src/HOL/Analysis/Radon_Nikodym.thy
author wenzelm
Mon Mar 25 17:21:26 2019 +0100 (2 months ago)
changeset 69981 3dced198b9ec
parent 69745 aec42cee2521
child 70136 f03a01a18c6e
permissions -rw-r--r--
more strict AFP properties;
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(*  Title:      HOL/Analysis/Radon_Nikodym.thy
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    Author:     Johannes Hölzl, TU München
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*)
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section \<open>Radon-Nikod{\'y}m Derivative\<close>
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theory Radon_Nikodym
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imports Bochner_Integration
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begin
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definition%important diff_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"
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where
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  "diff_measure M N = measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)"
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lemma
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  shows space_diff_measure[simp]: "space (diff_measure M N) = space M"
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    and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M"
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  by (auto simp: diff_measure_def)
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lemma emeasure_diff_measure:
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  assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N"
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  assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M"
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  shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?\<mu> A")
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  unfolding diff_measure_def
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proof (rule emeasure_measure_of_sigma)
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  show "sigma_algebra (space M) (sets M)" ..
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  show "positive (sets M) ?\<mu>"
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    using pos by (simp add: positive_def)
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  show "countably_additive (sets M) ?\<mu>"
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  proof (rule countably_additiveI)
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    fix A :: "nat \<Rightarrow> _"  assume A: "range A \<subseteq> sets M" and "disjoint_family A"
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    then have suminf:
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      "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
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      "(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)"
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      by (simp_all add: suminf_emeasure sets_eq)
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    with A have "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) =
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      (\<Sum>i. emeasure M (A i)) - (\<Sum>i. emeasure N (A i))"
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      using fin pos[of "A _"]
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      by (intro ennreal_suminf_minus)
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         (auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure)
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    then show "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) =
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      emeasure M (\<Union>i. A i) - emeasure N (\<Union>i. A i) "
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      by (simp add: suminf)
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  qed
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qed fact
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text \<open>An equivalent characterization of sigma-finite spaces is the existence of integrable
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positive functions (or, still equivalently, the existence of a probability measure which is in
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the same measure class as the original measure).\<close>
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proposition (in sigma_finite_measure) obtain_positive_integrable_function:
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  obtains f::"'a \<Rightarrow> real" where
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    "f \<in> borel_measurable M"
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    "\<And>x. f x > 0"
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    "\<And>x. f x \<le> 1"
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    "integrable M f"
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proof -
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  obtain A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
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    using sigma_finite by auto
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  then have [measurable]:"A n \<in> sets M" for n by auto
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  define g where "g = (\<lambda>x. (\<Sum>n. (1/2)^(Suc n) * indicator (A n) x / (1+ measure M (A n))))"
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  have [measurable]: "g \<in> borel_measurable M" unfolding g_def by auto
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  have *: "summable (\<lambda>n. (1/2)^(Suc n) * indicator (A n) x / (1+ measure M (A n)))" for x
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    apply (rule summable_comparison_test'[of "\<lambda>n. (1/2)^(Suc n)" 0])
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    using power_half_series summable_def by (auto simp add: indicator_def divide_simps)
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  have "g x \<le> (\<Sum>n. (1/2)^(Suc n))" for x unfolding g_def
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    apply (rule suminf_le) using * power_half_series summable_def by (auto simp add: indicator_def divide_simps)
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  then have g_le_1: "g x \<le> 1" for x using power_half_series sums_unique by fastforce
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  have g_pos: "g x > 0" if "x \<in> space M" for x
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  unfolding g_def proof (subst suminf_pos_iff[OF *[of x]], auto)
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    obtain i where "x \<in> A i" using \<open>(\<Union>i. A i) = space M\<close> \<open>x \<in> space M\<close> by auto
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    then have "0 < (1 / 2) ^ Suc i * indicator (A i) x / (1 + Sigma_Algebra.measure M (A i))"
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      unfolding indicator_def apply (auto simp add: divide_simps) using measure_nonneg[of M "A i"]
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      by (auto, meson add_nonneg_nonneg linorder_not_le mult_nonneg_nonneg zero_le_numeral zero_le_one zero_le_power)
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    then show "\<exists>i. 0 < (1 / 2) ^ i * indicator (A i) x / (2 + 2 * Sigma_Algebra.measure M (A i))"
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      by auto
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  qed
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  have "integrable M g"
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  unfolding g_def proof (rule integrable_suminf)
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    fix n
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    show "integrable M (\<lambda>x. (1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n)))"
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      using \<open>emeasure M (A n) \<noteq> \<infinity>\<close>
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      by (auto intro!: integrable_mult_right integrable_divide_zero integrable_real_indicator simp add: top.not_eq_extremum)
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  next
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    show "AE x in M. summable (\<lambda>n. norm ((1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n))))"
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      using * by auto
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    show "summable (\<lambda>n. (\<integral>x. norm ((1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n))) \<partial>M))"
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      apply (rule summable_comparison_test'[of "\<lambda>n. (1/2)^(Suc n)" 0], auto)
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      using power_half_series summable_def apply auto[1]
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      apply (auto simp add: divide_simps) using measure_nonneg[of M] not_less by fastforce
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  qed
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  define f where "f = (\<lambda>x. if x \<in> space M then g x else 1)"
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  have "f x > 0" for x unfolding f_def using g_pos by auto
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  moreover have "f x \<le> 1" for x unfolding f_def using g_le_1 by auto
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  moreover have [measurable]: "f \<in> borel_measurable M" unfolding f_def by auto
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  moreover have "integrable M f"
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    apply (subst integrable_cong[of _ _ _ g]) unfolding f_def using \<open>integrable M g\<close> by auto
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  ultimately show "(\<And>f. f \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 < f x) \<Longrightarrow> (\<And>x. f x \<le> 1) \<Longrightarrow> integrable M f \<Longrightarrow> thesis) \<Longrightarrow> thesis"
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    by (meson that)
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qed
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lemma (in sigma_finite_measure) Ex_finite_integrable_function:
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  "\<exists>h\<in>borel_measurable M. integral\<^sup>N M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>)"
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proof -
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  obtain A :: "nat \<Rightarrow> 'a set" where
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    range[measurable]: "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. emeasure M (A i) \<noteq> \<infinity>" and
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    disjoint: "disjoint_family A"
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    using sigma_finite_disjoint by blast
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  let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)"
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  have [measurable]: "\<And>i. A i \<in> sets M"
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    using range by fastforce+
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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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      using measure[of i]
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      by (auto intro!: dense simp: ennreal_inverse_positive ennreal_mult_eq_top_iff power_eq_top_ennreal)
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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 * emeasure M (A i))" by auto
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  { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this
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  let ?h = "\<lambda>x. \<Sum>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 "integral\<^sup>N M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos
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      by (simp add: nn_integral_suminf nn_integral_cmult_indicator)
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    also have "\<dots> \<le> (\<Sum>i. ennreal ((1/2)^Suc i))"
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    proof (intro suminf_le allI)
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      fix N
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      have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)"
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        using n[of N] by (intro mult_right_mono) auto
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      also have "\<dots> = (1/2)^Suc N * (inverse (emeasure M (A N)) * emeasure M (A N))"
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        using measure[of N]
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        by (simp add: ennreal_inverse_power divide_ennreal_def ennreal_inverse_mult
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                      power_eq_top_ennreal less_top[symmetric] mult_ac
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                 del: power_Suc)
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      also have "\<dots> \<le> inverse (ennreal 2) ^ Suc N"
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        using measure[of N]
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        by (cases "emeasure M (A N)"; cases "emeasure M (A N) = 0")
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           (auto simp: inverse_ennreal ennreal_mult[symmetric] divide_ennreal_def simp del: power_Suc)
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      also have "\<dots> = ennreal (inverse 2 ^ Suc N)"
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        by (subst ennreal_power[symmetric], simp) (simp add: inverse_ennreal)
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      finally show "n N * emeasure M (A N) \<le> ennreal ((1/2)^Suc N)"
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        by simp
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    qed auto
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    also have "\<dots> < top"
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      unfolding less_top[symmetric]
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      by (rule ennreal_suminf_neq_top)
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         (auto simp: summable_geometric summable_Suc_iff simp del: power_Suc)
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    finally show "integral\<^sup>N M ?h \<noteq> \<infinity>"
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      by (auto simp: top_unique)
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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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      with disjoint n have "?h x = n i"
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        by (auto intro!: suminf_cmult_indicator intro: less_imp_le)
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      then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by (auto simp: less_top[symmetric]) }
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    note pos = this
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  qed measurable
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qed
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subsection "Absolutely continuous"
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definition%important absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
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  "absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N"
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lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M"
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  unfolding absolutely_continuous_def by (auto simp: null_sets_count_space)
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lemma absolutely_continuousI_density:
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  "f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)"
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  by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in)
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lemma absolutely_continuousI_point_measure_finite:
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  "(\<And>x. \<lbrakk> x \<in> A ; f x \<le> 0 \<rbrakk> \<Longrightarrow> g x \<le> 0) \<Longrightarrow> absolutely_continuous (point_measure A f) (point_measure A g)"
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  unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff)
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lemma absolutely_continuousD:
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  "absolutely_continuous M N \<Longrightarrow> A \<in> sets M \<Longrightarrow> emeasure M A = 0 \<Longrightarrow> emeasure N A = 0"
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  by (auto simp: absolutely_continuous_def null_sets_def)
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lemma absolutely_continuous_AE:
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  assumes sets_eq: "sets M' = sets M"
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    and "absolutely_continuous M M'" "AE x in M. P x"
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   shows "AE x in M'. P x"
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proof -
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  from \<open>AE x in M. P x\<close> obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
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    unfolding eventually_ae_filter by auto
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  show "AE x in M'. P x"
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  proof (rule AE_I')
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    show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp
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    from \<open>absolutely_continuous M M'\<close> show "N \<in> null_sets M'"
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      using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto
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  qed
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qed
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subsection "Existence of the Radon-Nikodym derivative"
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proposition
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 (in finite_measure) Radon_Nikodym_finite_measure:
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  assumes "finite_measure N" and sets_eq[simp]: "sets N = sets M"
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  assumes "absolutely_continuous M N"
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  shows "\<exists>f \<in> borel_measurable M. density M f = N"
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proof -
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  interpret N: finite_measure N by fact
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  define G where "G = {g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A}"
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  have [measurable_dest]: "f \<in> G \<Longrightarrow> f \<in> borel_measurable M"
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    and G_D: "\<And>A. f \<in> G \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) \<le> N A" for f
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    by (auto simp: G_def)
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  note this[measurable_dest]
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  have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
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  hence "G \<noteq> {}" by auto
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  { fix f g assume f[measurable]: "f \<in> G" and g[measurable]: "g \<in> G"
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    have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
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    proof safe
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      let ?A = "{x \<in> space M. f x \<le> g x}"
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      have "?A \<in> sets M" using f g unfolding G_def by auto
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      fix A assume [measurable]: "A \<in> sets M"
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      have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
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        using sets.sets_into_space[OF \<open>A \<in> sets M\<close>] by auto
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      have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
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        g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
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        by (auto simp: indicator_def max_def)
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      hence "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) =
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        (\<integral>\<^sup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
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        (\<integral>\<^sup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
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        by (auto cong: nn_integral_cong intro!: nn_integral_add)
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      also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)"
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        using f g unfolding G_def by (auto intro!: add_mono)
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      also have "\<dots> = N A"
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        using union by (subst plus_emeasure) auto
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      finally show "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" .
