(* Title: HOL/Analysis/Radon_Nikodym.thy Author: Johannes Hölzl, TU München *) section ‹Radon-Nikod{\'y}m Derivative› theory Radon_Nikodym imports Bochner_Integration begin definition✐‹tag important› diff_measure :: "'a measure ⇒ 'a measure ⇒ 'a measure" where "diff_measure M N = measure_of (space M) (sets M) (λA. 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: "⋀A. A ∈ sets M ⟹ emeasure N A ≤ emeasure M A" and A: "A ∈ sets M" shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?μ A") unfolding diff_measure_def proof (rule emeasure_measure_of_sigma) show "sigma_algebra (space M) (sets M)" .. show "positive (sets M) ?μ" using pos by (simp add: positive_def) show "countably_additive (sets M) ?μ" proof (rule countably_additiveI) fix A :: "nat ⇒ _" assume A: "range A ⊆ sets M" and "disjoint_family A" then have suminf: "(∑i. emeasure M (A i)) = emeasure M (⋃i. A i)" "(∑i. emeasure N (A i)) = emeasure N (⋃i. A i)" by (simp_all add: suminf_emeasure sets_eq) with A have "(∑i. emeasure M (A i) - emeasure N (A i)) = (∑i. emeasure M (A i)) - (∑i. emeasure N (A i))" using fin pos[of "A _"] by (intro ennreal_suminf_minus) (auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure) then show "(∑i. emeasure M (A i) - emeasure N (A i)) = emeasure M (⋃i. A i) - emeasure N (⋃i. A i) " by (simp add: suminf) qed qed fact text ‹An equivalent characterization of sigma-finite spaces is the existence of integrable positive functions (or, still equivalently, the existence of a probability measure which is in the same measure class as the original measure).› proposition (in sigma_finite_measure) obtain_positive_integrable_function: obtains f::"'a ⇒ real" where "f ∈ borel_measurable M" "⋀x. f x > 0" "⋀x. f x ≤ 1" "integrable M f" proof - obtain A :: "nat ⇒ 'a set" where "range A ⊆ sets M" "(⋃i. A i) = space M" "⋀i. emeasure M (A i) ≠ ∞" using sigma_finite by auto then have [measurable]:"A n ∈ sets M" for n by auto define g where "g = (λx. (∑n. (1/2)^(Suc n) * indicator (A n) x / (1+ measure M (A n))))" have [measurable]: "g ∈ borel_measurable M" unfolding g_def by auto have *: "summable (λn. (1/2)^(Suc n) * indicator (A n) x / (1+ measure M (A n)))" for x apply (rule summable_comparison_test'[of "λn. (1/2)^(Suc n)" 0]) using power_half_series summable_def by (auto simp add: indicator_def divide_simps) have "g x ≤ (∑n. (1/2)^(Suc n))" for x unfolding g_def apply (rule suminf_le) using * power_half_series summable_def by (auto simp add: indicator_def divide_simps) then have g_le_1: "g x ≤ 1" for x using power_half_series sums_unique by fastforce have g_pos: "g x > 0" if "x ∈ space M" for x unfolding g_def proof (subst suminf_pos_iff[OF *[of x]], auto) obtain i where "x ∈ A i" using ‹(⋃i. A i) = space M› ‹x ∈ space M› by auto then have "0 < (1 / 2) ^ Suc i * indicator (A i) x / (1 + Sigma_Algebra.measure M (A i))" unfolding indicator_def apply (auto simp add: divide_simps) using measure_nonneg[of M "A i"] by (auto, meson add_nonneg_nonneg linorder_not_le mult_nonneg_nonneg zero_le_numeral zero_le_one zero_le_power) then show "∃i. 0 < (1 / 2) ^ i * indicator (A i) x / (2 + 2 * Sigma_Algebra.measure M (A i))" by auto qed have "integrable M g" unfolding g_def proof (rule integrable_suminf) fix n show "integrable M (λx. (1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n)))" using ‹emeasure M (A n) ≠ ∞› by (auto intro!: integrable_mult_right integrable_divide_zero integrable_real_indicator simp add: top.not_eq_extremum) next show "AE x in M. summable (λn. norm ((1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n))))" using * by auto show "summable (λn. (∫x. norm ((1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n))) ∂M))" apply (rule summable_comparison_test'[of "λn. (1/2)^(Suc n)" 0], auto) using power_half_series summable_def apply auto[1] apply (auto simp add: field_split_simps) using measure_nonneg[of M] not_less by fastforce qed define f where "f = (λx. if x ∈ space M then g x else 1)" have "f x > 0" for x unfolding f_def using g_pos by auto moreover have "f x ≤ 1" for x unfolding f_def using g_le_1 by auto moreover have [measurable]: "f ∈ borel_measurable M" unfolding f_def by auto moreover have "integrable M f" apply (subst integrable_cong[of _ _ _ g]) unfolding f_def using ‹integrable M g› by auto ultimately show "(⋀f. f ∈ borel_measurable M ⟹ (⋀x. 0 < f x) ⟹ (⋀x. f x ≤ 1) ⟹ integrable M f ⟹ thesis) ⟹ thesis" by (meson that) qed lemma (in sigma_finite_measure) Ex_finite_integrable_function: "∃h∈borel_measurable M. integral⇧^{N}M h ≠ ∞ ∧ (∀x∈space M. 0 < h x ∧ h x < ∞)" proof - obtain A :: "nat ⇒ 'a set" where range[measurable]: "range A ⊆ sets M" and space: "(⋃i. A i) = space M" and measure: "⋀i. emeasure M (A i) ≠ ∞" and disjoint: "disjoint_family A" using sigma_finite_disjoint by blast let ?B = "λi. 2^Suc i * emeasure M (A i)" have [measurable]: "⋀i. A i ∈ sets M" using range by fastforce+ have "∀i. ∃x. 0 < x ∧ x < inverse (?B i)" proof fix i show "∃x. 0 < x ∧ x < inverse (?B i)" using measure[of i] by (auto intro!: dense simp: ennreal_inverse_positive ennreal_mult_eq_top_iff power_eq_top_ennreal) qed from choice[OF this] obtain n where n: "⋀i. 0 < n i" "⋀i. n i < inverse (2^Suc i * emeasure M (A i))" by auto { fix i have "0 ≤ n i" using n(1)[of i] by auto } note pos = this let ?h = "λx. ∑i. n i * indicator (A i) x" show ?thesis proof (safe intro!: bexI[of _ ?h] del: notI) have "integral⇧^{N}M ?h = (∑i. n i * emeasure M (A i))" using pos by (simp add: nn_integral_suminf nn_integral_cmult_indicator) also have "… ≤ (∑i. ennreal ((1/2)^Suc i))" proof (intro suminf_le allI) fix N have "n N * emeasure M (A N) ≤ inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)" using n[of N] by (intro mult_right_mono) auto also have "… = (1/2)^Suc N * (inverse (emeasure M (A N)) * emeasure M (A N))" using measure[of N] by (simp add: ennreal_inverse_power divide_ennreal_def ennreal_inverse_mult power_eq_top_ennreal less_top[symmetric] mult_ac del: power_Suc) also have "… ≤ inverse (ennreal 2) ^ Suc N" using measure[of N] by (cases "emeasure M (A N)"; cases "emeasure M (A N) = 0") (auto simp: inverse_ennreal ennreal_mult[symmetric] divide_ennreal_def simp del: power_Suc) also have "… = ennreal (inverse 2 ^ Suc N)" by (subst ennreal_power[symmetric], simp) (simp add: inverse_ennreal) finally show "n N * emeasure M (A N) ≤ ennreal ((1/2)^Suc N)" by simp qed auto also have "… < top" unfolding less_top[symmetric] by (rule ennreal_suminf_neq_top) (auto simp: summable_geometric summable_Suc_iff simp del: power_Suc) finally show "integral⇧^{N}M ?h ≠ ∞" by (auto simp: top_unique) next { fix x assume "x ∈ space M" then obtain i where "x ∈ 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 < ∞" using n[of i] by (auto simp: less_top[symmetric]) } note pos = this qed measurable qed subsection "Absolutely continuous" definition✐‹tag important› absolutely_continuous :: "'a measure ⇒ 'a measure ⇒ bool" where "absolutely_continuous M N ⟷ null_sets M ⊆ 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 ∈ borel_measurable M ⟹ 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: "(⋀x. ⟦ x ∈ A ; f x ≤ 0 ⟧ ⟹ g x ≤ 0) ⟹ absolutely_continuous (point_measure A f) (point_measure A g)" unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff) lemma absolutely_continuousD: "absolutely_continuous M N ⟹ A ∈ sets M ⟹ emeasure M A = 0 ⟹ emeasure N A = 0" by (auto simp: absolutely_continuous_def null_sets_def) 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 ∈ null_sets M" "{x∈space M. ¬ P x} ⊆ N" unfolding eventually_ae_filter by auto show "AE x in M'. P x" proof (rule AE_I') show "{x∈space M'. ¬ P x} ⊆ N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp from ‹absolutely_continuous M M'› show "N ∈ 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" proposition (in finite_measure) Radon_Nikodym_finite_measure: assumes "finite_measure N" and sets_eq[simp]: "sets N = sets M" assumes "absolutely_continuous M N" shows "∃f ∈ borel_measurable M. density M f = N" proof - interpret N: finite_measure N by fact define G where "G = {g ∈ borel_measurable M. ∀A∈sets M. (∫⇧^{+}x. g x * indicator A x ∂M) ≤ N A}" have [measurable_dest]: "f ∈ G ⟹ f ∈ borel_measurable M" and G_D: "⋀A. f ∈ G ⟹ A ∈ sets M ⟹ (∫⇧^{+}x. f x * indicator A x ∂M) ≤ N A" for f by (auto simp: G_def) note this[measurable_dest] have "(λx. 0) ∈ G" unfolding G_def by auto hence "G ≠ {}" by auto { fix f g assume f[measurable]: "f ∈ G" and g[measurable]: "g ∈ G" have "(λx. max (g x) (f x)) ∈ G" (is "?max ∈ G") unfolding G_def proof safe let ?A = "{x ∈ space M. f x ≤ g x}" have "?A ∈ sets M" using f g unfolding G_def by auto fix A assume [measurable]: "A ∈ sets M" have union: "((?A ∩ A) ∪ ((space M - ?A) ∩ A)) = A" using sets.sets_into_space[OF ‹A ∈ sets M›] by auto have "⋀x. x ∈ space M ⟹ max (g x) (f x) * indicator A x = g x * indicator (?A ∩ A) x + f x * indicator ((space M - ?A) ∩ A) x" by (auto simp: indicator_def max_def) hence "(∫⇧^{+}x. max (g x) (f x) * indicator A x ∂M) = (∫⇧^{+}x. g x * indicator (?A ∩ A) x ∂M) + (∫⇧^{+}x. f x * indicator ((space M - ?A) ∩ A) x ∂M)" by (auto cong: nn_integral_cong intro!: nn_integral_add) also have "… ≤ N (?A ∩ A) + N ((space M - ?A) ∩ A)" using f g unfolding G_def by (auto intro!: add_mono) also have "… = N A" using union by (subst plus_emeasure) auto finally show "(∫⇧^{+}x. max (g x) (f x) * indicator A x ∂M) ≤ N A" . qed auto } note max_in_G = this { fix f assume "incseq f" and f: "⋀i. f i ∈ G" then have [measurable]: "⋀i. f i ∈ borel_measurable M" by (auto simp: G_def) have "(λx. SUP i. f i x) ∈ G" unfolding G_def proof safe show "(λx. SUP i. f i x) ∈ borel_measurable M" by measurable next fix A assume "A ∈ sets M" have "(∫⇧^{+}x. (SUP i. f i x) * indicator A x ∂M) = (∫⇧^{+}x. (SUP i. f i x * indicator A x) ∂M)" by (intro nn_integral_cong) (simp split: split_indicator) also have "… = (SUP i. (∫⇧^{+}x. f i x * indicator A x ∂M))" using ‹incseq f› f ‹A ∈ sets M› by (intro nn_integral_monotone_convergence_SUP) (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator) finally show "(∫⇧^{+}x. (SUP i. f i x) * indicator A x ∂M) ≤ N A" using f ‹A ∈ sets M› by (auto intro!: SUP_least simp: G_D) qed } note SUP_in_G = this let ?y = "SUP g ∈ G. integral⇧^{N}M g" have y_le: "?y ≤ N (space M)" unfolding G_def proof (safe intro!: SUP_least) fix g assume "∀A∈sets M. (∫⇧^{+}x. g x * indicator A x ∂M) ≤ N A" from this[THEN bspec, OF sets.top] show "integral⇧^{N}M g ≤ N (space M)" by (simp cong: nn_integral_cong) qed from ennreal_SUP_countable_SUP [OF ‹G ≠ {}›, of "integral⇧^{N}M"] obtain ys :: "nat ⇒ ennreal" where ys: "range ys ⊆ integral⇧^{N}M ` G ∧ Sup (integral⇧^{N}M ` G) = Sup (range ys)" by auto then have "∀n. ∃g. g∈G ∧ integral⇧^{N}M g = ys n" proof safe fix n assume "range ys ⊆ integral⇧^{N}M ` G" hence "ys n ∈ integral⇧^{N}M ` G" by auto thus "∃g. g∈G ∧ integral⇧^{N}M g = ys n" by auto qed from choice[OF this] obtain gs where "⋀i. gs i ∈ G" "⋀n. integral⇧^{N}M (gs n) = ys n" by auto hence y_eq: "?y = (SUP i. integral⇧^{N}M (gs i))" using ys by auto let ?g = "λi x. Max ((λn. gs n x) ` {..i})" define f where [abs_def]: "f x = (SUP i. ?g i x)" for x let ?F = "λA x. f x * indicator A x" have gs_not_empty: "⋀i x. (λn. gs n x) ` {..i} ≠ {}" by auto { fix i have "?g i ∈ G" proof (induct i) case 0 thus ?case by simp fact next case (Suc i) with Suc gs_not_empty ‹gs (Suc i) ∈ 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 ∈ G" unfolding f_def . then have [measurable]: "f ∈ borel_measurable M" unfolding G_def by auto have "integral⇧^{N}M f = (SUP i. integral⇧^{N}M (?g i))" unfolding f_def using g_in_G ‹incseq ?g› by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def) also have "… = ?y" proof (rule antisym) show "(SUP i. integral⇧^{N}M (?g i)) ≤ ?y" using g_in_G by (auto intro: SUP_mono) show "?y ≤ (SUP i. integral⇧^{N}M (?g i))" unfolding y_eq by (auto intro!: SUP_mono nn_integral_mono Max_ge) qed finally have int_f_eq_y: "integral⇧^{N}M f = ?y" . have upper_bound: "∀A∈sets M. N A ≤ density M f A" proof (rule ccontr) assume "¬ ?thesis" then obtain A where A[measurable]: "A ∈ sets M" and f_less_N: "density M f A < N A" by (auto simp: not_le) then have pos_A: "0 < M A" using ‹absolutely_continuous M N›[THEN absolutely_continuousD, OF A] by (auto simp: zero_less_iff_neq_zero) define b where "b = (N A - density M f A) / M A / 2" with f_less_N pos_A have "0 < b" "b ≠ top" by (auto intro!: diff_gr0_ennreal simp: zero_less_iff_neq_zero diff_eq_0_iff_ennreal ennreal_divide_eq_top_iff) let ?f = "λx. f x + b" have "nn_integral M f ≠ top" using ‹f ∈ G›[THEN G_D, of "space M"] by (auto simp: top_unique cong: nn_integral_cong) with ‹b ≠ top› interpret Mf: finite_measure "density M ?f" by (intro finite_measureI) (auto simp: field_simps mult_indicator_subset ennreal_mult_eq_top_iff emeasure_density nn_integral_cmult_indicator nn_integral_add cong: nn_integral_cong) from unsigned_Hahn_decomposition[of "density M ?f" N A] obtain Y where [measurable]: "Y ∈ sets M" and [simp]: "Y ⊆ A" and Y1: "⋀C. C ∈ sets M ⟹ C ⊆ Y ⟹ density M ?f C ≤ N C" and Y2: "⋀C. C ∈ sets M ⟹ C ⊆ A ⟹ C ∩ Y = {} ⟹ N C ≤ density M ?f C" by auto let ?f' = "λx. f x + b * indicator Y x" have "M Y ≠ 0" proof assume "M Y = 0" then have "N Y = 0" using ‹absolutely_continuous M N›[THEN absolutely_continuousD, of Y] by auto then have "N A = N (A - Y)" by (subst emeasure_Diff) auto also have "… ≤ density M ?f (A - Y)" by (rule Y2) auto also have "… ≤ density M ?f A - density M ?f Y" by (subst emeasure_Diff) auto also have "… ≤ density M ?f A - 0" by (intro ennreal_minus_mono) auto also have "density M ?f A = b * M A + density M f A" by (simp add: emeasure_density field_simps mult_indicator_subset nn_integral_add nn_integral_cmult_indicator) also have "… < N A" using f_less_N pos_A by (cases "density M f A"; cases "M A"; cases "N A") (auto simp: b_def ennreal_less_iff ennreal_minus divide_ennreal ennreal_numeral[symmetric] ennreal_plus[symmetric] ennreal_mult[symmetric] field_simps simp del: ennreal_numeral ennreal_plus) finally show False by simp qed then have "nn_integral M f < nn_integral M ?f'" using ‹0 < b› ‹nn_integral M f ≠ top› by (simp add: nn_integral_add nn_integral_cmult_indicator ennreal_zero_less_mult_iff zero_less_iff_neq_zero) moreover have "?f' ∈ G" unfolding G_def proof safe fix X assume [measurable]: "X ∈ sets M" have "(∫⇧^{+}x. ?f' x * indicator X x ∂M) = density M f (X - Y) + density M ?f (X ∩ Y)" by (auto simp add: emeasure_density nn_integral_add[symmetric] split: split_indicator intro!: nn_integral_cong) also have "… ≤ N (X - Y) + N (X ∩ Y)" using G_D[OF ‹f ∈ G›] by (intro add_mono Y1) (auto simp: emeasure_density) also have "… = N X" by (subst plus_emeasure) (auto intro!: arg_cong2[where f=emeasure]) finally show "(∫⇧^{+}x. ?f' x * indicator X x ∂M) ≤ N X" . qed simp then have "nn_integral M ?f' ≤ ?y" by (rule SUP_upper) ultimately show False by (simp add: int_f_eq_y) qed show ?thesis proof (intro bexI[of _ f] measure_eqI conjI antisym) fix A assume "A ∈ sets (density M f)" then show "emeasure (density M f) A ≤ emeasure N A" by (auto simp: emeasure_density intro!: G_D[OF ‹f ∈ G›]) next fix A assume A: "A ∈ sets (density M f)" then show "emeasure N A ≤ emeasure (density M f) A" using upper_bound by auto qed auto qed lemma (in finite_measure) split_space_into_finite_sets_and_rest: assumes ac: "absolutely_continuous M N" and sets_eq[simp]: "sets N = sets M" shows "∃B::nat⇒'a set. disjoint_family B ∧ range B ⊆ sets M ∧ (∀i. N (B i) ≠ ∞) ∧ (∀A∈sets M. A ∩ (⋃i. B i) = {} ⟶ (emeasure M A = 0 ∧ N A = 0) ∨ (emeasure M A > 0 ∧ N A = ∞))" proof - let ?Q = "{Q∈sets M. N Q ≠ ∞}" let ?a = "SUP Q∈?Q. emeasure M Q" have "{} ∈ ?Q" by auto then have Q_not_empty: "?Q ≠ {}" by blast have "?a ≤ emeasure M (space M)" using sets.sets_into_space by (auto intro!: SUP_least emeasure_mono) then have "?a ≠ ∞" using finite_emeasure_space by (auto simp: less_top[symmetric] top_unique simp del: SUP_eq_top_iff Sup_eq_top_iff) from ennreal_SUP_countable_SUP [OF Q_not_empty, of "emeasure M"] obtain Q'' where "range Q'' ⊆ emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)" by auto then have "∀i. ∃Q'. Q'' i = emeasure M Q' ∧ Q' ∈ ?Q" by auto from choice[OF this] obtain Q' where Q': "⋀i. Q'' i = emeasure M (Q' i)" "⋀i. Q' i ∈ ?Q" by auto then have a_Lim: "?a = (SUP i. emeasure M (Q' i))" using a by simp let ?O = "λn. ⋃i≤n. Q' i" have Union: "(SUP i. emeasure M (?O i)) = emeasure M (⋃i. ?O i)" proof (rule SUP_emeasure_incseq[of ?O]) show "range ?O ⊆ sets M" using Q' by auto show "incseq ?O" by (fastforce intro!: incseq_SucI) qed have Q'_sets[measurable]: "⋀i. Q' i ∈ sets M" using Q' by auto have O_sets: "⋀i. ?O i ∈ sets M" using Q' by auto then have O_in_G: "⋀i. ?O i ∈ ?Q" proof (safe del: notI) fix i have "Q' ` {..i} ⊆ sets M" using Q' by auto then have "N (?O i) ≤ (∑i≤i. N (Q' i))" by (simp add: emeasure_subadditive_finite) also have "… < ∞" using Q' by (simp add: less_top) finally show "N (?O i) ≠ ∞" by simp qed auto have O_mono: "⋀n. ?O n ⊆ ?O (Suc n)" by fastforce have a_eq: "?a = emeasure M (⋃i. ?O i)" unfolding Union[symmetric] proof (rule antisym) show "?a ≤ (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)) ≤ ?a" proof (safe intro!: Sup_mono, unfold bex_simps) fix i have *: "(⋃(Q' ` {..i})) = ?O i" by auto then show "∃x. (x ∈ sets M ∧ N x ≠ ∞) ∧ emeasure M (⋃(Q' ` {..i})) ≤ emeasure M x" using O_in_G[of i] by (auto intro!: exI[of _ "?O i"]) qed qed let ?O_0 = "(⋃i. ?O i)" have "?O_0 ∈ sets M" using Q' by auto have "disjointed Q' i ∈ sets M" for i using sets.range_disjointed_sets[of Q' M] using Q'_sets by (auto simp: subset_eq) note Q_sets = this show ?thesis proof (intro bexI exI conjI ballI impI allI) show "disjoint_family (disjointed Q')" by (rule disjoint_family_disjointed) show "range (disjointed Q') ⊆ sets M" using Q'_sets by (intro sets.range_disjointed_sets) auto { fix A assume A: "A ∈ sets M" "A ∩ (⋃i. disjointed Q' i) = {}" then have A1: "A ∩ (⋃i. Q' i) = {}" unfolding UN_disjointed_eq by auto show "emeasure M A = 0 ∧ N A = 0 ∨ 0 < emeasure M A ∧ N A = ∞" proof (rule disjCI, simp) assume *: "emeasure M A = 0 ∨ N A ≠ top" show "emeasure M A = 0 ∧ N A = 0" proof (cases "emeasure M A = 0") case True with ac A have "N A = 0" unfolding absolutely_continuous_def by auto with True show ?thesis by simp next case False with * have "N A ≠ ∞" by auto with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 ∪ A)" using Q' A1 by (auto intro!: plus_emeasure simp: set_eq_iff) also have "… = (SUP i. emeasure M (?O i ∪ A))" proof (rule SUP_emeasure_incseq[of "λi. ?O i ∪ A", symmetric, simplified]) show "range (λi. ?O i ∪ A) ⊆ sets M" using ‹N A ≠ ∞› O_sets A by auto qed (fastforce intro!: incseq_SucI) also have "… ≤ ?a" proof (safe intro!: SUP_least) fix i have "?O i ∪ A ∈ ?Q" proof (safe del: notI) show "?O i ∪ A ∈ sets M" using O_sets A by auto from O_in_G[of i] have "N (?O i ∪ A) ≤ N (?O i) + N A" using emeasure_subadditive[of "?O i" N A] A O_sets by auto with O_in_G[of i] show "N (?O i ∪ A) ≠ ∞" using ‹N A ≠ ∞› by (auto simp: top_unique) qed then show "emeasure M (?O i ∪ A) ≤ ?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 ≠ 0› show ?thesis by auto qed qed } { fix i have "N (disjointed Q' i) ≤ N (Q' i)" by (auto intro!: emeasure_mono simp: disjointed_def) then show "N (disjointed Q' i) ≠ ∞" using Q'(2)[of i] by (auto simp: top_unique) } qed qed proposition (in finite_measure) Radon_Nikodym_finite_measure_infinite: assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M" shows "∃f∈borel_measurable M. density M f = N" proof - from split_space_into_finite_sets_and_rest[OF assms] obtain Q :: "nat ⇒ 'a set" where Q: "disjoint_family Q" "range Q ⊆ sets M" and in_Q0: "⋀A. A ∈ sets M ⟹ A ∩ (⋃i. Q i) = {} ⟹ emeasure M A = 0 ∧ N A = 0 ∨ 0 < emeasure M A ∧ N A = ∞" and Q_fin: "⋀i. N (Q i) ≠ ∞" by force from Q have Q_sets: "⋀i. Q i ∈ sets M" by auto let ?N = "λi. density N (indicator (Q i))" and ?M = "λi. density M (indicator (Q i))" have "∀i. ∃f∈borel_measurable (?M i). 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: nn_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: nn_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]: "⋀A. A ∈ sets M ⟹ A ∈ sets N" by (simp add: sets_eq) show "absolutely_continuous (?M i) (?N i)" using ‹absolutely_continuous M N› ‹Q i ∈ 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: "⋀i. f i ∈ borel_measurable M" "⋀i x. 0 ≤ f i x" and f_density: "⋀i. density (?M i) (f i) = ?N i" by force { fix A i assume A: "A ∈ sets M" with Q borel have "(∫⇧^{+}x. f i x * indicator (Q i ∩ A) x ∂M) = emeasure (density (?M i) (f i)) A" by (auto simp add: emeasure_density nn_integral_density subset_eq intro!: nn_integral_cong split: split_indicator) also have "… = emeasure N (Q i ∩ A)" using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq) finally have "emeasure N (Q i ∩ A) = (∫⇧^{+}x. f i x * indicator (Q i ∩ A) x ∂M)" .. } note integral_eq = this let ?f = "λx. (∑i. f i x * indicator (Q i) x) + ∞ * indicator (space M - (⋃i. Q i)) x" show ?thesis proof (safe intro!: bexI[of _ ?f]) show "?f ∈ borel_measurable M" using borel Q_sets by (auto intro!: measurable_If) show "density M ?f = N" proof (rule measure_eqI) fix A assume "A ∈ sets (density M ?f)" then