src/HOL/Probability/Radon_Nikodym.thy
author hoelzl
Wed Dec 08 16:15:14 2010 +0100 (2010-12-08)
changeset 41095 c335d880ff82
parent 41023 9118eb4eb8dc
child 41097 a1abfa4e2b44
permissions -rw-r--r--
cleanup bijectivity btw. product spaces and pairs
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theory Radon_Nikodym
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imports Lebesgue_Integration
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begin
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lemma less_\<omega>_Ex_of_nat: "x < \<omega> \<longleftrightarrow> (\<exists>n. x < of_nat n)"
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proof safe
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  assume "x < \<omega>"
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  then obtain r where "0 \<le> r" "x = Real r" by (cases x) auto
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  moreover obtain n where "r < of_nat n" using ex_less_of_nat by auto
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  ultimately show "\<exists>n. x < of_nat n" by (auto simp: real_eq_of_nat)
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qed auto
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lemma (in sigma_finite_measure) Ex_finite_integrable_function:
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  shows "\<exists>h\<in>borel_measurable M. positive_integral h \<noteq> \<omega> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<omega>)"
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proof -
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  obtain A :: "nat \<Rightarrow> 'a set" where
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    range: "range A \<subseteq> sets M" and
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    space: "(\<Union>i. A i) = space M" and
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    measure: "\<And>i. \<mu> (A i) \<noteq> \<omega>" and
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    disjoint: "disjoint_family A"
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    using disjoint_sigma_finite by auto
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  let "?B i" = "2^Suc i * \<mu> (A i)"
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  have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
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  proof
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    fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
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    proof cases
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      assume "\<mu> (A i) = 0"
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      then show ?thesis by (auto intro!: exI[of _ 1])
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    next
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      assume not_0: "\<mu> (A i) \<noteq> 0"
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      then have "?B i \<noteq> \<omega>" using measure[of i] by auto
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      then have "inverse (?B i) \<noteq> 0" unfolding pextreal_inverse_eq_0 by simp
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      then show ?thesis using measure[of i] not_0
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        by (auto intro!: exI[of _ "inverse (?B i) / 2"]
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                 simp: pextreal_noteq_omega_Ex zero_le_mult_iff zero_less_mult_iff mult_le_0_iff power_le_zero_eq)
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    qed
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  qed
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  from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
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    "\<And>i. n i < inverse (2^Suc i * \<mu> (A i))" by auto
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  let "?h x" = "\<Sum>\<^isub>\<infinity> i. n i * indicator (A i) x"
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  show ?thesis
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  proof (safe intro!: bexI[of _ ?h] del: notI)
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    have "\<And>i. A i \<in> sets M"
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      using range by fastsimp+
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    then have "positive_integral ?h = (\<Sum>\<^isub>\<infinity> i. n i * \<mu> (A i))"
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      by (simp add: positive_integral_psuminf positive_integral_cmult_indicator)
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    also have "\<dots> \<le> (\<Sum>\<^isub>\<infinity> i. Real ((1 / 2)^Suc i))"
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    proof (rule psuminf_le)
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      fix N show "n N * \<mu> (A N) \<le> Real ((1 / 2) ^ Suc N)"
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        using measure[of N] n[of N]
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        by (cases "n N")
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           (auto simp: pextreal_noteq_omega_Ex field_simps zero_le_mult_iff
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                       mult_le_0_iff mult_less_0_iff power_less_zero_eq
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                       power_le_zero_eq inverse_eq_divide less_divide_eq
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                       power_divide split: split_if_asm)
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    qed
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    also have "\<dots> = Real 1"
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      by (rule suminf_imp_psuminf, rule power_half_series, auto)
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    finally show "positive_integral ?h \<noteq> \<omega>" by auto
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  next
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    fix x assume "x \<in> space M"
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    then obtain i where "x \<in> A i" using space[symmetric] by auto
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    from psuminf_cmult_indicator[OF disjoint, OF this]
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    have "?h x = n i" by simp
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    then show "0 < ?h x" and "?h x < \<omega>" using n[of i] by auto
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  next
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    show "?h \<in> borel_measurable M" using range
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      by (auto intro!: borel_measurable_psuminf borel_measurable_pextreal_times)
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  qed
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qed
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subsection "Absolutely continuous"
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definition (in measure_space)
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  "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pextreal))"
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lemma (in sigma_finite_measure) absolutely_continuous_AE:
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  assumes "measure_space M \<nu>" "absolutely_continuous \<nu>" "AE x. P x"
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  shows "measure_space.almost_everywhere M \<nu> P"
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proof -
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  interpret \<nu>: measure_space M \<nu> by fact
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  from `AE x. P x` obtain N where N: "N \<in> null_sets" and "{x\<in>space M. \<not> P x} \<subseteq> N"
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    unfolding almost_everywhere_def by auto
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  show "\<nu>.almost_everywhere P"
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  proof (rule \<nu>.AE_I')
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    show "{x\<in>space M. \<not> P x} \<subseteq> N" by fact
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    from `absolutely_continuous \<nu>` show "N \<in> \<nu>.null_sets"
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      using N unfolding absolutely_continuous_def by auto
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  qed
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qed
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lemma (in finite_measure_space) absolutely_continuousI:
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  assumes "finite_measure_space M \<nu>"
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  assumes v: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
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  shows "absolutely_continuous \<nu>"
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proof (unfold absolutely_continuous_def sets_eq_Pow, safe)
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  fix N assume "\<mu> N = 0" "N \<subseteq> space M"
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  interpret v: finite_measure_space M \<nu> by fact
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  have "\<nu> N = \<nu> (\<Union>x\<in>N. {x})" by simp
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  also have "\<dots> = (\<Sum>x\<in>N. \<nu> {x})"
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  proof (rule v.measure_finitely_additive''[symmetric])
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    show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset)
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    show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto
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    fix x assume "x \<in> N" thus "{x} \<in> sets M" using `N \<subseteq> space M` sets_eq_Pow by auto
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  qed
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  also have "\<dots> = 0"
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  proof (safe intro!: setsum_0')
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    fix x assume "x \<in> N"
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    hence "\<mu> {x} \<le> \<mu> N" using sets_eq_Pow `N \<subseteq> space M` by (auto intro!: measure_mono)
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    hence "\<mu> {x} = 0" using `\<mu> N = 0` by simp
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    thus "\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
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  qed
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  finally show "\<nu> N = 0" .
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qed
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lemma (in measure_space) density_is_absolutely_continuous:
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  assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
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  shows "absolutely_continuous \<nu>"
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  using assms unfolding absolutely_continuous_def
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  by (simp add: positive_integral_null_set)
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subsection "Existence of the Radon-Nikodym derivative"
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lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
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  fixes e :: real assumes "0 < e"
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  assumes "finite_measure M s"
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  shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (s (space M)) \<le>
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                    real (\<mu> A) - real (s A) \<and>
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                    (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < real (\<mu> B) - real (s B))"
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proof -
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  let "?d A" = "real (\<mu> A) - real (s A)"
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  interpret M': finite_measure M s by fact
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  let "?A A" = "if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
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    then {}
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    else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
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  def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}"
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  have A_simps[simp]:
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    "A 0 = {}"
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    "\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all
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  { fix A assume "A \<in> sets M"
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    have "?A A \<in> sets M"
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      by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) }
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  note A'_in_sets = this
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  { fix n have "A n \<in> sets M"
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    proof (induct n)
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      case (Suc n) thus "A (Suc n) \<in> sets M"
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        using A'_in_sets[of "A n"] by (auto split: split_if_asm)
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    qed (simp add: A_def) }
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  note A_in_sets = this
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  hence "range A \<subseteq> sets M" by auto
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  { fix n B
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    assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e"
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    hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less)
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    have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False]
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    proof (rule someI2_ex[OF Ex])
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      fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
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      hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
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      hence "?d (A n \<union> B) = ?d (A n) + ?d B"
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        using `A n \<in> sets M` real_finite_measure_Union M'.real_finite_measure_Union by simp
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      also have "\<dots> \<le> ?d (A n) - e" using dB by simp
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      finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
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    qed }
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  note dA_epsilon = this
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  { fix n have "?d (A (Suc n)) \<le> ?d (A n)"
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    proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
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      case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
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    next
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      case False
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      hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le)
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      thus ?thesis by simp
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    qed }
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  note dA_mono = this
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  show ?thesis
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  proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B")
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    case True then obtain n where B: "\<And>B. \<lbrakk> B \<in> sets M; B \<subseteq> space M - A n\<rbrakk> \<Longrightarrow> -e < ?d B" by blast
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    show ?thesis
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    proof (safe intro!: bexI[of _ "space M - A n"])
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      fix B assume "B \<in> sets M" "B \<subseteq> space M - A n"
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      from B[OF this] show "-e < ?d B" .
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    next
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      show "space M - A n \<in> sets M" by (rule compl_sets) fact
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    next
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      show "?d (space M) \<le> ?d (space M - A n)"
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      proof (induct n)
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        fix n assume "?d (space M) \<le> ?d (space M - A n)"
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        also have "\<dots> \<le> ?d (space M - A (Suc n))"
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          using A_in_sets sets_into_space dA_mono[of n]
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            real_finite_measure_Diff[of "space M"]
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            real_finite_measure_Diff[of "space M"]
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            M'.real_finite_measure_Diff[of "space M"]
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            M'.real_finite_measure_Diff[of "space M"]
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          by (simp del: A_simps)
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        finally show "?d (space M) \<le> ?d (space M - A (Suc n))" .
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      qed simp
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    qed
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  next
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    case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
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      by (auto simp add: not_less)
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    { fix n have "?d (A n) \<le> - real n * e"
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      proof (induct n)
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        case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
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      qed simp } note dA_less = this
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    have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq
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    proof (rule incseq_SucI)
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      fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
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    qed
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    from real_finite_continuity_from_below[of A] `range A \<subseteq> sets M`
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      M'.real_finite_continuity_from_below[of A]
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    have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
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      by (auto intro!: LIMSEQ_diff)
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    obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
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    moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
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    have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
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    ultimately show ?thesis by auto
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  qed
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qed
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lemma (in finite_measure) Radon_Nikodym_aux:
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  assumes "finite_measure M s"
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  shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (s (space M)) \<le>
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                    real (\<mu> A) - real (s A) \<and>
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                    (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> real (\<mu> B) - real (s B))"
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proof -
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  let "?d A" = "real (\<mu> A) - real (s A)"
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  let "?P A B n" = "A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)"
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  interpret M': finite_measure M s by fact
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  let "?r S" = "restricted_space S"
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  { fix S n
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    assume S: "S \<in> sets M"
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    hence **: "\<And>X. X \<in> op \<inter> S ` sets M \<longleftrightarrow> X \<in> sets M \<and> X \<subseteq> S" by auto
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    from M'.restricted_finite_measure[of S] restricted_finite_measure[of S] S
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    have "finite_measure (?r S) \<mu>" "0 < 1 / real (Suc n)"
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      "finite_measure (?r S) s" by auto
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    from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X ..