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    qed auto }
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  note max_in_G = this
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  { fix f assume  "incseq f" and f: "\<And>i. f i \<in> G"
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    then have [measurable]: "\<And>i. f i \<in> borel_measurable M" by (auto simp: G_def)
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    have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def
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   242
    proof safe
hoelzl@50003
   243
      show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" by measurable
hoelzl@41981
   244
    next
hoelzl@38656
   245
      fix A assume "A \<in> sets M"
wenzelm@53015
   246
      have "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
wenzelm@53015
   247
        (\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
hoelzl@56996
   248
        by (intro nn_integral_cong) (simp split: split_indicator)
wenzelm@53015
   249
      also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))"
wenzelm@61808
   250
        using \<open>incseq f\<close> f \<open>A \<in> sets M\<close>
hoelzl@56996
   251
        by (intro nn_integral_monotone_convergence_SUP)
hoelzl@41981
   252
           (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
wenzelm@53015
   253
      finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A"
hoelzl@63330
   254
        using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_D)
hoelzl@38656
   255
    qed }
hoelzl@38656
   256
  note SUP_in_G = this
haftmann@69260
   257
  let ?y = "SUP g \<in> G. integral\<^sup>N M g"
hoelzl@47694
   258
  have y_le: "?y \<le> N (space M)" unfolding G_def
hoelzl@44928
   259
  proof (safe intro!: SUP_least)
wenzelm@53015
   260
    fix g assume "\<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A"
hoelzl@56996
   261
    from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)"
hoelzl@56996
   262
      by (simp cong: nn_integral_cong)
hoelzl@38656
   263
  qed
hoelzl@62975
   264
  from ennreal_SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this
hoelzl@56996
   265
  then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n"
hoelzl@38656
   266
  proof safe
hoelzl@56996
   267
    fix n assume "range ys \<subseteq> integral\<^sup>N M ` G"
hoelzl@56996
   268
    hence "ys n \<in> integral\<^sup>N M ` G" by auto
hoelzl@56996
   269
    thus "\<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" by auto
hoelzl@38656
   270
  qed
hoelzl@56996
   271
  from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^sup>N M (gs n) = ys n" by auto
hoelzl@56996
   272
  hence y_eq: "?y = (SUP i. integral\<^sup>N M (gs i))" using ys by auto
wenzelm@46731
   273
  let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})"
wenzelm@63040
   274
  define f where [abs_def]: "f x = (SUP i. ?g i x)" for x
wenzelm@46731
   275
  let ?F = "\<lambda>A x. f x * indicator A x"
hoelzl@41981
   276
  have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
hoelzl@38656
   277
  { fix i have "?g i \<in> G"
hoelzl@38656
   278
    proof (induct i)
hoelzl@38656
   279
      case 0 thus ?case by simp fact
hoelzl@38656
   280
    next
hoelzl@38656
   281
      case (Suc i)
wenzelm@61808
   282
      with Suc gs_not_empty \<open>gs (Suc i) \<in> G\<close> show ?case
hoelzl@38656
   283
        by (auto simp add: atMost_Suc intro!: max_in_G)
hoelzl@38656
   284
    qed }
hoelzl@38656
   285
  note g_in_G = this
hoelzl@41981
   286
  have "incseq ?g" using gs_not_empty
hoelzl@41981
   287
    by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)
hoelzl@63330
   288
hoelzl@50003
   289
  from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def .
hoelzl@63330
   290
  then have [measurable]: "f \<in> borel_measurable M" unfolding G_def by auto
hoelzl@63330
   291
hoelzl@56996
   292
  have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def
hoelzl@63330
   293
    using g_in_G \<open>incseq ?g\<close> by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def)
hoelzl@38656
   294
  also have "\<dots> = ?y"
hoelzl@38656
   295
  proof (rule antisym)
hoelzl@56996
   296
    show "(SUP i. integral\<^sup>N M (?g i)) \<le> ?y"
haftmann@56166
   297
      using g_in_G by (auto intro: SUP_mono)
hoelzl@56996
   298
    show "?y \<le> (SUP i. integral\<^sup>N M (?g i))" unfolding y_eq
hoelzl@56996
   299
      by (auto intro!: SUP_mono nn_integral_mono Max_ge)
hoelzl@38656
   300
  qed
hoelzl@56996
   301
  finally have int_f_eq_y: "integral\<^sup>N M f = ?y" .
hoelzl@47694
   302
hoelzl@63330
   303
  have upper_bound: "\<forall>A\<in>sets M. N A \<le> density M f A"
hoelzl@38656
   304
  proof (rule ccontr)
hoelzl@38656
   305
    assume "\<not> ?thesis"
hoelzl@63330
   306
    then obtain A where A[measurable]: "A \<in> sets M" and f_less_N: "density M f A < N A"
hoelzl@63330
   307
      by (auto simp: not_le)
hoelzl@63330
   308
    then have pos_A: "0 < M A"
hoelzl@63330
   309
      using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, OF A]
hoelzl@62975
   310
      by (auto simp: zero_less_iff_neq_zero)
hoelzl@63330
   311
hoelzl@63330
   312
    define b where "b = (N A - density M f A) / M A / 2"
hoelzl@63330
   313
    with f_less_N pos_A have "0 < b" "b \<noteq> top"
hoelzl@63330
   314
      by (auto intro!: diff_gr0_ennreal simp: zero_less_iff_neq_zero diff_eq_0_iff_ennreal ennreal_divide_eq_top_iff)
hoelzl@63330
   315
hoelzl@63330
   316
    let ?f = "\<lambda>x. f x + b"
hoelzl@63330
   317
    have "nn_integral M f \<noteq> top"
wenzelm@64911
   318
      using \<open>f \<in> G\<close>[THEN G_D, of "space M"] by (auto simp: top_unique cong: nn_integral_cong)
hoelzl@63330
   319
    with \<open>b \<noteq> top\<close> interpret Mf: finite_measure "density M ?f"
hoelzl@63330
   320
      by (intro finite_measureI)
hoelzl@63330
   321
         (auto simp: field_simps mult_indicator_subset ennreal_mult_eq_top_iff
hoelzl@63330
   322
                     emeasure_density nn_integral_cmult_indicator nn_integral_add
hoelzl@63330
   323
               cong: nn_integral_cong)
hoelzl@63330
   324
hoelzl@63330
   325
    from unsigned_Hahn_decomposition[of "density M ?f" N A]
hoelzl@63330
   326
    obtain Y where [measurable]: "Y \<in> sets M" and [simp]: "Y \<subseteq> A"
hoelzl@63330
   327
       and Y1: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> density M ?f C \<le> N C"
hoelzl@63330
   328
       and Y2: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> A \<Longrightarrow> C \<inter> Y = {} \<Longrightarrow> N C \<le> density M ?f C"
hoelzl@63330
   329
       by auto
hoelzl@63330
   330
hoelzl@63330
   331
    let ?f' = "\<lambda>x. f x + b * indicator Y x"
hoelzl@63330
   332
    have "M Y \<noteq> 0"
hoelzl@63330
   333
    proof
hoelzl@63330
   334
      assume "M Y = 0"
hoelzl@63330
   335
      then have "N Y = 0"
hoelzl@63330
   336
        using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, of Y] by auto
hoelzl@63330
   337
      then have "N A = N (A - Y)"
hoelzl@63330
   338
        by (subst emeasure_Diff) auto
hoelzl@63330
   339
      also have "\<dots> \<le> density M ?f (A - Y)"
hoelzl@63330
   340
        by (rule Y2) auto
hoelzl@63330
   341
      also have "\<dots> \<le> density M ?f A - density M ?f Y"
hoelzl@63330
   342
        by (subst emeasure_Diff) auto
hoelzl@63330
   343
      also have "\<dots> \<le> density M ?f A - 0"
hoelzl@63330
   344
        by (intro ennreal_minus_mono) auto
hoelzl@63330
   345
