: "\<not> integrable M f \<Longrightarrow> integral\<^sup>L M f = 0"
hoelzl@56993
   918
  unfolding integrable.simps lebesgue_integral_def by (auto intro!: arg_cong[where f=The])
hoelzl@56993
   919
hoelzl@56993
   920
lemma integral_eq_cases:
hoelzl@56993
   921
  "integrable M f \<longleftrightarrow> integrable N g \<Longrightarrow>
hoelzl@56993
   922
    (integrable M f \<Longrightarrow> integrable N g \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g) \<Longrightarrow>
hoelzl@56993
   923
    integral\<^sup>L M f = integral\<^sup>L N g"
hoelzl@56993
   924
  by (metis not_integrable_integral_eq)
hoelzl@56993
   925
hoelzl@56993
   926
lemma borel_measurable_integrable[measurable_dest]: "integrable M f \<Longrightarrow> f \<in> borel_measurable M"
hoelzl@56993
   927
  by (auto elim: integrable.cases has_bochner_integral.cases)
hoelzl@56993
   928
hoelzl@59353
   929
lemma borel_measurable_integrable'[measurable_dest]:
hoelzl@59353
   930
  "integrable M f \<Longrightarrow> g \<in> measurable N M \<Longrightarrow> (\<lambda>x. f (g x)) \<in> borel_measurable N"
hoelzl@59353
   931
  using borel_measurable_integrable[measurable] by measurable
hoelzl@59353
   932
hoelzl@56993
   933
lemma integrable_cong:
hoelzl@56993
   934
  "M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable N g"
hoelzl@56993
   935
  using assms by (simp cong: has_bochner_integral_cong add: integrable.simps)
hoelzl@56993
   936
hoelzl@56993
   937
lemma integrable_cong_AE:
hoelzl@56993
   938
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow>
hoelzl@56993
   939
    integrable M f \<longleftrightarrow> integrable M g"
hoelzl@56993
   940
  unfolding integrable.simps
hoelzl@56993
   941
  by (intro has_bochner_integral_cong_AE arg_cong[where f=Ex] ext)
hoelzl@56993
   942
hoelzl@56993
   943
lemma integral_cong:
hoelzl@56993
   944
  "M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g"
hoelzl@57166
   945
  using assms by (simp cong: has_bochner_integral_cong cong del: if_cong add: lebesgue_integral_def)
hoelzl@56993
   946
hoelzl@56993
   947
lemma integral_cong_AE:
hoelzl@56993
   948
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow>
hoelzl@56993
   949
    integral\<^sup>L M f = integral\<^sup>L M g"
hoelzl@56993
   950
  unfolding lebesgue_integral_def
hoelzl@57166
   951
  by (rule arg_cong[where x="has_bochner_integral M f"]) (intro has_bochner_integral_cong_AE ext)
hoelzl@56993
   952
hoelzl@56993
   953
lemma integrable_add[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x + g x)"
hoelzl@57036
   954
  by (auto simp: integrable.simps)
hoelzl@56993
   955
hoelzl@56993
   956
lemma integrable_zero[simp, intro]: "integrable M (\<lambda>x. 0)"
hoelzl@56993
   957
  by (metis has_bochner_integral_zero integrable.simps) 
hoelzl@56993
   958
hoelzl@56993
   959
lemma integrable_setsum[simp, intro]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow> integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
hoelzl@56993
   960
  by (metis has_bochner_integral_setsum integrable.simps) 
hoelzl@56993
   961
hoelzl@56993
   962
lemma integrable_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
hoelzl@56993
   963
  integrable M (\<lambda>x. indicator A x *\<^sub>R c)"
hoelzl@56993
   964
  by (metis has_bochner_integral_indicator integrable.simps) 
hoelzl@56993
   965
hoelzl@56993
   966
lemma integrable_real_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
hoelzl@56993
   967
  integrable M (indicator A :: 'a \<Rightarrow> real)"
hoelzl@56993
   968
  by (metis has_bochner_integral_real_indicator integrable.simps)
hoelzl@56993
   969
hoelzl@56993
   970
lemma integrable_diff[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x - g x)"
hoelzl@56993
   971
  by (auto simp: integrable.simps intro: has_bochner_integral_diff)
hoelzl@56993
   972
  
hoelzl@56993
   973
lemma integrable_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. T (f x))"
hoelzl@56993
   974
  by (auto simp: integrable.simps intro: has_bochner_integral_bounded_linear)
hoelzl@56993
   975
hoelzl@56993
   976
lemma integrable_scaleR_left[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x *\<^sub>R c)"
hoelzl@56993
   977
  unfolding integrable.simps by fastforce
hoelzl@56993
   978
hoelzl@56993
   979
lemma integrable_scaleR_right[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c *\<^sub>R f x)"
hoelzl@56993
   980
  unfolding integrable.simps by fastforce
hoelzl@56993
   981
hoelzl@56993
   982
lemma integrable_mult_left[simp, intro]:
hoelzl@56993
   983
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
hoelzl@56993
   984
  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x * c)"
hoelzl@56993
   985
  unfolding integrable.simps by fastforce
hoelzl@56993
   986
hoelzl@56993
   987
lemma integrable_mult_right[simp, intro]:
hoelzl@56993
   988
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
hoelzl@56993
   989
  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c * f x)"
hoelzl@56993
   990
  unfolding integrable.simps by fastforce
hoelzl@56993
   991
hoelzl@56993
   992
lemma integrable_divide_zero[simp, intro]:
hoelzl@56993
   993
  fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
hoelzl@56993
   994
  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x / c)"
hoelzl@56993
   995
  unfolding integrable.simps by fastforce
hoelzl@56993
   996
hoelzl@56993
   997
lemma integrable_inner_left[simp, intro]:
hoelzl@56993
   998
  "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x \<bullet> c)"
hoelzl@56993
   999
  unfolding integrable.simps by fastforce
hoelzl@56993
  1000
hoelzl@56993
  1001
lemma integrable_inner_right[simp, intro]:
hoelzl@56993
  1002
  "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c \<bullet> f x)"
hoelzl@56993
  1003
  unfolding integrable.simps by fastforce
hoelzl@56993
  1004
hoelzl@56993
  1005
lemmas integrable_minus[simp, intro] =
hoelzl@56993
  1006
  integrable_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
hoelzl@56993
  1007
lemmas integrable_divide[simp, intro] =
hoelzl@56993
  1008
  integrable_bounded_linear[OF bounded_linear_divide]
hoelzl@56993
  1009
lemmas integrable_Re[simp, intro] =
hoelzl@56993
  1010
  integrable_bounded_linear[OF bounded_linear_Re]
hoelzl@56993
  1011
lemmas integrable_Im[simp, intro] =
hoelzl@56993
  1012
  integrable_bounded_linear[OF bounded_linear_Im]
hoelzl@56993
  1013
lemmas integrable_cnj[simp, intro] =
hoelzl@56993
  1014
  integrable_bounded_linear[OF bounded_linear_cnj]
hoelzl@56993
  1015
lemmas integrable_of_real[simp, intro] =
hoelzl@56993
  1016
  integrable_bounded_linear[OF bounded_linear_of_real]
hoelzl@56993
  1017
lemmas integrable_fst[simp, intro] =
hoelzl@56993
  1018
  integrable_bounded_linear[OF bounded_linear_fst]
hoelzl@56993
  1019
lemmas integrable_snd[simp, intro] =
hoelzl@56993
  1020
  integrable_bounded_linear[OF bounded_linear_snd]
hoelzl@56993
  1021
hoelzl@56993
  1022
lemma integral_zero[simp]: "integral\<^sup>L M (\<lambda>x. 0) = 0"
hoelzl@56993
  1023
  by (intro has_bochner_integral_integral_eq has_bochner_integral_zero)
hoelzl@56993
  1024
hoelzl@56993
  1025
lemma integral_add[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
hoelzl@56993
  1026
    integral\<^sup>L M (\<lambda>x. f x + g x) = integral\<^sup>L M f + integral\<^sup>L M g"
hoelzl@56993
  1027
  by (intro has_bochner_integral_integral_eq has_bochner_integral_add has_bochner_integral_integrable)
hoelzl@56993
  1028
hoelzl@56993
  1029
lemma integral_diff[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
hoelzl@56993
  1030
    integral\<^sup>L M (\<lambda>x. f x - g x) = integral\<^sup>L M f - integral\<^sup>L M g"
hoelzl@56993
  1031
  by (intro has_bochner_integral_integral_eq has_bochner_integral_diff has_bochner_integral_integrable)
hoelzl@56993
  1032
hoelzl@56993
  1033
lemma integral_setsum[simp]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow>
hoelzl@56993
  1034
  integral\<^sup>L M (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
hoelzl@56993
  1035
  by (intro has_bochner_integral_integral_eq has_bochner_integral_setsum has_bochner_integral_integrable)
hoelzl@56993
  1036
hoelzl@56993
  1037
lemma integral_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow>
hoelzl@56993
  1038
    integral\<^sup>L M (\<lambda>x. T (f x)) = T (integral\<^sup>L M f)"
hoelzl@56993
  1039
  by (metis has_bochner_integral_bounded_linear has_bochner_integral_integrable has_bochner_integral_integral_eq)
hoelzl@56993
  1040
hoelzl@57166
  1041
lemma integral_bounded_linear':
hoelzl@57166
  1042
  assumes T: "bounded_linear T" and T': "bounded_linear T'"
hoelzl@57166
  1043
  assumes *: "\<not> (\<forall>x. T x = 0) \<Longrightarrow> (\<forall>x. T' (T x) = x)"
hoelzl@57166
  1044
  shows "integral\<^sup>L M (\<lambda>x. T (f x)) = T (integral\<^sup>L M f)"
hoelzl@57166
  1045
proof cases
hoelzl@57166
  1046
  assume "(\<forall>x. T x = 0)" then show ?thesis
hoelzl@57166
  1047
    by simp
hoelzl@57166
  1048
next
hoelzl@57166
  1049
  assume **: "\<not> (\<forall>x. T x = 0)"
hoelzl@57166
  1050
  show ?thesis
hoelzl@57166
  1051
  proof cases
hoelzl@57166
  1052
    assume "integrable M f" with T show ?thesis
hoelzl@57166