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    hence "?P X S n"
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    proof (simp add: **, safe)
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      fix C assume C: "C \<in> sets M" "C \<subseteq> X" "X \<subseteq> S" and
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        *: "\<forall>B\<in>sets M. S \<inter> B \<subseteq> X \<longrightarrow> - (1 / real (Suc n)) < ?d (S \<inter> B)"
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      hence "C \<subseteq> S" "C \<subseteq> X" "S \<inter> C = C" by auto
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      with *[THEN bspec, OF `C \<in> sets M`]
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      show "- (1 / real (Suc n)) < ?d C" by auto
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    qed
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    hence "\<exists>A. ?P A S n" by auto }
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  note Ex_P = this
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  def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)"
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  have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def)
hoelzl@38656
   247
  have A_0[simp]: "A 0 = space M" unfolding A_def by simp
hoelzl@38656
   248
  { fix i have "A i \<in> sets M" unfolding A_def
hoelzl@38656
   249
    proof (induct i)
hoelzl@38656
   250
      case (Suc i)
hoelzl@38656
   251
      from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc
hoelzl@38656
   252
        by (rule someI2_ex) simp
hoelzl@38656
   253
    qed simp }
hoelzl@38656
   254
  note A_in_sets = this
hoelzl@38656
   255
  { fix n have "?P (A (Suc n)) (A n) n"
hoelzl@38656
   256
      using Ex_P[OF A_in_sets] unfolding A_Suc
hoelzl@38656
   257
      by (rule someI2_ex) simp }
hoelzl@38656
   258
  note P_A = this
hoelzl@38656
   259
  have "range A \<subseteq> sets M" using A_in_sets by auto
hoelzl@38656
   260
  have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp
hoelzl@38656
   261
  have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc)
hoelzl@38656
   262
  have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C"
hoelzl@38656
   263
      using P_A by auto
hoelzl@38656
   264
  show ?thesis
hoelzl@38656
   265
  proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
hoelzl@38656
   266
    show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
hoelzl@38656
   267
    from `range A \<subseteq> sets M` A_mono
hoelzl@38656
   268
      real_finite_continuity_from_above[of A]
hoelzl@38656
   269
      M'.real_finite_continuity_from_above[of A]
hoelzl@38656
   270
    have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: LIMSEQ_diff)
hoelzl@38656
   271
    thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
hoelzl@38656
   272
      by (rule_tac LIMSEQ_le_const) (auto intro!: exI)
hoelzl@38656
   273
  next
hoelzl@38656
   274
    fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)"
hoelzl@38656
   275
    show "0 \<le> ?d B"
hoelzl@38656
   276
    proof (rule ccontr)
hoelzl@38656
   277
      assume "\<not> 0 \<le> ?d B"
hoelzl@38656
   278
      hence "0 < - ?d B" by auto
hoelzl@38656
   279
      from ex_inverse_of_nat_Suc_less[OF this]
hoelzl@38656
   280
      obtain n where *: "?d B < - 1 / real (Suc n)"
hoelzl@38656
   281
        by (auto simp: real_eq_of_nat inverse_eq_divide field_simps)
hoelzl@38656
   282
      have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc)
hoelzl@38656
   283
      from epsilon[OF B(1) this] *
hoelzl@38656
   284
      show False by auto
hoelzl@38656
   285
    qed
hoelzl@38656
   286
  qed
hoelzl@38656
   287
qed
hoelzl@38656
   288
hoelzl@38656
   289
lemma (in finite_measure) Radon_Nikodym_finite_measure:
hoelzl@38656
   290
  assumes "finite_measure M \<nu>"
hoelzl@38656
   291
  assumes "absolutely_continuous \<nu>"
hoelzl@38656
   292
  shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@38656
   293
proof -
hoelzl@38656
   294
  interpret M': finite_measure M \<nu> using assms(1) .
hoelzl@38656
   295
  def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A}"
hoelzl@38656
   296
  have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
hoelzl@38656
   297
  hence "G \<noteq> {}" by auto
hoelzl@38656
   298
  { fix f g assume f: "f \<in> G" and g: "g \<in> G"
hoelzl@38656
   299
    have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
hoelzl@38656
   300
    proof safe
hoelzl@38656
   301
      show "?max \<in> borel_measurable M" using f g unfolding G_def by auto
hoelzl@38656
   302
      let ?A = "{x \<in> space M. f x \<le> g x}"
hoelzl@38656
   303
      have "?A \<in> sets M" using f g unfolding G_def by auto
hoelzl@38656
   304
      fix A assume "A \<in> sets M"
hoelzl@38656
   305
      hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
hoelzl@38656
   306
      have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
hoelzl@38656
   307
        using sets_into_space[OF `A \<in> sets M`] by auto
hoelzl@38656
   308
      have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
hoelzl@38656
   309
        g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
hoelzl@38656
   310
        by (auto simp: indicator_def max_def)
hoelzl@38656
   311
      hence "positive_integral (\<lambda>x. max (g x) (f x) * indicator A x) =
hoelzl@38656
   312
        positive_integral (\<lambda>x. g x * indicator (?A \<inter> A) x) +
hoelzl@38656
   313
        positive_integral (\<lambda>x. f x * indicator ((space M - ?A) \<inter> A) x)"
hoelzl@38656
   314
        using f g sets unfolding G_def
hoelzl@38656
   315
        by (auto cong: positive_integral_cong intro!: positive_integral_add borel_measurable_indicator)
hoelzl@38656
   316
      also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)"
hoelzl@38656
   317
        using f g sets unfolding G_def by (auto intro!: add_mono)
hoelzl@38656
   318
      also have "\<dots> = \<nu> A"
hoelzl@38656
   319
        using M'.measure_additive[OF sets] union by auto
hoelzl@38656
   320
      finally show "positive_integral (\<lambda>x. max (g x) (f x) * indicator A x) \<le> \<nu> A" .
hoelzl@38656
   321
    qed }
hoelzl@38656
   322
  note max_in_G = this
hoelzl@38656
   323
  { fix f g assume  "f \<up> g" and f: "\<And>i. f i \<in> G"
hoelzl@38656
   324
    have "g \<in> G" unfolding G_def
hoelzl@38656
   325
    proof safe
hoelzl@38656
   326
      from `f \<up> g` have [simp]: "g = (SUP i. f i)" unfolding isoton_def by simp
hoelzl@38656
   327
      have f_borel: "\<And>i. f i \<in> borel_measurable M" using f unfolding G_def by simp
hoelzl@38656
   328
      thus "g \<in> borel_measurable M" by (auto intro!: borel_measurable_SUP)
hoelzl@38656
   329
      fix A assume "A \<in> sets M"
hoelzl@38656
   330
      hence "\<And>i. (\<lambda>x. f i x * indicator A x) \<in> borel_measurable M"
hoelzl@38656
   331
        using f_borel by (auto intro!: borel_measurable_indicator)
hoelzl@38656
   332
      from positive_integral_isoton[OF isoton_indicator[OF `f \<up> g`] this]
hoelzl@38656
   333
      have SUP: "positive_integral (\<lambda>x. g x * indicator A x) =
hoelzl@38656
   334
          (SUP i. positive_integral (\<lambda>x. f i x * indicator A x))"
hoelzl@38656
   335
        unfolding isoton_def by simp
hoelzl@38656
   336
      show "positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A" unfolding SUP
hoelzl@38656
   337
        using f `A \<in> sets M` unfolding G_def by (auto intro!: SUP_leI)
hoelzl@38656
   338
    qed }
hoelzl@38656
   339
  note SUP_in_G = this
hoelzl@38656
   340
  let ?y = "SUP g : G. positive_integral g"
hoelzl@38656
   341
  have "?y \<le> \<nu> (space M)" unfolding G_def
hoelzl@38656
   342
  proof (safe intro!: SUP_leI)
hoelzl@38656
   343
    fix g assume "\<forall>A\<in>sets M. positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A"
hoelzl@38656
   344
    from this[THEN bspec, OF top] show "positive_integral g \<le> \<nu> (space M)"
hoelzl@38656
   345
      by (simp cong: positive_integral_cong)
hoelzl@38656
   346
  qed
hoelzl@38656
   347
  hence "?y \<noteq> \<omega>" using M'.finite_measure_of_space by auto
hoelzl@38656
   348
  from SUPR_countable_SUPR[OF this `G \<noteq> {}`] guess ys .. note ys = this
hoelzl@38656
   349
  hence "\<forall>n. \<exists>g. g\<in>G \<and> positive_integral g = ys n"
hoelzl@38656
   350
  proof safe
hoelzl@38656
   351
    fix n assume "range ys \<subseteq> positive_integral ` G"
hoelzl@38656
   352
    hence "ys n \<in> positive_integral ` G" by auto
hoelzl@38656
   353
    thus "\<exists>g. g\<in>G \<and> positive_integral g = ys n" by auto
hoelzl@38656
   354
  qed
hoelzl@38656
   355
  from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. positive_integral (gs n) = ys n" by auto
hoelzl@38656
   356
  hence y_eq: "?y = (SUP i. positive_integral (gs i))" using ys by auto
hoelzl@38656
   357
  let "?g i x" = "Max ((\<lambda>n. gs n x) ` {..i})"
hoelzl@38656
   358
  def f \<equiv> "SUP i. ?g i"
hoelzl@38656
   359
  have gs_not_empty: "\<And>i. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
hoelzl@38656
   360
  { fix i have "?g i \<in> G"
hoelzl@38656
   361
    proof (induct i)
hoelzl@38656
   362
      case 0 thus ?case by simp fact
hoelzl@38656
   363
    next
hoelzl@38656
   364
      case (Suc i)
hoelzl@38656
   365
      with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case
hoelzl@38656
   366
        by (auto simp add: atMost_Suc intro!: max_in_G)
hoelzl@38656
   367
    qed }
hoelzl@38656
   368
  note g_in_G = this
hoelzl@38656
   369
  have "\<And>x. \<forall>i. ?g i x \<le> ?g (Suc i) x"
hoelzl@38656
   370
    using gs_not_empty by (simp add: atMost_Suc)
hoelzl@38656
   371
  hence isoton_g: "?g \<up> f" by (simp add: isoton_def le_fun_def f_def)
hoelzl@38656
   372
  from SUP_in_G[OF this g_in_G] have "f \<in> G" .
hoelzl@38656
   373
  hence [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
hoelzl@38656
   374
  have "(\<lambda>i. positive_integral (?g i)) \<up> positive_integral f"
hoelzl@38656
   375
    using isoton_g g_in_G by (auto intro!: positive_integral_isoton simp: G_def f_def)
hoelzl@38656
   376
  hence "positive_integral f = (SUP i. positive_integral (?g i))"
hoelzl@38656
   377
    unfolding isoton_def by simp
hoelzl@38656
   378
  also have "\<dots> = ?y"
hoelzl@38656
   379
  proof (rule antisym)
hoelzl@38656
   380
    show "(SUP i. positive_integral (?g i)) \<le> ?y"
hoelzl@38656
   381
      using g_in_G by (auto intro!: exI Sup_mono simp: SUPR_def)
hoelzl@38656
   382
    show "?y \<le> (SUP i. positive_integral (?g i))" unfolding y_eq
hoelzl@38656
   383
      by (auto intro!: SUP_mono positive_integral_mono Max_ge)
hoelzl@38656
   384
  qed
hoelzl@38656
   385
  finally have int_f_eq_y: "positive_integral f = ?y" .
