src/HOL/Analysis/Radon_Nikodym.thy
author wenzelm
Tue Jan 17 13:59:10 2017 +0100 (2017-01-17)
changeset 64911 f0e07600de47
parent 64283 979cdfdf7a79
child 69173 38beaaebe736
permissions -rw-r--r--
isabelle update_cartouches -c -t;
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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 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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lemma (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 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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lemma (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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   240
    have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def
hoelzl@38656
   241
    proof safe
hoelzl@50003
   242
      show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" by measurable
hoelzl@41981
   243
    next
hoelzl@38656
   244
      fix A assume "A \<in> sets M"
wenzelm@53015
   245
      have "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
wenzelm@53015
   246
        (\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
hoelzl@56996
   247
        by (intro nn_integral_cong) (simp split: split_indicator)
wenzelm@53015
   248
      also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))"
wenzelm@61808
   249
        using \<open>incseq f\<close> f \<open>A \<in> sets M\<close>
hoelzl@56996
   250
        by (intro nn_integral_monotone_convergence_SUP)
hoelzl@41981
   251
           (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
wenzelm@53015
   252
      finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A"
hoelzl@63330
   253
        using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_D)
hoelzl@38656
   254
    qed }
hoelzl@38656
   255
  note SUP_in_G = this
hoelzl@56996
   256
  let ?y = "SUP g : G. integral\<^sup>N M g"
hoelzl@47694
   257
  have y_le: "?y \<le> N (space M)" unfolding G_def
hoelzl@44928
   258
  proof (safe intro!: SUP_least)
wenzelm@53015
   259
    fix g assume "\<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A"
hoelzl@56996
   260
    from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)"
hoelzl@56996
   261
      by (simp cong: nn_integral_cong)
hoelzl@38656
   262
  qed
hoelzl@62975
   263
  from ennreal_SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this
hoelzl@56996
   264
  then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n"
hoelzl@38656
   265
  proof safe
hoelzl@56996
   266
    fix n assume "range ys \<subseteq> integral\<^sup>N M ` G"
hoelzl@56996
   267
    hence "ys n \<in> integral\<^sup>N M ` G" by auto
hoelzl@56996
   268
    thus "\<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" by auto
hoelzl@38656
   269
  qed
hoelzl@56996
   270
  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
   271
  hence y_eq: "?y = (SUP i. integral\<^sup>N M (gs i))" using ys by auto
wenzelm@46731
   272
  let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})"
wenzelm@63040
   273
  define f where [abs_def]: "f x = (SUP i. ?g i x)" for x
wenzelm@46731
   274
  let ?F = "\<lambda>A x. f x * indicator A x"
hoelzl@41981
   275
  have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
hoelzl@38656
   276
  { fix i have "?g i \<in> G"
hoelzl@38656
   277
    proof (induct i)
hoelzl@38656
   278
      case 0 thus ?case by simp fact
hoelzl@38656
   279
    next
hoelzl@38656
   280
      case (Suc i)
wenzelm@61808
   281
      with Suc gs_not_empty \<open>gs (Suc i) \<in> G\<close> show ?case
hoelzl@38656
   282
        by (auto simp add: atMost_Suc intro!: max_in_G)
hoelzl@38656
   283
    qed }
hoelzl@38656
   284
  note g_in_G = this
hoelzl@41981
   285
  have "incseq ?g" using gs_not_empty
hoelzl@41981
   286
    by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)
hoelzl@63330
   287
hoelzl@50003
   288
  from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def .
hoelzl@63330
   289
  then have [measurable]: "f \<in> borel_measurable M" unfolding G_def by auto
hoelzl@63330
   290
hoelzl@56996
   291
  have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def
hoelzl@63330
   292
    using g_in_G \<open>incseq ?g\<close> by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def)
hoelzl@38656
   293
  also have "\<dots> = ?y"
hoelzl@38656
   294
  proof (rule antisym)
hoelzl@56996
   295
    show "(SUP i. integral\<^sup>N M (?g i)) \<le> ?y"
haftmann@56166
   296
      using g_in_G by (auto intro: SUP_mono)
hoelzl@56996
   297
    show "?y \<le> (SUP i. integral\<^sup>N M (?g i))" unfolding y_eq
hoelzl@56996
   298
      by (auto intro!: SUP_mono nn_integral_mono Max_ge)
hoelzl@38656
   299
  qed
hoelzl@56996
   300
  finally have int_f_eq_y: "integral\<^sup>N M f = ?y" .
hoelzl@47694
   301
hoelzl@63330
   302
  have upper_bound: "\<forall>A\<in>sets M. N A \<le> density M f A"
hoelzl@38656
   303
  proof (rule ccontr)
hoelzl@38656
   304
    assume "\<not> ?thesis"
hoelzl@63330
   305
    then obtain A where A[measurable]: "A \<in> sets M" and f_less_N: "density M f A < N A"
hoelzl@63330
   306
      by (auto simp: not_le)
hoelzl@63330
   307
    then have pos_A: "0 < M A"
hoelzl@63330
   308
      using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, OF A]
hoelzl@62975
   309
      by (auto simp: zero_less_iff_neq_zero)
hoelzl@63330
   310
hoelzl@63330
   311
    define b where "b = (N A - density M f A) / M A / 2"
hoelzl@63330
   312
    with f_less_N pos_A have "0 < b" "b \<noteq> top"
hoelzl@63330
   313
      by (auto intro!: diff_gr0_ennreal simp: zero_less_iff_neq_zero diff_eq_0_iff_ennreal ennreal_divide_eq_top_iff)
hoelzl@63330
   314
hoelzl@63330
   315
    let ?f = "\<lambda>x. f x + b"
hoelzl@63330
   316
    have "nn_integral M f \<noteq> top"
wenzelm@64911
   317
      using \<open>f \<in> G\<close>[THEN G_D, of "space M"] by (auto simp: top_unique cong: nn_integral_cong)
hoelzl@63330
   318
    with \<open>b \<noteq> top\<close> interpret Mf: finite_measure "density M ?f"
hoelzl@63330
   319
      by (intro finite_measureI)
hoelzl@63330
   320
         (auto simp: field_simps mult_indicator_subset ennreal_mult_eq_top_iff
hoelzl@63330
   321
                     emeasure_density nn_integral_cmult_indicator nn_integral_add
hoelzl@63330
   322
               cong: nn_integral_cong)
hoelzl@63330
   323
hoelzl@63330
   324
    from unsigned_Hahn_decomposition[of "density M ?f" N A]
hoelzl@63330
   325
    obtain Y where [measurable]: "Y \<in> sets M" and [simp]: "Y \<subseteq> A"
hoelzl@63330
   326
       and Y1: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> density M ?f C \<le> N C"
hoelzl@63330
   327
       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
   328
       by auto
hoelzl@63330
   329
hoelzl@63330
   330
    let ?f' = "\<lambda>x. f x + b * indicator Y x"
hoelzl@63330
   331
    have "M Y \<noteq> 0"
hoelzl@63330
   332
    proof
hoelzl@63330
   333
      assume "M Y = 0"
hoelzl@63330
   334
      then have "N Y = 0"
hoelzl@63330
   335
        using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, of Y] by auto
hoelzl@63330
   336
      then have "N A = N (A - Y)"
hoelzl@63330
   337
        by (subst emeasure_Diff) auto
hoelzl@63330
   338
      also have "\<dots> \<le> density M ?f (A - Y)"
hoelzl@63330
   339
        by (rule Y2) auto
hoelzl@63330
   340
      also have "\<dots> \<le> density M ?f A - density M ?f Y"
hoelzl@63330
   341
        by (subst emeasure_Diff) auto
hoelzl@63330
   342
      also have "\<dots> \<le> density M ?f A - 0"
hoelzl@63330