  1053
      by (rule integral_bounded_linear)
hoelzl@57166
  1054
  next
hoelzl@57166
  1055
    assume not: "\<not> integrable M f"
hoelzl@57166
  1056
    moreover have "\<not> integrable M (\<lambda>x. T (f x))"
hoelzl@57166
  1057
    proof
hoelzl@57166
  1058
      assume "integrable M (\<lambda>x. T (f x))"
hoelzl@57166
  1059
      from integrable_bounded_linear[OF T' this] not *[OF **]
hoelzl@57166
  1060
      show False
hoelzl@57166
  1061
        by auto
hoelzl@57166
  1062
    qed
hoelzl@57166
  1063
    ultimately show ?thesis
hoelzl@57166
  1064
      using T by (simp add: not_integrable_integral_eq linear_simps)
hoelzl@57166
  1065
  qed
hoelzl@57166
  1066
qed
hoelzl@56993
  1067
hoelzl@56993
  1068
lemma integral_scaleR_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x *\<^sub>R c \<partial>M) = integral\<^sup>L M f *\<^sub>R c"
hoelzl@56993
  1069
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_scaleR_left)
hoelzl@56993
  1070
hoelzl@57166
  1071
lemma integral_scaleR_right[simp]: "(\<integral> x. c *\<^sub>R f x \<partial>M) = c *\<^sub>R integral\<^sup>L M f"
hoelzl@57166
  1072
  by (rule integral_bounded_linear'[OF bounded_linear_scaleR_right bounded_linear_scaleR_right[of "1 / c"]]) simp
hoelzl@56993
  1073
hoelzl@56993
  1074
lemma integral_mult_left[simp]:
hoelzl@56993
  1075
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
hoelzl@56993
  1076
  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
hoelzl@56993
  1077
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_left)
hoelzl@56993
  1078
hoelzl@56993
  1079
lemma integral_mult_right[simp]:
hoelzl@56993
  1080
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
hoelzl@56993
  1081
  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c * f x \<partial>M) = c * integral\<^sup>L M f"
hoelzl@56993
  1082
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_right)
hoelzl@56993
  1083
hoelzl@57166
  1084
lemma integral_mult_left_zero[simp]:
hoelzl@57166
  1085
  fixes c :: "_::{real_normed_field,second_countable_topology}"
hoelzl@57166
  1086
  shows "(\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
hoelzl@57166
  1087
  by (rule integral_bounded_linear'[OF bounded_linear_mult_left bounded_linear_mult_left[of "1 / c"]]) simp
hoelzl@57166
  1088
hoelzl@57166
  1089
lemma integral_mult_right_zero[simp]:
hoelzl@57166
  1090
  fixes c :: "_::{real_normed_field,second_countable_topology}"
hoelzl@57166
  1091
  shows "(\<integral> x. c * f x \<partial>M) = c * integral\<^sup>L M f"
hoelzl@57166
  1092
  by (rule integral_bounded_linear'[OF bounded_linear_mult_right bounded_linear_mult_right[of "1 / c"]]) simp
hoelzl@57166
  1093
hoelzl@56993
  1094
lemma integral_inner_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x \<bullet> c \<partial>M) = integral\<^sup>L M f \<bullet> c"
hoelzl@56993
  1095
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_left)
hoelzl@56993
  1096
hoelzl@56993
  1097
lemma integral_inner_right[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c \<bullet> f x \<partial>M) = c \<bullet> integral\<^sup>L M f"
hoelzl@56993
  1098
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_right)
hoelzl@56993
  1099
hoelzl@56993
  1100
lemma integral_divide_zero[simp]:
hoelzl@56993
  1101
  fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
hoelzl@57166
  1102
  shows "integral\<^sup>L M (\<lambda>x. f x / c) = integral\<^sup>L M f / c"
hoelzl@57166
  1103
  by (rule integral_bounded_linear'[OF bounded_linear_divide bounded_linear_mult_left[of c]]) simp
hoelzl@57166
  1104
hoelzl@57166
  1105
lemma integral_minus[simp]: "integral\<^sup>L M (\<lambda>x. - f x) = - integral\<^sup>L M f"
hoelzl@57166
  1106
  by (rule integral_bounded_linear'[OF bounded_linear_minus[OF bounded_linear_ident] bounded_linear_minus[OF bounded_linear_ident]]) simp
hoelzl@56993
  1107
hoelzl@57166
  1108
lemma integral_complex_of_real[simp]: "integral\<^sup>L M (\<lambda>x. complex_of_real (f x)) = of_real (integral\<^sup>L M f)"
hoelzl@57166
  1109
  by (rule integral_bounded_linear'[OF bounded_linear_of_real bounded_linear_Re]) simp
hoelzl@57166
  1110
hoelzl@57166
  1111
lemma integral_cnj[simp]: "integral\<^sup>L M (\<lambda>x. cnj (f x)) = cnj (integral\<^sup>L M f)"
hoelzl@57166
  1112
  by (rule integral_bounded_linear'[OF bounded_linear_cnj bounded_linear_cnj]) simp
hoelzl@57166
  1113
hoelzl@56993
  1114
lemmas integral_divide[simp] =
hoelzl@56993
  1115
  integral_bounded_linear[OF bounded_linear_divide]
hoelzl@56993
  1116
lemmas integral_Re[simp] =
hoelzl@56993
  1117
  integral_bounded_linear[OF bounded_linear_Re]
hoelzl@56993
  1118
lemmas integral_Im[simp] =
hoelzl@56993
  1119
  integral_bounded_linear[OF bounded_linear_Im]
hoelzl@56993
  1120
lemmas integral_of_real[simp] =
hoelzl@56993
  1121
  integral_bounded_linear[OF bounded_linear_of_real]
hoelzl@56993
  1122
lemmas integral_fst[simp] =
hoelzl@56993
  1123
  integral_bounded_linear[OF bounded_linear_fst]
hoelzl@56993
  1124
lemmas integral_snd[simp] =
hoelzl@56993
  1125
  integral_bounded_linear[OF bounded_linear_snd]
hoelzl@56993
  1126
hoelzl@56993
  1127
lemma integral_norm_bound_ereal:
hoelzl@56993
  1128
  "integrable M f \<Longrightarrow> norm (integral\<^sup>L M f) \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
hoelzl@56993
  1129
  by (metis has_bochner_integral_integrable has_bochner_integral_norm_bound)
hoelzl@56993
  1130
hoelzl@56993
  1131
lemma integrableI_sequence:
hoelzl@56993
  1132
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1133
  assumes f[measurable]: "f \<in> borel_measurable M"
hoelzl@56993
  1134
  assumes s: "\<And>i. simple_bochner_integrable M (s i)"
hoelzl@56993
  1135
  assumes lim: "(\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) ----> 0" (is "?S ----> 0")
hoelzl@56993
  1136
  shows "integrable M f"
hoelzl@56993
  1137
proof -
hoelzl@56993
  1138
  let ?s = "\<lambda>n. simple_bochner_integral M (s n)"
hoelzl@56993
  1139
hoelzl@56993
  1140
  have "\<exists>x. ?s ----> x"
hoelzl@56993
  1141
    unfolding convergent_eq_cauchy
hoelzl@56993
  1142
  proof (rule metric_CauchyI)
hoelzl@56993
  1143
    fix e :: real assume "0 < e"
hoelzl@56993
  1144
    then have "0 < ereal (e / 2)" by auto
hoelzl@56993
  1145
    from order_tendstoD(2)[OF lim this]
hoelzl@56993
  1146
    obtain M where M: "\<And>n. M \<le> n \<Longrightarrow> ?S n < e / 2"
hoelzl@56993
  1147
      by (auto simp: eventually_sequentially)
hoelzl@56993
  1148
    show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (?s m) (?s n) < e"
hoelzl@56993
  1149
    proof (intro exI allI impI)
hoelzl@56993
  1150
      fix m n assume m: "M \<le> m" and n: "M \<le> n"
hoelzl@56993
  1151
      have "?S n \<noteq> \<infinity>"
hoelzl@56993
  1152
        using M[OF n] by auto
hoelzl@56993
  1153
      have "norm (?s n - ?s m) \<le> ?S n + ?S m"
hoelzl@56993
  1154
        by (intro simple_bochner_integral_bounded s f)
hoelzl@56993
  1155
      also have "\<dots> < ereal (e / 2) + e / 2"
hoelzl@56993
  1156
        using ereal_add_strict_mono[OF less_imp_le[OF M[OF n]] _ `?S n \<noteq> \<infinity>` M[OF m]]
hoelzl@56996
  1157
        by (auto simp: nn_integral_nonneg)
hoelzl@56993
  1158
      also have "\<dots> = e" by simp
hoelzl@56993
  1159
      finally show "dist (?s n) (?s m) < e"
hoelzl@56993
  1160
        by (simp add: dist_norm)
hoelzl@56993
  1161
    qed
hoelzl@56993
  1162
  qed
hoelzl@56993
  1163
  then obtain x where "?s ----> x" ..
hoelzl@56993
  1164
  show ?thesis
hoelzl@56993
  1165
    by (rule, rule) fact+
hoelzl@56993
  1166
qed
hoelzl@56993
  1167
hoelzl@56996
  1168
lemma nn_integral_dominated_convergence_norm:
hoelzl@56993
  1169
  fixes u' :: "_ \<Rightarrow> _::{real_normed_vector, second_countable_topology}"
hoelzl@56993
  1170
  assumes [measurable]:
hoelzl@56993
  1171
       "\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M"
hoelzl@56993
  1172
    and bound: "\<And>j. AE x in M. norm (u j x) \<le> w x"
hoelzl@56993
  1173
    and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
hoelzl@56993
  1174
    and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
hoelzl@56993
  1175
  shows "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) ----> 0"
hoelzl@56993
  1176
proof -
hoelzl@56993
  1177
  have "AE x in M. \<forall>j. norm (u j x) \<le> w x"
hoelzl@56993
  1178
    unfolding AE_all_countable by rule fact
hoelzl@56993
  1179
  with u' have bnd: "AE x in M. \<forall>j. norm (u' x - u j x) \<le> 2 * w x"
hoelzl@56993
  1180
  proof (eventually_elim, intro allI)
hoelzl@56993
  1181
    fix i x assume "(\<lambda>i. u i x) ----> u' x" "\<forall>j. norm (u j x) \<le> w x" "\<forall>j. norm (u j x) \<le> w x"
hoelzl@56993
  1182
    then have "norm (u' x) \<le> w x" "norm (u i x) \<le> w x"
hoelzl@56993
  1183
      by (auto intro: LIMSEQ_le_const2 tendsto_norm)
hoelzl@56993
  1184
    then have "norm (u' x) + norm (u i x) \<le> 2 * w x"
hoelzl@56993
  1185
      by simp
hoelzl@56993
  1186
    also have "norm (u' x - u i x) \<le> norm (u' x) + norm (u i x)"
hoelzl@56993
  1187
      by (rule norm_triangle_ineq4)
hoelzl@56993
  1188
    finally (xtrans) show "norm (u' x - u i x) \<le> 2 * w x" .