hoelzl@38656
   386
  let "?t A" = "\<nu> A - positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@38656
   387
  have "finite_measure M ?t"
hoelzl@38656
   388
  proof
hoelzl@38656
   389
    show "?t {} = 0" by simp
hoelzl@38656
   390
    show "?t (space M) \<noteq> \<omega>" using M'.finite_measure by simp
hoelzl@38656
   391
    show "countably_additive M ?t" unfolding countably_additive_def
hoelzl@38656
   392
    proof safe
hoelzl@38656
   393
      fix A :: "nat \<Rightarrow> 'a set"  assume A: "range A \<subseteq> sets M" "disjoint_family A"
hoelzl@38656
   394
      have "(\<Sum>\<^isub>\<infinity> n. positive_integral (\<lambda>x. f x * indicator (A n) x))
hoelzl@38656
   395
        = positive_integral (\<lambda>x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x))"
hoelzl@38656
   396
        using `range A \<subseteq> sets M`
hoelzl@38656
   397
        by (rule_tac positive_integral_psuminf[symmetric]) (auto intro!: borel_measurable_indicator)
hoelzl@38656
   398
      also have "\<dots> = positive_integral (\<lambda>x. f x * indicator (\<Union>n. A n) x)"
hoelzl@38656
   399
        apply (rule positive_integral_cong)
hoelzl@38656
   400
        apply (subst psuminf_cmult_right)
hoelzl@38656
   401
        unfolding psuminf_indicator[OF `disjoint_family A`] ..
hoelzl@38656
   402
      finally have "(\<Sum>\<^isub>\<infinity> n. positive_integral (\<lambda>x. f x * indicator (A n) x))
hoelzl@38656
   403
        = positive_integral (\<lambda>x. f x * indicator (\<Union>n. A n) x)" .
hoelzl@38656
   404
      moreover have "(\<Sum>\<^isub>\<infinity>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
hoelzl@38656
   405
        using M'.measure_countably_additive A by (simp add: comp_def)
hoelzl@38656
   406
      moreover have "\<And>i. positive_integral (\<lambda>x. f x * indicator (A i) x) \<le> \<nu> (A i)"
hoelzl@38656
   407
          using A `f \<in> G` unfolding G_def by auto
hoelzl@38656
   408
      moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<omega>" using M'.finite_measure A by (simp add: countable_UN)
hoelzl@38656
   409
      moreover {
hoelzl@38656
   410
        have "positive_integral (\<lambda>x. f x * indicator (\<Union>i. A i) x) \<le> \<nu> (\<Union>i. A i)"
hoelzl@38656
   411
          using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
hoelzl@41023
   412
        also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pextreal_less_\<omega>)
hoelzl@38656
   413
        finally have "positive_integral (\<lambda>x. f x * indicator (\<Union>i. A i) x) \<noteq> \<omega>"
hoelzl@41023
   414
          by (simp add: pextreal_less_\<omega>) }
hoelzl@38656
   415
      ultimately
hoelzl@38656
   416
      show "(\<Sum>\<^isub>\<infinity> n. ?t (A n)) = ?t (\<Union>i. A i)"
hoelzl@38656
   417
        apply (subst psuminf_minus) by simp_all
hoelzl@38656
   418
    qed
hoelzl@38656
   419
  qed
hoelzl@38656
   420
  then interpret M: finite_measure M ?t .
hoelzl@38656
   421
  have ac: "absolutely_continuous ?t" using `absolutely_continuous \<nu>` unfolding absolutely_continuous_def by auto
hoelzl@38656
   422
  have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0"
hoelzl@38656
   423
  proof (rule ccontr)
hoelzl@38656
   424
    assume "\<not> ?thesis"
hoelzl@38656
   425
    then obtain A where A: "A \<in> sets M" and pos: "0 < ?t A"
hoelzl@38656
   426
      by (auto simp: not_le)
hoelzl@38656
   427
    note pos
hoelzl@38656
   428
    also have "?t A \<le> ?t (space M)"
hoelzl@38656
   429
      using M.measure_mono[of A "space M"] A sets_into_space by simp
hoelzl@38656
   430
    finally have pos_t: "0 < ?t (space M)" by simp
hoelzl@38656
   431
    moreover
hoelzl@38656
   432
    hence pos_M: "0 < \<mu> (space M)"
hoelzl@38656
   433
      using ac top unfolding absolutely_continuous_def by auto
hoelzl@38656
   434
    moreover
hoelzl@38656
   435
    have "positive_integral (\<lambda>x. f x * indicator (space M) x) \<le> \<nu> (space M)"
hoelzl@38656
   436
      using `f \<in> G` unfolding G_def by auto
hoelzl@38656
   437
    hence "positive_integral (\<lambda>x. f x * indicator (space M) x) \<noteq> \<omega>"
hoelzl@38656
   438
      using M'.finite_measure_of_space by auto
hoelzl@38656
   439
    moreover
hoelzl@38656
   440
    def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
hoelzl@38656
   441
    ultimately have b: "b \<noteq> 0" "b \<noteq> \<omega>"
hoelzl@38656
   442
      using M'.finite_measure_of_space
hoelzl@41023
   443
      by (auto simp: pextreal_inverse_eq_0 finite_measure_of_space)
hoelzl@38656
   444
    have "finite_measure M (\<lambda>A. b * \<mu> A)" (is "finite_measure M ?b")
hoelzl@38656
   445
    proof
hoelzl@38656
   446
      show "?b {} = 0" by simp
hoelzl@38656
   447
      show "?b (space M) \<noteq> \<omega>" using finite_measure_of_space b by auto
hoelzl@38656
   448
      show "countably_additive M ?b"
hoelzl@38656
   449
        unfolding countably_additive_def psuminf_cmult_right
hoelzl@38656
   450
        using measure_countably_additive by auto
hoelzl@38656
   451
    qed
hoelzl@38656
   452
    from M.Radon_Nikodym_aux[OF this]
hoelzl@38656
   453
    obtain A0 where "A0 \<in> sets M" and
hoelzl@38656
   454
      space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and
hoelzl@38656
   455
      *: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)" by auto
hoelzl@38656
   456
    { fix B assume "B \<in> sets M" "B \<subseteq> A0"
hoelzl@38656
   457
      with *[OF this] have "b * \<mu> B \<le> ?t B"
hoelzl@38656
   458
        using M'.finite_measure b finite_measure
hoelzl@38656
   459
        by (cases "b * \<mu> B", cases "?t B") (auto simp: field_simps) }
hoelzl@38656
   460
    note bM_le_t = this
hoelzl@38656
   461
    let "?f0 x" = "f x + b * indicator A0 x"
hoelzl@38656
   462
    { fix A assume A: "A \<in> sets M"
hoelzl@38656
   463
      hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
hoelzl@38656
   464
      have "positive_integral (\<lambda>x. ?f0 x  * indicator A x) =
hoelzl@38656
   465
        positive_integral (\<lambda>x. f x * indicator A x + b * indicator (A \<inter> A0) x)"
hoelzl@38656
   466
        by (auto intro!: positive_integral_cong simp: field_simps indicator_inter_arith)
hoelzl@38656
   467
      hence "positive_integral (\<lambda>x. ?f0 x * indicator A x) =
hoelzl@38656
   468
          positive_integral (\<lambda>x. f x * indicator A x) + b * \<mu> (A \<inter> A0)"
hoelzl@38656
   469
        using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A
hoelzl@38656
   470
        by (simp add: borel_measurable_indicator positive_integral_add positive_integral_cmult_indicator) }
hoelzl@38656
   471
    note f0_eq = this
hoelzl@38656
   472
    { fix A assume A: "A \<in> sets M"
hoelzl@38656
   473
      hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
hoelzl@38656
   474
      have f_le_v: "positive_integral (\<lambda>x. f x * indicator A x) \<le> \<nu> A"
hoelzl@38656
   475
        using `f \<in> G` A unfolding G_def by auto
hoelzl@38656
   476
      note f0_eq[OF A]
hoelzl@38656
   477
      also have "positive_integral (\<lambda>x. f x * indicator A x) + b * \<mu> (A \<inter> A0) \<le>
hoelzl@38656
   478
          positive_integral (\<lambda>x. f x * indicator A x) + ?t (A \<inter> A0)"
hoelzl@38656
   479
        using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
hoelzl@38656
   480
        by (auto intro!: add_left_mono)
hoelzl@38656
   481
      also have "\<dots> \<le> positive_integral (\<lambda>x. f x * indicator A x) + ?t A"
hoelzl@38656
   482
        using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
hoelzl@38656
   483
        by (auto intro!: add_left_mono)
hoelzl@38656
   484
      also have "\<dots> \<le> \<nu> A"
hoelzl@38656
   485
        using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`]
hoelzl@38656
   486
        by (cases "positive_integral (\<lambda>x. f x * indicator A x)", cases "\<nu> A", auto)
hoelzl@38656
   487
      finally have "positive_integral (\<lambda>x. ?f0 x * indicator A x) \<le> \<nu> A" . }
hoelzl@38656
   488
    hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def
hoelzl@41023
   489
      by (auto intro!: borel_measurable_indicator borel_measurable_pextreal_add borel_measurable_pextreal_times)
hoelzl@38656
   490
    have real: "?t (space M) \<noteq> \<omega>" "?t A0 \<noteq> \<omega>"
hoelzl@38656
   491
      "b * \<mu> (space M) \<noteq> \<omega>" "b * \<mu> A0 \<noteq> \<omega>"
hoelzl@38656
   492
      using `A0 \<in> sets M` b
hoelzl@38656
   493
        finite_measure[of A0] M.finite_measure[of A0]
hoelzl@38656
   494
        finite_measure_of_space M.finite_measure_of_space
hoelzl@38656
   495
      by auto
hoelzl@38656
   496
    have int_f_finite: "positive_integral f \<noteq> \<omega>"
hoelzl@41023
   497
      using M'.finite_measure_of_space pos_t unfolding pextreal_zero_less_diff_iff
hoelzl@38656
   498
      by (auto cong: positive_integral_cong)
hoelzl@38656
   499
    have "?t (space M) > b * \<mu> (space M)" unfolding b_def
hoelzl@38656
   500
      apply (simp add: field_simps)
hoelzl@38656
   501
      apply (subst mult_assoc[symmetric])
hoelzl@41023
   502
      apply (subst pextreal_mult_inverse)
hoelzl@38656
   503
      using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M
hoelzl@41023
   504
      using pextreal_mult_strict_right_mono[of "Real (1/2)" 1 "?t (space M)"]
hoelzl@38656
   505
      by simp_all
hoelzl@38656
   506
    hence  "0 < ?t (space M) - b * \<mu> (space M)"
hoelzl@41023
   507
      by (simp add: pextreal_zero_less_diff_iff)
hoelzl@38656
   508
    also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
hoelzl@41023
   509
      using space_less_A0 pos_M pos_t b real[unfolded pextreal_noteq_omega_Ex] by auto
hoelzl@41023
   510
    finally have "b * \<mu> A0 < ?t A0" unfolding pextreal_zero_less_diff_iff .