have "A ∈ sets M" by simp have Qi: "⋀i. Q i ∈ sets M" using Q by auto have [intro,simp]: "⋀i. (λx. f i x * indicator (Q i ∩ A) x) ∈ borel_measurable M" "⋀i. AE x in M. 0 ≤ f i x * indicator (Q i ∩ A) x" using borel Qi ‹A ∈ sets M› by auto have "(∫⇧^{+}x. ?f x * indicator A x ∂M) = (∫⇧^{+}x. (∑i. f i x * indicator (Q i ∩ A) x) + ∞ * indicator ((space M - (⋃i. Q i)) ∩ A) x ∂M)" using borel by (intro nn_integral_cong) (auto simp: indicator_def) also have "… = (∫⇧^{+}x. (∑i. f i x * indicator (Q i ∩ A) x) ∂M) + ∞ * emeasure M ((space M - (⋃i. Q i)) ∩ A)" using borel Qi ‹A ∈ sets M› by (subst nn_integral_add) (auto simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le) also have "… = (∑i. N (Q i ∩ A)) + ∞ * emeasure M ((space M - (⋃i. Q i)) ∩ A)" by (subst integral_eq[OF ‹A ∈ sets M›], subst nn_integral_suminf) auto finally have "(∫⇧^{+}x. ?f x * indicator A x ∂M) = (∑i. N (Q i ∩ A)) + ∞ * emeasure M ((space M - (⋃i. Q i)) ∩ A)" . moreover have "(∑i. N (Q i ∩ A)) = N ((⋃i. Q i) ∩ A)" using Q Q_sets ‹A ∈ sets M› by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq) moreover have "(space M - (⋃x. Q x)) ∩ A ∩ (⋃x. Q x) = {}" by auto then have "∞ * emeasure M ((space M - (⋃i. Q i)) ∩ A) = N ((space M - (⋃i. Q i)) ∩ A)" using in_Q0[of "(space M - (⋃i. Q i)) ∩ A"] ‹A ∈ sets M› Q by (auto simp: ennreal_top_mult) moreover have "(space M - (⋃i. Q i)) ∩ A ∈ sets M" "((⋃i. Q i) ∩ A) ∈ sets M" using Q_sets ‹A ∈ sets M› by auto moreover have "((⋃i. Q i) ∩ A) ∪ ((space M - (⋃i. Q i)) ∩ A) = A" "((⋃i. Q i) ∩ A) ∩ ((space M - (⋃i. Q i)) ∩ A) = {}" using ‹A ∈ sets M› sets.sets_into_space by auto ultimately have "N A = (∫⇧^{+}x. ?f x * indicator A x ∂M)" using plus_emeasure[of "(⋃i. Q i) ∩ A" N "(space M - (⋃i. Q i)) ∩ A"] by (simp add: sets_eq) with ‹A ∈ sets M› borel Q show "emeasure (density M ?f) A = N A" by (auto simp: subset_eq emeasure_density) qed (simp add: sets_eq) qed qed theorem (in sigma_finite_measure) Radon_Nikodym: assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M" shows "∃f ∈ borel_measurable M. density M f = N" proof - from Ex_finite_integrable_function obtain h where finite: "integral⇧^{N}M h ≠ ∞" and borel: "h ∈ borel_measurable M" and nn: "⋀x. 0 ≤ h x" and pos: "⋀x. x ∈ space M ⟹ 0 < h x" and "⋀x. x ∈ space M ⟹ h x < ∞" by auto let ?T = "λA. (∫⇧^{+}x. h x * indicator A x ∂M)" let ?MT = "density M h" from borel finite nn interpret T: finite_measure ?MT by (auto intro!: finite_measureI cong: nn_integral_cong simp: emeasure_density) have "absolutely_continuous ?MT N" "sets N = sets ?MT" proof (unfold absolutely_continuous_def, safe) fix A assume "A ∈ null_sets ?MT" with borel have "A ∈ sets M" "AE x in M. x ∈ A ⟶ h x ≤ 0" by (auto simp add: null_sets_density_iff) with pos sets.sets_into_space have "AE x in M. x ∉ A" by (elim eventually_mono) (auto simp: not_le[symmetric]) then have "A ∈ null_sets M" using ‹A ∈ sets M› by (simp add: AE_iff_null_sets) with ac show "A ∈ 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 ∈ borel_measurable M" "density ?MT f = N" by auto with nn borel show ?thesis by (auto intro!: bexI[of _ "λx. h x * f x"] simp: density_density_eq) qed subsection ‹Uniqueness of densities› lemma finite_density_unique: assumes borel: "f ∈ borel_measurable M" "g ∈ borel_measurable M" assumes pos: "AE x in M. 0 ≤ f x" "AE x in M. 0 ≤ g x" and fin: "integral⇧^{N}M f ≠ ∞" shows "density M f = density M g ⟷ (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 = "λf A. ∫⇧^{+}x. f x * indicator A x ∂M" assume "density M f = density M g" with borel have eq: "∀A∈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⇧^{N}M g ≠ ∞" by (simp cong: nn_integral_cong) { fix f g assume borel: "f ∈ borel_measurable M" "g ∈ borel_measurable M" and pos: "AE x in M. 0 ≤ f x" "AE x in M. 0 ≤ g x" and g_fin: "integral⇧^{N}M g ≠ ∞" and eq: "∀A∈sets M. ?P f A = ?P g A" let ?N = "{x∈space M. g x < f x}" have N: "?N ∈ sets M" using borel by simp have "?P g ?N ≤ integral⇧^{N}M g" using pos by (intro nn_integral_mono_AE) (auto split: split_indicator) then have Pg_fin: "?P g ?N ≠ ∞" using g_fin by (auto simp: top_unique) have "?P (λx. (f x - g x)) ?N = (∫⇧^{+}x. f x * indicator ?N x - g x * indicator ?N x ∂M)" by (auto intro!: nn_integral_cong simp: indicator_def) also have "… = ?P f ?N - ?P g ?N" proof (rule nn_integral_diff) show "(λx. f x * indicator ?N x) ∈ borel_measurable M" "(λx. g x * indicator ?N x) ∈ borel_measurable M" using borel N by auto show "AE x in M. g x * indicator ?N x ≤ f x * indicator ?N x" using pos by (auto split: split_indicator) qed fact also have "… = 0" unfolding eq[THEN bspec, OF N] using Pg_fin by auto finally have "AE x in M. f x ≤ g x" using pos borel nn_integral_PInf_AE[OF borel(2) g_fin] by (subst (asm) nn_integral_0_iff_AE) (auto split: split_indicator simp: not_less ennreal_minus_eq_0) } 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 ∈ borel_measurable M" "f' ∈ borel_measurable M" assumes pos: "AE x in M. 0 ≤ f x" "AE x in M. 0 ≤ f' x" assumes f: "⋀A. A ∈ sets M ⟹ (∫⇧^{+}x. f x * indicator A x ∂M) = (∫⇧^{+}x. f' x * indicator A x ∂M)" (is "⋀A. A ∈ sets M ⟹ ?P f A = ?P f' A") shows "AE x in M. f x = f' x" proof - let ?D = "λf. density M f" let ?N = "λA. ?P f A" and ?N' = "λA. ?P f' A" let ?f = "λA x. f x * indicator A x" and ?f' = "λA 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 