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src/HOL/Probability/Radon_Nikodym.thy

author | haftmann |

Sat, 25 May 2013 15:44:29 +0200 | |

changeset 52141 | eff000cab70f |

parent 51329 | 4a3c453f99a1 |

child 53015 | a1119cf551e8 |

permissions | -rw-r--r-- |

weaker precendence of syntax for big intersection and union on sets

(* Title: HOL/Probability/Radon_Nikodym.thy Author: Johannes Hölzl, TU München *) header {*Radon-Nikod{\'y}m derivative*} theory Radon_Nikodym imports Lebesgue_Integration begin definition "diff_measure M N = measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)" lemma shows space_diff_measure[simp]: "space (diff_measure M N) = space M" and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M" by (auto simp: diff_measure_def) lemma emeasure_diff_measure: assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N" assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M" shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?\<mu> A") unfolding diff_measure_def proof (rule emeasure_measure_of_sigma) show "sigma_algebra (space M) (sets M)" .. show "positive (sets M) ?\<mu>" using pos by (simp add: positive_def ereal_diff_positive) show "countably_additive (sets M) ?\<mu>" proof (rule countably_additiveI) fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> sets M" and "disjoint_family A" then have suminf: "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)" "(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)" by (simp_all add: suminf_emeasure sets_eq) with A have "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) = (\<Sum>i. emeasure M (A i)) - (\<Sum>i. emeasure N (A i))" using fin by (intro suminf_ereal_minus pos emeasure_nonneg) (auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure) then show "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) = emeasure M (\<Union>i. A i) - emeasure N (\<Union>i. A i) " by (simp add: suminf) qed qed fact lemma (in sigma_finite_measure) Ex_finite_integrable_function: shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>) \<and> (\<forall>x. 0 \<le> h x)" proof - obtain A :: "nat \<Rightarrow> 'a set" where range[measurable]: "range A \<subseteq> sets M" and space: "(\<Union>i. A i) = space M" and measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and disjoint: "disjoint_family A" using sigma_finite_disjoint by auto let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)" have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)" proof fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)" using measure[of i] emeasure_nonneg[of M "A i"] by (auto intro!: dense simp: ereal_0_gt_inverse ereal_zero_le_0_iff) qed from choice[OF this] obtain n where n: "\<And>i. 0 < n i" "\<And>i. n i < inverse (2^Suc i * emeasure M (A i))" by auto { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x" show ?thesis proof (safe intro!: bexI[of _ ?h] del: notI) have "\<And>i. A i \<in> sets M" using range by fastforce+ then have "integral\<^isup>P M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos by (simp add: positive_integral_suminf positive_integral_cmult_indicator) also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)" proof (rule suminf_le_pos) fix N have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)" using n[of N] by (intro ereal_mult_right_mono) auto also have "\<dots> \<le> (1 / 2) ^ Suc N" using measure[of N] n[of N] by (cases rule: ereal2_cases[of "n N" "emeasure M (A N)"]) (simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide) finally show "n N * emeasure M (A N) \<le> (1 / 2) ^ Suc N" . show "0 \<le> n N * emeasure M (A N)" using n[of N] `A N \<in> sets M` by (simp add: emeasure_nonneg) qed finally show "integral\<^isup>P M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto next { fix x assume "x \<in> space M" then obtain i where "x \<in> A i" using space[symmetric] by auto with disjoint n have "?h x = n i" by (auto intro!: suminf_cmult_indicator intro: less_imp_le) then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by auto } note pos = this fix x show "0 \<le> ?h x" proof cases assume "x \<in> space M" then show "0 \<le> ?h x" using pos by (auto intro: less_imp_le) next assume "x \<notin> space M" then have "\<And>i. x \<notin> A i" using space by auto then show "0 \<le> ?h x" by auto qed qed measurable qed subsection "Absolutely continuous" definition absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where "absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N" lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M" unfolding absolutely_continuous_def by (auto simp: null_sets_count_space) lemma absolutely_continuousI_density: "f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)" by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in) lemma absolutely_continuousI_point_measure_finite: "(\<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)" unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff) lemma absolutely_continuous_AE: assumes sets_eq: "sets M' = sets M" and "absolutely_continuous M M'" "AE x in M. P x" shows "AE x in M'. P x" proof - from `AE x in M. P x` obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N" unfolding eventually_ae_filter by auto show "AE x in M'. P x" proof (rule AE_I') show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp from `absolutely_continuous M M'` show "N \<in> null_sets M'" using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto qed qed subsection "Existence of the Radon-Nikodym derivative" lemma (in finite_measure) Radon_Nikodym_aux_epsilon: fixes e :: real assumes "0 < e" assumes "finite_measure N" and sets_eq: "sets N = sets M" shows "\<exists>A\<in>sets M. measure M (space M) - measure N (space M) \<le> measure M A - measure N A \<and> (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < measure M B - measure N B)" proof - interpret M': finite_measure N by fact let ?d = "\<lambda>A. measure M A - measure N A" let ?A = "\<lambda>A. if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B) then {} else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)" def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}" have A_simps[simp]: "A 0 = {}" "\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all { fix A assume "A \<in> sets M" have "?A A \<in> sets M" by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) } note A'_in_sets = this { fix n have "A n \<in> sets M" proof (induct n) case (Suc n) thus "A (Suc n) \<in> sets M" using A'_in_sets[of "A n"] by (auto split: split_if_asm) qed (simp add: A_def) } note A_in_sets = this hence "range A \<subseteq> sets M" by auto { fix n B assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e" hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less) have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False] proof (rule someI2_ex[OF Ex]) fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e" hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto hence "?d (A n \<union> B) = ?d (A