hoelzl@38656
   511
    hence "0 < ?t A0" by auto
hoelzl@38656
   512
    hence "0 < \<mu> A0" using ac unfolding absolutely_continuous_def
hoelzl@38656
   513
      using `A0 \<in> sets M` by auto
hoelzl@38656
   514
    hence "0 < b * \<mu> A0" using b by auto
hoelzl@38656
   515
    from int_f_finite this
hoelzl@38656
   516
    have "?y + 0 < positive_integral f + b * \<mu> A0" unfolding int_f_eq_y
hoelzl@41023
   517
      by (rule pextreal_less_add)
hoelzl@38656
   518
    also have "\<dots> = positive_integral ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
hoelzl@38656
   519
      by (simp cong: positive_integral_cong)
hoelzl@38656
   520
    finally have "?y < positive_integral ?f0" by simp
hoelzl@38656
   521
    moreover from `?f0 \<in> G` have "positive_integral ?f0 \<le> ?y" by (auto intro!: le_SUPI)
hoelzl@38656
   522
    ultimately show False by auto
hoelzl@38656
   523
  qed
hoelzl@38656
   524
  show ?thesis
hoelzl@38656
   525
  proof (safe intro!: bexI[of _ f])
hoelzl@38656
   526
    fix A assume "A\<in>sets M"
hoelzl@38656
   527
    show "\<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@38656
   528
    proof (rule antisym)
hoelzl@38656
   529
      show "positive_integral (\<lambda>x. f x * indicator A x) \<le> \<nu> A"
hoelzl@38656
   530
        using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
hoelzl@38656
   531
      show "\<nu> A \<le> positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@38656
   532
        using upper_bound[THEN bspec, OF `A \<in> sets M`]
hoelzl@41023
   533
         by (simp add: pextreal_zero_le_diff)
hoelzl@38656
   534
    qed
hoelzl@38656
   535
  qed simp
hoelzl@38656
   536
qed
hoelzl@38656
   537
hoelzl@40859
   538
lemma (in finite_measure) split_space_into_finite_sets_and_rest:
hoelzl@38656
   539
  assumes "measure_space M \<nu>"
hoelzl@40859
   540
  assumes ac: "absolutely_continuous \<nu>"
hoelzl@40859
   541
  shows "\<exists>\<Omega>0\<in>sets M. \<exists>\<Omega>::nat\<Rightarrow>'a set. disjoint_family \<Omega> \<and> range \<Omega> \<subseteq> sets M \<and> \<Omega>0 = space M - (\<Union>i. \<Omega> i) \<and>
hoelzl@40859
   542
    (\<forall>A\<in>sets M. A \<subseteq> \<Omega>0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<omega>)) \<and>
hoelzl@40859
   543
    (\<forall>i. \<nu> (\<Omega> i) \<noteq> \<omega>)"
hoelzl@38656
   544
proof -
hoelzl@38656
   545
  interpret v: measure_space M \<nu> by fact
hoelzl@38656
   546
  let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<omega>}"
hoelzl@38656
   547
  let ?a = "SUP Q:?Q. \<mu> Q"
hoelzl@38656
   548
  have "{} \<in> ?Q" using v.empty_measure by auto
hoelzl@38656
   549
  then have Q_not_empty: "?Q \<noteq> {}" by blast
hoelzl@38656
   550
  have "?a \<le> \<mu> (space M)" using sets_into_space
hoelzl@38656
   551
    by (auto intro!: SUP_leI measure_mono top)
hoelzl@38656
   552
  then have "?a \<noteq> \<omega>" using finite_measure_of_space
hoelzl@38656
   553
    by auto
hoelzl@38656
   554
  from SUPR_countable_SUPR[OF this Q_not_empty]
hoelzl@38656
   555
  obtain Q'' where "range Q'' \<subseteq> \<mu> ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
hoelzl@38656
   556
    by auto
hoelzl@38656
   557
  then have "\<forall>i. \<exists>Q'. Q'' i = \<mu> Q' \<and> Q' \<in> ?Q" by auto
hoelzl@38656
   558
  from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = \<mu> (Q' i)" "\<And>i. Q' i \<in> ?Q"
hoelzl@38656
   559
    by auto
hoelzl@38656
   560
  then have a_Lim: "?a = (SUP i::nat. \<mu> (Q' i))" using a by simp
hoelzl@38656
   561
  let "?O n" = "\<Union>i\<le>n. Q' i"
hoelzl@38656
   562
  have Union: "(SUP i. \<mu> (?O i)) = \<mu> (\<Union>i. ?O i)"
hoelzl@38656
   563
  proof (rule continuity_from_below[of ?O])
hoelzl@38656
   564
    show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN)
hoelzl@38656
   565
    show "\<And>i. ?O i \<subseteq> ?O (Suc i)" by fastsimp
hoelzl@38656
   566
  qed
hoelzl@38656
   567
  have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
hoelzl@38656
   568
  have O_sets: "\<And>i. ?O i \<in> sets M"
hoelzl@38656
   569
     using Q' by (auto intro!: finite_UN Un)
hoelzl@38656
   570
  then have O_in_G: "\<And>i. ?O i \<in> ?Q"
hoelzl@38656
   571
  proof (safe del: notI)
hoelzl@38656
   572
    fix i have "Q' ` {..i} \<subseteq> sets M"
hoelzl@38656
   573
      using Q' by (auto intro: finite_UN)
hoelzl@38656
   574
    with v.measure_finitely_subadditive[of "{.. i}" Q']
hoelzl@38656
   575
    have "\<nu> (?O i) \<le> (\<Sum>i\<le>i. \<nu> (Q' i))" by auto
hoelzl@41023
   576
    also have "\<dots> < \<omega>" unfolding setsum_\<omega> pextreal_less_\<omega> using Q' by auto
hoelzl@41023
   577
    finally show "\<nu> (?O i) \<noteq> \<omega>" unfolding pextreal_less_\<omega> by auto
hoelzl@38656
   578
  qed auto
hoelzl@38656
   579
  have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastsimp
hoelzl@38656
   580
  have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric]
hoelzl@38656
   581
  proof (rule antisym)
hoelzl@38656
   582
    show "?a \<le> (SUP i. \<mu> (?O i))" unfolding a_Lim
hoelzl@38656
   583
      using Q' by (auto intro!: SUP_mono measure_mono finite_UN)
hoelzl@38656
   584
    show "(SUP i. \<mu> (?O i)) \<le> ?a" unfolding SUPR_def
hoelzl@38656
   585
    proof (safe intro!: Sup_mono, unfold bex_simps)
hoelzl@38656
   586
      fix i
hoelzl@38656
   587
      have *: "(\<Union>Q' ` {..i}) = ?O i" by auto
hoelzl@38656
   588
      then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<omega>) \<and>
hoelzl@38656
   589
        \<mu> (\<Union>Q' ` {..i}) \<le> \<mu> x"
hoelzl@38656
   590
        using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
hoelzl@38656
   591
    qed
hoelzl@38656
   592
  qed
hoelzl@38656
   593
  let "?O_0" = "(\<Union>i. ?O i)"
hoelzl@38656
   594
  have "?O_0 \<in> sets M" using Q' by auto
hoelzl@40859
   595
  def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n"
hoelzl@38656
   596
  { fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) }
hoelzl@38656
   597
  note Q_sets = this
hoelzl@40859
   598
  show ?thesis
hoelzl@40859
   599
  proof (intro bexI exI conjI ballI impI allI)
hoelzl@40859
   600
    show "disjoint_family Q"
hoelzl@40859
   601
      by (fastsimp simp: disjoint_family_on_def Q_def
hoelzl@40859
   602
        split: nat.split_asm)
hoelzl@40859
   603
    show "range Q \<subseteq> sets M"
hoelzl@40859
   604
      using Q_sets by auto
hoelzl@40859
   605
    { fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
hoelzl@40859
   606
      show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>"
hoelzl@40859
   607
      proof (rule disjCI, simp)
hoelzl@40859
   608
        assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<omega>"
hoelzl@40859
   609
        show "\<mu> A = 0 \<and> \<nu> A = 0"
hoelzl@40859
   610
        proof cases
hoelzl@40859
   611
          assume "\<mu> A = 0" moreover with ac A have "\<nu> A = 0"
hoelzl@40859
   612
            unfolding absolutely_continuous_def by auto
hoelzl@40859
   613
          ultimately show ?thesis by simp
hoelzl@40859
   614
        next
hoelzl@40859
   615
          assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<omega>" by auto
hoelzl@40859
   616
          with A have "\<mu> ?O_0 + \<mu> A = \<mu> (?O_0 \<union> A)"
hoelzl@40859
   617
            using Q' by (auto intro!: measure_additive countable_UN)
hoelzl@40859
   618
          also have "\<dots> = (SUP i. \<mu> (?O i \<union> A))"
hoelzl@40859
   619
          proof (rule continuity_from_below[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
hoelzl@40859
   620
            show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
hoelzl@40859
   621
              using `\<nu> A \<noteq> \<omega>` O_sets A by auto
hoelzl@40859
   622
          qed fastsimp
hoelzl@40859
   623
          also have "\<dots> \<le> ?a"
hoelzl@40859
   624
          proof (safe intro!: SUPR_bound)
hoelzl@40859
   625
            fix i have "?O i \<union> A \<in> ?Q"
hoelzl@40859
   626
            proof (safe del: notI)
hoelzl@40859
   627
              show "?O i \<union> A \<in> sets M" using O_sets A by auto
hoelzl@40859
   628
              from O_in_G[of i]
hoelzl@40859
   629
              moreover have "\<nu> (?O i \<union> A) \<le> \<nu> (?O i) + \<nu> A"
hoelzl@40859
   630
                using v.measure_subadditive[of "?O i" A] A O_sets by auto
hoelzl@40859
   631
              ultimately show "\<nu> (?O i \<union> A) \<noteq> \<omega>"
hoelzl@40859
   632
                using `\<nu> A \<noteq> \<omega>` by auto
hoelzl@40859
   633
            qed
hoelzl@40859
   634
            then show "\<mu> (?O i \<union> A) \<le> ?a" by (rule le_SUPI)
hoelzl@40859
   635
          qed
hoelzl@40859
   636
          finally have "\<mu> A = 0" unfolding a_eq using finite_measure[OF `?O_0 \<in> sets M`]
hoelzl@41023
   637
            by (cases "\<mu> A") (auto simp: pextreal_noteq_omega_Ex)
hoelzl@40859
   638
          with `\<mu> A \<noteq> 0` show ?thesis by auto
hoelzl@40859
   639
        qed
hoelzl@40859
   640
      qed }
hoelzl@40859
   641
    { fix i show "\<nu> (Q i) \<noteq> \<omega>"
hoelzl@40859
   642
      proof (cases i)
hoelzl@40859
   643
        case 0 then show ?thesis
hoelzl@40859
   644
          unfolding Q_def using Q'[of 0] by simp
hoelzl@40859
   645
      next
hoelzl@40859
   646
        case (Suc n)
hoelzl@40859
   647
        then show ?thesis unfolding Q_def
hoelzl@40859
   648
          using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q` O_mono
hoelzl@40859
   649
          using v.measure_Diff[of "?O n" "?O (Suc n)"] by auto
hoelzl@40859
   650
      qed }
hoelzl@40859
   651
    show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
hoelzl@40859
   652
    { fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
hoelzl@40859
   653
      proof (induct j)
hoelzl@40859
   654
        case 0 then show ?case by (simp add: Q_def)
hoelzl@40859
   655
      next
hoelzl@40859
   656
        case (Suc j)
hoelzl@40859
   657
        have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastsimp
hoelzl@40859
   658
        have "{..j} \<union> {..Suc j} = {..Suc j}" by auto
hoelzl@40859
   659
        then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)"
hoelzl@40859
   660
          by (simp add: UN_Un[symmetric] Q_def del: UN_Un)
hoelzl@40859
   661
        then show ?case using Suc by (auto simp add: eq atMost_Suc)
hoelzl@40859
   662
      qed }
hoelzl@40859
   663
    then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp
hoelzl@40859
   664
    then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastsimp
hoelzl@40859
   665
  qed
hoelzl@40859
   666
qed
hoelzl@40859
   667
hoelzl@40859
   668
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
hoelzl@40859
   669
  assumes "measure_space M \<nu>"
hoelzl@40859
   670
  assumes "absolutely_continuous \<nu>"
hoelzl@40859
   671
  shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@40859
   672
proof -
hoelzl@40859
   673
  interpret v: measure_space M \<nu> by fact
hoelzl@40859
   674
  from split_space_into_finite_sets_and_rest[OF assms]
hoelzl@40859
   675
  obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
hoelzl@40859
   676
    where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
hoelzl@40859
   677
    and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
hoelzl@40859
   678
    and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>"
hoelzl@40859
   679
    and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<omega>" by force
hoelzl@40859
   680
  from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
hoelzl@38656
   681
  have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
hoelzl@38656
   682
    \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f x * indicator (Q i \<inter> A) x))"
hoelzl@38656
   683
  proof
hoelzl@38656
   684
    fix i
hoelzl@41023
   685
    have indicator_eq: "\<And>f x A. (f x :: pextreal) * indicator (Q i \<inter> A) x * indicator (Q i) x
hoelzl@38656
   686
      = (f x * indicator (Q i) x) * indicator A x"
hoelzl@38656
   687
      unfolding indicator_def by auto
hoelzl@39092
   688
    have fm: "finite_measure (restricted_space (Q i)) \<mu>"
hoelzl@38656
   689
      (is "finite_measure ?R \<mu>") by (rule restricted_finite_measure[OF Q_sets[of i]])
hoelzl@38656
   690
    then interpret R: finite_measure ?R .