      also have "density M ?f A = b * M A + density M f A"
hoelzl@63330
   346
        by (simp add: emeasure_density field_simps mult_indicator_subset nn_integral_add nn_integral_cmult_indicator)
hoelzl@63330
   347
      also have "\<dots> < N A"
hoelzl@63330
   348
        using f_less_N pos_A
hoelzl@63330
   349
        by (cases "density M f A"; cases "M A"; cases "N A")
hoelzl@63330
   350
           (auto simp: b_def ennreal_less_iff ennreal_minus divide_ennreal ennreal_numeral[symmetric]
hoelzl@63330
   351
                       ennreal_plus[symmetric] ennreal_mult[symmetric] field_simps
hoelzl@63330
   352
                    simp del: ennreal_numeral ennreal_plus)
hoelzl@63330
   353
      finally show False
hoelzl@63330
   354
        by simp
hoelzl@63330
   355
    qed
hoelzl@63330
   356
    then have "nn_integral M f < nn_integral M ?f'"
hoelzl@63330
   357
      using \<open>0 < b\<close> \<open>nn_integral M f \<noteq> top\<close>
hoelzl@63330
   358
      by (simp add: nn_integral_add nn_integral_cmult_indicator ennreal_zero_less_mult_iff zero_less_iff_neq_zero)
hoelzl@38656
   359
    moreover
hoelzl@63330
   360
    have "?f' \<in> G"
hoelzl@63330
   361
      unfolding G_def
hoelzl@63330
   362
    proof safe
hoelzl@63330
   363
      fix X assume [measurable]: "X \<in> sets M"
hoelzl@63330
   364
      have "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) = density M f (X - Y) + density M ?f (X \<inter> Y)"
hoelzl@63330
   365
        by (auto simp add: emeasure_density nn_integral_add[symmetric] split: split_indicator intro!: nn_integral_cong)
hoelzl@63330
   366
      also have "\<dots> \<le> N (X - Y) + N (X \<inter> Y)"
hoelzl@63330
   367
        using G_D[OF \<open>f \<in> G\<close>] by (intro add_mono Y1) (auto simp: emeasure_density)
hoelzl@63330
   368
      also have "\<dots> = N X"
hoelzl@63330
   369
        by (subst plus_emeasure) (auto intro!: arg_cong2[where f=emeasure])
hoelzl@63330
   370
      finally show "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) \<le> N X" .
hoelzl@63330
   371
    qed simp
hoelzl@63330
   372
    then have "nn_integral M ?f' \<le> ?y"
hoelzl@63330
   373
      by (rule SUP_upper)
hoelzl@63330
   374
    ultimately show False
hoelzl@63330
   375
      by (simp add: int_f_eq_y)
hoelzl@38656
   376
  qed
hoelzl@38656
   377
  show ?thesis
hoelzl@63330
   378
  proof (intro bexI[of _ f] measure_eqI conjI antisym)
hoelzl@63330
   379
    fix A assume "A \<in> sets (density M f)" then show "emeasure (density M f) A \<le> emeasure N A"
hoelzl@63330
   380
      by (auto simp: emeasure_density intro!: G_D[OF \<open>f \<in> G\<close>])
hoelzl@63330
   381
  next
hoelzl@63330
   382
    fix A assume A: "A \<in> sets (density M f)" then show "emeasure N A \<le> emeasure (density M f) A"
hoelzl@63330
   383
      using upper_bound by auto
hoelzl@47694
   384
  qed auto
hoelzl@38656
   385
qed
hoelzl@38656
   386
ak2110@69730
   387
lemma (in finite_measure) split_space_into_finite_sets_and_rest:
hoelzl@63330
   388
  assumes ac: "absolutely_continuous M N" and sets_eq[simp]: "sets N = sets M"
hoelzl@63330
   389
  shows "\<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> (\<forall>i. N (B i) \<noteq> \<infinity>) \<and>
hoelzl@63330
   390
    (\<forall>A\<in>sets M. A \<inter> (\<Union>i. B i) = {} \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>))"
ak2110@69730
   391
proof -
hoelzl@47694
   392
  let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}"
haftmann@69260
   393
  let ?a = "SUP Q\<in>?Q. emeasure M Q"
hoelzl@47694
   394
  have "{} \<in> ?Q" by auto
hoelzl@38656
   395
  then have Q_not_empty: "?Q \<noteq> {}" by blast
immler@50244
   396
  have "?a \<le> emeasure M (space M)" using sets.sets_into_space
hoelzl@47694
   397
    by (auto intro!: SUP_least emeasure_mono)
hoelzl@62975
   398
  then have "?a \<noteq> \<infinity>"
hoelzl@62975
   399
    using finite_emeasure_space
hoelzl@62975
   400
    by (auto simp: less_top[symmetric] top_unique simp del: SUP_eq_top_iff Sup_eq_top_iff)
hoelzl@62975
   401
  from ennreal_SUP_countable_SUP [OF Q_not_empty, of "emeasure M"]
hoelzl@47694
   402
  obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
hoelzl@38656
   403
    by auto
hoelzl@47694
   404
  then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto
hoelzl@47694
   405
  from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q"
hoelzl@38656
   406
    by auto
haftmann@69260
   407
  then have a_Lim: "?a = (SUP i. emeasure M (Q' i))" using a by simp
wenzelm@46731
   408
  let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i"
hoelzl@47694
   409
  have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)"
hoelzl@47694
   410
  proof (rule SUP_emeasure_incseq[of ?O])
hoelzl@47694
   411
    show "range ?O \<subseteq> sets M" using Q' by auto
nipkow@44890
   412
    show "incseq ?O" by (fastforce intro!: incseq_SucI)
hoelzl@38656
   413
  qed
hoelzl@63330
   414
  have Q'_sets[measurable]: "\<And>i. Q' i \<in> sets M" using Q' by auto
hoelzl@47694
   415
  have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto
hoelzl@38656
   416
  then have O_in_G: "\<And>i. ?O i \<in> ?Q"
hoelzl@38656
   417
  proof (safe del: notI)
hoelzl@47694
   418
    fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto
hoelzl@47694
   419
    then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))"
hoelzl@63330
   420
      by (simp add: emeasure_subadditive_finite)
hoelzl@62975
   421
    also have "\<dots> < \<infinity>" using Q' by (simp add: less_top)
hoelzl@47694
   422
    finally show "N (?O i) \<noteq> \<infinity>" by simp
hoelzl@38656
   423
  qed auto
nipkow@44890
   424
  have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce
hoelzl@47694
   425
  have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric]
hoelzl@38656
   426
  proof (rule antisym)
hoelzl@47694
   427
    show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim
hoelzl@47694
   428
      using Q' by (auto intro!: SUP_mono emeasure_mono)
haftmann@62343
   429
    show "(SUP i. emeasure M (?O i)) \<le> ?a"
hoelzl@38656
   430
    proof (safe intro!: Sup_mono, unfold bex_simps)
hoelzl@38656
   431
      fix i
haftmann@52141
   432
      have *: "(\<Union>(Q' ` {..i})) = ?O i" by auto
hoelzl@47694
   433
      then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and>
haftmann@52141
   434
        emeasure M (\<Union>(Q' ` {..i})) \<le> emeasure M x"
hoelzl@38656
   435
        using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
hoelzl@38656
   436
    qed
hoelzl@38656
   437
  qed
wenzelm@46731
   438
  let ?O_0 = "(\<Union>i. ?O i)"
hoelzl@38656
   439
  have "?O_0 \<in> sets M" using Q' by auto
hoelzl@63330
   440
  have "disjointed Q' i \<in> sets M" for i
hoelzl@63330
   441
    using sets.range_disjointed_sets[of Q' M] using Q'_sets by (auto simp: subset_eq)
hoelzl@38656
   442
  note Q_sets = this
hoelzl@40859
   443
  show ?thesis
hoelzl@40859
   444
  proof (intro bexI exI conjI ballI impI allI)
hoelzl@63330
   445
    show "disjoint_family (disjointed Q')"
hoelzl@63330
   446
      by (rule disjoint_family_disjointed)
hoelzl@63330
   447
    show "range (disjointed Q') \<subseteq> sets M"
hoelzl@63330
   448
      using Q'_sets by (intro sets.range_disjointed_sets) auto
hoelzl@63330
   449
    { fix A assume A: "A \<in> sets M" "A \<inter> (\<Union>i. disjointed Q' i) = {}"
hoelzl@63330
   450
      then have A1: "A \<inter> (\<Union>i. Q' i) = {}"
hoelzl@63330
   451
        unfolding UN_disjointed_eq by auto
hoelzl@47694
   452
      show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