Q :: "nat ⇒ 'a set" where Q: "disjoint_family Q" "range Q ⊆ sets M" and in_Q0: "⋀A. A ∈ sets M ⟹ A ∩ (⋃i. Q i) = {} ⟹ emeasure M A = 0 ∧ ?D f A = 0 ∨ 0 < emeasure M A ∧ ?D f A = ∞" and Q_fin: "⋀i. ?D f (Q i) ≠ ∞" by force with borel pos have in_Q0: "⋀A. A ∈ sets M ⟹ A ∩ (⋃i. Q i) = {} ⟹ emeasure M A = 0 ∧ ?N A = 0 ∨ 0 < emeasure M A ∧ ?N A = ∞" and Q_fin: "⋀i. ?N (Q i) ≠ ∞" by (auto simp: emeasure_density subset_eq) from Q have Q_sets[measurable]: "⋀i. Q i ∈ sets M" by auto let ?D = "{x∈space M. f x ≠ f' x}" have "?D ∈ sets M" using borel by auto have *: "⋀i x A. ⋀y::ennreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i ∩ A) x" unfolding indicator_def by auto have "∀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 (space M - (⋃i. Q i)) x = ?f' (space M - (⋃i. Q i)) x" proof (rule AE_I') { fix f :: "'a ⇒ ennreal" assume borel: "f ∈ borel_measurable M" and eq: "⋀A. A ∈ sets M ⟹ ?N A = (∫⇧^{+}x. f x * indicator A x ∂M)" let ?A = "λi. (space M - (⋃i. Q i)) ∩ {x ∈ space M. f x < (i::nat)}" have "(⋃i. ?A i) ∈ null_sets M" proof (rule null_sets_UN) fix i ::nat have "?A i ∈ sets M" using borel by auto have "?N (?A i) ≤ (∫⇧^{+}x. (i::ennreal) * indicator (?A i) x ∂M)" unfolding eq[OF ‹?A i ∈ sets M›] by (auto intro!: nn_integral_mono simp: indicator_def) also have "… = i * emeasure M (?A i)" using ‹?A i ∈ sets M› by (auto intro!: nn_integral_cmult_indicator) also have "… < ∞" using emeasure_real[of "?A i"] by (auto simp: ennreal_mult_less_top of_nat_less_top) finally have "?N (?A i) ≠ ∞" by simp then show "?A i ∈ null_sets M" using in_Q0[OF ‹?A i ∈ sets M›] ‹?A i ∈ sets M› by auto qed also have "(⋃i. ?A i) = (space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞}" by (auto simp: ennreal_Ex_less_of_nat less_top[symmetric]) finally have "(space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞} ∈ null_sets M" by simp } from this[OF borel(1) refl] this[OF borel(2) f] have "(space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞} ∈ null_sets M" "(space M - (⋃i. Q i)) ∩ {x∈space M. f' x ≠ ∞} ∈ null_sets M" by simp_all then show "((space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞}) ∪ ((space M - (⋃i. Q i)) ∩ {x∈space M. f' x ≠ ∞}) ∈ null_sets M" by (rule null_sets.Un) show "{x ∈ space M. ?f (space M - (⋃i. Q i)) x ≠ ?f' (space M - (⋃i. Q i)) x} ⊆ ((space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞}) ∪ ((space M - (⋃i. Q i)) ∩ {x∈space M. f' x ≠ ∞})" by (auto simp: indicator_def) qed moreover have "AE x in M. (?f (space M - (⋃i. Q i)) x = ?f' (space M - (⋃i. Q i)) x) ⟶ (∀i. ?f (Q i) x = ?f' (Q i) x) ⟶ ?f (space M) x = ?f' (space M) x" by (auto simp: indicator_def) ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x" unfolding AE_all_countable[symmetric] by eventually_elim (auto split: if_split_asm simp: indicator_def) then show "AE x in M. f x = f' x" by auto qed proposition (in sigma_finite_measure) density_unique: assumes f: "f ∈ borel_measurable M" assumes f': "f' ∈ borel_measurable M" 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 ∈ borel_measurable M" and fin: "integral⇧^{N}M h ≠ ∞" and pos: "⋀x. x ∈ space M ⟹ 0 < h x ∧ h x < ∞" "⋀x. 0 ≤ h x" using Ex_finite_integrable_function by auto then have h_nn: "AE x in M. 0 ≤ 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: nn_integral_cong emeasure_density add: fin) let ?fM = "density M f" let ?f'M = "density M f'" { fix A assume "A ∈ sets M" then have "{x ∈ space M. h x * indicator A x ≠ 0} = A" using pos(1) sets.sets_into_space by (force simp: indicator_def) then have "(∫⇧^{+}x. h x * indicator A x ∂M) = 0 ⟷ A ∈ null_sets M" using h_borel ‹A ∈ sets M› h_nn by (subst nn_integral_0_iff) auto } note h_null_sets = this { fix A assume "A ∈ sets M" have "(∫⇧^{+}x. f x * (h x * indicator A x) ∂M) = (∫⇧^{+}x. h x * indicator A x ∂?fM)" using ‹A ∈ sets M› h_borel h_nn f f' by (intro nn_integral_density[symmetric]) auto also have "… = (∫⇧^{+}x. h x * indicator A x ∂?f'M)" by (simp_all add: density_eq) also have "… = (∫⇧^{+}x. f' x * (h x * indicator A x) ∂M)" using ‹A ∈ sets M› h_borel h_nn f f' by (intro nn_integral_density) auto finally have "(∫⇧^{+}x. h x * (f x * indicator A x) ∂M) = (∫⇧^{+}x. h x * (f' x * indicator A x) ∂M)" by (simp add: ac_simps) then have "(∫⇧^{+}x. (f x * indicator A x) ∂?H) = (∫⇧^{+}x. (f' x * indicator A x) ∂?H)" using ‹A ∈ sets M› h_borel h_nn f f' by (subst (asm) (1 2) nn_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 with AE_space[of M] pos show "AE x in M. f x = f' x" unfolding AE_density[OF h_borel] by auto qed lemma (in sigma_finite_measure) density_unique_iff: assumes f: "f ∈ borel_measurable M" and f': "f' ∈ borel_measurable M" shows "density M f = density M f' ⟷ (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 ∈ borel_measurable M" "g ∈ borel_measurable M" and fin: "sigma_finite_measure (density M f)" shows "density M f = density M g ⟷ (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 obtain A where cover: "range A ⊆ sets (density M f)" "⋃ (range A) = space (density M f)" "⋀i. emeasure (density M f) (A i) ≠ ∞" "incseq A" by auto have "AE x in M. ∀i. x ∈ A i ⟶ 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 "(∫⇧^{+}x. f x * indicator (A i) x ∂M) ≠ ∞" 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 cover(1) by (intro