hoelzl@38656
   691
    have fmv: "finite_measure ?R \<nu>"
hoelzl@38656
   692
      unfolding finite_measure_def finite_measure_axioms_def
hoelzl@38656
   693
    proof
hoelzl@38656
   694
      show "measure_space ?R \<nu>"
hoelzl@38656
   695
        using v.restricted_measure_space Q_sets[of i] by auto
hoelzl@38656
   696
      show "\<nu>  (space ?R) \<noteq> \<omega>"
hoelzl@40859
   697
        using Q_fin by simp
hoelzl@38656
   698
    qed
hoelzl@38656
   699
    have "R.absolutely_continuous \<nu>"
hoelzl@38656
   700
      using `absolutely_continuous \<nu>` `Q i \<in> sets M`
hoelzl@38656
   701
      by (auto simp: R.absolutely_continuous_def absolutely_continuous_def)
hoelzl@38656
   702
    from finite_measure.Radon_Nikodym_finite_measure[OF fm fmv this]
hoelzl@38656
   703
    obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M"
hoelzl@38656
   704
      and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. (f x * indicator (Q i) x) * indicator A x)"
hoelzl@38656
   705
      unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
hoelzl@38656
   706
        positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
hoelzl@38656
   707
    then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
hoelzl@38656
   708
      \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f x * indicator (Q i \<inter> A) x))"
hoelzl@38656
   709
      by (fastsimp intro!: exI[of _ "\<lambda>x. f x * indicator (Q i) x"] positive_integral_cong
hoelzl@38656
   710
          simp: indicator_def)
hoelzl@38656
   711
  qed
hoelzl@38656
   712
  from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M"
hoelzl@38656
   713
    and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
hoelzl@38656
   714
      \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f i x * indicator (Q i \<inter> A) x)"
hoelzl@38656
   715
    by auto
hoelzl@38656
   716
  let "?f x" =
hoelzl@40859
   717
    "(\<Sum>\<^isub>\<infinity> i. f i x * indicator (Q i) x) + \<omega> * indicator Q0 x"
hoelzl@38656
   718
  show ?thesis
hoelzl@38656
   719
  proof (safe intro!: bexI[of _ ?f])
hoelzl@38656
   720
    show "?f \<in> borel_measurable M"
hoelzl@41023
   721
      by (safe intro!: borel_measurable_psuminf borel_measurable_pextreal_times
hoelzl@41023
   722
        borel_measurable_pextreal_add borel_measurable_indicator
hoelzl@40859
   723
        borel_measurable_const borel Q_sets Q0 Diff countable_UN)
hoelzl@38656
   724
    fix A assume "A \<in> sets M"
hoelzl@40859
   725
    have *:
hoelzl@38656
   726
      "\<And>x i. indicator A x * (f i x * indicator (Q i) x) =
hoelzl@38656
   727
        f i x * indicator (Q i \<inter> A) x"
hoelzl@41023
   728
      "\<And>x i. (indicator A x * indicator Q0 x :: pextreal) =
hoelzl@40859
   729
        indicator (Q0 \<inter> A) x" by (auto simp: indicator_def)
hoelzl@38656
   730
    have "positive_integral (\<lambda>x. ?f x * indicator A x) =
hoelzl@40859
   731
      (\<Sum>\<^isub>\<infinity> i. \<nu> (Q i \<inter> A)) + \<omega> * \<mu> (Q0 \<inter> A)"
hoelzl@38656
   732
      unfolding f[OF `A \<in> sets M`]
hoelzl@41023
   733
      apply (simp del: pextreal_times(2) add: field_simps *)
hoelzl@38656
   734
      apply (subst positive_integral_add)
hoelzl@40859
   735
      apply (fastsimp intro: Q0 `A \<in> sets M`)
hoelzl@40859
   736
      apply (fastsimp intro: Q_sets `A \<in> sets M` borel_measurable_psuminf borel)
hoelzl@40859
   737
      apply (subst positive_integral_cmult_indicator)
hoelzl@40859
   738
      apply (fastsimp intro: Q0 `A \<in> sets M`)
hoelzl@38656
   739
      unfolding psuminf_cmult_right[symmetric]
hoelzl@38656
   740
      apply (subst positive_integral_psuminf)
hoelzl@40859
   741
      apply (fastsimp intro: `A \<in> sets M` Q_sets borel)
hoelzl@40859
   742
      apply (simp add: *)
hoelzl@40859
   743
      done
hoelzl@38656
   744
    moreover have "(\<Sum>\<^isub>\<infinity>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
hoelzl@40859
   745
      using Q Q_sets `A \<in> sets M`
hoelzl@40859
   746
      by (intro v.measure_countably_additive[of "\<lambda>i. Q i \<inter> A", unfolded comp_def, simplified])
hoelzl@40859
   747
         (auto simp: disjoint_family_on_def)
hoelzl@40859
   748
    moreover have "\<omega> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)"
hoelzl@40859
   749
    proof -
hoelzl@40859
   750
      have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
hoelzl@40859
   751
      from in_Q0[OF this] show ?thesis by auto
hoelzl@38656
   752
    qed
hoelzl@40859
   753
    moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
hoelzl@40859
   754
      using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN)
hoelzl@40859
   755
    moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
hoelzl@40859
   756
      using `A \<in> sets M` sets_into_space Q0 by auto
hoelzl@38656
   757
    ultimately show "\<nu> A = positive_integral (\<lambda>x. ?f x * indicator A x)"
hoelzl@40859
   758
      using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
hoelzl@40859
   759
      by simp
hoelzl@38656
   760
  qed
hoelzl@38656
   761
qed
hoelzl@38656
   762
hoelzl@38656
   763
lemma (in sigma_finite_measure) Radon_Nikodym:
hoelzl@38656
   764
  assumes "measure_space M \<nu>"
hoelzl@38656
   765
  assumes "absolutely_continuous \<nu>"
hoelzl@38656
   766
  shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@38656
   767
proof -
hoelzl@38656
   768
  from Ex_finite_integrable_function
hoelzl@38656
   769
  obtain h where finite: "positive_integral h \<noteq> \<omega>" and
hoelzl@38656
   770
    borel: "h \<in> borel_measurable M" and
hoelzl@38656
   771
    pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
hoelzl@38656
   772
    "\<And>x. x \<in> space M \<Longrightarrow> h x < \<omega>" by auto
hoelzl@38656
   773
  let "?T A" = "positive_integral (\<lambda>x. h x * indicator A x)"
hoelzl@38656
   774
  from measure_space_density[OF borel] finite
hoelzl@38656
   775
  interpret T: finite_measure M ?T
hoelzl@38656
   776
    unfolding finite_measure_def finite_measure_axioms_def
hoelzl@38656
   777
    by (simp cong: positive_integral_cong)
hoelzl@41023
   778
  have "\<And>N. N \<in> sets M \<Longrightarrow> {x \<in> space M. h x \<noteq> 0 \<and> indicator N x \<noteq> (0::pextreal)} = N"
hoelzl@38656
   779
    using sets_into_space pos by (force simp: indicator_def)
hoelzl@38656
   780
  then have "T.absolutely_continuous \<nu>" using assms(2) borel
hoelzl@38656
   781
    unfolding T.absolutely_continuous_def absolutely_continuous_def
hoelzl@38656
   782
    by (fastsimp simp: borel_measurable_indicator positive_integral_0_iff)
hoelzl@38656
   783
  from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this]
hoelzl@38656
   784
  obtain f where f_borel: "f \<in> borel_measurable M" and
hoelzl@38656
   785
    fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = T.positive_integral (\<lambda>x. f x * indicator A x)" by auto
hoelzl@38656
   786
  show ?thesis
hoelzl@38656
   787
  proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"])
hoelzl@38656
   788
    show "(\<lambda>x. h x * f x) \<in> borel_measurable M"
hoelzl@41023
   789
      using borel f_borel by (auto intro: borel_measurable_pextreal_times)
hoelzl@38656
   790
    fix A assume "A \<in> sets M"
hoelzl@38656
   791
    then have "(\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
hoelzl@41023
   792
      using f_borel by (auto intro: borel_measurable_pextreal_times borel_measurable_indicator)
hoelzl@38656
   793
    from positive_integral_translated_density[OF borel this]
hoelzl@38656
   794
    show "\<nu> A = positive_integral (\<lambda>x. h x * f x * indicator A x)"
hoelzl@38656
   795
      unfolding fT[OF `A \<in> sets M`] by (simp add: field_simps)
hoelzl@38656
   796
  qed
hoelzl@38656
   797
qed
hoelzl@38656
   798
hoelzl@40859
   799
section "Uniqueness of densities"
hoelzl@40859
   800
hoelzl@40859
   801
lemma (in measure_space) finite_density_unique:
hoelzl@40859
   802
  assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@40859
   803
  and fin: "positive_integral f < \<omega>"
hoelzl@40859
   804
  shows "(\<forall>A\<in>sets M. positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. g x * indicator A x))
hoelzl@40859
   805
    \<longleftrightarrow> (AE x. f x = g x)"
hoelzl@40859
   806
    (is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
hoelzl@40859
   807
proof (intro iffI ballI)
hoelzl@40859
   808
  fix A assume eq: "AE x. f x = g x"
hoelzl@40859
   809
  show "?P f A = ?P g A"
hoelzl@40859
   810
    by (rule positive_integral_cong_AE[OF AE_mp[OF eq]]) simp
hoelzl@40859
   811
next
hoelzl@40859
   812
  assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
hoelzl@40859
   813
  from this[THEN bspec, OF top] fin
hoelzl@40859
   814
  have g_fin: "positive_integral g < \<omega>" by (simp cong: positive_integral_cong)
hoelzl@40859
   815
  { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@40859
   816
      and g_fin: "positive_integral g < \<omega>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
hoelzl@40859
   817
    let ?N = "{x\<in>space M. g x < f x}"
hoelzl@40859
   818
    have N: "?N \<in> sets M" using borel by simp
hoelzl@40859
   819
    have "?P (\<lambda>x. (f x - g x)) ?N = positive_integral (\<lambda>x. f x * indicator ?N x - g x * indicator ?N x)"
hoelzl@40859
   820
      by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@40859
   821
    also have "\<dots> = ?P f ?N - ?P g ?N"
hoelzl@40859
   822
    proof (rule positive_integral_diff)
hoelzl@40859
   823
      show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
hoelzl@40859
   824
        using borel N by auto
hoelzl@40859
   825
      have "?P g ?N \<le> positive_integral g"
hoelzl@40859
   826
        by (auto intro!: positive_integral_mono simp: indicator_def)
hoelzl@40859
   827
      then show "?P g ?N \<noteq> \<omega>" using g_fin by auto
hoelzl@40859
   828
      fix x assume "x \<in> space M"
hoelzl@40859
   829
      show "g x * indicator ?N x \<le> f x * indicator ?N x"