hoelzl@40859
   453
      proof (rule disjCI, simp)
hoelzl@62975
   454
        assume *: "emeasure M A = 0 \<or> N A \<noteq> top"
hoelzl@47694
   455
        show "emeasure M A = 0 \<and> N A = 0"
wenzelm@53374
   456
        proof (cases "emeasure M A = 0")
wenzelm@53374
   457
          case True
wenzelm@53374
   458
          with ac A have "N A = 0"
hoelzl@40859
   459
            unfolding absolutely_continuous_def by auto
wenzelm@53374
   460
          with True show ?thesis by simp
hoelzl@40859
   461
        next
wenzelm@53374
   462
          case False
hoelzl@62975
   463
          with * have "N A \<noteq> \<infinity>" by auto
hoelzl@47694
   464
          with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)"
hoelzl@63330
   465
            using Q' A1 by (auto intro!: plus_emeasure simp: set_eq_iff)
hoelzl@47694
   466
          also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))"
hoelzl@47694
   467
          proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
hoelzl@40859
   468
            show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
wenzelm@61808
   469
              using \<open>N A \<noteq> \<infinity>\<close> O_sets A by auto
nipkow@44890
   470
          qed (fastforce intro!: incseq_SucI)
hoelzl@40859
   471
          also have "\<dots> \<le> ?a"
hoelzl@44928
   472
          proof (safe intro!: SUP_least)
hoelzl@40859
   473
            fix i have "?O i \<union> A \<in> ?Q"
hoelzl@40859
   474
            proof (safe del: notI)
hoelzl@40859
   475
              show "?O i \<union> A \<in> sets M" using O_sets A by auto
hoelzl@47694
   476
              from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A"
hoelzl@63330
   477
                using emeasure_subadditive[of "?O i" N A] A O_sets by auto
hoelzl@47694
   478
              with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>"
hoelzl@62975
   479
                using \<open>N A \<noteq> \<infinity>\<close> by (auto simp: top_unique)
hoelzl@40859
   480
            qed
hoelzl@47694
   481
            then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
hoelzl@40859
   482
          qed
hoelzl@47694
   483
          finally have "emeasure M A = 0"
hoelzl@47694
   484
            unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure)
wenzelm@61808
   485
          with \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto
hoelzl@40859
   486
        qed
hoelzl@40859
   487
      qed }
hoelzl@63330
   488
    { fix i
hoelzl@63330
   489
      have "N (disjointed Q' i) \<le> N (Q' i)"
hoelzl@63330
   490
        by (auto intro!: emeasure_mono simp: disjointed_def)
hoelzl@63330
   491
      then show "N (disjointed Q' i) \<noteq> \<infinity>"
hoelzl@63330
   492
        using Q'(2)[of i] by (auto simp: top_unique) }
hoelzl@40859
   493
  qed
hoelzl@40859
   494
qed
hoelzl@40859
   495
ak2110@69730
   496
proposition (in finite_measure) Radon_Nikodym_finite_measure_infinite:
hoelzl@47694
   497
  assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M"
hoelzl@63329
   498
  shows "\<exists>f\<in>borel_measurable M. density M f = N"
ak2110@69730
   499
proof -
hoelzl@40859
   500
  from split_space_into_finite_sets_and_rest[OF assms]
hoelzl@63330
   501
  obtain Q :: "nat \<Rightarrow> 'a set"
hoelzl@40859
   502
    where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
hoelzl@63330
   503
    and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
hoelzl@47694
   504
    and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force
hoelzl@40859
   505
  from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
hoelzl@47694
   506
  let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))"
hoelzl@63329
   507
  have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). density (?M i) f = ?N i"
hoelzl@47694
   508
  proof (intro allI finite_measure.Radon_Nikodym_finite_measure)
hoelzl@38656
   509
    fix i
hoelzl@47694
   510
    from Q show "finite_measure (?M i)"
hoelzl@56996
   511
      by (auto intro!: finite_measureI cong: nn_integral_cong
hoelzl@47694
   512
               simp add: emeasure_density subset_eq sets_eq)
hoelzl@47694
   513
    from Q have "emeasure (?N i) (space N) = emeasure N (Q i)"
hoelzl@56996
   514
      by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: nn_integral_cong)
hoelzl@47694
   515
    with Q_fin show "finite_measure (?N i)"
hoelzl@47694
   516
      by (auto intro!: finite_measureI)
hoelzl@47694
   517
    show "sets (?N i) = sets (?M i)" by (simp add: sets_eq)
hoelzl@50003
   518
    have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq)
hoelzl@47694
   519
    show "absolutely_continuous (?M i) (?N i)"
wenzelm@61808
   520
      using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close>
hoelzl@47694
   521
      by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq
hoelzl@47694
   522
               intro!: absolutely_continuous_AE[OF sets_eq])
hoelzl@38656
   523
  qed
hoelzl@47694
   524
  from choice[OF this[unfolded Bex_def]]
hoelzl@47694
   525
  obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
hoelzl@47694
   526
    and f_density: "\<And>i. density (?M i) (f i) = ?N i"
immler@54776
   527
    by force
hoelzl@47694
   528
  { fix A i assume A: "A \<in> sets M"
wenzelm@53015
   529
    with Q borel have "(\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A"
hoelzl@56996
   530
      by (auto simp add: emeasure_density nn_integral_density subset_eq
hoelzl@56996
   531
               intro!: nn_integral_cong split: split_indicator)
hoelzl@47694
   532
    also have "\<dots> = emeasure N (Q i \<inter> A)"
hoelzl@47694
   533
      using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq)
wenzelm@53015
   534
    finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. }
hoelzl@47694
   535
  note integral_eq = this
hoelzl@63330
   536
  let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator (space M - (\<Union>i. Q i)) x"
hoelzl@38656
   537
  show ?thesis
hoelzl@38656
   538
  proof (safe intro!: bexI[of _ ?f])
hoelzl@63330
   539
    show "?f \<in> borel_measurable M" using borel Q_sets
hoelzl@41981
   540
      by (auto intro!: measurable_If)
hoelzl@47694
   541
    show "density M ?f = N"
hoelzl@47694
   542
    proof (rule measure_eqI)
hoelzl@47694
   543
      fix A assume "A \<in> sets (density M ?f)"
hoelzl@47694
   544
      then have "A \<in> sets M" by simp
hoelzl@47694
   545
      have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
hoelzl@47694
   546
      have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
hoelzl@47694
   547
        "\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x"
hoelzl@63330
   548
        using borel Qi \<open>A \<in> sets M\<close> by auto
hoelzl@63330
   549
      have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator ((space M - (\<Union>i. Q i)) \<inter> A) x \<partial>M)"
hoelzl@56996
   550
        using borel by (intro nn_integral_cong) (auto simp: indicator_def)
hoelzl@63330
   551
      also have "\<dots> = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)"
hoelzl@63330
   552
        using borel Qi \<open>A \<in> sets M\<close>
hoelzl@62975
   553
        by (subst nn_integral_add)
hoelzl@62975
   554
           (auto simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le)
hoelzl@63330
   555
      also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)"
wenzelm@61808
   556
        by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto
hoelzl@63330
   557
      finally have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" .