finite_density_unique[THEN iffD1]) (auto simp: density_density_eq subset_eq) then show "AE x in M. x ∈ A i ⟶ 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 ∈ borel_measurable M" shows "sigma_finite_measure (density M f) ⟷ (AE x in M. f x ≠ ∞)" (is "sigma_finite_measure ?N ⟷ _") proof assume "sigma_finite_measure ?N" then interpret N: sigma_finite_measure ?N . from N.Ex_finite_integrable_function obtain h where h: "h ∈ borel_measurable M" "integral⇧^{N}?N h ≠ ∞" and fin: "∀x∈space M. 0 < h x ∧ h x < ∞" by auto have "AE x in M. f x * h x ≠ ∞" proof (rule AE_I') have "integral⇧^{N}?N h = (∫⇧^{+}x. f x * h x ∂M)" using f h by (auto intro!: nn_integral_density) then have "(∫⇧^{+}x. f x * h x ∂M) ≠ ∞" using h(2) by simp then show "(λx. f x * h x) -` {∞} ∩ space M ∈ null_sets M" using f h(1) by (auto intro!: nn_integral_PInf[unfolded infinity_ennreal_def] borel_measurable_vimage) qed auto then show "AE x in M. f x ≠ ∞" using fin by (auto elim!: AE_Ball_mp simp: less_top ennreal_mult_less_top) next assume AE: "AE x in M. f x ≠ ∞" from sigma_finite obtain Q :: "nat ⇒ 'a set" where Q: "range Q ⊆ sets M" "⋃ (range Q) = space M" "⋀i. emeasure M (Q i) ≠ ∞" by auto define A where "A i = f -` (case i of 0 ⇒ {∞} | Suc n ⇒ {.. ennreal(of_nat (Suc n))}) ∩ space M" for i { fix i j have "A i ∩ Q j ∈ 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 show "sigma_finite_measure ?N" proof (standard, intro exI conjI ballI) show "countable (range (λ(i, j). A i ∩ Q j))" by auto show "range (λ(i, j). A i ∩ Q j) ⊆ sets (density M f)" using A_in_sets by auto next have "⋃(range (λ(i, j). A i ∩ Q j)) = (⋃i j. A i ∩ Q j)" by auto also have "… = (⋃i. A i) ∩ space M" using Q by auto also have "(⋃i. A i) = space M" proof safe fix x assume x: "x ∈ space M" show "x ∈ (⋃i. A i)" proof (cases "f x" rule: ennreal_cases) case top with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0]) next case (real r) with ennreal_Ex_less_of_nat[of "f x"] obtain n :: nat where "f x < n" by auto also have "n < (Suc n :: ennreal)" by simp finally show ?thesis using x real by (auto simp: A_def ennreal_of_nat_eq_real_of_nat intro!: exI[of _ "Suc n"]) qed qed (auto simp: A_def) finally show "⋃(range (λ(i, j). A i ∩ Q j)) = space ?N" by simp next fix X assume "X ∈ range (λ(i, j). A i ∩ Q j)" then obtain i j where [simp]:"X = A i ∩ Q j" by auto have "(∫⇧^{+}x. f x * indicator (A i ∩ Q j) x ∂M) ≠ ∞" proof (cases i) case 0 have "AE x in M. f x * indicator (A i ∩ Q j) x = 0" using AE by (auto simp: A_def ‹i = 0›) from nn_integral_cong_AE[OF this] show ?thesis by simp next case (Suc n) then have "(∫⇧^{+}x. f x * indicator (A i ∩ Q j) x ∂M) ≤ (∫⇧^{+}x. (Suc n :: ennreal) * indicator (Q j) x ∂M)" by (auto intro!: nn_integral_mono simp: indicator_def A_def ennreal_of_nat_eq_real_of_nat) also have "… = Suc n * emeasure M (Q j)" using Q by (auto intro!: nn_integral_cmult_indicator) also have "… < ∞" using Q by (auto simp: ennreal_mult_less_top less_top of_nat_less_top) finally show ?thesis by simp qed then show "emeasure ?N X ≠ ∞" using A_in_sets Q f by (auto simp: emeasure_density) qed qed lemma (in sigma_finite_measure) sigma_finite_iff_density_finite: "f ∈ borel_measurable M ⟹ sigma_finite_measure (density M f) ⟷ (AE x in M. f x ≠ ∞)" by (subst sigma_finite_iff_density_finite') (auto simp: max_def intro!: measurable_If) subsection ‹Radon-Nikodym derivative› definition✐‹tag important› RN_deriv :: "'a measure ⇒ 'a measure ⇒ 'a ⇒ ennreal" where "RN_deriv M N = (if ∃f. f ∈ borel_measurable M ∧ density M f = N then SOME f. f ∈ borel_measurable M ∧ density M f = N else (λ_. 0))" lemma RN_derivI: assumes "f ∈ borel_measurable M" "density M f = N" shows "density M (RN_deriv M N) = N" proof - have *: "∃f. f ∈ borel_measurable M ∧ density M f = N" using assms by auto then have "density M (SOME f. f ∈ borel_measurable M ∧ density M f = N) = N" by (rule someI2_ex) auto with * show ?thesis by (auto simp: RN_deriv_def) qed lemma borel_measurable_RN_deriv[measurable]: "RN_deriv M N ∈ borel_measurable M" proof - { assume ex: "∃f. f ∈ borel_measurable M ∧ density M f = N" have 1: "(SOME f. f ∈ borel_measurable M ∧ density M f = N) ∈ borel_measurable M" using ex by (rule someI2_ex) auto } from this show ?thesis by (auto simp: RN_deriv_def) qed lemma density_RN_deriv_density: assumes f: "f ∈ borel_measurable M" shows "density M (RN_deriv M (density M f)) = density M f" by (rule RN_derivI[OF f]) simp lemma (in sigma_finite_measure) density_RN_deriv: "absolutely_continuous M N ⟹ sets N = sets M ⟹ density M (RN_deriv M N) = N" by (metis RN_derivI Radon_Nikodym) lemma (in sigma_finite_measure) RN_deriv_nn_integral: assumes N: "absolutely_continuous M N" "sets N = sets M" and f: "f ∈ borel_measurable M" shows "integral⇧^{N}N f = (∫⇧^{+}x. RN_deriv M N x * f x ∂M)" proof - have "integral⇧^{N}N f = integral⇧^{N}(density M (RN_deriv M N)) f" using N by (simp add: density_RN_deriv) also have "… = (∫⇧^{+}x. RN_deriv M N x * f x ∂M)" using f by (simp add: nn_integral_density) finally show ?thesis by simp qed lemma (in sigma_finite_measure) RN_deriv_unique: assumes f: "f ∈ borel_measurable M" 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[symmetric]) lemma RN_deriv_unique_sigma_finite: assumes f: "f ∈ borel_measurable M" 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[symmetric]) lemma (in sigma_finite_measure) RN_deriv_distr: fixes T :: "'a ⇒ 'b" assumes T: "T ∈ measurable M M'" and T': "T' ∈ measurable M' M" and inv: "∀x∈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]: "⋀M'. measurable N M' = measurable M M'" by (auto simp: measurable_def) { fix A assume "A ∈ sets M" with inv T T' sets.sets_into_space[OF this] have "T -` T' -` A ∩ T -` space M' ∩ space M = A" by (auto simp: measurable_def) } note eq = this[simp] { fix A assume "A ∈ sets M" with inv T T' sets.sets_into_space[OF this] have "(T' ∘ T) -` A ∩ 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_countable obtain F where F: "countable F ∧ F ⊆ sets M ∧ ⋃ F = space M ∧ (∀a∈F. emeasure M a ≠ ∞)" .. show "∃A. countable A ∧ A ⊆ sets (distr M M' T) ∧ ⋃A = space (distr M M' T) ∧ (∀a∈A. emeasure (distr M M' T) a ≠ ∞)" proof (intro exI conjI ballI) show *: "(λA. T' -` A ∩ space ?M') ` F ⊆ sets ?M'" using F T' by (auto simp: measurable_def) show "⋃((λA. T' -` A ∩ space ?M')`F) = space ?M'" using F T'[THEN measurable_space] by (auto simp: set_eq_iff) next fix X assume "X ∈ (λA. T' -` A ∩ space ?M')`F" then obtain A where [simp]: "X = T' -` A ∩ space ?M'" and "A ∈ F" by auto have "X ∈ sets M'" using F T' ‹A∈F› by auto moreover have Fi: "A ∈ sets M" using F ‹A∈F› by auto ultimately show "emeasure ?M' X ≠ ∞" using F T T' ‹A∈F› by (simp add: emeasure_distr) qed (use F in auto) qed have "(RN_deriv ?M' ?N') ∘ T ∈ borel_measurable M" using T ac by measurable then show "(λx. RN_deriv ?M' ?N' (T x)) ∈ borel_measurable M" by (simp add: comp_def) have "N = distr N M (T' ∘ 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 "… = distr (distr N M' T) M T'" using T T' by (simp add: distr_distr) also have "… = 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 "… = density M (RN_deriv (distr M M' T) (distr N M' T) ∘ T)" by (simp add: distr_density_distr[OF T T', OF inv]) finally show "density M (λx. 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 ≠ ∞" proof - interpret N: sigma_finite_measure N by fact from N show ?thesis using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N] density_RN_deriv[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 ∈ borel_measurable M" shows RN_deriv_integrable: "integrable N f ⟷ integrable M (λx. enn2real (RN_deriv M N x) * f x)" (is ?integrable) and RN_deriv_integral: "integral⇧^{L}N f = (∫x. enn2real (RN_deriv M N x) * f x ∂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 eq: "density M (RN_deriv M N) = density M (λx. enn2real (RN_deriv M N x))" proof (rule density_cong) from RN_deriv_finite[OF assms(1,2,3)] show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" by eventually_elim (auto simp: less_top) qed (insert ac, auto) show ?integrable apply (subst density_RN_deriv[OF ac, symmetric]) unfolding eq apply (intro integrable_real_density f AE_I2 enn2real_nonneg) apply (insert ac, auto) done show ?integral apply (subst density_RN_deriv[OF ac, symmetric]) unfolding eq apply (intro integral_real_density f AE_I2 enn2real_nonneg) apply (insert ac, auto) done qed proposition (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 ∈ borel_measurable M" and "AE x in M. RN_deriv M N x = ennreal (D x)" and "AE x in N. 0 < D x" and "⋀x. 0 ≤ D x" proof interpret N: finite_measure N by fact note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac] let ?RN = "λt. {x ∈ space M. RN_deriv M N x = t}" show "(λx. enn2real (RN_deriv M N x)) ∈ borel_measurable M" using RN by auto have "N (?RN ∞) = (∫⇧^{+}x. RN_deriv M N x * indicator (?RN ∞) x ∂M)" using RN(1) by (subst RN(2)[symmetric]) (auto simp: emeasure_density) also have "… = (∫⇧^{+}x. ∞ * indicator (?RN ∞) x ∂M)" by (intro nn_integral_cong) (auto simp: indicator_def) also have "… = ∞ * emeasure M (?RN ∞)" using RN by (intro nn_integral_cmult_indicator) auto finally have eq: "N (?RN ∞) = ∞ * emeasure M (?RN ∞)" . moreover have "emeasure M (?RN ∞) = 0" proof (rule ccontr) assume "emeasure M {x ∈ space M. RN_deriv M N x = ∞} ≠ 0" then have "0 < emeasure M {x ∈ space M. RN_deriv M N x = ∞}" by (auto simp: zero_less_iff_neq_zero) with eq have "N (?RN ∞) = ∞" by (simp add: ennreal_mult_eq_top_iff) with N.emeasure_finite[of "?RN ∞"] RN show False by auto qed ultimately have "AE x in M. RN_deriv M N x < ∞" using RN by (intro AE_iff_measurable[THEN iffD2]) (auto simp: less_top[symmetric]) then show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" by auto then have eq: "AE x in N. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" using ac absolutely_continuous_AE by auto have "N (?RN 0) = (∫⇧^{+}x. RN_deriv M N x * indicator (?RN 0) x ∂M)" by (subst RN(2)[symmetric]) (auto simp: emeasure_density) also have "… = (∫⇧^{+}x. 0 ∂M)" by (intro nn_integral_cong) (auto simp: indicator_def) finally have "AE x in N. RN_deriv M N x ≠ 0" using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq) with eq show "AE x in N. 0 < enn2real (RN_deriv M N x)" by (auto simp: enn2real_positive_iff less_top[symmetric] zero_less_iff_neq_zero) qed (rule enn2real_nonneg) lemma (in sigma_finite_measure) RN_deriv_singleton: assumes ac: "absolutely_continuous M N" "sets N = sets M" and x: "{x} ∈ sets M" shows "N {x} = RN_deriv M N x * emeasure M {x}" proof - from ‹{x} ∈ sets M› have "density M (RN_deriv M N) {x} = (∫⇧^{+}w. RN_deriv M N x * indicator {x} w ∂M)" by (