hoelzl@56993
  1189
  qed
hoelzl@56993
  1190
  
hoelzl@56993
  1191
  have "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) ----> (\<integral>\<^sup>+x. 0 \<partial>M)"
hoelzl@56996
  1192
  proof (rule nn_integral_dominated_convergence)  
hoelzl@56993
  1193
    show "(\<integral>\<^sup>+x. 2 * w x \<partial>M) < \<infinity>"
hoelzl@56996
  1194
      by (rule nn_integral_mult_bounded_inf[OF _ w, of 2]) auto
hoelzl@56993
  1195
    show "AE x in M. (\<lambda>i. ereal (norm (u' x - u i x))) ----> 0"
hoelzl@56993
  1196
      using u' 
hoelzl@56993
  1197
    proof eventually_elim
hoelzl@56993
  1198
      fix x assume "(\<lambda>i. u i x) ----> u' x"
hoelzl@56993
  1199
      from tendsto_diff[OF tendsto_const[of "u' x"] this]
hoelzl@56993
  1200
      show "(\<lambda>i. ereal (norm (u' x - u i x))) ----> 0"
hoelzl@56993
  1201
        by (simp add: zero_ereal_def tendsto_norm_zero_iff)
hoelzl@56993
  1202
    qed
hoelzl@56993
  1203
  qed (insert bnd, auto)
hoelzl@56993
  1204
  then show ?thesis by simp
hoelzl@56993
  1205
qed
hoelzl@56993
  1206
hoelzl@56993
  1207
lemma integrableI_bounded:
hoelzl@56993
  1208
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1209
  assumes f[measurable]: "f \<in> borel_measurable M" and fin: "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
hoelzl@56993
  1210
  shows "integrable M f"
hoelzl@56993
  1211
proof -
hoelzl@56993
  1212
  from borel_measurable_implies_sequence_metric[OF f, of 0] obtain s where
hoelzl@56993
  1213
    s: "\<And>i. simple_function M (s i)" and
hoelzl@56993
  1214
    pointwise: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x" and
hoelzl@56993
  1215
    bound: "\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)"
hoelzl@56993
  1216
    by (simp add: norm_conv_dist) metis
hoelzl@56993
  1217
  
hoelzl@56993
  1218
  show ?thesis
hoelzl@56993
  1219
  proof (rule integrableI_sequence)
hoelzl@56993
  1220
    { fix i
hoelzl@56993
  1221
      have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) \<le> (\<integral>\<^sup>+x. 2 * ereal (norm (f x)) \<partial>M)"
hoelzl@56996
  1222
        by (intro nn_integral_mono) (simp add: bound)
hoelzl@56993
  1223
      also have "\<dots> = 2 * (\<integral>\<^sup>+x. ereal (norm (f x)) \<partial>M)"
hoelzl@56996
  1224
        by (rule nn_integral_cmult) auto
hoelzl@56993
  1225
      finally have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>"
hoelzl@56993
  1226
        using fin by auto }
hoelzl@56993
  1227
    note fin_s = this
hoelzl@56993
  1228
hoelzl@56993
  1229
    show "\<And>i. simple_bochner_integrable M (s i)"
hoelzl@56993
  1230
      by (rule simple_bochner_integrableI_bounded) fact+
hoelzl@56993
  1231
hoelzl@56993
  1232
    show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
hoelzl@56996
  1233
    proof (rule nn_integral_dominated_convergence_norm)
hoelzl@56993
  1234
      show "\<And>j. AE x in M. norm (s j x) \<le> 2 * norm (f x)"
hoelzl@56993
  1235
        using bound by auto
hoelzl@56993
  1236
      show "\<And>i. s i \<in> borel_measurable M" "(\<lambda>x. 2 * norm (f x)) \<in> borel_measurable M"
hoelzl@56993
  1237
        using s by (auto intro: borel_measurable_simple_function)
hoelzl@56993
  1238
      show "(\<integral>\<^sup>+ x. ereal (2 * norm (f x)) \<partial>M) < \<infinity>"
hoelzl@56996
  1239
        using fin unfolding times_ereal.simps(1)[symmetric] by (subst nn_integral_cmult) auto
hoelzl@56993
  1240
      show "AE x in M. (\<lambda>i. s i x) ----> f x"
hoelzl@56993
  1241
        using pointwise by auto
hoelzl@56993
  1242
    qed fact
hoelzl@56993
  1243
  qed fact
hoelzl@56993
  1244
qed
hoelzl@56993
  1245
hoelzl@57275
  1246
lemma integrableI_bounded_set:
hoelzl@57275
  1247
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57275
  1248
  assumes [measurable]: "A \<in> sets M" "f \<in> borel_measurable M"
hoelzl@57275
  1249
  assumes finite: "emeasure M A < \<infinity>"
hoelzl@57275
  1250
    and bnd: "AE x in M. x \<in> A \<longrightarrow> norm (f x) \<le> B"
hoelzl@57275
  1251
    and null: "AE x in M. x \<notin> A \<longrightarrow> f x = 0"
hoelzl@57275
  1252
  shows "integrable M f"
hoelzl@57275
  1253
proof (rule integrableI_bounded)
hoelzl@57275
  1254
  { fix x :: 'b have "norm x \<le> B \<Longrightarrow> 0 \<le> B"
hoelzl@57275
  1255
      using norm_ge_zero[of x] by arith }
hoelzl@57275
  1256
  with bnd null have "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (max 0 B) * indicator A x \<partial>M)"
hoelzl@57275
  1257
    by (intro nn_integral_mono_AE) (auto split: split_indicator split_max)
hoelzl@57275
  1258
  also have "\<dots> < \<infinity>"
hoelzl@57275
  1259
    using finite by (subst nn_integral_cmult_indicator) (auto simp: max_def)
hoelzl@57275
  1260
  finally show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" .
hoelzl@57275
  1261
qed simp
hoelzl@57275
  1262
hoelzl@57275
  1263
lemma integrableI_bounded_set_indicator:
hoelzl@57275
  1264
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57275
  1265
  shows "A \<in> sets M \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow>
hoelzl@57275
  1266
    emeasure M A < \<infinity> \<Longrightarrow> (AE x in M. x \<in> A \<longrightarrow> norm (f x) \<le> B) \<Longrightarrow>
hoelzl@57275
  1267
    integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
hoelzl@57275
  1268
  by (rule integrableI_bounded_set[where A=A]) auto
hoelzl@57275
  1269
hoelzl@56993
  1270
lemma integrableI_nonneg:
hoelzl@56993
  1271
  fixes f :: "'a \<Rightarrow> real"
hoelzl@56993
  1272
  assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>"
hoelzl@56993
  1273
  shows "integrable M f"
hoelzl@56993
  1274
proof -
hoelzl@56993
  1275
  have "(\<integral>\<^sup>+x. norm (f x) \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M)"
hoelzl@56996
  1276
    using assms by (intro nn_integral_cong_AE) auto
hoelzl@56993
  1277
  then show ?thesis
hoelzl@56993
  1278
    using assms by (intro integrableI_bounded) auto
hoelzl@56993
  1279
qed
hoelzl@56993
  1280
hoelzl@56993
  1281
lemma integrable_iff_bounded:
hoelzl@56993
  1282
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1283
  shows "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
hoelzl@56993
  1284
  using integrableI_bounded[of f M] has_bochner_integral_implies_finite_norm[of M f]
hoelzl@56993
  1285
  unfolding integrable.simps has_bochner_integral.simps[abs_def] by auto
hoelzl@56993
  1286
hoelzl@56993
  1287
lemma integrable_bound:
hoelzl@56993
  1288
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1289
    and g :: "'a \<Rightarrow> 'c::{banach, second_countable_topology}"
hoelzl@56993
  1290
  shows "integrable M f \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. norm (g x) \<le> norm (f x)) \<Longrightarrow>
hoelzl@56993
  1291
    integrable M g"
hoelzl@56993
  1292
  unfolding integrable_iff_bounded
hoelzl@56993
  1293
proof safe
hoelzl@56993
  1294
  assume "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@56993
  1295
  assume "AE x in M. norm (g x) \<le> norm (f x)"
hoelzl@56993
  1296
  then have "(\<integral>\<^sup>+ x. ereal (norm (g x)) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
hoelzl@56996
  1297
    by  (intro nn_integral_mono_AE) auto
hoelzl@56993
  1298
  also assume "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>"
hoelzl@56993
  1299
  finally show "(\<integral>\<^sup>+ x. ereal (norm (g x)) \<partial>M) < \<infinity>" .
hoelzl@56993
  1300
qed 
hoelzl@56993
  1301
hoelzl@57275
  1302
lemma integrable_mult_indicator:
hoelzl@57275
  1303
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57275
  1304
  shows "A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
hoelzl@57275
  1305
  by (rule integrable_bound[of M f]) (auto split: split_indicator)
hoelzl@57275
  1306
hoelzl@59000
  1307
lemma integrable_real_mult_indicator:
hoelzl@59000
  1308
  fixes f :: "'a \<Rightarrow> real"
hoelzl@59000
  1309
  shows "A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. f x * indicator A x)"
hoelzl@59000
  1310
  using integrable_mult_indicator[of A M f] by (simp add: mult_ac)
hoelzl@59000
  1311
hoelzl@56993
  1312
lemma integrable_abs[simp, intro]:
hoelzl@56993
  1313
  fixes f :: "'a \<Rightarrow> real"
hoelzl@56993
  1314
  assumes [measurable]: "integrable M f" shows "integrable M (\<lambda>x. \<bar>f x\<bar>)"
hoelzl@56993
  1315
  using assms by (rule integrable_bound) auto
hoelzl@56993
  1316
hoelzl@56993
  1317
lemma integrable_norm[simp, intro]:
hoelzl@56993
  1318
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1319
  assumes [measurable]: "integrable M f" shows "integrable M (\<lambda>x. norm (f x))"
hoelzl@56993
  1320
  using assms by (rule integrable_bound) auto
hoelzl@56993
  1321
  
hoelzl@56993
  1322
lemma integrable_norm_cancel:
hoelzl@56993
  1323
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1324
  assumes [measurable]: "integrable M (\<lambda>x. norm (f x))" "f \<in> borel_measurable M" shows "integrable M f"
hoelzl@56993
  1325
  using assms by (rule integrable_bound) auto
hoelzl@56993
  1326
hoelzl@57275
  1327
lemma integrable_norm_iff:
hoelzl@57275
  1328
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57275
  1329
  shows "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda>x. norm (f x)) \<longleftrightarrow> integrable M f"
hoelzl@57275
  1330
  by (auto intro: integrable_norm_cancel)
hoelzl@57275
  1331
hoelzl@56993
  1332
lemma integrable_abs_cancel:
hoelzl@56993
  1333
  fixes f :: "'a \<Rightarrow> real"
hoelzl@56993
  1334
  assumes [measurable]: "integrable M (\<lambda>x. \<bar>f x\<bar>)" "f \<in> borel_measurable M" shows "integrable M f"
hoelzl@56993
  1335
  using assms by (rule integrable_bound) auto
hoelzl@56993
  1336
hoelzl@57275
  1337
lemma integrable_abs_iff:
hoelzl@57275
  1338
  fixes f :: "'a \<Rightarrow> real"
hoelzl@57275
  1339
  shows "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda>x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
hoelzl@57275
  1340
  by (auto intro: integrable_abs_cancel)
hoelzl@57275
  1341
hoelzl@56993
  1342
lemma integrable_max[simp, intro]:
hoelzl@56993
  1343
  fixes f :: "'a \<Rightarrow> real"
hoelzl@56993
  1344
  assumes fg[measurable]: "integrable M f" "integrable M g"
hoelzl@56993
  1345
  shows "integrable M (\<lambda>x. max (f x) (g x))"
hoelzl@56993
  1346
  using integrable_add[OF integrable_norm[OF fg(1)] integrable_norm[OF fg(2)]]
hoelzl@56993
  1347
  by (rule integrable_bound) auto
hoelzl@56993
  1348
hoelzl@56993
  1349
lemma integrable_min[simp, intro]:
hoelzl@56993
  1350
  fixes f :: "'a \<Rightarrow> real"
hoelzl@56993
  1351
  assumes fg[measurable]: "integrable M f" "integrable M g"
hoelzl@56993
  1352
  shows "integrable M (\<lambda>x. min (f x) (g x))"
hoelzl@56993
  1353
  using integrable_add[OF integrable_norm[OF fg(1)] integrable_norm[OF fg(2)]]
hoelzl@56993
  1354
  by (rule integrable_bound) auto
hoelzl@56993
  1355
hoelzl@56993
  1356
lemma integral_minus_iff[simp]:
hoelzl@56993
  1357
  "integrable M (\<lambda>x. - f x ::'a::{banach, second_countable_topology}) \<longleftrightarrow> integrable M f"
hoelzl@56993
  1358
  unfolding integrable_iff_bounded
hoelzl@56993
  1359
  by (auto intro: borel_measurable_uminus[of "\<lambda>x. - f x" M, simplified])
hoelzl@56993
  1360
hoelzl@56993
  1361
lemma integrable_indicator_iff:
hoelzl@56993
  1362
  "integrable M (indicator A::_ \<Rightarrow> real) \<longleftrightarrow> A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>"
hoelzl@56996
  1363
  by (simp add: integrable_iff_bounded borel_measurable_indicator_iff ereal_indicator nn_integral_indicator'
hoelzl@56993
  1364
           cong: conj_cong)
hoelzl@56993
  1365
hoelzl@57166
  1366
lemma integral_indicator[simp]: "integral\<^sup>L M (indicator A) = measure M (A \<inter> space M)"
hoelzl@57166
  1367
proof cases
hoelzl@57166
  1368
  assume *: "A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>"
hoelzl@57166
  1369
  have "integral\<^sup>L M (indicator A) = integral\<^sup>L M (indicator (A \<inter> space M))"
hoelzl@57166
  1370
    by (intro integral_cong) (auto split: split_indicator)
hoelzl@57166
  1371
  also have "\<dots> = measure M (A \<inter> space M)"
hoelzl@57166
  1372
    using * by (intro has_bochner_integral_integral_eq has_bochner_integral_real_indicator) auto
hoelzl@57166
  1373
  finally show ?thesis .