hoelzl@47694
   558
      moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)"
wenzelm@61808
   559
        using Q Q_sets \<open>A \<in> sets M\<close>
hoelzl@47694
   560
        by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
hoelzl@63330
   561
      moreover
hoelzl@63330
   562
      have "(space M - (\<Union>x. Q x)) \<inter> A \<inter> (\<Union>x. Q x) = {}"
hoelzl@63330
   563
        by auto
hoelzl@63330
   564
      then have "\<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A) = N ((space M - (\<Union>i. Q i)) \<inter> A)"
hoelzl@63330
   565
        using in_Q0[of "(space M - (\<Union>i. Q i)) \<inter> A"] \<open>A \<in> sets M\<close> Q by (auto simp: ennreal_top_mult)
hoelzl@63330
   566
      moreover have "(space M - (\<Union>i. Q i)) \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
hoelzl@63330
   567
        using Q_sets \<open>A \<in> sets M\<close> by auto
hoelzl@63330
   568
      moreover have "((\<Union>i. Q i) \<inter> A) \<union> ((space M - (\<Union>i. Q i)) \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> ((space M - (\<Union>i. Q i)) \<inter> A) = {}"
hoelzl@63330
   569
        using \<open>A \<in> sets M\<close> sets.sets_into_space by auto
wenzelm@53015
   570
      ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)"
hoelzl@63330
   571
        using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "(space M - (\<Union>i. Q i)) \<inter> A"] by (simp add: sets_eq)
hoelzl@63330
   572
      with \<open>A \<in> sets M\<close> borel Q show "emeasure (density M ?f) A = N A"
hoelzl@50003
   573
        by (auto simp: subset_eq emeasure_density)
hoelzl@47694
   574
    qed (simp add: sets_eq)
hoelzl@38656
   575
  qed
hoelzl@38656
   576
qed
hoelzl@38656
   577
ak2110@69730
   578
theorem (in sigma_finite_measure) Radon_Nikodym:
hoelzl@47694
   579
  assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M"
hoelzl@63329
   580
  shows "\<exists>f \<in> borel_measurable M. density M f = N"
hoelzl@38656
   581
proof -
hoelzl@38656
   582
  from Ex_finite_integrable_function
hoelzl@56996
   583
  obtain h where finite: "integral\<^sup>N M h \<noteq> \<infinity>" and
hoelzl@38656
   584
    borel: "h \<in> borel_measurable M" and
hoelzl@41981
   585
    nn: "\<And>x. 0 \<le> h x" and
hoelzl@38656
   586
    pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
hoelzl@41981
   587
    "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
wenzelm@53015
   588
  let ?T = "\<lambda>A. (\<integral>\<^sup>+x. h x * indicator A x \<partial>M)"
hoelzl@47694
   589
  let ?MT = "density M h"
hoelzl@47694
   590
  from borel finite nn interpret T: finite_measure ?MT
hoelzl@56996
   591
    by (auto intro!: finite_measureI cong: nn_integral_cong simp: emeasure_density)
hoelzl@47694
   592
  have "absolutely_continuous ?MT N" "sets N = sets ?MT"
hoelzl@47694
   593
  proof (unfold absolutely_continuous_def, safe)
hoelzl@47694
   594
    fix A assume "A \<in> null_sets ?MT"
hoelzl@47694
   595
    with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0"
hoelzl@47694
   596
      by (auto simp add: null_sets_density_iff)
immler@50244
   597
    with pos sets.sets_into_space have "AE x in M. x \<notin> A"
lp15@61810
   598
      by (elim eventually_mono) (auto simp: not_le[symmetric])
hoelzl@47694
   599
    then have "A \<in> null_sets M"
wenzelm@61808
   600
      using \<open>A \<in> sets M\<close> by (simp add: AE_iff_null_sets)
hoelzl@47694
   601
    with ac show "A \<in> null_sets N"
hoelzl@47694
   602
      by (auto simp: absolutely_continuous_def)
hoelzl@47694
   603
  qed (auto simp add: sets_eq)
hoelzl@47694
   604
  from T.Radon_Nikodym_finite_measure_infinite[OF this]
hoelzl@63329
   605
  obtain f where f_borel: "f \<in> borel_measurable M" "density ?MT f = N" by auto
hoelzl@47694
   606
  with nn borel show ?thesis
hoelzl@47694
   607
    by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq)
hoelzl@38656
   608
qed
hoelzl@38656
   609
immler@69683
   610
subsection \<open>Uniqueness of densities\<close>
hoelzl@40859
   611
ak2110@69730
   612
lemma finite_density_unique:
hoelzl@40859
   613
  assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@47694
   614
  assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
hoelzl@56996
   615
  and fin: "integral\<^sup>N M f \<noteq> \<infinity>"
hoelzl@49785
   616
  shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
ak2110@69730
   617
proof (intro iffI ballI)
hoelzl@47694
   618
  fix A assume eq: "AE x in M. f x = g x"
hoelzl@49785
   619
  with borel show "density M f = density M g"
hoelzl@49785
   620
    by (auto intro: density_cong)
hoelzl@40859
   621
next
wenzelm@53015
   622
  let ?P = "\<lambda>f A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M"
hoelzl@49785
   623
  assume "density M f = density M g"
hoelzl@49785
   624
  with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
hoelzl@49785
   625
    by (simp add: emeasure_density[symmetric])
immler@50244
   626
  from this[THEN bspec, OF sets.top] fin
hoelzl@56996
   627
  have g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" by (simp cong: nn_integral_cong)
hoelzl@40859
   628
  { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@47694
   629
      and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
hoelzl@56996
   630
      and g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
hoelzl@40859
   631
    let ?N = "{x\<in>space M. g x < f x}"
hoelzl@40859
   632
    have N: "?N \<in> sets M" using borel by simp
hoelzl@56996
   633
    have "?P g ?N \<le> integral\<^sup>N M g" using pos
hoelzl@56996
   634
      by (intro nn_integral_mono_AE) (auto split: split_indicator)
hoelzl@62975
   635
    then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by (auto simp: top_unique)
wenzelm@53015
   636
    have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^sup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
hoelzl@56996
   637
      by (auto intro!: nn_integral_cong simp: indicator_def)
hoelzl@40859
   638
    also have "\<dots> = ?P f ?N - ?P g ?N"
hoelzl@56996
   639
    proof (rule nn_integral_diff)
hoelzl@40859
   640
      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
   641
        using borel N by auto
hoelzl@47694
   642
      show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x"
hoelzl@41981
   643
        using pos by (auto split: split_indicator)
hoelzl@41981
   644
    qed fact
hoelzl@40859
   645
    also have "\<dots> = 0"
hoelzl@62975
   646
      unfolding eq[THEN bspec, OF N] using Pg_fin by auto
hoelzl@47694
   647
    finally have "AE x in M. f x \<le> g x"
hoelzl@56996
   648
      using pos borel nn_integral_PInf_AE[OF borel(2) g_fin]
hoelzl@56996
   649
      by (subst (asm) nn_integral_0_iff_AE)
hoelzl@62975
   650
         (auto split: split_indicator simp: not_less ennreal_minus_eq_0) }
hoelzl@41981
   651
  from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
hoelzl@47694
   652
  show "AE x in M. f x = g x" by auto
hoelzl@40859
   653
qed
hoelzl@40859
   654
ak2110@69730
   655
lemma (in finite_measure) density_unique_finite_measure:
hoelzl@40859
   656
  assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
hoelzl@47694
   657
  assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x"
wenzelm@53015
   658
  assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. f' x * indicator A x \<partial>M)"
hoelzl@40859
   659
    (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
hoelzl@47694
   660
  shows "AE x in M. f x = f' x"
ak2110@69730
   661
proof -
hoelzl@47694
   662
  let ?D = "\<lambda>f. density M f"
hoelzl@47694
   663
  let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A"
wenzelm@46731
   664
  let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
hoelzl@47694
   665
hoelzl@47694
   666
  have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M"
lp15@61609
   667
    using borel by (auto intro!: absolutely_continuousI_density)
hoelzl@47694
   668
  from split_space_into_finite_sets_and_rest[OF this]
hoelzl@63330
   669
  obtain Q :: "nat \<Rightarrow> 'a set"
hoelzl@40859
   670
    where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
hoelzl@63330
   671
    and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>"
hoelzl@47694
   672
    and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force
hoelzl@63330
   673
  with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>"
hoelzl@47694
   674
    and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq)
hoelzl@47694
   675
hoelzl@63330
   676
  from Q have Q_sets[measurable]: "\<And>i. Q i \<in> sets M" by auto
hoelzl@47694
   677
  let ?D = "{x\<in>space M. f x \<noteq> f' x}"
hoelzl@47694
   678
  have "?D \<in> sets M" using borel by auto
hoelzl@62975
   679
  have *: "\<And>i x A. \<And>y::ennreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
hoelzl@40859
   680
    unfolding indicator_def by auto
hoelzl@47694
   681
  have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
hoelzl@40859
   682
    by (intro finite_density_unique[THEN iffD1] allI)
hoelzl@50003
   683
       (auto intro!: f measure_eqI simp: emeasure_density * subset_eq)
hoelzl@63330
   684
  moreover have "AE x in M. ?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x"
hoelzl@40859
   685
  proof (rule AE_I')
hoelzl@62975
   686