hoelzl@40859
   830
        by (auto simp: indicator_def)
hoelzl@40859
   831
    qed
hoelzl@40859
   832
    also have "\<dots> = 0"
hoelzl@40859
   833
      using eq[THEN bspec, OF N] by simp
hoelzl@40859
   834
    finally have "\<mu> {x\<in>space M. (f x - g x) * indicator ?N x \<noteq> 0} = 0"
hoelzl@40859
   835
      using borel N by (subst (asm) positive_integral_0_iff) auto
hoelzl@40859
   836
    moreover have "{x\<in>space M. (f x - g x) * indicator ?N x \<noteq> 0} = ?N"
hoelzl@41023
   837
      by (auto simp: pextreal_zero_le_diff)
hoelzl@40859
   838
    ultimately have "?N \<in> null_sets" using N by simp }
hoelzl@40859
   839
  from this[OF borel g_fin eq] this[OF borel(2,1) fin]
hoelzl@40859
   840
  have "{x\<in>space M. g x < f x} \<union> {x\<in>space M. f x < g x} \<in> null_sets"
hoelzl@40859
   841
    using eq by (intro null_sets_Un) auto
hoelzl@40859
   842
  also have "{x\<in>space M. g x < f x} \<union> {x\<in>space M. f x < g x} = {x\<in>space M. f x \<noteq> g x}"
hoelzl@40859
   843
    by auto
hoelzl@40859
   844
  finally show "AE x. f x = g x"
hoelzl@40859
   845
    unfolding almost_everywhere_def by auto
hoelzl@40859
   846
qed
hoelzl@40859
   847
hoelzl@40859
   848
lemma (in finite_measure) density_unique_finite_measure:
hoelzl@40859
   849
  assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
hoelzl@40859
   850
  assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. f' x * indicator A x)"
hoelzl@40859
   851
    (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
hoelzl@40859
   852
  shows "AE x. f x = f' x"
hoelzl@40859
   853
proof -
hoelzl@40859
   854
  let "?\<nu> A" = "?P f A" and "?\<nu>' A" = "?P f' A"
hoelzl@40859
   855
  let "?f A x" = "f x * indicator A x" and "?f' A x" = "f' x * indicator A x"
hoelzl@40859
   856
  interpret M: measure_space M ?\<nu>
hoelzl@40859
   857
    using borel(1) by (rule measure_space_density)
hoelzl@40859
   858
  have ac: "absolutely_continuous ?\<nu>"
hoelzl@40859
   859
    using f by (rule density_is_absolutely_continuous)
hoelzl@40859
   860
  from split_space_into_finite_sets_and_rest[OF `measure_space M ?\<nu>` ac]
hoelzl@40859
   861
  obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
hoelzl@40859
   862
    where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
hoelzl@40859
   863
    and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
hoelzl@40859
   864
    and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> ?\<nu> A = 0 \<or> 0 < \<mu> A \<and> ?\<nu> A = \<omega>"
hoelzl@40859
   865
    and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<omega>" by force
hoelzl@40859
   866
  from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
hoelzl@40859
   867
  let ?N = "{x\<in>space M. f x \<noteq> f' x}"
hoelzl@40859
   868
  have "?N \<in> sets M" using borel by auto
hoelzl@41023
   869
  have *: "\<And>i x A. \<And>y::pextreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
hoelzl@40859
   870
    unfolding indicator_def by auto
hoelzl@40859
   871
  have 1: "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x"
hoelzl@40859
   872
    using borel Q_fin Q
hoelzl@40859
   873
    by (intro finite_density_unique[THEN iffD1] allI)
hoelzl@41023
   874
       (auto intro!: borel_measurable_pextreal_times f Int simp: *)
hoelzl@40859
   875
  have 2: "AE x. ?f Q0 x = ?f' Q0 x"
hoelzl@40859
   876
  proof (rule AE_I')
hoelzl@41023
   877
    { fix f :: "'a \<Rightarrow> pextreal" assume borel: "f \<in> borel_measurable M"
hoelzl@40859
   878
        and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@40859
   879
      let "?A i" = "Q0 \<inter> {x \<in> space M. f x < of_nat i}"
hoelzl@40859
   880
      have "(\<Union>i. ?A i) \<in> null_sets"
hoelzl@40859
   881
      proof (rule null_sets_UN)
hoelzl@40859
   882
        fix i have "?A i \<in> sets M"
hoelzl@40859
   883
          using borel Q0(1) by auto
hoelzl@40859
   884
        have "?\<nu> (?A i) \<le> positive_integral (\<lambda>x. of_nat i * indicator (?A i) x)"
hoelzl@40859
   885
          unfolding eq[OF `?A i \<in> sets M`]
hoelzl@40859
   886
          by (auto intro!: positive_integral_mono simp: indicator_def)
hoelzl@40859
   887
        also have "\<dots> = of_nat i * \<mu> (?A i)"
hoelzl@40859
   888
          using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
hoelzl@40859
   889
        also have "\<dots> < \<omega>"
hoelzl@40859
   890
          using `?A i \<in> sets M`[THEN finite_measure] by auto
hoelzl@40859
   891
        finally have "?\<nu> (?A i) \<noteq> \<omega>" by simp
hoelzl@40859
   892
        then show "?A i \<in> null_sets" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
hoelzl@40859
   893
      qed
hoelzl@40859
   894
      also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x < \<omega>}"
hoelzl@40859
   895
        by (auto simp: less_\<omega>_Ex_of_nat)
hoelzl@41023
   896
      finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" by (simp add: pextreal_less_\<omega>) }
hoelzl@40859
   897
    from this[OF borel(1) refl] this[OF borel(2) f]
hoelzl@40859
   898
    have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>} \<in> null_sets" by simp_all
hoelzl@40859
   899
    then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>}) \<in> null_sets" by (rule null_sets_Un)
hoelzl@40859
   900
    show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
hoelzl@40859
   901
      (Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>})" by (auto simp: indicator_def)
hoelzl@40859
   902
  qed
hoelzl@40859
   903
  have **: "\<And>x. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
hoelzl@40859
   904
    ?f (space M) x = ?f' (space M) x"
hoelzl@40859
   905
    by (auto simp: indicator_def Q0)
hoelzl@40859
   906
  have 3: "AE x. ?f (space M) x = ?f' (space M) x"
hoelzl@40859
   907
    by (rule AE_mp[OF 1[unfolded all_AE_countable] AE_mp[OF 2]]) (simp add: **)
hoelzl@40859
   908
  then show "AE x. f x = f' x"
hoelzl@40859
   909
    by (rule AE_mp) (auto intro!: AE_cong simp: indicator_def)
hoelzl@40859
   910
qed
hoelzl@40859
   911
hoelzl@40859
   912
lemma (in sigma_finite_measure) density_unique:
hoelzl@40859
   913
  assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
hoelzl@40859
   914
  assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. f' x * indicator A x)"
hoelzl@40859
   915
    (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
hoelzl@40859
   916
  shows "AE x. f x = f' x"
hoelzl@40859
   917
proof -
hoelzl@40859
   918
  obtain h where h_borel: "h \<in> borel_measurable M"
hoelzl@40859
   919
    and fin: "positive_integral h \<noteq> \<omega>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>"
hoelzl@40859
   920
    using Ex_finite_integrable_function by auto
hoelzl@40859
   921
  interpret h: measure_space M "\<lambda>A. positive_integral (\<lambda>x. h x * indicator A x)"
hoelzl@40859
   922
    using h_borel by (rule measure_space_density)
hoelzl@40859
   923
  interpret h: finite_measure M "\<lambda>A. positive_integral (\<lambda>x. h x * indicator A x)"
hoelzl@40859
   924
    by default (simp cong: positive_integral_cong add: fin)
hoelzl@40859
   925
  interpret f: measure_space M "\<lambda>A. positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@40859
   926
    using borel(1) by (rule measure_space_density)
hoelzl@40859
   927
  interpret f': measure_space M "\<lambda>A. positive_integral (\<lambda>x. f' x * indicator A x)"
hoelzl@40859
   928
    using borel(2) by (rule measure_space_density)
hoelzl@40859
   929
  { fix A assume "A \<in> sets M"
hoelzl@41023
   930
    then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pextreal)} = A"
hoelzl@40859
   931
      using pos sets_into_space by (force simp: indicator_def)
hoelzl@40859
   932
    then have "positive_integral (\<lambda>xa. h xa * indicator A xa) = 0 \<longleftrightarrow> A \<in> null_sets"
hoelzl@40859
   933
      using h_borel `A \<in> sets M` by (simp add: positive_integral_0_iff) }
hoelzl@40859
   934
  note h_null_sets = this
hoelzl@40859
   935
  { fix A assume "A \<in> sets M"
hoelzl@40859
   936
    have "positive_integral (\<lambda>x. h x * (f x * indicator A x)) =
hoelzl@40859
   937
      f.positive_integral (\<lambda>x. h x * indicator A x)"
hoelzl@40859
   938
      using `A \<in> sets M` h_borel borel
hoelzl@40859
   939
      by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
hoelzl@40859
   940
    also have "\<dots> = f'.positive_integral (\<lambda>x. h x * indicator A x)"
hoelzl@40859
   941
      by (rule f'.positive_integral_cong_measure) (rule f)
hoelzl@40859
   942
    also have "\<dots> = positive_integral (\<lambda>x. h x * (f' x * indicator A x))"
hoelzl@40859
   943
      using `A \<in> sets M` h_borel borel
hoelzl@40859
   944
      by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
hoelzl@40859
   945
    finally have "positive_integral (\<lambda>x. h x * (f x * indicator A x)) = positive_integral (\<lambda>x. h x * (f' x * indicator A x))" . }
hoelzl@40859
   946
  then have "h.almost_everywhere (\<lambda>x. f x = f' x)"
hoelzl@40859
   947
    using h_borel borel
hoelzl@40859
   948
    by (intro h.density_unique_finite_measure[OF borel])
hoelzl@40859
   949
       (simp add: positive_integral_translated_density)
hoelzl@40859
   950
  then show "AE x. f x = f' x"
hoelzl@40859
   951
    unfolding h.almost_everywhere_def almost_everywhere_def
hoelzl@40859
   952
    by (auto simp add: h_null_sets)
hoelzl@40859
   953
qed
hoelzl@40859
   954
hoelzl@40859
   955
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
hoelzl@40859
   956
  assumes \<nu>: "measure_space M \<nu>" and f: "f \<in> borel_measurable M"
hoelzl@40859
   957
    and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@40859
   958
  shows "sigma_finite_measure M \<nu> \<longleftrightarrow> (AE x. f x \<noteq> \<omega>)"
hoelzl@40859
   959
proof
hoelzl@40859
   960
  assume "sigma_finite_measure M \<nu>"
hoelzl@40859
   961
  then interpret \<nu>: sigma_finite_measure M \<nu> .