hoelzl@57166
  1374
next
hoelzl@57166
  1375
  assume *: "\<not> (A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>)"
hoelzl@57166
  1376
  have "integral\<^sup>L M (indicator A) = integral\<^sup>L M (indicator (A \<inter> space M) :: _ \<Rightarrow> real)"
hoelzl@57166
  1377
    by (intro integral_cong) (auto split: split_indicator)
hoelzl@57166
  1378
  also have "\<dots> = 0"
hoelzl@57166
  1379
    using * by (subst not_integrable_integral_eq) (auto simp: integrable_indicator_iff)
hoelzl@57166
  1380
  also have "\<dots> = measure M (A \<inter> space M)"
hoelzl@57166
  1381
    using * by (auto simp: measure_def emeasure_notin_sets)
hoelzl@57166
  1382
  finally show ?thesis .
hoelzl@57166
  1383
qed
hoelzl@57166
  1384
hoelzl@57275
  1385
lemma integrable_discrete_difference:
hoelzl@57275
  1386
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57275
  1387
  assumes X: "countable X"
hoelzl@57275
  1388
  assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" 
hoelzl@57275
  1389
  assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
hoelzl@57275
  1390
  assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
hoelzl@57275
  1391
  shows "integrable M f \<longleftrightarrow> integrable M g"
hoelzl@57275
  1392
  unfolding integrable_iff_bounded
hoelzl@57275
  1393
proof (rule conj_cong)
hoelzl@57275
  1394
  { assume "f \<in> borel_measurable M" then have "g \<in> borel_measurable M"
hoelzl@57275
  1395
      by (rule measurable_discrete_difference[where X=X]) (auto simp: assms) }
hoelzl@57275
  1396
  moreover
hoelzl@57275
  1397
  { assume "g \<in> borel_measurable M" then have "f \<in> borel_measurable M"
hoelzl@57275
  1398
      by (rule measurable_discrete_difference[where X=X]) (auto simp: assms) }
hoelzl@57275
  1399
  ultimately show "f \<in> borel_measurable M \<longleftrightarrow> g \<in> borel_measurable M" ..
hoelzl@57275
  1400
next
hoelzl@57275
  1401
  have "AE x in M. x \<notin> X"
hoelzl@57275
  1402
    by (rule AE_discrete_difference) fact+
hoelzl@57275
  1403
  then have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) = (\<integral>\<^sup>+ x. norm (g x) \<partial>M)"
hoelzl@57275
  1404
    by (intro nn_integral_cong_AE) (auto simp: eq)
hoelzl@57275
  1405
  then show "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) < \<infinity> \<longleftrightarrow> (\<integral>\<^sup>+ x. norm (g x) \<partial>M) < \<infinity>"
hoelzl@57275
  1406
    by simp
hoelzl@57275
  1407
qed
hoelzl@57275
  1408
hoelzl@57275
  1409
lemma integral_discrete_difference:
hoelzl@57275
  1410
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57275
  1411
  assumes X: "countable X"
hoelzl@57275
  1412
  assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" 
hoelzl@57275
  1413
  assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
hoelzl@57275
  1414
  assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
hoelzl@57275
  1415
  shows "integral\<^sup>L M f = integral\<^sup>L M g"
hoelzl@57275
  1416
proof (rule integral_eq_cases)
hoelzl@57275
  1417
  show eq: "integrable M f \<longleftrightarrow> integrable M g"
hoelzl@57275
  1418
    by (rule integrable_discrete_difference[where X=X]) fact+
hoelzl@57275
  1419
hoelzl@57275
  1420
  assume f: "integrable M f"
hoelzl@57275
  1421
  show "integral\<^sup>L M f = integral\<^sup>L M g"
hoelzl@57275
  1422
  proof (rule integral_cong_AE)
hoelzl@57275
  1423
    show "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@57275
  1424
      using f eq by (auto intro: borel_measurable_integrable)
hoelzl@57275
  1425
hoelzl@57275
  1426
    have "AE x in M. x \<notin> X"
hoelzl@57275
  1427
      by (rule AE_discrete_difference) fact+
hoelzl@57275
  1428
    with AE_space show "AE x in M. f x = g x"
hoelzl@57275
  1429
      by eventually_elim fact
hoelzl@57275
  1430
  qed
hoelzl@57275
  1431
qed
hoelzl@57275
  1432
hoelzl@57275
  1433
lemma has_bochner_integral_discrete_difference:
hoelzl@57275
  1434
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57275
  1435
  assumes X: "countable X"
hoelzl@57275
  1436
  assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" 
hoelzl@57275
  1437
  assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
hoelzl@57275
  1438
  assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
hoelzl@57275
  1439
  shows "has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral M g x"
hoelzl@57275
  1440
  using integrable_discrete_difference[of X M f g, OF assms]
hoelzl@57275
  1441
  using integral_discrete_difference[of X M f g, OF assms]
hoelzl@57275
  1442
  by (metis has_bochner_integral_iff)
hoelzl@57275
  1443
hoelzl@57137
  1444
lemma
hoelzl@56993
  1445
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and w :: "'a \<Rightarrow> real"
hoelzl@56993
  1446
  assumes "f \<in> borel_measurable M" "\<And>i. s i \<in> borel_measurable M" "integrable M w"
hoelzl@56993
  1447
  assumes lim: "AE x in M. (\<lambda>i. s i x) ----> f x"
hoelzl@56993
  1448
  assumes bound: "\<And>i. AE x in M. norm (s i x) \<le> w x"
hoelzl@57137
  1449
  shows integrable_dominated_convergence: "integrable M f"
hoelzl@57137
  1450
    and integrable_dominated_convergence2: "\<And>i. integrable M (s i)"
hoelzl@57137
  1451
    and integral_dominated_convergence: "(\<lambda>i. integral\<^sup>L M (s i)) ----> integral\<^sup>L M f"
hoelzl@56993
  1452
proof -
hoelzl@56993
  1453
  have "AE x in M. 0 \<le> w x"
hoelzl@56993
  1454
    using bound[of 0] by eventually_elim (auto intro: norm_ge_zero order_trans)
hoelzl@56993
  1455
  then have "(\<integral>\<^sup>+x. w x \<partial>M) = (\<integral>\<^sup>+x. norm (w x) \<partial>M)"
hoelzl@56996
  1456
    by (intro nn_integral_cong_AE) auto
hoelzl@56993
  1457
  with `integrable M w` have w: "w \<in> borel_measurable M" "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
hoelzl@56993
  1458
    unfolding integrable_iff_bounded by auto
hoelzl@56993
  1459
hoelzl@56993
  1460
  show int_s: "\<And>i. integrable M (s i)"
hoelzl@56993
  1461
    unfolding integrable_iff_bounded
hoelzl@56993
  1462
  proof
hoelzl@56993
  1463
    fix i 
hoelzl@56993
  1464
    have "(\<integral>\<^sup>+ x. ereal (norm (s i x)) \<partial>M) \<le> (\<integral>\<^sup>+x. w x \<partial>M)"
hoelzl@56996
  1465
      using bound by (intro nn_integral_mono_AE) auto
hoelzl@56993
  1466
    with w show "(\<integral>\<^sup>+ x. ereal (norm (s i x)) \<partial>M) < \<infinity>" by auto
hoelzl@56993
  1467
  qed fact
hoelzl@56993
  1468
hoelzl@56993
  1469
  have all_bound: "AE x in M. \<forall>i. norm (s i x) \<le> w x"
hoelzl@56993
  1470
    using bound unfolding AE_all_countable by auto
hoelzl@56993
  1471
hoelzl@56993
  1472
  show int_f: "integrable M f"
hoelzl@56993
  1473
    unfolding integrable_iff_bounded
hoelzl@56993
  1474
  proof
hoelzl@56993
  1475
    have "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) \<le> (\<integral>\<^sup>+x. w x \<partial>M)"
hoelzl@56993
  1476
      using all_bound lim
hoelzl@56996
  1477
    proof (intro nn_integral_mono_AE, eventually_elim)
hoelzl@56993
  1478
      fix x assume "\<forall>i. norm (s i x) \<le> w x" "(\<lambda>i. s i x) ----> f x"
hoelzl@56993
  1479
      then show "ereal (norm (f x)) \<le> ereal (w x)"
hoelzl@56993
  1480
        by (intro LIMSEQ_le_const2[where X="\<lambda>i. ereal (norm (s i x))"] tendsto_intros lim_ereal[THEN iffD2]) auto
hoelzl@56993
  1481
    qed
hoelzl@56993
  1482
    with w show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" by auto
hoelzl@56993
  1483
  qed fact
hoelzl@56993
  1484
hoelzl@56993
  1485
  have "(\<lambda>n. ereal (norm (integral\<^sup>L M (s n) - integral\<^sup>L M f))) ----> ereal 0" (is "?d ----> ereal 0")
hoelzl@56993
  1486
  proof (rule tendsto_sandwich)
hoelzl@56993
  1487
    show "eventually (\<lambda>n. ereal 0 \<le> ?d n) sequentially" "(\<lambda>_. ereal 0) ----> ereal 0" by auto
hoelzl@56993
  1488
    show "eventually (\<lambda>n. ?d n \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)) sequentially"
hoelzl@56993
  1489
    proof (intro always_eventually allI)
hoelzl@56993
  1490
      fix n
hoelzl@56993
  1491
      have "?d n = norm (integral\<^sup>L M (\<lambda>x. s n x - f x))"
hoelzl@56993
  1492
        using int_f int_s by simp
hoelzl@56993
  1493
      also have "\<dots> \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)"
hoelzl@56993
  1494
        by (intro int_f int_s integrable_diff integral_norm_bound_ereal)
hoelzl@56993
  1495
      finally show "?d n \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)" .