    { fix f :: "'a \<Rightarrow> ennreal" assume borel: "f \<in> borel_measurable M"
wenzelm@53015
   687
        and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
hoelzl@63330
   688
      let ?A = "\<lambda>i. (space M - (\<Union>i. Q i)) \<inter> {x \<in> space M. f x < (i::nat)}"
hoelzl@47694
   689
      have "(\<Union>i. ?A i) \<in> null_sets M"
hoelzl@40859
   690
      proof (rule null_sets_UN)
hoelzl@43923
   691
        fix i ::nat have "?A i \<in> sets M"
hoelzl@63330
   692
          using borel by auto
hoelzl@62975
   693
        have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ennreal) * indicator (?A i) x \<partial>M)"
wenzelm@61808
   694
          unfolding eq[OF \<open>?A i \<in> sets M\<close>]
hoelzl@56996
   695
          by (auto intro!: nn_integral_mono simp: indicator_def)
hoelzl@47694
   696
        also have "\<dots> = i * emeasure M (?A i)"
wenzelm@61808
   697
          using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator)
hoelzl@62975
   698
        also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by (auto simp: ennreal_mult_less_top of_nat_less_top)
hoelzl@47694
   699
        finally have "?N (?A i) \<noteq> \<infinity>" by simp
wenzelm@61808
   700
        then show "?A i \<in> null_sets M" using in_Q0[OF \<open>?A i \<in> sets M\<close>] \<open>?A i \<in> sets M\<close> by auto
hoelzl@40859
   701
      qed
hoelzl@63330
   702
      also have "(\<Union>i. ?A i) = (space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
hoelzl@62975
   703
        by (auto simp: ennreal_Ex_less_of_nat less_top[symmetric])
hoelzl@63330
   704
      finally have "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp }
hoelzl@40859
   705
    from this[OF borel(1) refl] this[OF borel(2) f]
hoelzl@63330
   706
    have "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all
hoelzl@63330
   707
    then show "((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un)
hoelzl@63330
   708
    show "{x \<in> space M. ?f (space M - (\<Union>i. Q i)) x \<noteq> ?f' (space M - (\<Union>i. Q i)) x} \<subseteq>
hoelzl@63330
   709
      ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def)
hoelzl@40859
   710
  qed
hoelzl@63330
   711
  moreover have "AE x in M. (?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
hoelzl@40859
   712
    ?f (space M) x = ?f' (space M) x"
hoelzl@63330
   713
    by (auto simp: indicator_def)
hoelzl@47694
   714
  ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x"
hoelzl@47694
   715
    unfolding AE_all_countable[symmetric]
hoelzl@63330
   716
    by eventually_elim (auto split: if_split_asm simp: indicator_def)
hoelzl@47694
   717
  then show "AE x in M. f x = f' x" by auto
hoelzl@40859
   718
qed
hoelzl@40859
   719
ak2110@69730
   720
proposition (in sigma_finite_measure) density_unique:
hoelzl@62975
   721
  assumes f: "f \<in> borel_measurable M"
hoelzl@62975
   722
  assumes f': "f' \<in> borel_measurable M"
hoelzl@47694
   723
  assumes density_eq: "density M f = density M f'"
hoelzl@47694
   724
  shows "AE x in M. f x = f' x"
hoelzl@40859
   725
proof -
hoelzl@40859
   726
  obtain h where h_borel: "h \<in> borel_measurable M"
hoelzl@56996
   727
    and fin: "integral\<^sup>N M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x"
hoelzl@40859
   728
    using Ex_finite_integrable_function by auto
hoelzl@47694
   729
  then have h_nn: "AE x in M. 0 \<le> h x" by auto
hoelzl@47694
   730
  let ?H = "density M h"
hoelzl@47694
   731
  interpret h: finite_measure ?H
hoelzl@47694
   732
    using fin h_borel pos
hoelzl@56996
   733
    by (intro finite_measureI) (simp cong: nn_integral_cong emeasure_density add: fin)
hoelzl@47694
   734
  let ?fM = "density M f"
hoelzl@47694
   735
  let ?f'M = "density M f'"
hoelzl@40859
   736
  { fix A assume "A \<in> sets M"
hoelzl@41981
   737
    then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
immler@50244
   738
      using pos(1) sets.sets_into_space by (force simp: indicator_def)
wenzelm@53015
   739
    then have "(\<integral>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
wenzelm@61808
   740
      using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto }
hoelzl@40859
   741
  note h_null_sets = this
hoelzl@40859
   742
  { fix A assume "A \<in> sets M"
wenzelm@53015
   743
    have "(\<integral>\<^sup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?fM)"
wenzelm@61808
   744
      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
hoelzl@56996
   745
      by (intro nn_integral_density[symmetric]) auto
wenzelm@53015
   746
    also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)"
hoelzl@47694
   747
      by (simp_all add: density_eq)
wenzelm@53015
   748
    also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)"
wenzelm@61808
   749
      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
hoelzl@56996
   750
      by (intro nn_integral_density) auto
wenzelm@53015
   751
    finally have "(\<integral>\<^sup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * (f' x * indicator A x) \<partial>M)"
hoelzl@41981
   752
      by (simp add: ac_simps)
wenzelm@53015
   753
    then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)"
wenzelm@61808
   754
      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
hoelzl@56996
   755
      by (subst (asm) (1 2) nn_integral_density[symmetric]) auto }
hoelzl@41981
   756
  then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
hoelzl@62975
   757
    by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) auto
hoelzl@62975
   758
  with AE_space[of M] pos show "AE x in M. f x = f' x"
hoelzl@62975
   759
    unfolding AE_density[OF h_borel] by auto
hoelzl@40859
   760
qed
hoelzl@40859
   761
hoelzl@47694
   762
lemma (in sigma_finite_measure) density_unique_iff:
hoelzl@62975
   763
  assumes f: "f \<in> borel_measurable M" and f': "f' \<in> borel_measurable M"
hoelzl@47694
   764
  shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)"
hoelzl@47694
   765
  using density_unique[OF assms] density_cong[OF f f'] by auto
hoelzl@47694
   766
ak2110@69730
   767
lemma sigma_finite_density_unique:
hoelzl@49785
   768
  assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@49785
   769
  and fin: "sigma_finite_measure (density M f)"
hoelzl@49785
   770
  shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
ak2110@69730
   771
proof
lp15@61609
   772
  assume "AE x in M. f x = g x" with borel show "density M f = density M g"
hoelzl@49785
   773
    by (auto intro: density_cong)
hoelzl@49785
   774
next
hoelzl@49785
   775
  assume eq: "density M f = density M g"
wenzelm@61605
   776
  interpret f: sigma_finite_measure "density M f" by fact
hoelzl@49785
   777
  from f.sigma_finite_incseq guess A . note cover = this
hoelzl@49785
   778
hoelzl@49785
   779
  have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x"
hoelzl@49785
   780
    unfolding AE_all_countable
hoelzl@49785
   781
  proof
hoelzl@49785
   782
    fix i
hoelzl@49785
   783
    have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))"
hoelzl@49785
   784
      unfolding eq ..
wenzelm@53015
   785
    moreover have "(\<integral>\<^sup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>"
hoelzl@49785
   786
      using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq)
hoelzl@49785
   787
    ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x"
hoelzl@62975
   788
      using borel cover(1)
hoelzl@62975
   789
      by (intro finite_density_unique[THEN iffD1]) (auto simp: density_density_eq subset_eq)
hoelzl@49785
   790
    then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x"
hoelzl@49785
   791
      by auto
hoelzl@49785
   792
  qed
hoelzl@49785
   793
  with AE_space show "AE x in M. f x = g x"
hoelzl@49785
   794
    apply eventually_elim
hoelzl@49785
   795
    using cover(2)[symmetric]
hoelzl@49785
   796
    apply auto
hoelzl@49785
   797
    done
hoelzl@49785
   798
qed
hoelzl@49785
   799
ak2110@69730
   800
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite':
hoelzl@62975
   801
  assumes f: "f \<in> borel_measurable M"
hoelzl@47694
   802
  shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
hoelzl@47694
   803
    (is "sigma_finite_measure ?N \<longleftrightarrow> _")
ak2110@69730
   804
proof
hoelzl@41689
   805
  assume "sigma_finite_measure ?N"
hoelzl@47694
   806
  then interpret N: sigma_finite_measure ?N .
hoelzl@47694
   807
  from N.Ex_finite_integrable_function obtain h where
hoelzl@56996
   808
    h: "h \<in> borel_measurable M" "integral\<^sup>N ?N h \<noteq> \<infinity>" and
hoelzl@62975
   809
    fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>"
hoelzl@62975
   810
    by auto
hoelzl@47694
   811
  have "AE x in M. f x * h x \<noteq> \<infinity>"
hoelzl@40859
   812
  proof (rule AE_I')
hoelzl@62975
   813
    have "integral\<^sup>N ?N h = (\<integral>\<^sup>+x. f x * h x \<partial>M)"
hoelzl@62975
   814
      using f h by (auto intro!: nn_integral_density)
wenzelm@53015
   815
    then have "(\<integral>\<^sup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
hoelzl@40859
   816
      using h(2) by simp
hoelzl@47694
   817
    then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M"
hoelzl@62975
   818
      using f h(1) by (auto intro!: nn_integral_PInf[unfolded infinity_ennreal_def] borel_measurable_vimage)
hoelzl@40859
   819
  qed auto
hoelzl@47694