   343
        by (intro ennreal_minus_mono) auto
hoelzl@63330
   344
      also have "density M ?f A = b * M A + density M f A"
hoelzl@63330
   345
        by (simp add: emeasure_density field_simps mult_indicator_subset nn_integral_add nn_integral_cmult_indicator)
hoelzl@63330
   346
      also have "\<dots> < N A"
hoelzl@63330
   347
        using f_less_N pos_A
hoelzl@63330
   348
        by (cases "density M f A"; cases "M A"; cases "N A")
hoelzl@63330
   349
           (auto simp: b_def ennreal_less_iff ennreal_minus divide_ennreal ennreal_numeral[symmetric]
hoelzl@63330
   350
                       ennreal_plus[symmetric] ennreal_mult[symmetric] field_simps
hoelzl@63330
   351
                    simp del: ennreal_numeral ennreal_plus)
hoelzl@63330
   352
      finally show False
hoelzl@63330
   353
        by simp
hoelzl@63330
   354
    qed
hoelzl@63330
   355
    then have "nn_integral M f < nn_integral M ?f'"
hoelzl@63330
   356
      using \<open>0 < b\<close> \<open>nn_integral M f \<noteq> top\<close>
hoelzl@63330
   357
      by (simp add: nn_integral_add nn_integral_cmult_indicator ennreal_zero_less_mult_iff zero_less_iff_neq_zero)
hoelzl@38656
   358
    moreover
hoelzl@63330
   359
    have "?f' \<in> G"
hoelzl@63330
   360
      unfolding G_def
hoelzl@63330
   361
    proof safe
hoelzl@63330
   362
      fix X assume [measurable]: "X \<in> sets M"
hoelzl@63330
   363
      have "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) = density M f (X - Y) + density M ?f (X \<inter> Y)"
hoelzl@63330
   364
        by (auto simp add: emeasure_density nn_integral_add[symmetric] split: split_indicator intro!: nn_integral_cong)
hoelzl@63330
   365
      also have "\<dots> \<le> N (X - Y) + N (X \<inter> Y)"
hoelzl@63330
   366
        using G_D[OF \<open>f \<in> G\<close>] by (intro add_mono Y1) (auto simp: emeasure_density)
hoelzl@63330
   367
      also have "\<dots> = N X"
hoelzl@63330
   368
        by (subst plus_emeasure) (auto intro!: arg_cong2[where f=emeasure])
hoelzl@63330
   369
      finally show "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) \<le> N X" .
hoelzl@63330
   370
    qed simp
hoelzl@63330
   371
    then have "nn_integral M ?f' \<le> ?y"
hoelzl@63330
   372
      by (rule SUP_upper)
hoelzl@63330
   373
    ultimately show False
hoelzl@63330
   374
      by (simp add: int_f_eq_y)
hoelzl@38656
   375
  qed
hoelzl@38656
   376
  show ?thesis
hoelzl@63330
   377
  proof (intro bexI[of _ f] measure_eqI conjI antisym)
hoelzl@63330
   378
    fix A assume "A \<in> sets (density M f)" then show "emeasure (density M f) A \<le> emeasure N A"
hoelzl@63330
   379
      by (auto simp: emeasure_density intro!: G_D[OF \<open>f \<in> G\<close>])
hoelzl@63330
   380
  next
hoelzl@63330
   381
    fix A assume A: "A \<in> sets (density M f)" then show "emeasure N A \<le> emeasure (density M f) A"
hoelzl@63330
   382
      using upper_bound by auto
hoelzl@47694
   383
  qed auto
hoelzl@38656
   384
qed
hoelzl@38656
   385
hoelzl@40859
   386
lemma (in finite_measure) split_space_into_finite_sets_and_rest:
hoelzl@63330
   387
  assumes ac: "absolutely_continuous M N" and sets_eq[simp]: "sets N = sets M"
hoelzl@63330
   388
  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
   389
    (\<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>))"
hoelzl@38656
   390
proof -
hoelzl@47694
   391
  let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}"
hoelzl@47694
   392
  let ?a = "SUP Q:?Q. emeasure M Q"
hoelzl@47694
   393
  have "{} \<in> ?Q" by auto
hoelzl@38656
   394
  then have Q_not_empty: "?Q \<noteq> {}" by blast
immler@50244
   395
  have "?a \<le> emeasure M (space M)" using sets.sets_into_space
hoelzl@47694
   396
    by (auto intro!: SUP_least emeasure_mono)
hoelzl@62975
   397
  then have "?a \<noteq> \<infinity>"
hoelzl@62975
   398
    using finite_emeasure_space
hoelzl@62975
   399
    by (auto simp: less_top[symmetric] top_unique simp del: SUP_eq_top_iff Sup_eq_top_iff)
hoelzl@62975
   400
  from ennreal_SUP_countable_SUP [OF Q_not_empty, of "emeasure M"]
hoelzl@47694
   401
  obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
hoelzl@38656
   402
    by auto
hoelzl@47694
   403
  then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto
hoelzl@47694
   404
  from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q"
hoelzl@38656
   405
    by auto
hoelzl@47694
   406
  then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp
wenzelm@46731
   407
  let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i"
hoelzl@47694
   408
  have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)"
hoelzl@47694
   409
  proof (rule SUP_emeasure_incseq[of ?O])
hoelzl@47694
   410
    show "range ?O \<subseteq> sets M" using Q' by auto
nipkow@44890
   411
    show "incseq ?O" by (fastforce intro!: incseq_SucI)
hoelzl@38656
   412
  qed
hoelzl@63330
   413
  have Q'_sets[measurable]: "\<And>i. Q' i \<in> sets M" using Q' by auto
hoelzl@47694
   414
  have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto
hoelzl@38656
   415
  then have O_in_G: "\<And>i. ?O i \<in> ?Q"
hoelzl@38656
   416
  proof (safe del: notI)
hoelzl@47694
   417
    fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto
hoelzl@47694
   418
    then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))"
hoelzl@63330
   419
      by (simp add: emeasure_subadditive_finite)
hoelzl@62975
   420
    also have "\<dots> < \<infinity>" using Q' by (simp add: less_top)
hoelzl@47694
   421
    finally show "N (?O i) \<noteq> \<infinity>" by simp
hoelzl@38656
   422
  qed auto
nipkow@44890
   423
  have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce
hoelzl@47694
   424
  have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric]
hoelzl@38656
   425
  proof (rule antisym)
hoelzl@47694
   426
    show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim
hoelzl@47694
   427
      using Q' by (auto intro!: SUP_mono emeasure_mono)
haftmann@62343
   428
    show "(SUP i. emeasure M (?O i)) \<le> ?a"
hoelzl@38656
   429
    proof (safe intro!: Sup_mono, unfold bex_simps)
hoelzl@38656
   430
      fix i
haftmann@52141
   431
      have *: "(\<Union>(Q' ` {..i})) = ?O i" by auto
hoelzl@47694
   432
      then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and>
haftmann@52141
   433
        emeasure M (\<Union>(Q' ` {..i})) \<le> emeasure M x"
hoelzl@38656
   434
        using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
hoelzl@38656
   435
    qed
hoelzl@38656
   436
  qed
wenzelm@46731
   437
  let ?O_0 = "(\<Union>i. ?O i)"
hoelzl@38656
   438
  have "?O_0 \<in> sets M" using Q' by auto
hoelzl@63330
   439
  have "disjointed Q' i \<in> sets M" for i
hoelzl@63330
   440
    using sets.range_disjointed_sets[of Q' M] using Q'_sets by (auto simp: subset_eq)
hoelzl@38656
   441
  note Q_sets = this
hoelzl@40859
   442
  show ?thesis
hoelzl@40859
   443
  proof (intro bexI exI conjI ballI impI allI)
hoelzl@63330
   444
    show "disjoint_family (disjointed Q')"
hoelzl@63330
   445
      by (rule disjoint_family_disjointed)
hoelzl@63330
   446
    show "range (disjointed Q') \<subseteq> sets M"
hoelzl@63330
   447
      using Q'_sets by (intro sets.range_disjointed_sets) auto
hoelzl@63330
   448
    { fix A assume A: "A \<in> sets M" "A \<inter> (\<Union>i. disjointed Q' i) = {}"
hoelzl@63330
   449
      then have A1: "A \<inter> (\<Union>i. Q' i) = {}"
hoelzl@63330
   450
        unfolding UN_disjointed_eq by auto
hoelzl@47694
   451
      show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
hoelzl@40859
   452