n) + ?d B" using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq) also have "\<dots> \<le> ?d (A n) - e" using dB by simp finally show "?d (A n \<union> B) \<le> ?d (A n) - e" . qed } note dA_epsilon = this { fix n have "?d (A (Suc n)) \<le> ?d (A n)" proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e") case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp next case False hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le) thus ?thesis by simp qed } note dA_mono = this show ?thesis proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B") case True then obtain n where B: "\<And>B. \<lbrakk> B \<in> sets M; B \<subseteq> space M - A n\<rbrakk> \<Longrightarrow> -e < ?d B" by blast show ?thesis proof (safe intro!: bexI[of _ "space M - A n"]) fix B assume "B \<in> sets M" "B \<subseteq> space M - A n" from B[OF this] show "-e < ?d B" . next show "space M - A n \<in> sets M" by (rule sets.compl_sets) fact next show "?d (space M) \<le> ?d (space M - A n)" proof (induct n) fix n assume "?d (space M) \<le> ?d (space M - A n)" also have "\<dots> \<le> ?d (space M - A (Suc n))" using A_in_sets sets.sets_into_space dA_mono[of n] finite_measure_compl M'.finite_measure_compl by (simp del: A_simps add: sets_eq sets_eq_imp_space_eq[OF sets_eq]) finally show "?d (space M) \<le> ?d (space M - A (Suc n))" . qed simp qed next case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e" by (auto simp add: not_less) { fix n have "?d (A n) \<le> - real n * e" proof (induct n) case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps) next case 0 with measure_empty show ?case by (simp add: zero_ereal_def) qed } note dA_less = this have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq proof (rule incseq_SucI) fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto qed have A: "incseq A" by (auto intro!: incseq_SucI) from finite_Lim_measure_incseq[OF _ A] `range A \<subseteq> sets M` M'.finite_Lim_measure_incseq[OF _ A] have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq) obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less] have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps) ultimately show ?thesis by auto qed qed lemma (in finite_measure) Radon_Nikodym_aux: assumes "finite_measure N" and sets_eq: "sets N = sets M" shows "\<exists>A\<in>sets M. measure M (space M) - measure N (space M) \<le> measure M A - measure N A \<and> (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> measure M B - measure N B)" proof - interpret N: finite_measure N by fact let ?d = "\<lambda>A. measure M A - measure N A" let ?P = "\<lambda>A B n. A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)" let ?r = "\<lambda>S. restricted_space S" { fix S n assume S: "S \<in> sets M" then have "finite_measure (density M (indicator S))" "0 < 1 / real (Suc n)" "finite_measure (density N (indicator S))" "sets (density N (indicator S)) = sets (density M (indicator S))" by (auto simp: finite_measure_restricted N.finite_measure_restricted sets_eq) from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this with S have "?P (S \<inter> X) S n" by (simp add: measure_restricted sets_eq sets.Int) (metis inf_absorb2) hence "\<exists>A. ?P A S n" .. } note Ex_P = this def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)" have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def) have A_0[simp]: "A 0 = space M" unfolding A_def by simp { fix i have "A i \<in> sets M" unfolding A_def proof (induct i) case (Suc i) from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc by (rule someI2_ex) simp qed simp } note A_in_sets = this { fix n have "?P (A (Suc n)) (A n) n" using Ex_P[OF A_in_sets] unfolding A_Suc by (rule someI2_ex) simp } note P_A = this have "range A \<subseteq> sets M" using A_in_sets by auto have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc) have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C" using P_A by auto show ?thesis proof (safe intro!: bexI[of _ "\<Inter>i. A i"]) show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto have A: "decseq A" using A_mono by (auto intro!: decseq_SucI) from `range A \<subseteq> sets M` finite_Lim_measure_decseq[OF _ A] N.finite_Lim_measure_decseq[OF _ A] have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq) thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _] by (rule_tac LIMSEQ_le_const) auto next fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)" show "0 \<le> ?d B" proof (rule ccontr) assume "\<not> 0 \<le> ?d B" hence "0 < - ?d B" by auto from ex_inverse_of_nat_Suc_less[OF this] obtain n where *: "?d B < - 1 / real (Suc n)" by (auto simp: real_eq_of_nat inverse_eq_divide field_simps) have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc) from epsilon[OF B(1) this] * show False by auto qed qed qed lemma (in finite_measure) Radon_Nikodym_finite_measure: assumes "finite_measure N" and sets_eq: "sets N = sets M" assumes "absolutely_continuous M N" shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" proof - interpret N: finite_measure N by fact def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> N A)}" { fix f have "f \<in> G \<Longrightarrow> f \<in> borel_measurable M" by (auto simp: G_def) } note this[measurable_dest] have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto hence "G \<noteq> {}" by auto { fix f g assume f: "f \<in> G" and g: "g \<in> G" have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def proof safe show "?max \<in> borel_measurable M" using f g unfolding G_def by auto let ?A = "{x \<in> space M. f x \<le> g x}" have "?A \<in> sets M" using f g unfolding G_def by auto fix A assume "A \<in> sets M" hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto hence sets': "?A \<inter> A \<in> sets N" "(space M - ?A) \<inter> A \<in> sets N" by (auto simp: sets_eq) have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A" using sets.sets_into_space[OF `A \<in> sets M`] by auto have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x = g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x" by (auto simp: indicator_def max_def) hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) = (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x \<partial>M) + (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)" using f g sets unfolding G_def by (auto cong: positive_integral_cong intro!: positive_integral_add) also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)" using f g sets unfolding G_def by (auto intro!: add_mono) also have "\<dots> = N A" using plus_emeasure[OF sets'] union by auto finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" . next fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max) qed } note max_in_G = this { fix f assume "incseq f" and f: "\<And>i. f i \<in> G" then have [measurable]: "\<And>i. f i \<in> borel_measurable M" by (auto simp: G_def) have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def proof safe show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" by measurable { fix x