   820
  then show "AE x in M. f x \<noteq> \<infinity>"
hoelzl@62975
   821
    using fin by (auto elim!: AE_Ball_mp simp: less_top ennreal_mult_less_top)
hoelzl@40859
   822
next
hoelzl@47694
   823
  assume AE: "AE x in M. f x \<noteq> \<infinity>"
hoelzl@57447
   824
  from sigma_finite guess Q . note Q = this
wenzelm@63040
   825
  define A where "A i =
wenzelm@63040
   826
    f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ennreal(of_nat (Suc n))}) \<inter> space M" for i
hoelzl@40859
   827
  { fix i j have "A i \<inter> Q j \<in> sets M"
hoelzl@40859
   828
    unfolding A_def using f Q
immler@50244
   829
    apply (rule_tac sets.Int)
hoelzl@41981
   830
    by (cases i) (auto intro: measurable_sets[OF f(1)]) }
hoelzl@40859
   831
  note A_in_sets = this
hoelzl@57447
   832
hoelzl@41689
   833
  show "sigma_finite_measure ?N"
wenzelm@61169
   834
  proof (standard, intro exI conjI ballI)
hoelzl@57447
   835
    show "countable (range (\<lambda>(i, j). A i \<inter> Q j))"
hoelzl@57447
   836
      by auto
hoelzl@57447
   837
    show "range (\<lambda>(i, j). A i \<inter> Q j) \<subseteq> sets (density M f)"
hoelzl@57447
   838
      using A_in_sets by auto
hoelzl@40859
   839
  next
nipkow@69745
   840
    have "\<Union>(range (\<lambda>(i, j). A i \<inter> Q j)) = (\<Union>i j. A i \<inter> Q j)"
hoelzl@57447
   841
      by auto
hoelzl@40859
   842
    also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
hoelzl@40859
   843
    also have "(\<Union>i. A i) = space M"
hoelzl@40859
   844
    proof safe
hoelzl@40859
   845
      fix x assume x: "x \<in> space M"
hoelzl@40859
   846
      show "x \<in> (\<Union>i. A i)"
hoelzl@62975
   847
      proof (cases "f x" rule: ennreal_cases)
hoelzl@62975
   848
        case top with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
hoelzl@40859
   849
      next
hoelzl@41981
   850
        case (real r)
hoelzl@62975
   851
        with ennreal_Ex_less_of_nat[of "f x"] obtain n :: nat where "f x < n"
hoelzl@62975
   852
          by auto
hoelzl@62975
   853
        also have "n < (Suc n :: ennreal)"
hoelzl@62975
   854
          by simp
hoelzl@62975
   855
        finally show ?thesis
hoelzl@62975
   856
          using x real by (auto simp: A_def ennreal_of_nat_eq_real_of_nat intro!: exI[of _ "Suc n"])
hoelzl@40859
   857
      qed
hoelzl@40859
   858
    qed (auto simp: A_def)
nipkow@69745
   859
    finally show "\<Union>(range (\<lambda>(i, j). A i \<inter> Q j)) = space ?N" by simp
hoelzl@40859
   860
  next
hoelzl@57447
   861
    fix X assume "X \<in> range (\<lambda>(i, j). A i \<inter> Q j)"
hoelzl@57447
   862
    then obtain i j where [simp]:"X = A i \<inter> Q j" by auto
wenzelm@53015
   863
    have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
hoelzl@40859
   864
    proof (cases i)
hoelzl@40859
   865
      case 0
hoelzl@47694
   866
      have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0"
wenzelm@61808
   867
        using AE by (auto simp: A_def \<open>i = 0\<close>)
hoelzl@56996
   868
      from nn_integral_cong_AE[OF this] show ?thesis by simp
hoelzl@40859
   869
    next
hoelzl@40859
   870
      case (Suc n)
wenzelm@53015
   871
      then have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
hoelzl@62975
   872
        (\<integral>\<^sup>+x. (Suc n :: ennreal) * indicator (Q j) x \<partial>M)"
hoelzl@62975
   873
        by (auto intro!: nn_integral_mono simp: indicator_def A_def ennreal_of_nat_eq_real_of_nat)
hoelzl@47694
   874
      also have "\<dots> = Suc n * emeasure M (Q j)"
hoelzl@56996
   875
        using Q by (auto intro!: nn_integral_cmult_indicator)
hoelzl@41981
   876
      also have "\<dots> < \<infinity>"
hoelzl@62975
   877
        using Q by (auto simp: ennreal_mult_less_top less_top of_nat_less_top)
hoelzl@40859
   878
      finally show ?thesis by simp
hoelzl@40859
   879
    qed
hoelzl@57447
   880
    then show "emeasure ?N X \<noteq> \<infinity>"
hoelzl@47694
   881
      using A_in_sets Q f by (auto simp: emeasure_density)
hoelzl@40859
   882
  qed
hoelzl@40859
   883
qed
hoelzl@40859
   884
hoelzl@49778
   885
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
hoelzl@49778
   886
  "f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
hoelzl@62975
   887
  by (subst sigma_finite_iff_density_finite')
hoelzl@62975
   888
     (auto simp: max_def intro!: measurable_If)
hoelzl@49778
   889
immler@69683
   890
subsection \<open>Radon-Nikodym derivative\<close>
hoelzl@38656
   891
ak2110@69173
   892
definition%important RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ennreal" where
hoelzl@56993
   893
  "RN_deriv M N =
hoelzl@62975
   894
    (if \<exists>f. f \<in> borel_measurable M \<and> density M f = N
hoelzl@62975
   895
       then SOME f. f \<in> borel_measurable M \<and> density M f = N
hoelzl@56993
   896
       else (\<lambda>_. 0))"
hoelzl@38656
   897
ak2110@69730
   898
lemma RN_derivI:
hoelzl@62975
   899
  assumes "f \<in> borel_measurable M" "density M f = N"
hoelzl@56993
   900
  shows "density M (RN_deriv M N) = N"
ak2110@69730
   901
proof -
wenzelm@63540
   902
  have *: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N"
hoelzl@56993
   903
    using assms by auto
wenzelm@63540
   904
  then have "density M (SOME f. f \<in> borel_measurable M \<and> density M f = N) = N"
hoelzl@56993
   905
    by (rule someI2_ex) auto
wenzelm@63540
   906
  with * show ?thesis
hoelzl@56993
   907
    by (auto simp: RN_deriv_def)
hoelzl@40859
   908
qed
hoelzl@40859
   909
hoelzl@62975
   910
lemma borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M"
hoelzl@38656
   911
proof -
hoelzl@62975
   912
  { assume ex: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N"
hoelzl@62975
   913
    have 1: "(SOME f. f \<in> borel_measurable M \<and> density M f = N) \<in> borel_measurable M"
hoelzl@62975
   914
      using ex by (rule someI2_ex) auto }
hoelzl@62975
   915
  from this show ?thesis
hoelzl@56993
   916
    by (auto simp: RN_deriv_def)
hoelzl@38656
   917
qed
hoelzl@38656
   918
hoelzl@56993
   919
lemma density_RN_deriv_density:
hoelzl@62975
   920
  assumes f: "f \<in> borel_measurable M"
hoelzl@56993
   921
  shows "density M (RN_deriv M (density M f)) = density M f"
hoelzl@62975
   922
  by (rule RN_derivI[OF f]) simp
hoelzl@56993
   923
hoelzl@56993
   924
lemma (in sigma_finite_measure) density_RN_deriv:
hoelzl@56993
   925
  "absolutely_continuous M N \<Longrightarrow> sets N = sets M \<Longrightarrow> density M (RN_deriv M N) = N"
hoelzl@56993
   926
  by (metis RN_derivI Radon_Nikodym)
hoelzl@56993
   927
ak2110@69730
   928
lemma (in sigma_finite_measure) RN_deriv_nn_integral:
hoelzl@47694
   929
  assumes N: "absolutely_continuous M N" "sets N = sets M"
hoelzl@40859
   930
    and f: "f \<in> borel_measurable M"
hoelzl@56996
   931
  shows "integral\<^sup>N N f = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
ak2110@69730
   932
proof -
hoelzl@56996
   933
  have "integral\<^sup>N N f = integral\<^sup>N (density M (RN_deriv M N)) f"
hoelzl@47694
   934
    using N by (simp add: density_RN_deriv)
wenzelm@53015
   935
  also have "\<dots> = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
hoelzl@62975
   936
    using f by (simp add: nn_integral_density)
hoelzl@47694
   937
  finally show ?thesis by simp
hoelzl@40859
   938
qed
hoelzl@40859
   939
hoelzl@47694
   940
lemma (in sigma_finite_measure) RN_deriv_unique:
hoelzl@62975
   941
  assumes f: "f \<in> borel_measurable M"
hoelzl@47694
   942
  and eq: "density M f = N"
hoelzl@47694
   943
  shows "AE x in M. f x = RN_deriv M N x"
hoelzl@49785
   944
  unfolding eq[symmetric]
hoelzl@56993
   945
  by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv
hoelzl@62975
   946
            density_RN_deriv_density[symmetric])
hoelzl@49785
   947
hoelzl@49785
   948
lemma RN_deriv_unique_sigma_finite:
hoelzl@62975
   949
  assumes f: "f \<in> borel_measurable M"
hoelzl@49785
   950
  and eq: "density M f = N" and fin: "sigma_finite_measure N"
hoelzl@49785
   951
  shows "AE x in M. f x = RN_deriv M N x"
hoelzl@49785
   952
  using fin unfolding eq[symmetric]
hoelzl@56993
   953
  by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv
hoelzl@62975
   954
            density_RN_deriv_density[symmetric])
hoelzl@47694
   955
ak2110@69730
   956
lemma (in sigma_finite_measure) RN_deriv_distr:
hoelzl@47694
   957
  fixes T :: "'a \<Rightarrow> 'b"
hoelzl@47694
   958
  assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
hoelzl@47694
   959
    and inv: "\<forall>x\<in>space M. T' (T x) = x"
hoelzl@50021
   960
  and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)"
hoelzl@47694
   961
  and N: "sets N = sets M"
hoelzl@47694
   962
  shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x"
ak2110@69730
   963
proof (rule RN_deriv_unique)
hoelzl@47694
   964
  have [simp]: "sets N = sets M" by fact
hoelzl@47694
   965
  note sets_eq_imp_space_eq[OF N, simp]
hoelzl@47694
   966
  have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def)
hoelzl@47694
   967
  { fix A assume "A \<in> sets M"
immler@50244
   968
    with inv T T' sets.sets_into_space[OF this]
hoelzl@47694
   969
    have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A"
hoelzl@47694
   970
      by (auto simp: measurable_def) }
hoelzl@47694
   971
  note eq = this[simp]
hoelzl@47694