      proof (rule disjCI, simp)
hoelzl@62975
   453
        assume *: "emeasure M A = 0 \<or> N A \<noteq> top"
hoelzl@47694
   454
        show "emeasure M A = 0 \<and> N A = 0"
wenzelm@53374
   455
        proof (cases "emeasure M A = 0")
wenzelm@53374
   456
          case True
wenzelm@53374
   457
          with ac A have "N A = 0"
hoelzl@40859
   458
            unfolding absolutely_continuous_def by auto
wenzelm@53374
   459
          with True show ?thesis by simp
hoelzl@40859
   460
        next
wenzelm@53374
   461
          case False
hoelzl@62975
   462
          with * have "N A \<noteq> \<infinity>" by auto
hoelzl@47694
   463
          with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)"
hoelzl@63330
   464
            using Q' A1 by (auto intro!: plus_emeasure simp: set_eq_iff)
hoelzl@47694
   465
          also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))"
hoelzl@47694
   466
          proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
hoelzl@40859
   467
            show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
wenzelm@61808
   468
              using \<open>N A \<noteq> \<infinity>\<close> O_sets A by auto
nipkow@44890
   469
          qed (fastforce intro!: incseq_SucI)
hoelzl@40859
   470
          also have "\<dots> \<le> ?a"
hoelzl@44928
   471
          proof (safe intro!: SUP_least)
hoelzl@40859
   472
            fix i have "?O i \<union> A \<in> ?Q"
hoelzl@40859
   473
            proof (safe del: notI)
hoelzl@40859
   474
              show "?O i \<union> A \<in> sets M" using O_sets A by auto
hoelzl@47694
   475
              from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A"
hoelzl@63330
   476
                using emeasure_subadditive[of "?O i" N A] A O_sets by auto
hoelzl@47694
   477
              with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>"
hoelzl@62975
   478
                using \<open>N A \<noteq> \<infinity>\<close> by (auto simp: top_unique)
hoelzl@40859
   479
            qed
hoelzl@47694
   480
            then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
hoelzl@40859
   481
          qed
hoelzl@47694
   482
          finally have "emeasure M A = 0"
hoelzl@47694
   483
            unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure)
wenzelm@61808
   484
          with \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto
hoelzl@40859
   485
        qed
hoelzl@40859
   486
      qed }
hoelzl@63330
   487
    { fix i
hoelzl@63330
   488
      have "N (disjointed Q' i) \<le> N (Q' i)"
hoelzl@63330
   489
        by (auto intro!: emeasure_mono simp: disjointed_def)
hoelzl@63330
   490
      then show "N (disjointed Q' i) \<noteq> \<infinity>"
hoelzl@63330
   491
        using Q'(2)[of i] by (auto simp: top_unique) }
hoelzl@40859
   492
  qed
hoelzl@40859
   493
qed
hoelzl@40859
   494
hoelzl@40859
   495
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
hoelzl@47694
   496
  assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M"
hoelzl@63329
   497
  shows "\<exists>f\<in>borel_measurable M. density M f = N"
hoelzl@40859
   498
proof -
hoelzl@40859
   499
  from split_space_into_finite_sets_and_rest[OF assms]
hoelzl@63330
   500
  obtain Q :: "nat \<Rightarrow> 'a set"
hoelzl@40859
   501
    where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
hoelzl@63330
   502
    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
   503
    and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force
hoelzl@40859
   504
  from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
hoelzl@47694
   505
  let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))"
hoelzl@63329
   506
  have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). density (?M i) f = ?N i"
hoelzl@47694
   507
  proof (intro allI finite_measure.Radon_Nikodym_finite_measure)
hoelzl@38656
   508
    fix i
hoelzl@47694
   509
    from Q show "finite_measure (?M i)"
hoelzl@56996
   510
      by (auto intro!: finite_measureI cong: nn_integral_cong
hoelzl@47694
   511
               simp add: emeasure_density subset_eq sets_eq)
hoelzl@47694
   512
    from Q have "emeasure (?N i) (space N) = emeasure N (Q i)"
hoelzl@56996
   513
      by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: nn_integral_cong)
hoelzl@47694
   514
    with Q_fin show "finite_measure (?N i)"
hoelzl@47694
   515
      by (auto intro!: finite_measureI)
hoelzl@47694
   516
    show "sets (?N i) = sets (?M i)" by (simp add: sets_eq)
hoelzl@50003
   517
    have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq)
hoelzl@47694
   518
    show "absolutely_continuous (?M i) (?N i)"
wenzelm@61808
   519
      using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close>
hoelzl@47694
   520
      by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq
hoelzl@47694
   521
               intro!: absolutely_continuous_AE[OF sets_eq])
hoelzl@38656
   522
  qed
hoelzl@47694
   523
  from choice[OF this[unfolded Bex_def]]
hoelzl@47694
   524
  obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
hoelzl@47694
   525
    and f_density: "\<And>i. density (?M i) (f i) = ?N i"
immler@54776
   526
    by force
hoelzl@47694
   527
  { fix A i assume A: "A \<in> sets M"
wenzelm@53015
   528
    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
   529
      by (auto simp add: emeasure_density nn_integral_density subset_eq
hoelzl@56996
   530
               intro!: nn_integral_cong split: split_indicator)
hoelzl@47694
   531
    also have "\<dots> = emeasure N (Q i \<inter> A)"
hoelzl@47694
   532
      using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq)
wenzelm@53015
   533
    finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. }
hoelzl@47694
   534
  note integral_eq = this
hoelzl@63330
   535
  let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator (space M - (\<Union>i. Q i)) x"
hoelzl@38656
   536
  show ?thesis
hoelzl@38656
   537
  proof (safe intro!: bexI[of _ ?f])
hoelzl@63330
   538
    show "?f \<in> borel_measurable M" using borel Q_sets
hoelzl@41981
   539
      by (auto intro!: measurable_If)
hoelzl@47694
   540
    show "density M ?f = N"
hoelzl@47694
   541
    proof (rule measure_eqI)
hoelzl@47694
   542
      fix A assume "A \<in> sets (density M ?f)"
hoelzl@47694
   543
      then have "A \<in> sets M" by simp
hoelzl@47694
   544
      have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
hoelzl@47694
   545
      have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
hoelzl@47694
   546
        "\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x"
hoelzl@63330
   547
        using borel Qi \<open>A \<in> sets M\<close> by auto
hoelzl@63330
   548
      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
   549
        using borel by (intro nn_integral_cong) (auto simp: indicator_def)
hoelzl@63330
   550
      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
   551
        using borel Qi \<open>A \<in> sets M\<close>
hoelzl@62975
   552
        by (subst nn_integral_add)
hoelzl@62975
   553
           (auto simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le)
hoelzl@63330
   554
      also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)"
wenzelm@61808
   555
        by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto
hoelzl@63330
   556
      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
   557
      moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)"
wenzelm@61808
   558
        using Q Q_sets \<open>A \<in> sets M\<close>
hoelzl@47694
   559
        by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