show "0 \<le> (SUP i. f i x)" using f by (auto simp: G_def intro: SUP_upper2) } next fix A assume "A \<in> sets M" have "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) = (\<integral>\<^isup>+x. (SUP i. f i x * indicator A x) \<partial>M)" by (intro positive_integral_cong) (simp split: split_indicator) also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))" using `incseq f` f `A \<in> sets M` by (intro positive_integral_monotone_convergence_SUP) (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator) finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A" using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def) qed } note SUP_in_G = this let ?y = "SUP g : G. integral\<^isup>P M g" have y_le: "?y \<le> N (space M)" unfolding G_def proof (safe intro!: SUP_least) fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> N A" from this[THEN bspec, OF sets.top] show "integral\<^isup>P M g \<le> N (space M)" by (simp cong: positive_integral_cong) qed from SUPR_countable_SUPR[OF `G \<noteq> {}`, of "integral\<^isup>P M"] guess ys .. note ys = this then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" proof safe fix n assume "range ys \<subseteq> integral\<^isup>P M ` G" hence "ys n \<in> integral\<^isup>P M ` G" by auto thus "\<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" by auto qed from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^isup>P M (gs n) = ys n" by auto hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})" def f \<equiv> "\<lambda>x. SUP i. ?g i x" let ?F = "\<lambda>A x. f x * indicator A x" have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto { fix i have "?g i \<in> G" proof (induct i) case 0 thus ?case by simp fact next case (Suc i) with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case by (auto simp add: atMost_Suc intro!: max_in_G) qed } note g_in_G = this have "incseq ?g" using gs_not_empty by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc) from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def . then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto have "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))" unfolding f_def using g_in_G `incseq ?g` by (auto intro!: positive_integral_monotone_convergence_SUP simp: G_def) also have "\<dots> = ?y" proof (rule antisym) show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y" using g_in_G by (auto intro: Sup_mono simp: SUP_def) show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq by (auto intro!: SUP_mono positive_integral_mono Max_ge) qed finally have int_f_eq_y: "integral\<^isup>P M f = ?y" . have "\<And>x. 0 \<le> f x" unfolding f_def using `\<And>i. gs i \<in> G` by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def) let ?t = "\<lambda>A. N A - (\<integral>\<^isup>+x. ?F A x \<partial>M)" let ?M = "diff_measure N (density M f)" have f_le_N: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` unfolding G_def by auto have emeasure_M: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure ?M A = ?t A" proof (subst emeasure_diff_measure) from f_le_N[of "space M"] show "finite_measure N" "finite_measure (density M f)" by (auto intro!: finite_measureI simp: emeasure_density cong: positive_integral_cong) next fix B assume "B \<in> sets N" with f_le_N[of B] show "emeasure (density M f) B \<le> emeasure N B" by (auto simp: sets_eq emeasure_density cong: positive_integral_cong) qed (auto simp: sets_eq emeasure_density) from emeasure_M[of "space M"] N.finite_emeasure_space positive_integral_positive[of M "?F (space M)"] interpret M': finite_measure ?M by (auto intro!: finite_measureI simp: sets_eq_imp_space_eq[OF sets_eq] N.emeasure_eq_measure ereal_minus_eq_PInfty_iff) have ac: "absolutely_continuous M ?M" unfolding absolutely_continuous_def proof fix A assume A: "A \<in> null_sets M" with `absolutely_continuous M N` have "A \<in> null_sets N" unfolding absolutely_continuous_def by auto moreover with A have "(\<integral>\<^isup>+ x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` by (auto simp: G_def) ultimately have "N A - (\<integral>\<^isup>+ x. ?F A x \<partial>M) = 0" using positive_integral_positive[of M] by (auto intro!: antisym) then show "A \<in> null_sets ?M" using A by (simp add: emeasure_M null_sets_def sets_eq) qed have upper_bound: "\<forall>A\<in>sets M. ?M A \<le> 0" proof (rule ccontr) assume "\<not> ?thesis" then obtain A where A: "A \<in> sets M" and pos: "0 < ?M A" by (auto simp: not_le) note pos also have "?M A \<le> ?M (space M)" using emeasure_space[of ?M A] by (simp add: sets_eq[THEN sets_eq_imp_space_eq]) finally have pos_t: "0 < ?M (space M)" by simp moreover then have "emeasure M (space M) \<noteq> 0" using ac unfolding absolutely_continuous_def by (auto simp: null_sets_def) then have pos_M: "0 < emeasure M (space M)" using emeasure_nonneg[of M "space M"] by (simp add: le_less) moreover have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> N (space M)" using `f \<in> G` unfolding G_def by auto hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>" using M'.finite_emeasure_space by auto moreover def b \<equiv> "?M (space M) / emeasure M (space M) / 2" ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>" by (auto simp: ereal_divide_eq) then have b: "b \<noteq> 0" "0 \<le> b" "0 < b" "b \<noteq> \<infinity>" by auto let ?Mb = "density M (\<lambda>_. b)" have Mb: "finite_measure ?Mb" "sets ?Mb = sets ?M" using b by (auto simp: emeasure_density_const sets_eq intro!: finite_measureI) from M'.Radon_Nikodym_aux[OF this] guess A0 .. then have "A0 \<in> sets M" and space_less_A0: "measure ?M (space M) - real b * measure M (space M) \<le> measure ?M A0 - real b * measure M A0" and *: "\<And>B. B \<in> sets M \<Longrightarrow> B \<subseteq> A0 \<Longrightarrow> 0 \<le> measure ?M B - real b * measure M B" using b by (simp_all add: measure_density_const sets_eq_imp_space_eq[OF sets_eq] sets_eq) { fix B assume B: "B \<in> sets M" "B \<subseteq> A0" with *[OF this] have "b * emeasure M B \<le> ?M B" using b unfolding M'.emeasure_eq_measure emeasure_eq_measure by (cases b) auto } note bM_le_t = this let ?f0 = "\<lambda>x. f x + b * indicator A0 x" { fix A assume A: "A \<in> sets M" hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)" by (auto intro!: positive_integral_cong split: split_indicator) hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * emeasure M (A \<inter> A0)" using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G` by (simp add: positive_integral_add positive_integral_cmult_indicator G_def) } note f0_eq = this { fix A assume A: "A \<in> sets M" hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto have f_le_v: "(\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` A unfolding G_def by auto note f0_eq[OF A] also have "(\<integral>\<^isup>+x. ?F A x \<partial>M) + b * emeasure M (A \<inter> A0) \<le> (\<integral>\<^isup>+x. ?F A x \<partial>M) + ?M (A \<inter> A0)" using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M` by (auto intro!: add_left_mono) also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?M A" using emeasure_mono[of "A \<inter> A0" A ?M] `A \<in> sets M` `A0 \<in> sets M` by (auto intro!: add_left_mono simp: sets_eq) also have "\<dots> \<le> N A" unfolding emeasure_M[OF `A \<in> sets M`] using f_le_v N.emeasure_eq_measure[of A] positive_integral_positive[of M "?F A"] by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "N A") auto finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> N A" . } hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` by (auto intro!: ereal_add_nonneg_nonneg simp: G_def) have int_f_finite: "integral\<^isup>P M f \<noteq> \<infinity>" by (metis N.emeasure_finite ereal_infty_less_eq2(1) int_f_eq_y y_le) have "0 < ?M (space M) - emeasure ?Mb (space M)" using pos_t by (simp add: b emeasure_density_const) (simp add: M'.emeasure_eq_measure emeasure_eq_measure pos_M b_def) also have "\<dots> \<le> ?M A0 - b * emeasure M A0" using space_less_A0 `A0 \<in> sets M` b by (cases b) (auto simp add: b emeasure_density_const sets_eq M'.emeasure_eq_measure emeasure_eq_measure) finally have 1: "b * emeasure M A0 < ?M A0" by (metis M'.emeasure_real `A0 \<in> sets M` bM_le_t diff_self ereal_less(1) ereal_minus(1) less_eq_ereal_def mult_zero_left not_square_less_zero subset_refl zero_ereal_def) with b have "0 < ?M A0" by (metis M'.emeasure_real MInfty_neq_PInfty(1) emeasure_real ereal_less_eq(5) ereal_zero_times ereal_mult_eq_MInfty ereal_mult_eq_PInfty ereal_zero_less_0_iff less_eq_ereal_def) then have "emeasure M A0 \<noteq> 0" using ac `A0 \<in> sets M` by (auto simp: absolutely_continuous_def null_sets_def) then have "0 < emeasure M A0" using emeasure_nonneg[of M A0] by auto hence "0 < b * emeasure M A0" using b by (auto simp: ereal_zero_less_0_iff) with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * emeasure M A0" unfolding int_f_eq_y using `f \<in> G` by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 positive_integral_positive) also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF sets.top] `A0 \<in> sets M` sets.sets_into_space by (simp cong: positive_integral_cong) finally have "?y < integral\<^isup>P M ?f0" by simp moreover from `?f0 \<in> G` have "integral\<^isup>P M ?f0 \<le> ?y" by (auto intro!: SUP_upper) ultimately show False by auto qed let ?f = "\<lambda>x. max 0 (f x)" show ?thesis proof (intro bexI[of _ ?f] measure_eqI conjI) show "sets (density M ?f) = sets N" by (simp add: sets_eq) fix A assume A: "A\<in>sets (density M ?f)" then show "emeasure (density M ?f) A = emeasure N A" using `f \<in> G` A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A] by (cases "integral\<^isup>P M (?F A)") (auto intro!: antisym simp add: emeasure_density G_def emeasure_M density_max_0[symmetric]) qed auto qed lemma (in finite_measure) split_space_into_finite_sets_and_rest: assumes ac: "absolutely_continuous M N" and sets_eq: "sets N = sets M" shows "\<exists>A0\<in>sets M. \<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> A0 = space M - (\<Union>i. B i) \<and> (\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>)) \<and> (\<forall>i. N (B i) \<noteq> \<infinity>)" proof - let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}" let ?a = "SUP Q:?Q. emeasure M Q" have "{} \<in> ?Q" by auto then have Q_not_empty: "?Q \<noteq> {}" by blast have "?a \<le> emeasure M (space M)" using sets.sets_into_space by (auto intro!: SUP_least emeasure_mono) then have "?a \<noteq> \<infinity>" using finite_emeasure_space by auto from SUPR_countable_SUPR[OF Q_not_empty, of "emeasure M"] obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)" by auto then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q" by auto then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i" have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)" proof (rule SUP_emeasure_incseq[of ?O]) show "range ?O \<subseteq> sets M" using Q' by auto show "incseq ?O" by (fastforce intro!: incseq_SucI) qed have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto then have O_in_G: "\<And>i. ?O i \<in> ?Q" proof (safe del: notI) fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))" by (simp add: sets_eq emeasure_subadditive_finite) also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty) finally show "N (?O i) \<noteq> \<infinity>" by simp qed auto have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric] proof (rule antisym) show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim using Q' by (auto intro!: SUP_mono emeasure_mono) show "(SUP i. emeasure M (?O i)) \<le> ?a" unfolding SUP_def proof (safe intro!: Sup_mono, unfold bex_simps) fix i have *: "(\<Union>(Q' ` {..i})) = ?O i" by auto then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and> emeasure M (\<Union>(Q' ` {..i})) \<le> emeasure M x" using O_in_G[of i] by (auto intro!: exI[of _ "?O i"]) qed qed let ?O_0 = "(\<Union>i. ?O i)" have "?O_0 \<in> sets M" using Q' by auto def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n" { fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) } note Q_sets = this show ?thesis proof (intro bexI exI conjI ballI impI allI) show "disjoint_family Q" by (fastforce simp: disjoint_family_on_def Q_def split: nat.split_asm) show "range Q \<subseteq> sets M" using Q_sets by auto { fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0" show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" proof (rule disjCI, simp) assume *: "0 < emeasure M A \<longrightarrow> N A \<noteq> \<infinity>" show "emeasure M A = 0 \<and> N A = 0" proof cases assume "emeasure M A = 0" moreover with ac A have "N A = 0" unfolding absolutely_continuous_def by auto ultimately show ?thesis by simp next assume "emeasure M A \<noteq> 0" with * have "N A \<noteq> \<infinity>" using emeasure_nonneg[of M A] by auto with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)" using Q' by (auto intro!: plus_emeasure sets.countable_UN) also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))" proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified]) show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M" using `N A \<noteq> \<infinity>` O_sets A by auto qed (fastforce intro!: incseq_SucI) also have "\<dots> \<le> ?a" proof (safe intro!: SUP_least) fix i have "?O i \<union> A \<in> ?Q" proof (safe del: notI) show "?O i \<union> A \<in> sets M" using O_sets A by auto from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A" using emeasure_subadditive[of "?O i" N A] A O_sets by (auto simp: sets_eq) with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>" using `N A \<noteq> \<infinity>` by auto qed then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper) qed finally have "emeasure M A = 0" unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure) with `emeasure M A \<noteq> 0` show ?thesis by auto qed qed } { fix i show "N (Q i) \<noteq> \<infinity>" proof (cases i) case 0 then show ?thesis unfolding Q_def using Q'[of 0] by simp next case (Suc n) with `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q` emeasure_Diff[OF _ _ _ O_mono, of N n] emeasure_nonneg[of N "(\<Union> x\<le>n. Q' x)"] show ?thesis by (auto simp: sets_eq ereal_minus_eq_PInfty_iff Q_def) qed } show "space M - ?O_0 \<in> sets M" using Q'_sets by auto { fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)" proof (induct j) case 0 then show ?case by (simp add: Q_def) next case (Suc j) have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastforce have "{..j} \<union> {..Suc j} = {..Suc j}" by auto then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)" by (simp add: UN_Un[symmetric] Q_def del: UN_Un) then show ?case using Suc by (auto simp add: eq atMost_Suc) qed } then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastforce qed qed lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite: assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M" shows "\<exists>f\<in>borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" proof - from split_space_into_finite_sets_and_rest[OF assms] obtain Q0 and Q :: "nat \<Rightarrow> 'a set" where Q: "disjoint_family Q" "range Q \<subseteq> sets M" and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)" and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))" have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). (\<forall>x. 0 \<le> f x) \<and> density (?M i) f = ?N i" proof (intro allI finite_measure.Radon_Nikodym_finite_measure) fix i from Q show "finite_measure (?M i)" by (auto intro!: finite_measureI cong: positive_integral_cong simp add: emeasure_density subset_eq sets_eq) from Q have "emeasure (?N i) (space N) = emeasure N (Q i)" by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: positive_integral_cong) with Q_fin show "finite_measure (?N i)" by (auto intro!: finite_measureI) show "sets (?N i) = sets (?M i)" by (simp add: sets_eq) have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq) show "absolutely_continuous (?M i) (?N i)" using `absolutely_continuous M N` `Q i \<in> sets M` by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq intro!: absolutely_continuous_AE[OF sets_eq]) qed from choice[OF this[unfolded Bex_def]] obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x" and f_density: "\<And>i. density (?M i) (f i) = ?N i" by auto { fix A i assume A: "A \<in> sets M" with Q borel have "(\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A" by (auto simp add: emeasure_density positive_integral_density subset_eq intro!: positive_integral_cong split: split_indicator) also have "\<dots> = emeasure N (Q i \<inter> A)" using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq) finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. } note integral_eq = this let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x" show ?thesis proof (safe intro!: bexI[of _ ?f]) show "?f \<in> borel_measurable M" using Q0 borel Q_sets by (auto intro!: measurable_If) show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def) show "density M ?f = N" proof (rule measure_eqI) fix A assume "A \<in> sets (density M ?f)" then have "A \<in> sets M" by simp have Qi: "\<And>i. Q i \<in> sets M" using Q by auto have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x" using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times) have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)" using borel by (intro positive_integral_cong) (auto simp: indicator_def) also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M (Q0 \<inter> A)" using borel Qi Q0(1) `A \<in> sets M` by (subst positive_integral_add) (auto simp del: ereal_infty_mult simp add: positive_integral_cmult_indicator sets.Int intro!: suminf_0_le) also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" by (subst integral_eq[OF `A \<in> sets M`], subst positive_integral_suminf) auto finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" . moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)" using Q Q_sets `A \<in> sets M` by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq) moreover have "\<infinity> * emeasure M (Q0 \<inter> A) = N (Q0 \<inter> A)" proof - have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast from in_Q0[OF this] show ?thesis by auto qed moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M" using Q_sets `A \<in> sets M` Q0(1) by auto moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}" using `A \<in> sets M` sets.sets_into_space Q0 by auto ultimately have "N A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)" using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "Q0 \<inter> A"] by (simp add: sets_eq) with `A \<in> sets M` borel Q Q0(1) show "emeasure (density M ?f) A = N A" by (auto simp: subset_eq emeasure_density) qed (simp add: sets_eq) qed qed lemma (in sigma_finite_measure) Radon_Nikodym: assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M" shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" proof - from Ex_finite_integrable_function obtain h where finite: "integral\<^isup>P M h \<noteq> \<infinity>" and borel: "h \<in> borel_measurable M" and nn: "\<And>x. 0 \<le> h x" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto let ?T = "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M)" let ?MT = "density M h" from borel finite nn interpret T: finite_measure ?MT by (auto intro!: finite_measureI cong: positive_integral_cong simp: emeasure_density) have "absolutely_continuous ?MT N" "sets N = sets ?MT" proof (unfold absolutely_continuous_def, safe) fix A assume "A \<in> null_sets ?MT" with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0" by (auto simp add: null_sets_density_iff) with pos sets.sets_into_space have "AE x in M. x \<notin> A" by (elim eventually_elim1) (auto simp: not_le[symmetric]) then have "A \<in> null_sets M" using `A \<in> sets M` by (simp add: AE_iff_null_sets) with ac show "A \<in> null_sets N" by (auto simp: absolutely_continuous_def) qed (auto simp add: sets_eq) from T.Radon_Nikodym_finite_measure_infinite[OF this] obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "density ?MT f = N" by auto with nn borel show ?thesis by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq) qed section "Uniqueness of densities" lemma finite_density_unique: assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" and fin: "integral\<^isup>P M f \<noteq> \<infinity>" shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" proof (intro iffI ballI) fix A assume eq: "AE x in M. f x = g x" with borel show "density M f = density M g" by (auto intro: density_cong) next let ?P = "\<lambda>f A. \<integral>\<^isup>+ x. f x * indicator A x \<partial>M" assume "density M f = density M g" with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" by (simp add: emeasure_density[symmetric]) from this[THEN bspec, OF sets.top] fin have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong) { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" let ?N = "{x\<in>space M. g x < f x}" have N: "?N \<in> sets M" using borel by simp have "?P g ?N \<le> integral\<^isup>P M g" using pos by (intro positive_integral_mono_AE) (auto split: split_indicator) then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)" by (auto intro!: positive_integral_cong simp: indicator_def) also have "\<dots> = ?P f ?N - ?P g ?N" proof (rule positive_integral_diff) show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M" using borel N by auto show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x" "AE x in M. 0 \<le> g x * indicator ?N x" using pos by (auto split: split_indicator) qed fact also have "\<dots> = 0" unfolding eq[THEN bspec, OF N] using positive_integral_positive[of M] Pg_fin by auto finally have "AE x in M. f x \<le> g x" using pos borel positive_integral_PInf_AE[OF borel(2) g_fin] by (subst (asm) positive_integral_0_iff_AE) (auto split: split_indicator simp: not_less ereal_minus_le_iff) } from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq show "AE x in M. f x = g x" by auto qed lemma (in finite_measure) density_unique_finite_measure: assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M" assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x" assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)" (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A") shows "AE x in M. f x = f' x" proof - let ?D = "\<lambda>f. density M f" let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A" let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x" have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M" using borel by (auto intro!: absolutely_continuousI_density) from split_space_into_finite_sets_and_rest[OF this] obtain Q0 and Q :: "nat \<Rightarrow> 'a set" where Q: "disjoint_family Q" "range Q \<subseteq> sets M" and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)" and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>" and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>" and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq) from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto let ?D = "{x\<in>space M. f x \<noteq> f' x}" have "?D \<in> sets M" using borel by auto have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x" unfolding indicator_def by auto have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos by (intro finite_density_unique[THEN iffD1] allI) (auto intro!: f measure_eqI simp: emeasure_density * subset_eq) moreover have "AE x in M. ?f Q0 x = ?f' Q0 x" proof (rule AE_I') { fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)" let ?A = "\<lambda>i. Q0 \<inter> {x \<in> space M. f x < (i::nat)}" have "(\<Union>i. ?A i) \<in> null_sets M" proof (rule null_sets_UN) fix i ::nat have "?A i \<in> sets M" using borel Q0(1) by auto have "?N (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)" unfolding eq[OF `?A i \<in> sets M`] by (auto intro!: positive_integral_mono simp: indicator_def) also have "\<dots> = i * emeasure M (?A i)" using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator) also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by simp finally have "?N (?A i) \<noteq> \<infinity>" by simp then show "?A i \<in> null_sets M" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto qed also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}" by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat) finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp } from this[OF borel(1) refl] this[OF borel(2) f] have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un) show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq> (Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def) qed moreover have "AE x in M. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow> ?f (space M) x = ?f' (space M) x" by (auto simp: indicator_def Q0) ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x" unfolding AE_all_countable[symmetric] by eventually_elim (auto intro!: AE_I2 split: split_if_asm simp: indicator_def) then show "AE x in M. f x = f' x" by auto qed lemma (in sigma_finite_measure) density_unique: assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" assumes f': "f' \<in> borel_measurable M" "AE x in M. 0 \<le> f' x" assumes density_eq: "density M f = density M f'" shows "AE x in M. f x = f' x" proof - obtain h where h_borel: "h \<in> borel_measurable M" and fin: "integral\<^isup>P 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" using Ex_finite_integrable_function by auto then have h_nn: "AE x in M. 0 \<le> h x" by auto let ?H = "density M h" interpret h: finite_measure ?H using fin h_borel pos by (intro finite_measureI) (simp cong: positive_integral_cong emeasure_density add: fin) let ?fM = "density M f" let ?f'M = "density M f'" { fix A assume "A \<in> sets M" then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A" using pos(1) sets.sets_into_space by (force simp: indicator_def) then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M" using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto } note h_null_sets = this { fix A assume "A \<in> sets M" have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)" using `A \<in> sets M` h_borel h_nn f f' by (intro positive_integral_density[symmetric]) auto also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)" by (simp_all add: density_eq) also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)" using `A \<in> sets M` h_borel h_nn f f' by (intro positive_integral_density) auto finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)" by (simp add: ac_simps) then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)" using `A \<in> sets M` h_borel h_nn f f' by (subst (asm) (1 2) positive_integral_density[symmetric]) auto } then have "AE x in ?H. f x = f' x" using h_borel h_nn f f' by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) (auto simp add: AE_density) then show "AE x in M. f x = f' x" unfolding eventually_ae_filter using h_borel pos by (auto simp add: h_null_sets null_sets_density_iff not_less[symmetric] AE_iff_null_sets[symmetric]) blast qed lemma (in sigma_finite_measure) density_unique_iff: assumes f: "f \<in> borel_measurable M" and "AE x in M. 0 \<le> f x" assumes f': "f' \<in> borel_measurable M" and "AE x in M. 0 \<le> f' x" shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)" using density_unique[OF assms] density_cong[OF f f'] by auto lemma sigma_finite_density_unique: assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" and fin: "sigma_finite_measure (density M f)" shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" proof assume "AE x in M. f x = g x" with borel show "density M f = density M g" by (auto intro: density_cong) next assume eq: "density M f = density M g" interpret f!: sigma_finite_measure "density M f" by fact from f.sigma_finite_incseq guess A . note cover = this have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x" unfolding AE_all_countable proof fix i have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))" unfolding eq .. moreover have "(\<integral>\<^isup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>" using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq) ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x" using borel pos cover(1) pos by (intro finite_density_unique[THEN iffD1]) (auto simp: density_density_eq subset_eq) then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x" by auto qed with AE_space show "AE x in M. f x = g x" apply eventually_elim using cover(2)[symmetric] apply auto done qed lemma (in sigma_finite_measure) sigma_finite_iff_density_finite': assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" (is "sigma_finite_measure ?N \<longleftrightarrow> _") proof assume "sigma_finite_measure ?N" then interpret N: sigma_finite_measure ?N . from N.Ex_finite_integrable_function obtain h where h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and h_nn: "\<And>x. 0 \<le> h x" and fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto have "AE x in M. f x * h x \<noteq> \<infinity>" proof (rule AE_I') have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn by (auto intro!: positive_integral_density) then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>" using h(2) by simp then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M" using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage) qed auto then show "AE x in M. f x \<noteq> \<infinity>" using fin by (auto elim!: AE_Ball_mp) next assume AE: "AE x in M. f x \<noteq> \<infinity>" from sigma_finite guess Q .. note Q = this def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M" { fix i j have "A i \<inter> Q j \<in> sets M" unfolding A_def using f Q apply (rule_tac sets.Int) by (cases i) (auto intro: measurable_sets[OF f(1)]) } note A_in_sets = this let ?A = "\<lambda>n. case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j" show "sigma_finite_measure ?N" proof (default, intro exI conjI subsetI allI) fix x assume "x \<in> range ?A" then obtain n where n: "x = ?A n" by auto then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto next have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)" proof safe fix x i j assume "x \<in> A i" "x \<in> Q j" then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)" by (intro UN_I[of "prod_encode (i,j)"]) auto qed auto also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto also have "(\<Union>i. A i) = space M" proof safe fix x assume x: "x \<in> space M" show "x \<in> (\<Union>i. A i)" proof (cases "f x") case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0]) next case (real r) with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by (auto simp: real_eq_of_nat) then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"] simp: real_eq_of_nat) next case MInf with x show ?thesis unfolding A_def by (auto intro!: exI[of _ "Suc 0"]) qed qed (auto simp: A_def) finally show "(\<Union>i. ?A i) = space ?N" by simp next fix n obtain i j where [simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>" proof (cases i) case 0 have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0" using AE by (auto simp: A_def `i = 0`) from positive_integral_cong_AE[OF this] show ?thesis by simp next case (Suc n) then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le> (\<integral>\<^isup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)" by (auto intro!: positive_integral_mono simp: indicator_def A_def real_eq_of_nat) also have "\<dots> = Suc n * emeasure M (Q j)" using Q by (auto intro!: positive_integral_cmult_indicator) also have "\<dots> < \<infinity>" using Q by (auto simp: real_eq_of_nat[symmetric]) finally show ?thesis by simp qed then show "emeasure ?N (?A n) \<noteq> \<infinity>" using A_in_sets Q f by (auto simp: emeasure_density) qed qed lemma (in sigma_finite_measure) sigma_finite_iff_density_finite: "f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" apply (subst density_max_0) apply (subst sigma_finite_iff_density_finite') apply (auto simp: max_def intro!: measurable_If) done section "Radon-Nikodym derivative" definition "RN_deriv M N \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N" lemma assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" shows borel_measurable_RN_deriv_density: "RN_deriv M (density M f) \<in> borel_measurable M" (is ?borel) and density_RN_deriv_density: "density M (RN_deriv M (density M f)) = density M f" (is ?density) and RN_deriv_density_nonneg: "0 \<le> RN_deriv M (density M f) x" (is ?pos) proof - let ?f = "\<lambda>x. max 0 (f x)" let ?P = "\<lambda>g. g \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> g x) \<and> density M g = density M f" from f have "?P ?f" using f by (auto intro!: density_cong simp: split: split_max) then have "?P (RN_deriv M (density M f))" unfolding RN_deriv_def by (rule someI[where P="?P"]) then show ?borel ?density ?pos by auto qed lemma (in sigma_finite_measure) RN_deriv: assumes "absolutely_continuous M N" "sets N = sets M" shows borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M" (is ?borel) and density_RN_deriv: "density M (RN_deriv M N) = N" (is ?density) and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?pos) proof - note Ex = Radon_Nikodym[OF assms, unfolded Bex_def] from Ex show ?borel unfolding RN_deriv_def by (rule someI2_ex) simp from Ex show ?density unfolding RN_deriv_def by (rule someI2_ex) simp from Ex show ?pos unfolding RN_deriv_def by (rule someI2_ex) simp qed lemma (in sigma_finite_measure) RN_deriv_positive_integral: assumes N: "absolutely_continuous M N" "sets N = sets M" and f: "f \<in> borel_measurable M" shows "integral\<^isup>P N f = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)" proof - have "integral\<^isup>P N f = integral\<^isup>P (density M (RN_deriv M N)) f" using N by (simp add: density_RN_deriv) also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)" using RN_deriv(1,3)[OF N] f by (simp add: positive_integral_density) finally show ?thesis by simp qed lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N" using AE_iff_null_sets[of N M] by auto lemma (in sigma_finite_measure) RN_deriv_unique: assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" and eq: "density M f = N" shows "AE x in M. f x = RN_deriv M N x" unfolding eq[symmetric] by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv_density RN_deriv_density_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric]) lemma RN_deriv_unique_sigma_finite: assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" and eq: "density M f = N" and fin: "sigma_finite_measure N" shows "AE x in M. f x = RN_deriv M N x" using fin unfolding eq[symmetric] by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv_density RN_deriv_density_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric]) lemma (in sigma_finite_measure) RN_deriv_distr: fixes T :: "'a \<Rightarrow> 'b" assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M" and inv: "\<forall>x\<in>space M. T' (T x) = x" and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)" and N: "sets N = sets M" shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x" proof (rule RN_deriv_unique) have [simp]: "sets N = sets M" by fact note sets_eq_imp_space_eq[OF N, simp] have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def) { fix A assume "A \<in> sets M" with inv T T' sets.sets_into_space[OF this] have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A" by (auto simp: measurable_def) } note eq = this[simp] { fix A assume "A \<in> sets M" with inv T T' sets.sets_into_space[OF this] have "(T' \<circ> T) -` A \<inter> space M = A" by (auto simp: measurable_def) } note eq2 = this[simp] let ?M' = "distr M M' T" and ?N' = "distr N M' T" interpret M': sigma_finite_measure ?M' proof from sigma_finite guess F .. note F = this show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets ?M' \<and> (\<Union>i. A i) = space ?M' \<and> (\<forall>i. emeasure ?M' (A i) \<noteq> \<infinity>)" proof (intro exI conjI allI) show *: "range (\<lambda>i. T' -` F i \<inter> space ?M') \<subseteq> sets ?M'" using F T' by (auto simp: measurable_def) show "(\<Union>i. T' -` F i \<inter> space ?M') = space ?M'" using F T' by (force simp: measurable_def) fix i have "T' -` F i \<inter> space M' \<in> sets M'" using * by auto moreover have Fi: "F i \<in> sets M" using F by auto ultimately show "emeasure ?M' (T' -` F i \<inter> space ?M') \<noteq> \<infinity>" using F T T' by (simp add: emeasure_distr) qed qed have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M" using T ac by measurable then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M" by (simp add: comp_def) show "AE x in M. 0 \<le> RN_deriv ?M' ?N' (T x)" using M'.RN_deriv_nonneg[OF ac] by auto have "N = distr N M (T' \<circ> T)" by (subst measure_of_of_measure[of N, symmetric]) (auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed) also have "\<dots> = distr (distr N M' T) M T'" using T T' by (simp add: distr_distr) also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'" using ac by (simp add: M'.density_RN_deriv) also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)" using M'.borel_measurable_RN_deriv[OF ac] by (simp add: distr_density_distr[OF T T', OF inv]) finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N" by (simp add: comp_def) qed lemma (in sigma_finite_measure) RN_deriv_finite: assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>" proof - interpret N: sigma_finite_measure N by fact from N show ?thesis using sigma_finite_iff_density_finite[OF RN_deriv(1)[OF ac]] RN_deriv(2,3)[OF ac] by simp qed lemma (in sigma_finite_measure) assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" and f: "f \<in> borel_measurable M" shows RN_deriv_integrable: "integrable N f \<longleftrightarrow> integrable M (\<lambda>x. real (RN_deriv M N x) * f x)" (is ?integrable) and RN_deriv_integral: "integral\<^isup>L N f = (\<integral>x. real (RN_deriv M N x) * f x \<partial>M)" (is ?integral) proof - note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp] interpret N: sigma_finite_measure N by fact have minus_cong: "\<And>A B A' B'::ereal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto have Nf: "f \<in> borel_measurable N" using f by (simp add: measurable_def) { fix f :: "'a \<Rightarrow> real" { fix x assume *: "RN_deriv M N x \<noteq> \<infinity>" have "ereal (real (RN_deriv M N x)) * ereal (f x) = ereal (real (RN_deriv M N x) * f x)" by (simp add: mult_le_0_iff) then have "RN_deriv M N x * ereal (f x) = ereal (real (RN_deriv M N x) * f x)" using RN_deriv(3)[OF ac] * by (auto simp add: ereal_real split: split_if_asm) } then have "(\<integral>\<^isup>+x. ereal (real (RN_deriv M N x) * f x) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M N x * ereal (f x) \<partial>M)" "(\<integral>\<^isup>+x. ereal (- (real (RN_deriv M N x) * f x)) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M N x * ereal (- f x) \<partial>M)" using RN_deriv_finite[OF N ac] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric] by (auto intro!: positive_integral_cong_AE) } note * = this show ?integral ?integrable unfolding lebesgue_integral_def integrable_def * using Nf f RN_deriv(1)[OF ac] by (auto simp: RN_deriv_positive_integral[OF ac]) qed lemma (in sigma_finite_measure) real_RN_deriv: assumes "finite_measure N" assumes ac: "absolutely_continuous M N" "sets N = sets M" obtains D where "D \<in> borel_measurable M" and "AE x in M. RN_deriv M N x = ereal (D x)" and "AE x in N. 0 < D x" and "\<And>x. 0 \<le> D x" proof interpret N: finite_measure N by fact note RN = RN_deriv[OF ac] let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}" show "(\<lambda>x. real (RN_deriv M N x)) \<in> borel_measurable M" using RN by auto have "N (?RN \<infinity>) = (\<integral>\<^isup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)" using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density) also have "\<dots> = (\<integral>\<^isup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)" by (intro positive_integral_cong) (auto simp: indicator_def) also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)" using RN by (intro positive_integral_cmult_indicator) auto finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" . moreover have "emeasure M (?RN \<infinity>) = 0" proof (rule ccontr) assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0" moreover from RN have "0 \<le> emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto ultimately have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto with eq have "N (?RN \<infinity>) = \<infinity>" by simp with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto qed ultimately have "AE x in M. RN_deriv M N x < \<infinity>" using RN by (intro AE_iff_measurable[THEN iffD2]) auto then show "AE x in M. RN_deriv M N x = ereal (real (RN_deriv M N x))" using RN(3) by (auto simp: ereal_real) then have eq: "AE x in N. RN_deriv M N x = ereal (real (RN_deriv M N x))" using ac absolutely_continuous_AE by auto show "\<And>x. 0 \<le> real (RN_deriv M N x)" using RN by (auto intro: real_of_ereal_pos) have "N (?RN 0) = (\<integral>\<^isup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)" using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density) also have "\<dots> = (\<integral>\<^isup>+ x. 0 \<partial>M)" by (intro positive_integral_cong) (auto simp: indicator_def) finally have "AE x in N. RN_deriv M N x \<noteq> 0" using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq) with RN(3) eq show "AE x in N. 0 < real (RN_deriv M N x)" by (auto simp: zero_less_real_of_ereal le_less) qed lemma (in sigma_finite_measure) RN_deriv_singleton: assumes ac: "absolutely_continuous M N" "sets N = sets M" and x: "{x} \<in> sets M" shows "N {x} = RN_deriv M N x * emeasure M {x}" proof - note deriv = RN_deriv[OF ac] from deriv(1,3) `{x} \<in> sets M` have "density M (RN_deriv M N) {x} = (\<integral>\<^isup>+w. RN_deriv M N x * indicator {x} w \<partial>M)" by (auto simp: indicator_def emeasure_density intro!: positive_integral_cong) with x deriv show ?thesis by (auto simp: positive_integral_cmult_indicator) qed end