hoelzl@56993
  1496
    qed
hoelzl@56993
  1497
    show "(\<lambda>n. \<integral>\<^sup>+x. norm (s n x - f x) \<partial>M) ----> ereal 0"
hoelzl@56993
  1498
      unfolding zero_ereal_def[symmetric]
hoelzl@56993
  1499
      apply (subst norm_minus_commute)
hoelzl@56996
  1500
    proof (rule nn_integral_dominated_convergence_norm[where w=w])
hoelzl@56993
  1501
      show "\<And>n. s n \<in> borel_measurable M"
hoelzl@56993
  1502
        using int_s unfolding integrable_iff_bounded by auto
hoelzl@56993
  1503
    qed fact+
hoelzl@56993
  1504
  qed
hoelzl@56993
  1505
  then have "(\<lambda>n. integral\<^sup>L M (s n) - integral\<^sup>L M f) ----> 0"
hoelzl@56993
  1506
    unfolding lim_ereal tendsto_norm_zero_iff .
hoelzl@56993
  1507
  from tendsto_add[OF this tendsto_const[of "integral\<^sup>L M f"]]
hoelzl@56993
  1508
  show "(\<lambda>i. integral\<^sup>L M (s i)) ----> integral\<^sup>L M f"  by simp
hoelzl@56993
  1509
qed
hoelzl@56993
  1510
hoelzl@56993
  1511
lemma integrable_mult_left_iff:
hoelzl@56993
  1512
  fixes f :: "'a \<Rightarrow> real"
hoelzl@56993
  1513
  shows "integrable M (\<lambda>x. c * f x) \<longleftrightarrow> c = 0 \<or> integrable M f"
hoelzl@56993
  1514
  using integrable_mult_left[of c M f] integrable_mult_left[of "1 / c" M "\<lambda>x. c * f x"]
hoelzl@56993
  1515
  by (cases "c = 0") auto
hoelzl@56993
  1516
hoelzl@56996
  1517
lemma nn_integral_eq_integral:
hoelzl@56993
  1518
  assumes f: "integrable M f"
hoelzl@56993
  1519
  assumes nonneg: "AE x in M. 0 \<le> f x" 
hoelzl@56993
  1520
  shows "(\<integral>\<^sup>+ x. f x \<partial>M) = integral\<^sup>L M f"
hoelzl@56993
  1521
proof -
hoelzl@56993
  1522
  { fix f :: "'a \<Rightarrow> real" assume f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "integrable M f"
hoelzl@56993
  1523
    then have "(\<integral>\<^sup>+ x. f x \<partial>M) = integral\<^sup>L M f"
hoelzl@56993
  1524
    proof (induct rule: borel_measurable_induct_real)
hoelzl@56993
  1525
      case (set A) then show ?case
hoelzl@56993
  1526
        by (simp add: integrable_indicator_iff ereal_indicator emeasure_eq_ereal_measure)
hoelzl@56993
  1527
    next
hoelzl@56993
  1528
      case (mult f c) then show ?case
hoelzl@56993
  1529
        unfolding times_ereal.simps(1)[symmetric]
hoelzl@56996
  1530
        by (subst nn_integral_cmult)
hoelzl@56993
  1531
           (auto simp add: integrable_mult_left_iff zero_ereal_def[symmetric])
hoelzl@56993
  1532
    next
hoelzl@56993
  1533
      case (add g f)
hoelzl@56993
  1534
      then have "integrable M f" "integrable M g"
hoelzl@56993
  1535
        by (auto intro!: integrable_bound[OF add(8)])
hoelzl@56993
  1536
      with add show ?case
hoelzl@56993
  1537
        unfolding plus_ereal.simps(1)[symmetric]
hoelzl@56996
  1538
        by (subst nn_integral_add) auto
hoelzl@56993
  1539
    next
hoelzl@56993
  1540
      case (seq s)
hoelzl@56993
  1541
      { fix i x assume "x \<in> space M" with seq(4) have "s i x \<le> f x"
hoelzl@56993
  1542
          by (intro LIMSEQ_le_const[OF seq(5)] exI[of _ i]) (auto simp: incseq_def le_fun_def) }
hoelzl@56993
  1543
      note s_le_f = this
hoelzl@56993
  1544
hoelzl@56993
  1545
      show ?case
hoelzl@56993
  1546
      proof (rule LIMSEQ_unique)
hoelzl@56993
  1547
        show "(\<lambda>i. ereal (integral\<^sup>L M (s i))) ----> ereal (integral\<^sup>L M f)"
hoelzl@56993
  1548
          unfolding lim_ereal
hoelzl@56993
  1549
        proof (rule integral_dominated_convergence[where w=f])
hoelzl@56993
  1550
          show "integrable M f" by fact
hoelzl@56993
  1551
          from s_le_f seq show "\<And>i. AE x in M. norm (s i x) \<le> f x"
hoelzl@56993
  1552
            by auto
hoelzl@56993
  1553
        qed (insert seq, auto)
hoelzl@56993
  1554
        have int_s: "\<And>i. integrable M (s i)"
hoelzl@56993
  1555
          using seq f s_le_f by (intro integrable_bound[OF f(3)]) auto
hoelzl@56993
  1556
        have "(\<lambda>i. \<integral>\<^sup>+ x. s i x \<partial>M) ----> \<integral>\<^sup>+ x. f x \<partial>M"
hoelzl@56993
  1557
          using seq s_le_f f
hoelzl@56996
  1558
          by (intro nn_integral_dominated_convergence[where w=f])
hoelzl@56993
  1559
             (auto simp: integrable_iff_bounded)
hoelzl@56993
  1560
        also have "(\<lambda>i. \<integral>\<^sup>+x. s i x \<partial>M) = (\<lambda>i. \<integral>x. s i x \<partial>M)"
hoelzl@56993
  1561
          using seq int_s by simp
hoelzl@56993
  1562
        finally show "(\<lambda>i. \<integral>x. s i x \<partial>M) ----> \<integral>\<^sup>+x. f x \<partial>M"
hoelzl@56993
  1563
          by simp
hoelzl@56993
  1564
      qed
hoelzl@56993
  1565
    qed }
hoelzl@56993
  1566
  from this[of "\<lambda>x. max 0 (f x)"] assms have "(\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) = integral\<^sup>L M (\<lambda>x. max 0 (f x))"
hoelzl@56993
  1567
    by simp
hoelzl@56993
  1568
  also have "\<dots> = integral\<^sup>L M f"
hoelzl@56993
  1569
    using assms by (auto intro!: integral_cong_AE)
hoelzl@56993
  1570
  also have "(\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) = (\<integral>\<^sup>+ x. f x \<partial>M)"
hoelzl@56996
  1571
    using assms by (auto intro!: nn_integral_cong_AE simp: max_def)
hoelzl@56993
  1572
  finally show ?thesis .
hoelzl@56993
  1573
qed
hoelzl@56993
  1574
hoelzl@57036
  1575
lemma
hoelzl@57036
  1576
  fixes f :: "_ \<Rightarrow> _ \<Rightarrow> 'a :: {banach, second_countable_topology}"
hoelzl@57036
  1577
  assumes integrable[measurable]: "\<And>i. integrable M (f i)"
hoelzl@57036
  1578
  and summable: "AE x in M. summable (\<lambda>i. norm (f i x))"
hoelzl@57036
  1579
  and sums: "summable (\<lambda>i. (\<integral>x. norm (f i x) \<partial>M))"
hoelzl@57036
  1580
  shows integrable_suminf: "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
hoelzl@57036
  1581
    and sums_integral: "(\<lambda>i. integral\<^sup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is "?f sums ?x")
hoelzl@57036
  1582
    and integral_suminf: "(\<integral>x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>L M (f i))"
hoelzl@57036
  1583
    and summable_integral: "summable (\<lambda>i. integral\<^sup>L M (f i))"
hoelzl@57036
  1584
proof -
hoelzl@57036
  1585
  have 1: "integrable M (\<lambda>x. \<Sum>i. norm (f i x))"
hoelzl@57036
  1586
  proof (rule integrableI_bounded)
hoelzl@57036
  1587
    have "(\<integral>\<^sup>+ x. ereal (norm (\<Sum>i. norm (f i x))) \<partial>M) = (\<integral>\<^sup>+ x. (\<Sum>i. ereal (norm (f i x))) \<partial>M)"
hoelzl@57036
  1588
      apply (intro nn_integral_cong_AE) 
hoelzl@57036
  1589
      using summable
hoelzl@57036
  1590
      apply eventually_elim
hoelzl@57036
  1591
      apply (simp add: suminf_ereal' suminf_nonneg)
hoelzl@57036
  1592
      done
hoelzl@57036
  1593
    also have "\<dots> = (\<Sum>i. \<integral>\<^sup>+ x. norm (f i x) \<partial>M)"
hoelzl@57036
  1594
      by (intro nn_integral_suminf) auto
hoelzl@57036
  1595
    also have "\<dots> = (\<Sum>i. ereal (\<integral>x. norm (f i x) \<partial>M))"
hoelzl@57036
  1596
      by (intro arg_cong[where f=suminf] ext nn_integral_eq_integral integrable_norm integrable) auto
hoelzl@57036
  1597
    finally show "(\<integral>\<^sup>+ x. ereal (norm (\<Sum>i. norm (f i x))) \<partial>M) < \<infinity>"
hoelzl@57036
  1598
      by (simp add: suminf_ereal' sums)
hoelzl@57036
  1599
  qed simp
hoelzl@57036
  1600
hoelzl@57036
  1601
  have 2: "AE x in M. (\<lambda>n. \<Sum>i<n. f i x) ----> (\<Sum>i. f i x)"
hoelzl@57036
  1602
    using summable by eventually_elim (auto intro: summable_LIMSEQ summable_norm_cancel)
hoelzl@57036
  1603
hoelzl@57036
  1604
  have 3: "\<And>j. AE x in M. norm (\<Sum>i<j. f i x) \<le> (\<Sum>i. norm (f i x))"
hoelzl@57036
  1605
    using summable
hoelzl@57036
  1606
  proof eventually_elim
hoelzl@57036
  1607
    fix j x assume [simp]: "summable (\<lambda>i. norm (f i x))"
hoelzl@57036
  1608
    have "norm (\<Sum>i<j. f i x) \<le> (\<Sum>i<j. norm (f i x))" by (rule norm_setsum)
hoelzl@57036
  1609
    also have "\<dots> \<le> (\<Sum>i. norm (f i x))"
hoelzl@57036
  1610
      using setsum_le_suminf[of "\<lambda>i. norm (f i x)"] unfolding sums_iff by auto
hoelzl@57036
  1611
    finally show "norm (\<Sum>i<j. f i x) \<le> (\<Sum>i. norm (f i x))" by simp
hoelzl@57036
  1612
  qed
hoelzl@57036
  1613
hoelzl@57137
  1614
  note ibl = integrable_dominated_convergence[OF _ _ 1 2 3]
hoelzl@57137
  1615
  note int = integral_dominated_convergence[OF _ _ 1 2 3]
hoelzl@57036
  1616
hoelzl@57036
  1617
  show "integrable M ?S"
hoelzl@57137
  1618
    by (rule ibl) measurable
hoelzl@57036
  1619
hoelzl@57036
  1620
  show "?f sums ?x" unfolding sums_def
hoelzl@57137
  1621
    using int by (simp add: integrable)
hoelzl@57036
  1622
  then show "?x = suminf ?f" "summable ?f"
hoelzl@57036
  1623
    unfolding sums_iff by auto
hoelzl@57036
  1624
qed
hoelzl@57036
  1625
hoelzl@56993
  1626
lemma integral_norm_bound:
hoelzl@56993
  1627
  fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
hoelzl@56993
  1628