hoelzl@63330
   560
      moreover
hoelzl@63330
   561
      have "(space M - (\<Union>x. Q x)) \<inter> A \<inter> (\<Union>x. Q x) = {}"
hoelzl@63330
   562
        by auto
hoelzl@63330
   563
      then have "\<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A) = N ((space M - (\<Union>i. Q i)) \<inter> A)"
hoelzl@63330
   564
        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
   565
      moreover have "(space M - (\<Union>i. Q i)) \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
hoelzl@63330
   566
        using Q_sets \<open>A \<in> sets M\<close> by auto
hoelzl@63330
   567
      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
   568
        using \<open>A \<in> sets M\<close> sets.sets_into_space by auto
wenzelm@53015
   569
      ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)"
hoelzl@63330
   570
        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
   571
      with \<open>A \<in> sets M\<close> borel Q show "emeasure (density M ?f) A = N A"
hoelzl@50003
   572
        by (auto simp: subset_eq emeasure_density)
hoelzl@47694
   573
    qed (simp add: sets_eq)
hoelzl@38656
   574
  qed
hoelzl@38656
   575
qed
hoelzl@38656
   576
hoelzl@38656
   577
lemma (in sigma_finite_measure) Radon_Nikodym:
hoelzl@47694
   578
  assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M"
hoelzl@63329
   579
  shows "\<exists>f \<in> borel_measurable M. density M f = N"
hoelzl@38656
   580
proof -
hoelzl@38656
   581
  from Ex_finite_integrable_function
hoelzl@56996
   582
  obtain h where finite: "integral\<^sup>N M h \<noteq> \<infinity>" and
hoelzl@38656
   583
    borel: "h \<in> borel_measurable M" and
hoelzl@41981
   584
    nn: "\<And>x. 0 \<le> h x" and
hoelzl@38656
   585
    pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
hoelzl@41981
   586
    "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
wenzelm@53015
   587
  let ?T = "\<lambda>A. (\<integral>\<^sup>+x. h x * indicator A x \<partial>M)"
hoelzl@47694
   588
  let ?MT = "density M h"
hoelzl@47694
   589
  from borel finite nn interpret T: finite_measure ?MT
hoelzl@56996
   590
    by (auto intro!: finite_measureI cong: nn_integral_cong simp: emeasure_density)
hoelzl@47694
   591
  have "absolutely_continuous ?MT N" "sets N = sets ?MT"
hoelzl@47694
   592
  proof (unfold absolutely_continuous_def, safe)
hoelzl@47694
   593
    fix A assume "A \<in> null_sets ?MT"
hoelzl@47694
   594
    with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0"
hoelzl@47694
   595
      by (auto simp add: null_sets_density_iff)
immler@50244
   596
    with pos sets.sets_into_space have "AE x in M. x \<notin> A"
lp15@61810
   597
      by (elim eventually_mono) (auto simp: not_le[symmetric])
hoelzl@47694
   598
    then have "A \<in> null_sets M"
wenzelm@61808
   599
      using \<open>A \<in> sets M\<close> by (simp add: AE_iff_null_sets)
hoelzl@47694
   600
    with ac show "A \<in> null_sets N"
hoelzl@47694
   601
      by (auto simp: absolutely_continuous_def)
hoelzl@47694
   602
  qed (auto simp add: sets_eq)
hoelzl@47694
   603
  from T.Radon_Nikodym_finite_measure_infinite[OF this]
hoelzl@63329
   604
  obtain f where f_borel: "f \<in> borel_measurable M" "density ?MT f = N" by auto
hoelzl@47694
   605
  with nn borel show ?thesis
hoelzl@47694
   606
    by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq)
hoelzl@38656
   607
qed
hoelzl@38656
   608
wenzelm@61808
   609
subsection \<open>Uniqueness of densities\<close>
hoelzl@40859
   610
hoelzl@47694
   611
lemma finite_density_unique:
hoelzl@40859
   612
  assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@47694
   613
  assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
hoelzl@56996
   614
  and fin: "integral\<^sup>N M f \<noteq> \<infinity>"
hoelzl@49785
   615
  shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
hoelzl@40859
   616
proof (intro iffI ballI)
hoelzl@47694
   617
  fix A assume eq: "AE x in M. f x = g x"
hoelzl@49785
   618
  with borel show "density M f = density M g"
hoelzl@49785
   619
    by (auto intro: density_cong)
hoelzl@40859
   620
next
wenzelm@53015
   621
  let ?P = "\<lambda>f A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M"
hoelzl@49785
   622
  assume "density M f = density M g"
hoelzl@49785
   623
  with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
hoelzl@49785
   624
    by (simp add: emeasure_density[symmetric])
immler@50244
   625
  from this[THEN bspec, OF sets.top] fin
hoelzl@56996
   626
  have g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" by (simp cong: nn_integral_cong)
hoelzl@40859
   627
  { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@47694
   628
      and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
hoelzl@56996
   629
      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
   630
    let ?N = "{x\<in>space M. g x < f x}"
hoelzl@40859
   631
    have N: "?N \<in> sets M" using borel by simp
hoelzl@56996
   632
    have "?P g ?N \<le> integral\<^sup>N M g" using pos
hoelzl@56996
   633
      by (intro nn_integral_mono_AE) (auto split: split_indicator)
hoelzl@62975
   634
    then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by (auto simp: top_unique)
wenzelm@53015
   635
    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
   636
      by (auto intro!: nn_integral_cong simp: indicator_def)
hoelzl@40859
   637
    also have "\<dots> = ?P f ?N - ?P g ?N"
hoelzl@56996
   638
    proof (rule nn_integral_diff)
hoelzl@40859
   639
      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
   640
        using borel N by auto
hoelzl@47694
   641
      show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x"
hoelzl@41981
   642
        using pos by (auto split: split_indicator)
hoelzl@41981
   643
    qed fact
hoelzl@40859
   644
    also have "\<dots> = 0"
hoelzl@62975
   645
      unfolding eq[THEN bspec, OF N] using Pg_fin by auto
hoelzl@47694
   646
    finally have "AE x in M. f x \<le> g x"
hoelzl@56996
   647
      using pos borel nn_integral_PInf_AE[OF borel(2) g_fin]
hoelzl@56996
   648
      by (subst (asm) nn_integral_0_iff_AE)
hoelzl@62975
   649
         (auto split: split_indicator simp: not_less ennreal_minus_eq_0) }
hoelzl@41981
   650
  from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
hoelzl@47694
   651
  show "AE x in M. f x = g x" by auto
hoelzl@40859
   652
qed
hoelzl@40859
   653
hoelzl@40859
   654
lemma (in finite_measure) density_unique_finite_measure:
hoelzl@40859
   655
  assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
hoelzl@47694
   656
  assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x"
wenzelm@53015
   657
  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
   658
    (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
hoelzl@47694
   659
  shows "AE x in M. f x = f' x"
hoelzl@40859
   660
proof -
hoelzl@47694
   661
  let ?D = "\<lambda>f. density M f"
hoelzl@47694
   662
  let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A"
wenzelm@46731
   663
  let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
hoelzl@47694
   664
hoelzl@47694
   665
  have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M"
lp15@61609
   666
    using borel by (auto intro!: absolutely_continuousI_density)
hoelzl@47694
   667
  from split_space_into_finite_sets_and_rest[OF this]
hoelzl@63330
   668
  obtain Q :: "nat \<Rightarrow> 'a set"
hoelzl@40859
   669
    where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
hoelzl@63330
   670
    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
   671