hoelzl@40859
   962
  from \<nu>.Ex_finite_integrable_function obtain h where
hoelzl@40859
   963
    h: "h \<in> borel_measurable M" "\<nu>.positive_integral h \<noteq> \<omega>"
hoelzl@40859
   964
    and fin: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>" by auto
hoelzl@40859
   965
  have "AE x. f x * h x \<noteq> \<omega>"
hoelzl@40859
   966
  proof (rule AE_I')
hoelzl@40859
   967
    have "\<nu>.positive_integral h = positive_integral (\<lambda>x. f x * h x)"
hoelzl@40859
   968
      by (simp add: \<nu>.positive_integral_cong_measure[symmetric, OF eq[symmetric]])
hoelzl@40859
   969
         (intro positive_integral_translated_density f h)
hoelzl@40859
   970
    then have "positive_integral (\<lambda>x. f x * h x) \<noteq> \<omega>"
hoelzl@40859
   971
      using h(2) by simp
hoelzl@40859
   972
    then show "(\<lambda>x. f x * h x) -` {\<omega>} \<inter> space M \<in> null_sets"
hoelzl@40859
   973
      using f h(1) by (auto intro!: positive_integral_omega borel_measurable_vimage)
hoelzl@40859
   974
  qed auto
hoelzl@40859
   975
  then show "AE x. f x \<noteq> \<omega>"
hoelzl@40859
   976
  proof (rule AE_mp, intro AE_cong)
hoelzl@40859
   977
    fix x assume "x \<in> space M" from this[THEN fin]
hoelzl@40859
   978
    show "f x * h x \<noteq> \<omega> \<longrightarrow> f x \<noteq> \<omega>" by auto
hoelzl@40859
   979
  qed
hoelzl@40859
   980
next
hoelzl@40859
   981
  assume AE: "AE x. f x \<noteq> \<omega>"
hoelzl@40859
   982
  from sigma_finite guess Q .. note Q = this
hoelzl@40859
   983
  interpret \<nu>: measure_space M \<nu> by fact
hoelzl@40859
   984
  def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<omega>} | Suc n \<Rightarrow> {.. of_nat (Suc n)}) \<inter> space M"
hoelzl@40859
   985
  { fix i j have "A i \<inter> Q j \<in> sets M"
hoelzl@40859
   986
    unfolding A_def using f Q
hoelzl@40859
   987
    apply (rule_tac Int)
hoelzl@40859
   988
    by (cases i) (auto intro: measurable_sets[OF f]) }
hoelzl@40859
   989
  note A_in_sets = this
hoelzl@40859
   990
  let "?A n" = "case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
hoelzl@40859
   991
  show "sigma_finite_measure M \<nu>"
hoelzl@40859
   992
  proof (default, intro exI conjI subsetI allI)
hoelzl@40859
   993
    fix x assume "x \<in> range ?A"
hoelzl@40859
   994
    then obtain n where n: "x = ?A n" by auto
hoelzl@40859
   995
    then show "x \<in> sets M" using A_in_sets by (cases "prod_decode n") auto
hoelzl@40859
   996
  next
hoelzl@40859
   997
    have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)"
hoelzl@40859
   998
    proof safe
hoelzl@40859
   999
      fix x i j assume "x \<in> A i" "x \<in> Q j"
hoelzl@40859
  1000
      then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)"
hoelzl@40859
  1001
        by (intro UN_I[of "prod_encode (i,j)"]) auto
hoelzl@40859
  1002
    qed auto
hoelzl@40859
  1003
    also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
hoelzl@40859
  1004
    also have "(\<Union>i. A i) = space M"
hoelzl@40859
  1005
    proof safe
hoelzl@40859
  1006
      fix x assume x: "x \<in> space M"
hoelzl@40859
  1007
      show "x \<in> (\<Union>i. A i)"
hoelzl@40859
  1008
      proof (cases "f x")
hoelzl@40859
  1009
        case infinite then show ?thesis using x unfolding A_def by (auto intro: exI[of _ 0])
hoelzl@40859
  1010
      next
hoelzl@40859
  1011
        case (preal r)
hoelzl@40859
  1012
        with less_\<omega>_Ex_of_nat[of "f x"] obtain n where "f x < of_nat n" by auto
hoelzl@40859
  1013
        then show ?thesis using x preal unfolding A_def by (auto intro!: exI[of _ "Suc n"])
hoelzl@40859
  1014
      qed
hoelzl@40859
  1015
    qed (auto simp: A_def)
hoelzl@40859
  1016
    finally show "(\<Union>i. ?A i) = space M" by simp
hoelzl@40859
  1017
  next
hoelzl@40859
  1018
    fix n obtain i j where
hoelzl@40859
  1019
      [simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
hoelzl@40859
  1020
    have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) \<noteq> \<omega>"
hoelzl@40859
  1021
    proof (cases i)
hoelzl@40859
  1022
      case 0
hoelzl@40859
  1023
      have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
hoelzl@40859
  1024
        using AE by (rule AE_mp) (auto intro!: AE_cong simp: A_def `i = 0`)
hoelzl@40859
  1025
      then have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) = 0"
hoelzl@40859
  1026
        using A_in_sets f
hoelzl@40859
  1027
        apply (subst positive_integral_0_iff)
hoelzl@40859
  1028
        apply fast
hoelzl@40859
  1029
        apply (subst (asm) AE_iff_null_set)
hoelzl@41023
  1030
        apply (intro borel_measurable_pextreal_neq_const)
hoelzl@40859
  1031
        apply fast
hoelzl@40859
  1032
        by simp
hoelzl@40859
  1033
      then show ?thesis by simp
hoelzl@40859
  1034
    next
hoelzl@40859
  1035
      case (Suc n)
hoelzl@40859
  1036
      then have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) \<le>
hoelzl@40859
  1037
        positive_integral (\<lambda>x. of_nat (Suc n) * indicator (Q j) x)"
hoelzl@40859
  1038
        by (auto intro!: positive_integral_mono simp: indicator_def A_def)
hoelzl@40859
  1039
      also have "\<dots> = of_nat (Suc n) * \<mu> (Q j)"
hoelzl@40859
  1040
        using Q by (auto intro!: positive_integral_cmult_indicator)
hoelzl@40859
  1041
      also have "\<dots> < \<omega>"
hoelzl@40859
  1042
        using Q by auto
hoelzl@40859
  1043
      finally show ?thesis by simp
hoelzl@40859
  1044
    qed
hoelzl@40859
  1045
    then show "\<nu> (?A n) \<noteq> \<omega>"
hoelzl@40859
  1046
      using A_in_sets Q eq by auto
hoelzl@40859
  1047
  qed
hoelzl@40859
  1048
qed
hoelzl@40859
  1049
hoelzl@40871
  1050
section "Radon-Nikodym derivative"
hoelzl@38656
  1051
hoelzl@38656
  1052
definition (in sigma_finite_measure)
hoelzl@38656
  1053
  "RN_deriv \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and>
hoelzl@38656
  1054
    (\<forall>A \<in> sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x))"
hoelzl@38656
  1055
hoelzl@40859
  1056
lemma (in sigma_finite_measure) RN_deriv_cong:
hoelzl@40859
  1057
  assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> \<mu>' A = \<mu> A" "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
hoelzl@40859
  1058
  shows "sigma_finite_measure.RN_deriv M \<mu>' \<nu>' x = RN_deriv \<nu> x"
hoelzl@40859
  1059
proof -
hoelzl@40859
  1060
  interpret \<mu>': sigma_finite_measure M \<mu>'
hoelzl@40859
  1061
    using cong(1) by (rule sigma_finite_measure_cong)
hoelzl@40859
  1062
  show ?thesis
hoelzl@40859
  1063
    unfolding RN_deriv_def \<mu>'.RN_deriv_def
hoelzl@40859
  1064
    by (simp add: cong positive_integral_cong_measure[OF cong(1)])
hoelzl@40859
  1065
qed
hoelzl@40859
  1066
hoelzl@38656
  1067
lemma (in sigma_finite_measure) RN_deriv:
hoelzl@38656
  1068
  assumes "measure_space M \<nu>"
hoelzl@38656
  1069
  assumes "absolutely_continuous \<nu>"
hoelzl@38656
  1070
  shows "RN_deriv \<nu> \<in> borel_measurable M" (is ?borel)
hoelzl@38656
  1071
  and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)"
hoelzl@38656
  1072
    (is "\<And>A. _ \<Longrightarrow> ?int A")
hoelzl@38656
  1073
proof -
hoelzl@38656
  1074
  note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
hoelzl@38656
  1075
  thus ?borel unfolding RN_deriv_def by (rule someI2_ex) auto
hoelzl@38656
  1076
  fix A assume "A \<in> sets M"
hoelzl@38656
  1077
  from Ex show "?int A" unfolding RN_deriv_def
hoelzl@38656
  1078
    by (rule someI2_ex) (simp add: `A \<in> sets M`)
hoelzl@38656
  1079
qed
hoelzl@38656
  1080
hoelzl@40859
  1081
lemma (in sigma_finite_measure) RN_deriv_positive_integral:
hoelzl@40859
  1082
  assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
hoelzl@40859
  1083
    and f: "f \<in> borel_measurable M"
hoelzl@40859
  1084
  shows "measure_space.positive_integral M \<nu> f = positive_integral (\<lambda>x. RN_deriv \<nu> x * f x)"
hoelzl@40859
  1085
proof -
hoelzl@40859
  1086
  interpret \<nu>: measure_space M \<nu> by fact
hoelzl@40859
  1087
  have "\<nu>.positive_integral f =
hoelzl@40859
  1088
    measure_space.positive_integral M (\<lambda>A. positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)) f"
hoelzl@40859
  1089
    by (intro \<nu>.positive_integral_cong_measure[symmetric] RN_deriv(2)[OF \<nu>, symmetric])
hoelzl@40859
  1090
  also have "\<dots> = positive_integral (\<lambda>x. RN_deriv \<nu> x * f x)"
hoelzl@40859
  1091
    by (intro positive_integral_translated_density RN_deriv[OF \<nu>] f)
hoelzl@40859
  1092
  finally show ?thesis .