  shows "integrable M f \<Longrightarrow> norm (integral\<^sup>L M f) \<le> (\<integral>x. norm (f x) \<partial>M)"
hoelzl@56996
  1629
  using nn_integral_eq_integral[of M "\<lambda>x. norm (f x)"]
hoelzl@56993
  1630
  using integral_norm_bound_ereal[of M f] by simp
hoelzl@56993
  1631
  
hoelzl@57235
  1632
lemma integrableI_nn_integral_finite:
hoelzl@57235
  1633
  assumes [measurable]: "f \<in> borel_measurable M"
hoelzl@57235
  1634
    and nonneg: "AE x in M. 0 \<le> f x"
hoelzl@57235
  1635
    and finite: "(\<integral>\<^sup>+x. f x \<partial>M) = ereal x"
hoelzl@57235
  1636
  shows "integrable M f"
hoelzl@57235
  1637
proof (rule integrableI_bounded)
hoelzl@57235
  1638
  have "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) = (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
hoelzl@57235
  1639
    using nonneg by (intro nn_integral_cong_AE) auto
hoelzl@57235
  1640
  with finite show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>"
hoelzl@57235
  1641
    by auto
hoelzl@57235
  1642
qed simp
hoelzl@57235
  1643
hoelzl@56996
  1644
lemma integral_eq_nn_integral:
hoelzl@57235
  1645
  assumes [measurable]: "f \<in> borel_measurable M"
hoelzl@57235
  1646
  assumes nonneg: "AE x in M. 0 \<le> f x"
hoelzl@57235
  1647
  shows "integral\<^sup>L M f = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
hoelzl@57235
  1648
proof cases
hoelzl@57235
  1649
  assume *: "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = \<infinity>"
hoelzl@57235
  1650
  also have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
hoelzl@57235
  1651
    using nonneg by (intro nn_integral_cong_AE) auto
hoelzl@57235
  1652
  finally have "\<not> integrable M f"
hoelzl@57235
  1653
    by (auto simp: integrable_iff_bounded)
hoelzl@57235
  1654
  then show ?thesis
hoelzl@57235
  1655
    by (simp add: * not_integrable_integral_eq)
hoelzl@57235
  1656
next
hoelzl@57235
  1657
  assume "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>"
hoelzl@57235
  1658
  then have "integrable M f"
hoelzl@57235
  1659
    by (cases "\<integral>\<^sup>+ x. ereal (f x) \<partial>M") (auto intro!: integrableI_nn_integral_finite assms)
hoelzl@57235
  1660
  from nn_integral_eq_integral[OF this nonneg] show ?thesis
hoelzl@57235
  1661
    by simp
hoelzl@57235
  1662
qed
hoelzl@56993
  1663
  
hoelzl@57447
  1664
lemma has_bochner_integral_nn_integral:
hoelzl@57447
  1665
  assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
hoelzl@57447
  1666
  assumes "(\<integral>\<^sup>+x. f x \<partial>M) = ereal x"
hoelzl@57447
  1667
  shows "has_bochner_integral M f x"
hoelzl@57447
  1668
  unfolding has_bochner_integral_iff
hoelzl@57447
  1669
  using assms by (auto simp: assms integral_eq_nn_integral intro: integrableI_nn_integral_finite)
hoelzl@57447
  1670
hoelzl@56993
  1671
lemma integrableI_simple_bochner_integrable:
hoelzl@56993
  1672
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1673
  shows "simple_bochner_integrable M f \<Longrightarrow> integrable M f"
hoelzl@56993
  1674
  by (intro integrableI_sequence[where s="\<lambda>_. f"] borel_measurable_simple_function)
hoelzl@56993
  1675
     (auto simp: zero_ereal_def[symmetric] simple_bochner_integrable.simps)
hoelzl@56993
  1676
hoelzl@56993
  1677
lemma integrable_induct[consumes 1, case_names base add lim, induct pred: integrable]:
hoelzl@56993
  1678
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1679
  assumes "integrable M f"
hoelzl@56993
  1680
  assumes base: "\<And>A c. A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> P (\<lambda>x. indicator A x *\<^sub>R c)"
hoelzl@56993
  1681
  assumes add: "\<And>f g. integrable M f \<Longrightarrow> P f \<Longrightarrow> integrable M g \<Longrightarrow> P g \<Longrightarrow> P (\<lambda>x. f x + g x)"
hoelzl@56993
  1682
  assumes lim: "\<And>f s. (\<And>i. integrable M (s i)) \<Longrightarrow> (\<And>i. P (s i)) \<Longrightarrow>
hoelzl@56993
  1683
   (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x) \<Longrightarrow>
hoelzl@56993
  1684
   (\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)) \<Longrightarrow> integrable M f \<Longrightarrow> P f"
hoelzl@56993
  1685
  shows "P f"
hoelzl@56993
  1686
proof -
hoelzl@56993
  1687
  from `integrable M f` have f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
hoelzl@56993
  1688
    unfolding integrable_iff_bounded by auto
hoelzl@56993
  1689
  from borel_measurable_implies_sequence_metric[OF f(1)]
hoelzl@56993
  1690
  obtain s where s: "\<And>i. simple_function M (s i)" "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x"
hoelzl@56993
  1691
    "\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)"
hoelzl@56993
  1692
    unfolding norm_conv_dist by metis
hoelzl@56993
  1693
hoelzl@56993
  1694
  { fix f A 
hoelzl@56993
  1695
    have [simp]: "P (\<lambda>x. 0)"
hoelzl@56993
  1696
      using base[of "{}" undefined] by simp
hoelzl@56993
  1697
    have "(\<And>i::'b. i \<in> A \<Longrightarrow> integrable M (f i::'a \<Rightarrow> 'b)) \<Longrightarrow>
hoelzl@56993
  1698
    (\<And>i. i \<in> A \<Longrightarrow> P (f i)) \<Longrightarrow> P (\<lambda>x. \<Sum>i\<in>A. f i x)"
hoelzl@56993
  1699
    by (induct A rule: infinite_finite_induct) (auto intro!: add) }
hoelzl@56993
  1700
  note setsum = this
hoelzl@56993
  1701
hoelzl@56993
  1702
  def s' \<equiv> "\<lambda>i z. indicator (space M) z *\<^sub>R s i z"
hoelzl@56993
  1703
  then have s'_eq_s: "\<And>i x. x \<in> space M \<Longrightarrow> s' i x = s i x"
hoelzl@56993
  1704
    by simp
hoelzl@56993
  1705
hoelzl@56993
  1706
  have sf[measurable]: "\<And>i. simple_function M (s' i)"
hoelzl@56993
  1707
    unfolding s'_def using s(1)
hoelzl@56993
  1708
    by (intro simple_function_compose2[where h="op *\<^sub>R"] simple_function_indicator) auto
hoelzl@56993
  1709
hoelzl@56993
  1710
  { fix i 
hoelzl@56993
  1711
    have "\<And>z. {y. s' i z = y \<and> y \<in> s' i ` space M \<and> y \<noteq> 0 \<and> z \<in> space M} =
hoelzl@56993
  1712
        (if z \<in> space M \<and> s' i z \<noteq> 0 then {s' i z} else {})"
hoelzl@56993
  1713
      by (auto simp add: s'_def split: split_indicator)
hoelzl@56993
  1714
    then have "\<And>z. s' i = (\<lambda>z. \<Sum>y\<in>s' i`space M - {0}. indicator {x\<in>space M. s' i x = y} z *\<^sub>R y)"
hoelzl@56993
  1715
      using sf by (auto simp: fun_eq_iff simple_function_def s'_def) }
hoelzl@56993
  1716
  note s'_eq = this
hoelzl@56993
  1717
hoelzl@56993
  1718
  show "P f"
hoelzl@56993
  1719
  proof (rule lim)
hoelzl@56993
  1720
    fix i
hoelzl@56993
  1721
hoelzl@56993
  1722
    have "(\<integral>\<^sup>+x. norm (s' i x) \<partial>M) \<le> (\<integral>\<^sup>+x. 2 * ereal (norm (f x)) \<partial>M)"
hoelzl@56996
  1723
      using s by (intro nn_integral_mono) (auto simp: s'_eq_s)
hoelzl@56993
  1724
    also have "\<dots> < \<infinity>"
hoelzl@56996
  1725
      using f by (subst nn_integral_cmult) auto
hoelzl@56993
  1726
    finally have sbi: "simple_bochner_integrable M (s' i)"
hoelzl@56993
  1727
      using sf by (intro simple_bochner_integrableI_bounded) auto
hoelzl@56993
  1728
    then show "integrable M (s' i)"
hoelzl@56993
  1729
      by (rule integrableI_simple_bochner_integrable)
hoelzl@56993
  1730
hoelzl@56993
  1731
    { fix x assume"x \<in> space M" "s' i x \<noteq> 0"
hoelzl@56993
  1732
      then have "emeasure M {y \<in> space M. s' i y = s' i x} \<le> emeasure M {y \<in> space M. s' i y \<noteq> 0}"
hoelzl@56993
  1733
        by (intro emeasure_mono) auto
hoelzl@56993
  1734
      also have "\<dots> < \<infinity>"
hoelzl@56993
  1735
        using sbi by (auto elim: simple_bochner_integrable.cases)
hoelzl@56993
  1736
      finally have "emeasure M {y \<in> space M. s' i y = s' i x} \<noteq> \<infinity>" by simp }
hoelzl@56993
  1737
    then show "P (s' i)"
hoelzl@56993
  1738
      by (subst s'_eq) (auto intro!: setsum base)
hoelzl@56993
  1739
hoelzl@56993
  1740
    fix x assume "x \<in> space M" with s show "(\<lambda>i. s' i x) ----> f x"
hoelzl@56993
  1741
      by (simp add: s'_eq_s)
hoelzl@56993
  1742
    show "norm (s' i x) \<le> 2 * norm (f x)"
hoelzl@56993
  1743
      using `x \<in> space M` s by (simp add: s'_eq_s)
hoelzl@56993
  1744
  qed fact
hoelzl@56993
  1745
qed
hoelzl@56993
  1746
hoelzl@56993
  1747
lemma integral_nonneg_AE:
hoelzl@56993
  1748
  fixes f :: "'a \<Rightarrow> real"
hoelzl@57166
  1749
  assumes [measurable]: "AE x in M. 0 \<le> f x"
hoelzl@56993
  1750
  shows "0 \<le> integral\<^sup>L M f"
hoelzl@57166
  1751
proof cases
hoelzl@57166
  1752
  assume [measurable]: "integrable M f"
hoelzl@57166
  1753
  then have "0 \<le> ereal (integral\<^sup>L M (\<lambda>x. max 0 (f x)))"
hoelzl@57036
  1754
    by (subst integral_eq_nn_integral) (auto intro: real_of_ereal_pos nn_integral_nonneg assms)
hoelzl@56993
  1755
  also have "integral\<^sup>L M (\<lambda>x. max 0 (f x)) = integral\<^sup>L M f"
hoelzl@57166
  1756
    using assms by (intro integral_cong_AE assms integrable_max) auto
hoelzl@56993
  1757
  finally show ?thesis
hoelzl@56993
  1758
    by simp
hoelzl@57166
  1759
qed (simp add: not_integrable_integral_eq)