    and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force
hoelzl@63330
   672
  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
   673
    and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq)
hoelzl@47694
   674
hoelzl@63330
   675
  from Q have Q_sets[measurable]: "\<And>i. Q i \<in> sets M" by auto
hoelzl@47694
   676
  let ?D = "{x\<in>space M. f x \<noteq> f' x}"
hoelzl@47694
   677
  have "?D \<in> sets M" using borel by auto
hoelzl@62975
   678
  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
   679
    unfolding indicator_def by auto
hoelzl@47694
   680
  have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
hoelzl@40859
   681
    by (intro finite_density_unique[THEN iffD1] allI)
hoelzl@50003
   682
       (auto intro!: f measure_eqI simp: emeasure_density * subset_eq)
hoelzl@63330
   683
  moreover have "AE x in M. ?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x"
hoelzl@40859
   684
  proof (rule AE_I')
hoelzl@62975
   685
    { fix f :: "'a \<Rightarrow> ennreal" assume borel: "f \<in> borel_measurable M"
wenzelm@53015
   686
        and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
hoelzl@63330
   687
      let ?A = "\<lambda>i. (space M - (\<Union>i. Q i)) \<inter> {x \<in> space M. f x < (i::nat)}"
hoelzl@47694
   688
      have "(\<Union>i. ?A i) \<in> null_sets M"
hoelzl@40859
   689
      proof (rule null_sets_UN)
hoelzl@43923
   690
        fix i ::nat have "?A i \<in> sets M"
hoelzl@63330
   691
          using borel by auto
hoelzl@62975
   692
        have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ennreal) * indicator (?A i) x \<partial>M)"
wenzelm@61808
   693
          unfolding eq[OF \<open>?A i \<in> sets M\<close>]
hoelzl@56996
   694
          by (auto intro!: nn_integral_mono simp: indicator_def)
hoelzl@47694
   695
        also have "\<dots> = i * emeasure M (?A i)"
wenzelm@61808
   696
          using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator)
hoelzl@62975
   697
        also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by (auto simp: ennreal_mult_less_top of_nat_less_top)
hoelzl@47694
   698
        finally have "?N (?A i) \<noteq> \<infinity>" by simp
wenzelm@61808
   699
        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
   700
      qed
hoelzl@63330
   701
      also have "(\<Union>i. ?A i) = (space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
hoelzl@62975
   702
        by (auto simp: ennreal_Ex_less_of_nat less_top[symmetric])
hoelzl@63330
   703
      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
   704
    from this[OF borel(1) refl] this[OF borel(2) f]
hoelzl@63330
   705
    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
   706
    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
   707
    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
   708
      ((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
   709
  qed
hoelzl@63330
   710
  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
   711
    ?f (space M) x = ?f' (space M) x"
hoelzl@63330
   712
    by (auto simp: indicator_def)
hoelzl@47694
   713
  ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x"
hoelzl@47694
   714
    unfolding AE_all_countable[symmetric]
hoelzl@63330
   715
    by eventually_elim (auto split: if_split_asm simp: indicator_def)
hoelzl@47694
   716
  then show "AE x in M. f x = f' x" by auto
hoelzl@40859
   717
qed
hoelzl@40859
   718
hoelzl@40859
   719
lemma (in sigma_finite_measure) density_unique:
hoelzl@62975
   720
  assumes f: "f \<in> borel_measurable M"
hoelzl@62975
   721
  assumes f': "f' \<in> borel_measurable M"
hoelzl@47694
   722
  assumes density_eq: "density M f = density M f'"
hoelzl@47694
   723
  shows "AE x in M. f x = f' x"
hoelzl@40859
   724
proof -
hoelzl@40859
   725
  obtain h where h_borel: "h \<in> borel_measurable M"
hoelzl@56996
   726
    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
   727
    using Ex_finite_integrable_function by auto
hoelzl@47694
   728
  then have h_nn: "AE x in M. 0 \<le> h x" by auto
hoelzl@47694
   729
  let ?H = "density M h"
hoelzl@47694
   730
  interpret h: finite_measure ?H
hoelzl@47694
   731
    using fin h_borel pos
hoelzl@56996
   732
    by (intro finite_measureI) (simp cong: nn_integral_cong emeasure_density add: fin)
hoelzl@47694
   733
  let ?fM = "density M f"
hoelzl@47694
   734
  let ?f'M = "density M f'"
hoelzl@40859
   735
  { fix A assume "A \<in> sets M"
hoelzl@41981
   736
    then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
immler@50244
   737
      using pos(1) sets.sets_into_space by (force simp: indicator_def)
wenzelm@53015
   738
    then have "(\<integral>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
wenzelm@61808
   739
      using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto }
hoelzl@40859
   740
  note h_null_sets = this
hoelzl@40859
   741
  { fix A assume "A \<in> sets M"
wenzelm@53015
   742
    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
   743
      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
hoelzl@56996
   744
      by (intro nn_integral_density[symmetric]) auto
wenzelm@53015
   745
    also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)"
hoelzl@47694
   746
      by (simp_all add: density_eq)
wenzelm@53015
   747
    also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)"
wenzelm@61808
   748
      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
hoelzl@56996
   749
      by (intro nn_integral_density) auto
wenzelm@53015
   750
    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
   751
      by (simp add: ac_simps)
wenzelm@53015
   752
    then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)"
wenzelm@61808
   753
      using \<open>A \<in> sets M\<close> h_borel h_nn f f'
hoelzl@56996
   754
      by (subst (asm) (1 2) nn_integral_density[symmetric]) auto }
hoelzl@41981
   755
  then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
hoelzl@62975
   756
    by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) auto
hoelzl@62975
   757
  with AE_space[of M] pos show "AE x in M. f x = f' x"
hoelzl@62975
   758
    unfolding AE_density[OF h_borel] by auto
hoelzl@40859
   759
qed
hoelzl@40859
   760
hoelzl@47694
   761
lemma (in sigma_finite_measure) density_unique_iff:
hoelzl@62975
   762
  assumes f: "f \<in> borel_measurable M" and f': "f' \<in> borel_measurable M"
hoelzl@47694
   763
  shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)"
hoelzl@47694
   764
  using density_unique[OF assms] density_cong[OF f f'] by auto
hoelzl@47694
   765
hoelzl@49785
   766
lemma sigma_finite_density_unique:
hoelzl@49785
   767
  assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@49785
   768
  and fin: "sigma_finite_measure (density M f)"
hoelzl@49785
   769
  shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
hoelzl@49785
   770
proof
lp15@61609
   771
  assume "AE x in M. f x = g x" with borel show "density M f = density M g"
hoelzl@49785
   772
    by (auto intro: density_cong)
hoelzl@49785
   773
next
hoelzl@49785
   774
  assume eq: "density M f = density M g"
wenzelm@61605
   775
  interpret f: sigma_finite_measure "density M f" by fact
hoelzl@49785
   776
  from f.sigma_finite_incseq guess A . note cover = this
hoelzl@49785
   777
hoelzl@49785
   778
  have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x"
hoelzl@49785
   779
    unfolding AE_all_countable
hoelzl@49785
   780
  proof
hoelzl@49785
   781
    fix i
hoelzl@49785
   782
    have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))"
hoelzl@49785
   783
      unfolding eq ..