hoelzl@40859
  1093
qed
hoelzl@40859
  1094
hoelzl@40859
  1095
lemma (in sigma_finite_measure) RN_deriv_unique:
hoelzl@40859
  1096
  assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
hoelzl@40859
  1097
  and f: "f \<in> borel_measurable M"
hoelzl@40859
  1098
  and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@40859
  1099
  shows "AE x. f x = RN_deriv \<nu> x"
hoelzl@40859
  1100
proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
hoelzl@40859
  1101
  fix A assume A: "A \<in> sets M"
hoelzl@40859
  1102
  show "positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)"
hoelzl@40859
  1103
    unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
hoelzl@40859
  1104
qed
hoelzl@40859
  1105
hoelzl@40859
  1106
lemma (in sigma_finite_measure) RN_deriv_vimage:
hoelzl@40859
  1107
  fixes f :: "'b \<Rightarrow> 'a"
hoelzl@41095
  1108
  assumes f: "bij_inv S (space M) f g"
hoelzl@40859
  1109
  assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
hoelzl@40859
  1110
  shows "AE x.
hoelzl@41095
  1111
    sigma_finite_measure.RN_deriv (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>A. \<nu> (f ` A)) (g x) = RN_deriv \<nu> x"
hoelzl@40859
  1112
proof (rule RN_deriv_unique[OF \<nu>])
hoelzl@40859
  1113
  interpret sf: sigma_finite_measure "vimage_algebra S f" "\<lambda>A. \<mu> (f ` A)"
hoelzl@41095
  1114
    using f by (rule sigma_finite_measure_isomorphic[OF bij_inv_bij_betw(1)])
hoelzl@40859
  1115
  interpret \<nu>: measure_space M \<nu> using \<nu>(1) .
hoelzl@40859
  1116
  have \<nu>': "measure_space (vimage_algebra S f) (\<lambda>A. \<nu> (f ` A))"
hoelzl@41095
  1117
    using f by (rule \<nu>.measure_space_isomorphic[OF bij_inv_bij_betw(1)])
hoelzl@40859
  1118
  { fix A assume "A \<in> sets M" then have "f ` (f -` A \<inter> S) = A"
hoelzl@41095
  1119
      using sets_into_space f[THEN bij_inv_bij_betw(1), unfolded bij_betw_def]
hoelzl@40859
  1120
      by (intro image_vimage_inter_eq[where T="space M"]) auto }
hoelzl@40859
  1121
  note A_f = this
hoelzl@40859
  1122
  then have ac: "sf.absolutely_continuous (\<lambda>A. \<nu> (f ` A))"
hoelzl@40859
  1123
    using \<nu>(2) by (auto simp: sf.absolutely_continuous_def absolutely_continuous_def)
hoelzl@41095
  1124
  show "(\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x)) \<in> borel_measurable M"
hoelzl@40859
  1125
    using sf.RN_deriv(1)[OF \<nu>' ac]
hoelzl@40859
  1126
    unfolding measurable_vimage_iff_inv[OF f] comp_def .
hoelzl@40859
  1127
  fix A assume "A \<in> sets M"
hoelzl@41095
  1128
  then have *: "\<And>x. x \<in> space M \<Longrightarrow> indicator (f -` A \<inter> S) (g x) = (indicator A x :: pextreal)"
hoelzl@41095
  1129
    using f by (auto simp: bij_inv_def indicator_def)
hoelzl@40859
  1130
  have "\<nu> (f ` (f -` A \<inter> S)) = sf.positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) x * indicator (f -` A \<inter> S) x)"
hoelzl@40859
  1131
    using `A \<in> sets M` by (force intro!: sf.RN_deriv(2)[OF \<nu>' ac])
hoelzl@41095
  1132
  also have "\<dots> = positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)"
hoelzl@40859
  1133
    unfolding positive_integral_vimage_inv[OF f]
hoelzl@40859
  1134
    by (simp add: * cong: positive_integral_cong)
hoelzl@41095
  1135
  finally show "\<nu> A = positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)"
hoelzl@40859
  1136
    unfolding A_f[OF `A \<in> sets M`] .
hoelzl@40859
  1137
qed
hoelzl@40859
  1138
hoelzl@40859
  1139
lemma (in sigma_finite_measure) RN_deriv_finite:
hoelzl@40859
  1140
  assumes sfm: "sigma_finite_measure M \<nu>" and ac: "absolutely_continuous \<nu>"
hoelzl@40859
  1141
  shows "AE x. RN_deriv \<nu> x \<noteq> \<omega>"
hoelzl@40859
  1142
proof -
hoelzl@40859
  1143
  interpret \<nu>: sigma_finite_measure M \<nu> by fact
hoelzl@40859
  1144
  have \<nu>: "measure_space M \<nu>" by default
hoelzl@40859
  1145
  from sfm show ?thesis
hoelzl@40859
  1146
    using sigma_finite_iff_density_finite[OF \<nu> RN_deriv[OF \<nu> ac]] by simp
hoelzl@40859
  1147
qed
hoelzl@40859
  1148
hoelzl@40859
  1149
lemma (in sigma_finite_measure)
hoelzl@40859
  1150
  assumes \<nu>: "sigma_finite_measure M \<nu>" "absolutely_continuous \<nu>"
hoelzl@40859
  1151
    and f: "f \<in> borel_measurable M"
hoelzl@40859
  1152
  shows RN_deriv_integral: "measure_space.integral M \<nu> f = integral (\<lambda>x. real (RN_deriv \<nu> x) * f x)" (is ?integral)
hoelzl@40859
  1153
    and RN_deriv_integrable: "measure_space.integrable M \<nu> f \<longleftrightarrow> integrable (\<lambda>x. real (RN_deriv \<nu> x) * f x)" (is ?integrable)
hoelzl@40859
  1154
proof -
hoelzl@40859
  1155
  interpret \<nu>: sigma_finite_measure M \<nu> by fact
hoelzl@40859
  1156
  have ms: "measure_space M \<nu>" by default
hoelzl@41023
  1157
  have minus_cong: "\<And>A B A' B'::pextreal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp
hoelzl@40859
  1158
  have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
hoelzl@40859
  1159
  { fix f :: "'a \<Rightarrow> real" assume "f \<in> borel_measurable M"
hoelzl@40859
  1160
    { fix x assume *: "RN_deriv \<nu> x \<noteq> \<omega>"
hoelzl@40859
  1161
      have "Real (real (RN_deriv \<nu> x)) * Real (f x) = Real (real (RN_deriv \<nu> x) * f x)"
hoelzl@40859
  1162
        by (simp add: mult_le_0_iff)
hoelzl@40859
  1163
      then have "RN_deriv \<nu> x * Real (f x) = Real (real (RN_deriv \<nu> x) * f x)"
hoelzl@40859
  1164
        using * by (simp add: Real_real) }
hoelzl@40859
  1165
    note * = this
hoelzl@40859
  1166
    have "positive_integral (\<lambda>x. RN_deriv \<nu> x * Real (f x)) = positive_integral (\<lambda>x. Real (real (RN_deriv \<nu> x) * f x))"
hoelzl@40859
  1167
      apply (rule positive_integral_cong_AE)
hoelzl@40859
  1168
      apply (rule AE_mp[OF RN_deriv_finite[OF \<nu>]])
hoelzl@40859
  1169
      by (auto intro!: AE_cong simp: *) }
hoelzl@40859
  1170
  with this[OF f] this[OF f'] f f'
hoelzl@40859
  1171
  show ?integral ?integrable
hoelzl@40859
  1172
    unfolding \<nu>.integral_def integral_def \<nu>.integrable_def integrable_def
hoelzl@40859
  1173
    by (auto intro!: RN_deriv(1)[OF ms \<nu>(2)] minus_cong simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
hoelzl@40859
  1174
qed
hoelzl@40859
  1175
hoelzl@38656
  1176
lemma (in sigma_finite_measure) RN_deriv_singleton:
hoelzl@38656
  1177
  assumes "measure_space M \<nu>"
hoelzl@38656
  1178
  and ac: "absolutely_continuous \<nu>"
hoelzl@38656
  1179
  and "{x} \<in> sets M"
hoelzl@38656
  1180
  shows "\<nu> {x} = RN_deriv \<nu> x * \<mu> {x}"
hoelzl@38656
  1181
proof -
hoelzl@38656
  1182
  note deriv = RN_deriv[OF assms(1, 2)]
hoelzl@38656
  1183
  from deriv(2)[OF `{x} \<in> sets M`]
hoelzl@38656
  1184
  have "\<nu> {x} = positive_integral (\<lambda>w. RN_deriv \<nu> x * indicator {x} w)"
hoelzl@38656
  1185
    by (auto simp: indicator_def intro!: positive_integral_cong)
hoelzl@38656
  1186
  thus ?thesis using positive_integral_cmult_indicator[OF `{x} \<in> sets M`]
hoelzl@38656
  1187
    by auto
hoelzl@38656
  1188
qed
hoelzl@38656
  1189
hoelzl@38656
  1190
theorem (in finite_measure_space) RN_deriv_finite_measure:
hoelzl@38656
  1191
  assumes "measure_space M \<nu>"
hoelzl@38656
  1192
  and ac: "absolutely_continuous \<nu>"
hoelzl@38656
  1193
  and "x \<in> space M"
hoelzl@38656
  1194
  shows "\<nu> {x} = RN_deriv \<nu> x * \<mu> {x}"
hoelzl@38656
  1195
proof -
hoelzl@38656
  1196
  have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
hoelzl@38656
  1197
  from RN_deriv_singleton[OF assms(1,2) this] show ?thesis .
hoelzl@38656
  1198
qed
hoelzl@38656
  1199
hoelzl@38656
  1200
end