hoelzl@56993
  1760
hoelzl@56993
  1761
lemma integral_eq_zero_AE:
hoelzl@57166
  1762
  "(AE x in M. f x = 0) \<Longrightarrow> integral\<^sup>L M f = 0"
hoelzl@57166
  1763
  using integral_cong_AE[of f M "\<lambda>_. 0"]
hoelzl@57166
  1764
  by (cases "integrable M f") (simp_all add: not_integrable_integral_eq)
hoelzl@56993
  1765
hoelzl@56993
  1766
lemma integral_nonneg_eq_0_iff_AE:
hoelzl@56993
  1767
  fixes f :: "_ \<Rightarrow> real"
hoelzl@56993
  1768
  assumes f[measurable]: "integrable M f" and nonneg: "AE x in M. 0 \<le> f x"
hoelzl@56993
  1769
  shows "integral\<^sup>L M f = 0 \<longleftrightarrow> (AE x in M. f x = 0)"
hoelzl@56993
  1770
proof
hoelzl@56993
  1771
  assume "integral\<^sup>L M f = 0"
hoelzl@56996
  1772
  then have "integral\<^sup>N M f = 0"
hoelzl@56996
  1773
    using nn_integral_eq_integral[OF f nonneg] by simp
hoelzl@56993
  1774
  then have "AE x in M. ereal (f x) \<le> 0"
hoelzl@56996
  1775
    by (simp add: nn_integral_0_iff_AE)
hoelzl@56993
  1776
  with nonneg show "AE x in M. f x = 0"
hoelzl@56993
  1777
    by auto
hoelzl@56993
  1778
qed (auto simp add: integral_eq_zero_AE)
hoelzl@56993
  1779
hoelzl@56993
  1780
lemma integral_mono_AE:
hoelzl@56993
  1781
  fixes f :: "'a \<Rightarrow> real"
hoelzl@56993
  1782
  assumes "integrable M f" "integrable M g" "AE x in M. f x \<le> g x"
hoelzl@56993
  1783
  shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
hoelzl@56993
  1784
proof -
hoelzl@56993
  1785
  have "0 \<le> integral\<^sup>L M (\<lambda>x. g x - f x)"
hoelzl@56993
  1786
    using assms by (intro integral_nonneg_AE integrable_diff assms) auto
hoelzl@56993
  1787
  also have "\<dots> = integral\<^sup>L M g - integral\<^sup>L M f"
hoelzl@56993
  1788
    by (intro integral_diff assms)
hoelzl@56993
  1789
  finally show ?thesis by simp
hoelzl@56993
  1790
qed
hoelzl@56993
  1791
hoelzl@56993
  1792
lemma integral_mono:
hoelzl@56993
  1793
  fixes f :: "'a \<Rightarrow> real"
hoelzl@56993
  1794
  shows "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x) \<Longrightarrow> 
hoelzl@56993
  1795
    integral\<^sup>L M f \<le> integral\<^sup>L M g"
hoelzl@56993
  1796
  by (intro integral_mono_AE) auto
hoelzl@56993
  1797
hoelzl@57025
  1798
lemma (in finite_measure) integrable_measure: 
hoelzl@57025
  1799
  assumes I: "disjoint_family_on X I" "countable I"
hoelzl@57025
  1800
  shows "integrable (count_space I) (\<lambda>i. measure M (X i))"
hoelzl@57025
  1801
proof -
hoelzl@57025
  1802
  have "(\<integral>\<^sup>+i. measure M (X i) \<partial>count_space I) = (\<integral>\<^sup>+i. measure M (if X i \<in> sets M then X i else {}) \<partial>count_space I)"
hoelzl@57025
  1803
    by (auto intro!: nn_integral_cong measure_notin_sets)
hoelzl@57025
  1804
  also have "\<dots> = measure M (\<Union>i\<in>I. if X i \<in> sets M then X i else {})"
hoelzl@57025
  1805
    using I unfolding emeasure_eq_measure[symmetric]
hoelzl@57025
  1806
    by (subst emeasure_UN_countable) (auto simp: disjoint_family_on_def)
hoelzl@57025
  1807
  finally show ?thesis
hoelzl@57025
  1808
    by (auto intro!: integrableI_bounded simp: measure_nonneg)
hoelzl@57025
  1809
qed
hoelzl@57025
  1810
hoelzl@57025
  1811
lemma integrableI_real_bounded:
hoelzl@57025
  1812
  assumes f: "f \<in> borel_measurable M" and ae: "AE x in M. 0 \<le> f x" and fin: "integral\<^sup>N M f < \<infinity>"
hoelzl@57025
  1813
  shows "integrable M f"
hoelzl@57025
  1814
proof (rule integrableI_bounded)
hoelzl@57025
  1815
  have "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) = \<integral>\<^sup>+ x. ereal (f x) \<partial>M"
hoelzl@57025
  1816
    using ae by (auto intro: nn_integral_cong_AE)
hoelzl@57025
  1817
  also note fin
hoelzl@57025
  1818
  finally show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" .
hoelzl@57025
  1819
qed fact
hoelzl@57025
  1820
hoelzl@57025
  1821
lemma integral_real_bounded:
hoelzl@57166
  1822
  assumes "AE x in M. 0 \<le> f x" "integral\<^sup>N M f \<le> ereal r"
hoelzl@57025
  1823
  shows "integral\<^sup>L M f \<le> r"
hoelzl@57166
  1824
proof cases
hoelzl@57166
  1825
  assume "integrable M f" from nn_integral_eq_integral[OF this] assms show ?thesis
hoelzl@57025
  1826
    by simp
hoelzl@57166
  1827
next
hoelzl@57166
  1828
  assume "\<not> integrable M f"
hoelzl@57166
  1829
  moreover have "0 \<le> ereal r"
hoelzl@57166
  1830
    using nn_integral_nonneg assms(2) by (rule order_trans)
hoelzl@57166
  1831
  ultimately show ?thesis
hoelzl@57166
  1832
    by (simp add: not_integrable_integral_eq)
hoelzl@57025
  1833
qed
hoelzl@57025
  1834
hoelzl@57137
  1835
subsection {* Restricted measure spaces *}
hoelzl@57137
  1836
hoelzl@57137
  1837
lemma integrable_restrict_space:
hoelzl@57137
  1838
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57137
  1839
  assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
hoelzl@57137
  1840
  shows "integrable (restrict_space M \<Omega>) f \<longleftrightarrow> integrable M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
hoelzl@57137
  1841
  unfolding integrable_iff_bounded
hoelzl@57137
  1842
    borel_measurable_restrict_space_iff[OF \<Omega>]
hoelzl@57137
  1843
    nn_integral_restrict_space[OF \<Omega>]
hoelzl@57137
  1844
  by (simp add: ac_simps ereal_indicator times_ereal.simps(1)[symmetric] del: times_ereal.simps)
hoelzl@57137
  1845
hoelzl@57137
  1846
lemma integral_restrict_space:
hoelzl@57137
  1847
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@57137
  1848
  assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
hoelzl@57137
  1849
  shows "integral\<^sup>L (restrict_space M \<Omega>) f = integral\<^sup>L M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
hoelzl@57137
  1850
proof (rule integral_eq_cases)
hoelzl@57137
  1851
  assume "integrable (restrict_space M \<Omega>) f"
hoelzl@57137
  1852
  then show ?thesis
hoelzl@57137
  1853
  proof induct
hoelzl@57137
  1854
    case (base A c) then show ?case
hoelzl@57137
  1855
      by (simp add: indicator_inter_arith[symmetric] sets_restrict_space_iff
hoelzl@57137
  1856
                    emeasure_restrict_space Int_absorb1 measure_restrict_space)
hoelzl@57137
  1857
  next
hoelzl@57137
  1858
    case (add g f) then show ?case
hoelzl@57137
  1859
      by (simp add: scaleR_add_right integrable_restrict_space)
hoelzl@57137
  1860
  next
hoelzl@57137
  1861
    case (lim f s)
hoelzl@57137
  1862
    show ?case
hoelzl@57137
  1863
    proof (rule LIMSEQ_unique)
hoelzl@57137
  1864
      show "(\<lambda>i. integral\<^sup>L (restrict_space M \<Omega>) (s i)) ----> integral\<^sup>L (restrict_space M \<Omega>) f"
hoelzl@57137
  1865
        using lim by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) simp_all
hoelzl@57137
  1866
      
hoelzl@57137
  1867
      show "(\<lambda>i. integral\<^sup>L (restrict_space M \<Omega>) (s i)) ----> (\<integral> x. indicator \<Omega> x *\<^sub>R f x \<partial>M)"
hoelzl@57137
  1868
        unfolding lim
hoelzl@57137
  1869
        using lim
hoelzl@57137
  1870
        by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (indicator \<Omega> x *\<^sub>R f x)"])
hoelzl@57137
  1871
           (auto simp add: space_restrict_space integrable_restrict_space
hoelzl@57137
  1872
                 simp del: norm_scaleR
hoelzl@57137
  1873
                 split: split_indicator)
hoelzl@57137
  1874
    qed
hoelzl@57137
  1875
  qed
hoelzl@57137
  1876
qed (simp add: integrable_restrict_space)
hoelzl@57137
  1877
hoelzl@56994
  1878
subsection {* Measure spaces with an associated density *}
hoelzl@56993
  1879
hoelzl@56993
  1880
lemma integrable_density:
hoelzl@56993
  1881
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
hoelzl@56993
  1882
  assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@56993
  1883
    and nn: "AE x in M. 0 \<le> g x"
hoelzl@56993
  1884
  shows "integrable (density M g) f \<longleftrightarrow> integrable M (\<lambda>x. g x *\<^sub>R f x)"
hoelzl@56993
  1885
  unfolding integrable_iff_bounded using nn
hoelzl@56996
  1886
  apply (simp add: nn_integral_density )
hoelzl@56996
  1887
  apply (intro arg_cong2[where f="op ="] refl nn_integral_cong_AE)
hoelzl@56993
  1888
  apply auto
hoelzl@56993
  1889
  done
hoelzl@56993
  1890
hoelzl@56993
  1891
lemma integral_density:
hoelzl@56993
  1892
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
hoelzl@56993
  1893
  assumes f: "f \<in> borel_measurable M"
hoelzl@56993
  1894
    and g[measurable]: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
hoelzl@56993
  1895
  shows "integral\<^sup>L (density M g) f = integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
hoelzl@56993
  1896
proof (rule integral_eq_cases)
hoelzl@56993
  1897
  assume "integrable (density M g) f"
hoelzl@56993
  1898
  then show ?thesis
hoelzl@56993
  1899
  proof induct
hoelzl@56993
  1900
    case (base A c)
hoelzl@56993
  1901
    then have [measurable]: "A \<in> sets M" by auto
hoelzl@56993
  1902
  
hoelzl@56993
  1903
    have int: "integrable M (\<lambda>x. g x * indicator A x)"
hoelzl@56993
  1904
      using g base integrable_density[of "indicator A :: 'a \<Rightarrow> real" M g] by simp