wenzelm@53015
   784
    moreover have "(\<integral>\<^sup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>"
hoelzl@49785
   785
      using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq)
hoelzl@49785
   786
    ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x"
hoelzl@62975
   787
      using borel cover(1)
hoelzl@62975
   788
      by (intro finite_density_unique[THEN iffD1]) (auto simp: density_density_eq subset_eq)
hoelzl@49785
   789
    then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x"
hoelzl@49785
   790
      by auto
hoelzl@49785
   791
  qed
hoelzl@49785
   792
  with AE_space show "AE x in M. f x = g x"
hoelzl@49785
   793
    apply eventually_elim
hoelzl@49785
   794
    using cover(2)[symmetric]
hoelzl@49785
   795
    apply auto
hoelzl@49785
   796
    done
hoelzl@49785
   797
qed
hoelzl@49785
   798
hoelzl@49778
   799
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite':
hoelzl@62975
   800
  assumes f: "f \<in> borel_measurable M"
hoelzl@47694
   801
  shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
hoelzl@47694
   802
    (is "sigma_finite_measure ?N \<longleftrightarrow> _")
hoelzl@40859
   803
proof
hoelzl@41689
   804
  assume "sigma_finite_measure ?N"
hoelzl@47694
   805
  then interpret N: sigma_finite_measure ?N .
hoelzl@47694
   806
  from N.Ex_finite_integrable_function obtain h where
hoelzl@56996
   807
    h: "h \<in> borel_measurable M" "integral\<^sup>N ?N h \<noteq> \<infinity>" and
hoelzl@62975
   808
    fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>"
hoelzl@62975
   809
    by auto
hoelzl@47694
   810
  have "AE x in M. f x * h x \<noteq> \<infinity>"
hoelzl@40859
   811
  proof (rule AE_I')
hoelzl@62975
   812
    have "integral\<^sup>N ?N h = (\<integral>\<^sup>+x. f x * h x \<partial>M)"
hoelzl@62975
   813
      using f h by (auto intro!: nn_integral_density)
wenzelm@53015
   814
    then have "(\<integral>\<^sup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
hoelzl@40859
   815
      using h(2) by simp
hoelzl@47694
   816
    then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M"
hoelzl@62975
   817
      using f h(1) by (auto intro!: nn_integral_PInf[unfolded infinity_ennreal_def] borel_measurable_vimage)
hoelzl@40859
   818
  qed auto
hoelzl@47694
   819
  then show "AE x in M. f x \<noteq> \<infinity>"
hoelzl@62975
   820
    using fin by (auto elim!: AE_Ball_mp simp: less_top ennreal_mult_less_top)
hoelzl@40859
   821
next
hoelzl@47694
   822
  assume AE: "AE x in M. f x \<noteq> \<infinity>"
hoelzl@57447
   823
  from sigma_finite guess Q . note Q = this
wenzelm@63040
   824
  define A where "A i =
wenzelm@63040
   825
    f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ennreal(of_nat (Suc n))}) \<inter> space M" for i
hoelzl@40859
   826
  { fix i j have "A i \<inter> Q j \<in> sets M"
hoelzl@40859
   827
    unfolding A_def using f Q
immler@50244
   828
    apply (rule_tac sets.Int)
hoelzl@41981
   829
    by (cases i) (auto intro: measurable_sets[OF f(1)]) }
hoelzl@40859
   830
  note A_in_sets = this
hoelzl@57447
   831
hoelzl@41689
   832
  show "sigma_finite_measure ?N"
wenzelm@61169
   833
  proof (standard, intro exI conjI ballI)
hoelzl@57447
   834
    show "countable (range (\<lambda>(i, j). A i \<inter> Q j))"
hoelzl@57447
   835
      by auto
hoelzl@57447
   836
    show "range (\<lambda>(i, j). A i \<inter> Q j) \<subseteq> sets (density M f)"
hoelzl@57447
   837
      using A_in_sets by auto
hoelzl@40859
   838
  next
hoelzl@57447
   839
    have "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = (\<Union>i j. A i \<inter> Q j)"
hoelzl@57447
   840
      by auto
hoelzl@40859
   841
    also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
hoelzl@40859
   842
    also have "(\<Union>i. A i) = space M"
hoelzl@40859
   843
    proof safe
hoelzl@40859
   844
      fix x assume x: "x \<in> space M"
hoelzl@40859
   845
      show "x \<in> (\<Union>i. A i)"
hoelzl@62975
   846
      proof (cases "f x" rule: ennreal_cases)
hoelzl@62975
   847
        case top with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
hoelzl@40859
   848
      next
hoelzl@41981
   849
        case (real r)
hoelzl@62975
   850
        with ennreal_Ex_less_of_nat[of "f x"] obtain n :: nat where "f x < n"
hoelzl@62975
   851
          by auto
hoelzl@62975
   852
        also have "n < (Suc n :: ennreal)"
hoelzl@62975
   853
          by simp
hoelzl@62975
   854
        finally show ?thesis
hoelzl@62975
   855
          using x real by (auto simp: A_def ennreal_of_nat_eq_real_of_nat intro!: exI[of _ "Suc n"])
hoelzl@40859
   856
      qed
hoelzl@40859
   857
    qed (auto simp: A_def)
hoelzl@57447
   858
    finally show "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = space ?N" by simp
hoelzl@40859
   859
  next
hoelzl@57447
   860
    fix X assume "X \<in> range (\<lambda>(i, j). A i \<inter> Q j)"
hoelzl@57447
   861
    then obtain i j where [simp]:"X = A i \<inter> Q j" by auto
wenzelm@53015
   862
    have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
hoelzl@40859
   863
    proof (cases i)
hoelzl@40859
   864
      case 0
hoelzl@47694
   865
      have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0"
wenzelm@61808
   866
        using AE by (auto simp: A_def \<open>i = 0\<close>)
hoelzl@56996
   867
      from nn_integral_cong_AE[OF this] show ?thesis by simp
hoelzl@40859
   868
    next
hoelzl@40859
   869
      case (Suc n)
wenzelm@53015
   870
      then have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
hoelzl@62975
   871
        (\<integral>\<^sup>+x. (Suc n :: ennreal) * indicator (Q j) x \<partial>M)"
hoelzl@62975
   872
        by (auto intro!: nn_integral_mono simp: indicator_def A_def ennreal_of_nat_eq_real_of_nat)
hoelzl@47694
   873
      also have "\<dots> = Suc n * emeasure M (Q j)"
hoelzl@56996
   874
        using Q by (auto intro!: nn_integral_cmult_indicator)
hoelzl@41981
   875
      also have "\<dots> < \<infinity>"
hoelzl@62975
   876
        using Q by (auto simp: ennreal_mult_less_top less_top of_nat_less_top)
hoelzl@40859
   877
      finally show ?thesis by simp
hoelzl@40859
   878
    qed
hoelzl@57447
   879
    then show "emeasure ?N X \<noteq> \<infinity>"
hoelzl@47694
   880
      using A_in_sets Q f by (auto simp: emeasure_density)
hoelzl@40859
   881
  qed
hoelzl@40859
   882
qed
hoelzl@40859
   883
hoelzl@49778
   884
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
hoelzl@49778
   885
  "f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
hoelzl@62975
   886
  by (subst sigma_finite_iff_density_finite')
hoelzl@62975
   887
     (auto simp: max_def intro!: measurable_If)
hoelzl@49778
   888
wenzelm@61808
   889
subsection \<open>Radon-Nikodym derivative\<close>
hoelzl@38656
   890
hoelzl@62975
   891
definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ennreal" where
hoelzl@56993
   892
  "RN_deriv M N =
hoelzl@62975
   893