   972
  { fix A assume "A \<in> sets M"
immler@50244
   973
    with inv T T' sets.sets_into_space[OF this]
hoelzl@47694
   974
    have "(T' \<circ> T) -` A \<inter> space M = A"
hoelzl@47694
   975
      by (auto simp: measurable_def) }
hoelzl@47694
   976
  note eq2 = this[simp]
hoelzl@47694
   977
  let ?M' = "distr M M' T" and ?N' = "distr N M' T"
hoelzl@47694
   978
  interpret M': sigma_finite_measure ?M'
hoelzl@41832
   979
  proof
hoelzl@57447
   980
    from sigma_finite_countable guess F .. note F = this
hoelzl@57447
   981
    show "\<exists>A. countable A \<and> A \<subseteq> sets (distr M M' T) \<and> \<Union>A = space (distr M M' T) \<and> (\<forall>a\<in>A. emeasure (distr M M' T) a \<noteq> \<infinity>)"
hoelzl@57447
   982
    proof (intro exI conjI ballI)
hoelzl@57447
   983
      show *: "(\<lambda>A. T' -` A \<inter> space ?M') ` F \<subseteq> sets ?M'"
hoelzl@47694
   984
        using F T' by (auto simp: measurable_def)
hoelzl@57447
   985
      show "\<Union>((\<lambda>A. T' -` A \<inter> space ?M')`F) = space ?M'"
hoelzl@57447
   986
        using F T'[THEN measurable_space] by (auto simp: set_eq_iff)
hoelzl@57447
   987
    next
hoelzl@57447
   988
      fix X assume "X \<in> (\<lambda>A. T' -` A \<inter> space ?M')`F"
hoelzl@57447
   989
      then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto
wenzelm@61808
   990
      have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto
hoelzl@41832
   991
      moreover
wenzelm@61808
   992
      have Fi: "A \<in> sets M" using F \<open>A\<in>F\<close> by auto
hoelzl@57447
   993
      ultimately show "emeasure ?M' X \<noteq> \<infinity>"
wenzelm@61808
   994
        using F T T' \<open>A\<in>F\<close> by (simp add: emeasure_distr)
hoelzl@57447
   995
    qed (insert F, auto)
hoelzl@41832
   996
  qed
hoelzl@47694
   997
  have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M"
hoelzl@50021
   998
    using T ac by measurable
hoelzl@47694
   999
  then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M"
hoelzl@41832
  1000
    by (simp add: comp_def)
hoelzl@47694
  1001
hoelzl@47694
  1002
  have "N = distr N M (T' \<circ> T)"
hoelzl@47694
  1003
    by (subst measure_of_of_measure[of N, symmetric])
immler@50244
  1004
       (auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed)
hoelzl@47694
  1005
  also have "\<dots> = distr (distr N M' T) M T'"
hoelzl@47694
  1006
    using T T' by (simp add: distr_distr)
hoelzl@47694
  1007
  also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'"
hoelzl@47694
  1008
    using ac by (simp add: M'.density_RN_deriv)
hoelzl@47694
  1009
  also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)"
hoelzl@56993
  1010
    by (simp add: distr_density_distr[OF T T', OF inv])
hoelzl@47694
  1011
  finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N"
hoelzl@47694
  1012
    by (simp add: comp_def)
hoelzl@41832
  1013
qed
hoelzl@41832
  1014
ak2110@69730
  1015
lemma (in sigma_finite_measure) RN_deriv_finite:
hoelzl@47694
  1016
  assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@47694
  1017
  shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>"
ak2110@69730
  1018
proof -
hoelzl@47694
  1019
  interpret N: sigma_finite_measure N by fact
hoelzl@47694
  1020
  from N show ?thesis
hoelzl@62975
  1021
    using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N] density_RN_deriv[OF ac]
hoelzl@62975
  1022
    by simp
hoelzl@40859
  1023
qed
hoelzl@40859
  1024
nipkow@69739
  1025
lemma (in sigma_finite_measure)
hoelzl@47694
  1026
  assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@40859
  1027
    and f: "f \<in> borel_measurable M"
hoelzl@47694
  1028
  shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
hoelzl@62975
  1029
      integrable M (\<lambda>x. enn2real (RN_deriv M N x) * f x)" (is ?integrable)
hoelzl@62975
  1030
    and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. enn2real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
ak2110@69730
  1031
proof -
hoelzl@47694
  1032
  note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
hoelzl@47694
  1033
  interpret N: sigma_finite_measure N by fact
hoelzl@56993
  1034
hoelzl@62975
  1035
  have eq: "density M (RN_deriv M N) = density M (\<lambda>x. enn2real (RN_deriv M N x))"
hoelzl@56993
  1036
  proof (rule density_cong)
hoelzl@56993
  1037
    from RN_deriv_finite[OF assms(1,2,3)]
hoelzl@62975
  1038
    show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
hoelzl@62975
  1039
      by eventually_elim (auto simp: less_top)
hoelzl@56993
  1040
  qed (insert ac, auto)
hoelzl@56993
  1041
hoelzl@56993
  1042
  show ?integrable
hoelzl@56993
  1043
    apply (subst density_RN_deriv[OF ac, symmetric])
hoelzl@56993
  1044
    unfolding eq
hoelzl@62975
  1045
    apply (intro integrable_real_density f AE_I2 enn2real_nonneg)
hoelzl@56993
  1046
    apply (insert ac, auto)
hoelzl@56993
  1047
    done
hoelzl@56993
  1048
hoelzl@56993
  1049
  show ?integral
hoelzl@56993
  1050
    apply (subst density_RN_deriv[OF ac, symmetric])
hoelzl@56993
  1051
    unfolding eq
hoelzl@62975
  1052
    apply (intro integral_real_density f AE_I2 enn2real_nonneg)
hoelzl@56993
  1053
    apply (insert ac, auto)
hoelzl@56993
  1054
    done
hoelzl@40859
  1055
qed
hoelzl@40859
  1056
ak2110@69730
  1057
proposition (in sigma_finite_measure) real_RN_deriv:
hoelzl@47694
  1058
  assumes "finite_measure N"
hoelzl@47694
  1059
  assumes ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@43340
  1060
  obtains D where "D \<in> borel_measurable M"
hoelzl@62975
  1061
    and "AE x in M. RN_deriv M N x = ennreal (D x)"
hoelzl@47694
  1062
    and "AE x in N. 0 < D x"
hoelzl@43340
  1063
    and "\<And>x. 0 \<le> D x"
ak2110@69730
  1064
proof
hoelzl@47694
  1065
  interpret N: finite_measure N by fact
lp15@61609
  1066
hoelzl@62975
  1067
  note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac]
hoelzl@43340
  1068
hoelzl@47694
  1069
  let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
hoelzl@43340
  1070
hoelzl@62975
  1071
  show "(\<lambda>x. enn2real (RN_deriv M N x)) \<in> borel_measurable M"
hoelzl@43340
  1072
    using RN by auto
hoelzl@43340
  1073
wenzelm@53015
  1074
  have "N (?RN \<infinity>) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)"
hoelzl@62975
  1075
    using RN(1) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
wenzelm@53015
  1076
  also have "\<dots> = (\<integral>\<^sup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
hoelzl@56996
  1077
    by (intro nn_integral_cong) (auto simp: indicator_def)
hoelzl@47694
  1078
  also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)"
hoelzl@56996
  1079
    using RN by (intro nn_integral_cmult_indicator) auto
hoelzl@47694
  1080
  finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" .
hoelzl@43340
  1081
  moreover
hoelzl@47694
  1082
  have "emeasure M (?RN \<infinity>) = 0"
hoelzl@43340
  1083
  proof (rule ccontr)
hoelzl@47694
  1084
    assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0"
hoelzl@62975
  1085
    then have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}"
hoelzl@62975
  1086
      by (auto simp: zero_less_iff_neq_zero)
hoelzl@62975
  1087
    with eq have "N (?RN \<infinity>) = \<infinity>" by (simp add: ennreal_mult_eq_top_iff)
hoelzl@47694
  1088
    with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto
hoelzl@43340
  1089
  qed
hoelzl@47694
  1090
  ultimately have "AE x in M. RN_deriv M N x < \<infinity>"
hoelzl@62975
  1091
    using RN by (intro AE_iff_measurable[THEN iffD2]) (auto simp: less_top[symmetric])
hoelzl@62975
  1092
  then show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
hoelzl@62975
  1093
    by auto
hoelzl@62975
  1094
  then have eq: "AE x in N. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
hoelzl@47694
  1095
    using ac absolutely_continuous_AE by auto
hoelzl@43340
  1096
hoelzl@43340
  1097
wenzelm@53015
  1098
  have "N (?RN 0) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)"
hoelzl@62975
  1099
    by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
wenzelm@53015
  1100
  also have "\<dots> = (\<integral>\<^sup>+ x. 0 \<partial>M)"
hoelzl@56996
  1101
    by (intro nn_integral_cong) (auto simp: indicator_def)
hoelzl@47694
  1102
  finally have "AE x in N. RN_deriv M N x \<noteq> 0"
hoelzl@47694
  1103
    using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq)
hoelzl@62975
  1104
  with eq show "AE x in N. 0 < enn2real (RN_deriv M N x)"
hoelzl@62975
  1105
    by (auto simp: enn2real_positive_iff less_top[symmetric] zero_less_iff_neq_zero)
hoelzl@62975
  1106
qed (rule enn2real_nonneg)
hoelzl@43340
  1107
hoelzl@38656
  1108
lemma (in sigma_finite_measure) RN_deriv_singleton:
hoelzl@47694
  1109
  assumes ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@47694
  1110
  and x: "{x} \<in> sets M"
hoelzl@47694
  1111
  shows "N {x} = RN_deriv M N x * emeasure M {x}"
hoelzl@38656
  1112
proof -
wenzelm@61808
  1113
  from \<open>{x} \<in> sets M\<close>
wenzelm@53015
  1114
  have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
hoelzl@56996
  1115
    by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong)
hoelzl@62975
  1116
  with x density_RN_deriv[OF ac] show ?thesis
hoelzl@62083
  1117
    by (auto simp: max_def)
hoelzl@38656
  1118
qed
hoelzl@38656
  1119
hoelzl@38656
  1120
end