hoelzl@56993
  1905
    then have "integral\<^sup>L M (\<lambda>x. g x * indicator A x) = (\<integral>\<^sup>+ x. ereal (g x * indicator A x) \<partial>M)"
hoelzl@56996
  1906
      using g by (subst nn_integral_eq_integral) auto
hoelzl@56993
  1907
    also have "\<dots> = (\<integral>\<^sup>+ x. ereal (g x) * indicator A x \<partial>M)"
hoelzl@56996
  1908
      by (intro nn_integral_cong) (auto split: split_indicator)
hoelzl@56993
  1909
    also have "\<dots> = emeasure (density M g) A"
hoelzl@56993
  1910
      by (rule emeasure_density[symmetric]) auto
hoelzl@56993
  1911
    also have "\<dots> = ereal (measure (density M g) A)"
hoelzl@56993
  1912
      using base by (auto intro: emeasure_eq_ereal_measure)
hoelzl@56993
  1913
    also have "\<dots> = integral\<^sup>L (density M g) (indicator A)"
hoelzl@56993
  1914
      using base by simp
hoelzl@56993
  1915
    finally show ?case
hoelzl@56993
  1916
      using base by (simp add: int)
hoelzl@56993
  1917
  next
hoelzl@56993
  1918
    case (add f h)
hoelzl@56993
  1919
    then have [measurable]: "f \<in> borel_measurable M" "h \<in> borel_measurable M"
hoelzl@56993
  1920
      by (auto dest!: borel_measurable_integrable)
hoelzl@56993
  1921
    from add g show ?case
hoelzl@56993
  1922
      by (simp add: scaleR_add_right integrable_density)
hoelzl@56993
  1923
  next
hoelzl@56993
  1924
    case (lim f s)
hoelzl@56993
  1925
    have [measurable]: "f \<in> borel_measurable M" "\<And>i. s i \<in> borel_measurable M"
hoelzl@56993
  1926
      using lim(1,5)[THEN borel_measurable_integrable] by auto
hoelzl@56993
  1927
  
hoelzl@56993
  1928
    show ?case
hoelzl@56993
  1929
    proof (rule LIMSEQ_unique)
hoelzl@56993
  1930
      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) ----> integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
hoelzl@57137
  1931
      proof (rule integral_dominated_convergence)
hoelzl@56993
  1932
        show "integrable M (\<lambda>x. 2 * norm (g x *\<^sub>R f x))"
hoelzl@56993
  1933
          by (intro integrable_mult_right integrable_norm integrable_density[THEN iffD1] lim g) auto
hoelzl@56993
  1934
        show "AE x in M. (\<lambda>i. g x *\<^sub>R s i x) ----> g x *\<^sub>R f x"
hoelzl@56993
  1935
          using lim(3) by (auto intro!: tendsto_scaleR AE_I2[of M])
hoelzl@56993
  1936
        show "\<And>i. AE x in M. norm (g x *\<^sub>R s i x) \<le> 2 * norm (g x *\<^sub>R f x)"
hoelzl@56993
  1937
          using lim(4) g by (auto intro!: AE_I2[of M] mult_left_mono simp: field_simps)
hoelzl@56993
  1938
      qed auto
hoelzl@56993
  1939
      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) ----> integral\<^sup>L (density M g) f"
hoelzl@56993
  1940
        unfolding lim(2)[symmetric]
hoelzl@57137
  1941
        by (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
hoelzl@57036
  1942
           (insert lim(3-5), auto)
hoelzl@56993
  1943
    qed
hoelzl@56993
  1944
  qed
hoelzl@56993
  1945
qed (simp add: f g integrable_density)
hoelzl@56993
  1946
hoelzl@56993
  1947
lemma
hoelzl@56993
  1948
  fixes g :: "'a \<Rightarrow> real"
hoelzl@56993
  1949
  assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "g \<in> borel_measurable M"
hoelzl@56993
  1950
  shows integral_real_density: "integral\<^sup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
hoelzl@56993
  1951
    and integrable_real_density: "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
hoelzl@56993
  1952
  using assms integral_density[of g M f] integrable_density[of g M f] by auto
hoelzl@56993
  1953
hoelzl@56993
  1954
lemma has_bochner_integral_density:
hoelzl@56993
  1955
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
hoelzl@56993
  1956
  shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. 0 \<le> g x) \<Longrightarrow>
hoelzl@56993
  1957
    has_bochner_integral M (\<lambda>x. g x *\<^sub>R f x) x \<Longrightarrow> has_bochner_integral (density M g) f x"
hoelzl@56993
  1958
  by (simp add: has_bochner_integral_iff integrable_density integral_density)
hoelzl@56993
  1959
hoelzl@56993
  1960
subsection {* Distributions *}
hoelzl@56993
  1961
hoelzl@56993
  1962
lemma integrable_distr_eq:
hoelzl@56993
  1963
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1964
  assumes [measurable]: "g \<in> measurable M N" "f \<in> borel_measurable N"
hoelzl@56993
  1965
  shows "integrable (distr M N g) f \<longleftrightarrow> integrable M (\<lambda>x. f (g x))"
hoelzl@56996
  1966
  unfolding integrable_iff_bounded by (simp_all add: nn_integral_distr)
hoelzl@56993
  1967
hoelzl@56993
  1968
lemma integrable_distr:
hoelzl@56993
  1969
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1970
  shows "T \<in> measurable M M' \<Longrightarrow> integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))"
hoelzl@56993
  1971
  by (subst integrable_distr_eq[symmetric, where g=T])
hoelzl@56993
  1972
     (auto dest: borel_measurable_integrable)
hoelzl@56993
  1973
hoelzl@56993
  1974
lemma integral_distr:
hoelzl@56993
  1975
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  1976
  assumes g[measurable]: "g \<in> measurable M N" and f: "f \<in> borel_measurable N"
hoelzl@56993
  1977
  shows "integral\<^sup>L (distr M N g) f = integral\<^sup>L M (\<lambda>x. f (g x))"
hoelzl@56993
  1978
proof (rule integral_eq_cases)
hoelzl@56993
  1979
  assume "integrable (distr M N g) f"
hoelzl@56993
  1980
  then show ?thesis
hoelzl@56993
  1981
  proof induct
hoelzl@56993
  1982
    case (base A c)
hoelzl@56993
  1983
    then have [measurable]: "A \<in> sets N" by auto
hoelzl@56993
  1984
    from base have int: "integrable (distr M N g) (\<lambda>a. indicator A a *\<^sub>R c)"
hoelzl@56993
  1985
      by (intro integrable_indicator)
hoelzl@56993
  1986
  
hoelzl@56993
  1987
    have "integral\<^sup>L (distr M N g) (\<lambda>a. indicator A a *\<^sub>R c) = measure (distr M N g) A *\<^sub>R c"
hoelzl@57166
  1988
      using base by auto
hoelzl@56993
  1989
    also have "\<dots> = measure M (g -` A \<inter> space M) *\<^sub>R c"
hoelzl@56993
  1990
      by (subst measure_distr) auto
hoelzl@56993
  1991
    also have "\<dots> = integral\<^sup>L M (\<lambda>a. indicator (g -` A \<inter> space M) a *\<^sub>R c)"
hoelzl@57166
  1992
      using base by (auto simp: emeasure_distr)
hoelzl@56993
  1993
    also have "\<dots> = integral\<^sup>L M (\<lambda>a. indicator A (g a) *\<^sub>R c)"
hoelzl@56993
  1994
      using int base by (intro integral_cong_AE) (auto simp: emeasure_distr split: split_indicator)
hoelzl@56993
  1995
    finally show ?case .
hoelzl@56993
  1996
  next
hoelzl@56993
  1997
    case (add f h)
hoelzl@56993
  1998
    then have [measurable]: "f \<in> borel_measurable N" "h \<in> borel_measurable N"
hoelzl@56993
  1999
      by (auto dest!: borel_measurable_integrable)
hoelzl@56993
  2000
    from add g show ?case
hoelzl@56993
  2001
      by (simp add: scaleR_add_right integrable_distr_eq)
hoelzl@56993
  2002
  next
hoelzl@56993
  2003
    case (lim f s)
hoelzl@56993
  2004
    have [measurable]: "f \<in> borel_measurable N" "\<And>i. s i \<in> borel_measurable N"
hoelzl@56993
  2005
      using lim(1,5)[THEN borel_measurable_integrable] by auto
hoelzl@56993
  2006
  
hoelzl@56993
  2007
    show ?case
hoelzl@56993
  2008
    proof (rule LIMSEQ_unique)
hoelzl@56993
  2009
      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) ----> integral\<^sup>L M (\<lambda>x. f (g x))"
hoelzl@57137
  2010
      proof (rule integral_dominated_convergence)
hoelzl@56993
  2011
        show "integrable M (\<lambda>x. 2 * norm (f (g x)))"
hoelzl@57036
  2012
          using lim by (auto simp: integrable_distr_eq) 
hoelzl@56993
  2013
        show "AE x in M. (\<lambda>i. s i (g x)) ----> f (g x)"
hoelzl@56993
  2014
          using lim(3) g[THEN measurable_space] by auto
hoelzl@56993
  2015
        show "\<And>i. AE x in M. norm (s i (g x)) \<le> 2 * norm (f (g x))"
hoelzl@56993
  2016
          using lim(4) g[THEN measurable_space] by auto
hoelzl@56993
  2017
      qed auto
hoelzl@56993
  2018
      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) ----> integral\<^sup>L (distr M N g) f"
hoelzl@56993
  2019
        unfolding lim(2)[symmetric]
hoelzl@57137
  2020
        by (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
hoelzl@57036
  2021
           (insert lim(3-5), auto)
hoelzl@56993
  2022
    qed
hoelzl@56993
  2023
  qed
hoelzl@56993
  2024
qed (simp add: f g integrable_distr_eq)
hoelzl@56993
  2025
hoelzl@56993
  2026
lemma has_bochner_integral_distr:
hoelzl@56993
  2027
  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
hoelzl@56993
  2028
  shows "f \<in> borel_measurable N \<Longrightarrow> g \<in> measurable M N \<Longrightarrow>
hoelzl@56993
  2029
    has_bochner_integral M (\<lambda>x. f (g x)) x \<Longrightarrow> has_bochner_integral (distr M N g) f x"
hoelzl@56993
  2030
  by (simp add: has_bochner_integral_iff integrable_distr_eq integral_distr)
hoelzl@56993
  2031
hoelzl@56994
  2032
subsection {* Lebesgue integration on @{const count_space} *}
hoelzl@56993
  2033
hoelzl@56993
  2034