    (if \<exists>f. f \<in> borel_measurable M \<and> density M f = N
hoelzl@62975
   894
       then SOME f. f \<in> borel_measurable M \<and> density M f = N
hoelzl@56993
   895
       else (\<lambda>_. 0))"
hoelzl@38656
   896
lp15@61609
   897
lemma RN_derivI:
hoelzl@62975
   898
  assumes "f \<in> borel_measurable M" "density M f = N"
hoelzl@56993
   899
  shows "density M (RN_deriv M N) = N"
hoelzl@40859
   900
proof -
wenzelm@63540
   901
  have *: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N"
hoelzl@56993
   902
    using assms by auto
wenzelm@63540
   903
  then have "density M (SOME f. f \<in> borel_measurable M \<and> density M f = N) = N"
hoelzl@56993
   904
    by (rule someI2_ex) auto
wenzelm@63540
   905
  with * show ?thesis
hoelzl@56993
   906
    by (auto simp: RN_deriv_def)
hoelzl@40859
   907
qed
hoelzl@40859
   908
hoelzl@62975
   909
lemma borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M"
hoelzl@38656
   910
proof -
hoelzl@62975
   911
  { assume ex: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N"
hoelzl@62975
   912
    have 1: "(SOME f. f \<in> borel_measurable M \<and> density M f = N) \<in> borel_measurable M"
hoelzl@62975
   913
      using ex by (rule someI2_ex) auto }
hoelzl@62975
   914
  from this show ?thesis
hoelzl@56993
   915
    by (auto simp: RN_deriv_def)
hoelzl@38656
   916
qed
hoelzl@38656
   917
hoelzl@56993
   918
lemma density_RN_deriv_density:
hoelzl@62975
   919
  assumes f: "f \<in> borel_measurable M"
hoelzl@56993
   920
  shows "density M (RN_deriv M (density M f)) = density M f"
hoelzl@62975
   921
  by (rule RN_derivI[OF f]) simp
hoelzl@56993
   922
hoelzl@56993
   923
lemma (in sigma_finite_measure) density_RN_deriv:
hoelzl@56993
   924
  "absolutely_continuous M N \<Longrightarrow> sets N = sets M \<Longrightarrow> density M (RN_deriv M N) = N"
hoelzl@56993
   925
  by (metis RN_derivI Radon_Nikodym)
hoelzl@56993
   926
hoelzl@56996
   927
lemma (in sigma_finite_measure) RN_deriv_nn_integral:
hoelzl@47694
   928
  assumes N: "absolutely_continuous M N" "sets N = sets M"
hoelzl@40859
   929
    and f: "f \<in> borel_measurable M"
hoelzl@56996
   930
  shows "integral\<^sup>N N f = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
hoelzl@40859
   931
proof -
hoelzl@56996
   932
  have "integral\<^sup>N N f = integral\<^sup>N (density M (RN_deriv M N)) f"
hoelzl@47694
   933
    using N by (simp add: density_RN_deriv)
wenzelm@53015
   934
  also have "\<dots> = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
hoelzl@62975
   935
    using f by (simp add: nn_integral_density)
hoelzl@47694
   936
  finally show ?thesis by simp
hoelzl@40859
   937
qed
hoelzl@40859
   938
hoelzl@47694
   939
lemma (in sigma_finite_measure) RN_deriv_unique:
hoelzl@62975
   940
  assumes f: "f \<in> borel_measurable M"
hoelzl@47694
   941
  and eq: "density M f = N"
hoelzl@47694
   942
  shows "AE x in M. f x = RN_deriv M N x"
hoelzl@49785
   943
  unfolding eq[symmetric]
hoelzl@56993
   944
  by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv
hoelzl@62975
   945
            density_RN_deriv_density[symmetric])
hoelzl@49785
   946
hoelzl@49785
   947
lemma RN_deriv_unique_sigma_finite:
hoelzl@62975
   948
  assumes f: "f \<in> borel_measurable M"
hoelzl@49785
   949
  and eq: "density M f = N" and fin: "sigma_finite_measure N"
hoelzl@49785
   950
  shows "AE x in M. f x = RN_deriv M N x"
hoelzl@49785
   951
  using fin unfolding eq[symmetric]
hoelzl@56993
   952
  by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv
hoelzl@62975
   953
            density_RN_deriv_density[symmetric])
hoelzl@47694
   954
hoelzl@47694
   955
lemma (in sigma_finite_measure) RN_deriv_distr:
hoelzl@47694
   956
  fixes T :: "'a \<Rightarrow> 'b"
hoelzl@47694
   957
  assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
hoelzl@47694
   958
    and inv: "\<forall>x\<in>space M. T' (T x) = x"
hoelzl@50021
   959
  and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)"
hoelzl@47694
   960
  and N: "sets N = sets M"
hoelzl@47694
   961
  shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x"
hoelzl@41832
   962
proof (rule RN_deriv_unique)
hoelzl@47694
   963
  have [simp]: "sets N = sets M" by fact
hoelzl@47694
   964
  note sets_eq_imp_space_eq[OF N, simp]
hoelzl@47694
   965
  have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def)
hoelzl@47694
   966
  { fix A assume "A \<in> sets M"
immler@50244
   967
    with inv T T' sets.sets_into_space[OF this]
hoelzl@47694
   968
    have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A"
hoelzl@47694
   969
      by (auto simp: measurable_def) }
hoelzl@47694
   970
  note eq = this[simp]
hoelzl@47694
   971
  { fix A assume "A \<in> sets M"
immler@50244
   972
    with inv T T' sets.sets_into_space[OF this]
hoelzl@47694
   973
    have "(T' \<circ> T) -` A \<inter> space M = A"
hoelzl@47694
   974
      by (auto simp: measurable_def) }
hoelzl@47694
   975
  note eq2 = this[simp]
hoelzl@47694
   976
  let ?M' = "distr M M' T" and ?N' = "distr N M' T"
hoelzl@47694
   977
  interpret M': sigma_finite_measure ?M'
hoelzl@41832
   978
  proof
hoelzl@57447
   979
    from sigma_finite_countable guess F .. note F = this
hoelzl@57447
   980
    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
   981
    proof (intro exI conjI ballI)
hoelzl@57447
   982
      show *: "(\<lambda>A. T' -` A \<inter> space ?M') ` F \<subseteq> sets ?M'"
hoelzl@47694
   983
        using F T' by (auto simp: measurable_def)
hoelzl@57447
   984
      show "\<Union>((\<lambda>A. T' -` A \<inter> space ?M')`F) = space ?M'"
hoelzl@57447
   985
        using F T'[THEN measurable_space] by (auto simp: set_eq_iff)
hoelzl@57447
   986
    next
hoelzl@57447
   987
      fix X assume "X \<in> (\<lambda>A. T' -` A \<inter> space ?M')`F"
hoelzl@57447
   988
      then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto
wenzelm@61808
   989
      have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto
hoelzl@41832
   990
      moreover
wenzelm@61808
   991
      have Fi: "A \<in> sets M" using F \<open>A\<in>F\<close> by auto
hoelzl@57447
   992
      ultimately show "emeasure ?M' X \<noteq> \<infinity>"
wenzelm@61808
   993
        using F T T' \<open>A\<in>F\<close> by (simp add: emeasure_distr)
hoelzl@57447
   994
    qed (insert F, auto)
hoelzl@41832
   995
  qed
hoelzl@47694
   996
  have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M"
hoelzl@50021
   997
    using T ac by measurable
hoelzl@47694
   998
  then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M"
hoelzl@41832
   999
    by (simp add: comp_def)
hoelzl@47694
  1000
hoelzl@47694
  1001
  have "N = distr N M (T' \<circ> T)"
hoelzl@47694
  1002
    by (subst measure_of_of_measure[of N, symmetric])
immler@50244
  1003
       (auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed)
hoelzl@47694
  1004
  also have "\<dots> = distr (distr N M' T) M T'"
hoelzl@47694
  1005
    using T T' by (simp add: distr_distr)
hoelzl@47694
  1006
  also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'"
hoelzl@47694
  1007
    using ac by (simp add: M'.density_RN_deriv)
hoelzl@47694
  1008
  also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)"
hoelzl@56993
  1009
    by (simp add: distr_density_distr[OF T T', OF inv])
hoelzl@47694
  1010
  finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N"
hoelzl@47694
  1011
    by (simp add: comp_def)
hoelzl@41832
  1012
qed
hoelzl@41832
  1013
hoelzl@40859
  1014
lemma (in sigma_finite_measure) RN_deriv_finite:
hoelzl@47694
  1015
  assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@47694
  1016
  shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>"
hoelzl@40859
  1017
proof -
hoelzl@47694
  1018
  interpret N: sigma_finite_measure N by fact
hoelzl@47694
  1019
  from N show ?thesis
hoelzl@62975
  1020
    using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N] density_RN_deriv[OF ac]
hoelzl@62975
  1021
    by simp
hoelzl@40859
  1022
qed
hoelzl@40859
  1023
hoelzl@40859
  1024
lemma (in sigma_finite_measure)
hoelzl@47694
  1025
  assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@40859
  1026
    and f: "f \<in> borel_measurable M"
hoelzl@47694
  1027
  shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
hoelzl@62975
  1028
      integrable M (\<lambda>x. enn2real (RN_deriv M N x) * f x)" (is ?integrable)
hoelzl@62975
  1029
    and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. enn2real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
hoelzl@40859
  1030
proof -
hoelzl@47694
  1031
  note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
hoelzl@47694
  1032
  interpret N: sigma_finite_measure N by fact
hoelzl@56993
  1033
hoelzl@62975
  1034
  have eq: "density M (RN_deriv M N) = density M (\<lambda>x. enn2real (RN_deriv M N x))"
hoelzl@56993
  1035
  proof (rule density_cong)
hoelzl@56993
  1036
    from RN_deriv_finite[OF assms(1,2,3)]
hoelzl@62975
  1037
    show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
hoelzl@62975
  1038
      by eventually_elim (auto simp: less_top)
hoelzl@56993
  1039
  qed (insert ac, auto)
hoelzl@56993
  1040
hoelzl@56993
  1041
  show ?integrable
hoelzl@56993
  1042
    apply (subst density_RN_deriv[OF ac, symmetric])
hoelzl@56993
  1043
    unfolding eq
hoelzl@62975
  1044
    apply (intro integrable_real_density f AE_I2 enn2real_nonneg)
hoelzl@56993
  1045
    apply (insert ac, auto)
hoelzl@56993
  1046
    done
hoelzl@56993
  1047
hoelzl@56993
  1048
  show ?integral
hoelzl@56993
  1049
    apply (subst density_RN_deriv[OF ac, symmetric])
hoelzl@56993
  1050
    unfolding eq
hoelzl@62975
  1051
    apply (intro integral_real_density f AE_I2 enn2real_nonneg)
hoelzl@56993
  1052
    apply (insert ac, auto)
hoelzl@56993
  1053
    done
hoelzl@40859
  1054
qed
hoelzl@40859
  1055
hoelzl@43340
  1056
lemma (in sigma_finite_measure) real_RN_deriv:
hoelzl@47694
  1057
  assumes "finite_measure N"
hoelzl@47694
  1058
  assumes ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@43340
  1059
  obtains D where "D \<in> borel_measurable M"
hoelzl@62975
  1060
    and "AE x in M. RN_deriv M N x = ennreal (D x)"
hoelzl@47694
  1061
    and "AE x in N. 0 < D x"
hoelzl@43340
  1062
    and "\<And>x. 0 \<le> D x"
hoelzl@43340
  1063
proof
hoelzl@47694
  1064
  interpret N: finite_measure N by fact
lp15@61609
  1065
hoelzl@62975
  1066
  note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac]
hoelzl@43340
  1067
hoelzl@47694
  1068
  let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
hoelzl@43340
  1069
hoelzl@62975
  1070
  show "(\<lambda>x. enn2real (RN_deriv M N x)) \<in> borel_measurable M"
hoelzl@43340
  1071
    using RN by auto
hoelzl@43340
  1072
wenzelm@53015
  1073
  have "N (?RN \<infinity>) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)"
hoelzl@62975
  1074
    using RN(1) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
wenzelm@53015
  1075
  also have "\<dots> = (\<integral>\<^sup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
hoelzl@56996
  1076
    by (intro nn_integral_cong) (auto simp: indicator_def)
hoelzl@47694
  1077
  also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)"
hoelzl@56996
  1078
    using RN by (intro nn_integral_cmult_indicator) auto
hoelzl@47694
  1079
  finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" .
hoelzl@43340
  1080
  moreover
hoelzl@47694
  1081
  have "emeasure M (?RN \<infinity>) = 0"
hoelzl@43340
  1082
  proof (rule ccontr)
hoelzl@47694
  1083
    assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0"
hoelzl@62975
  1084
    then have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}"
hoelzl@62975
  1085
      by (auto simp: zero_less_iff_neq_zero)
hoelzl@62975
  1086
    with eq have "N (?RN \<infinity>) = \<infinity>" by (simp add: ennreal_mult_eq_top_iff)
hoelzl@47694
  1087
    with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto
hoelzl@43340
  1088
  qed
hoelzl@47694
  1089
  ultimately have "AE x in M. RN_deriv M N x < \<infinity>"
hoelzl@62975
  1090
    using RN by (intro AE_iff_measurable[THEN iffD2]) (auto simp: less_top[symmetric])
hoelzl@62975
  1091
  then show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
hoelzl@62975
  1092
    by auto
hoelzl@62975
  1093
  then have eq: "AE x in N. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
hoelzl@47694
  1094
    using ac absolutely_continuous_AE by auto
hoelzl@43340
  1095
hoelzl@43340
  1096
wenzelm@53015
  1097
  have "N (?RN 0) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)"
hoelzl@62975
  1098
    by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
wenzelm@53015
  1099
  also have "\<dots> = (\<integral>\<^sup>+ x. 0 \<partial>M)"
hoelzl@56996
  1100
    by (intro nn_integral_cong) (auto simp: indicator_def)
hoelzl@47694
  1101
  finally have "AE x in N. RN_deriv M N x \<noteq> 0"
hoelzl@47694
  1102
    using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq)
hoelzl@62975
  1103
  with eq show "AE x in N. 0 < enn2real (RN_deriv M N x)"
hoelzl@62975
  1104
    by (auto simp: enn2real_positive_iff less_top[symmetric] zero_less_iff_neq_zero)
hoelzl@62975
  1105
qed (rule enn2real_nonneg)
hoelzl@43340
  1106
hoelzl@38656
  1107
lemma (in sigma_finite_measure) RN_deriv_singleton:
hoelzl@47694
  1108
  assumes ac: "absolutely_continuous M N" "sets N = sets M"
hoelzl@47694
  1109
  and x: "{x} \<in> sets M"
hoelzl@47694
  1110
  shows "N {x} = RN_deriv M N x * emeasure M {x}"
hoelzl@38656
  1111
proof -
wenzelm@61808
  1112
  from \<open>{x} \<in> sets M\<close>
wenzelm@53015
  1113
  have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
hoelzl@56996
  1114
    by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong)
hoelzl@62975
  1115
  with x density_RN_deriv[OF ac] show ?thesis
hoelzl@62083
  1116
    by (auto simp: max_def)
hoelzl@38656
  1117
qed
hoelzl@38656
  1118
hoelzl@38656
  1119
end