(* Title: HOL/Probability/Radon_Nikodym.thy
Author: Johannes Hölzl, TU München
*)
section \<open>Radon-Nikod{\'y}m derivative\<close>
theory Radon_Nikodym
imports Bochner_Integration
begin
definition diff_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"
where
"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)
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 pos[of "A _"]
by (intro ennreal_suminf_minus)
(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:
"\<exists>h\<in>borel_measurable M. integral\<^sup>N M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>)"
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 blast
let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)"
have [measurable]: "\<And>i. A i \<in> sets M"
using range by fastforce+
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]
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: "\<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 "integral\<^sup>N M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos
by (simp add: nn_integral_suminf nn_integral_cmult_indicator)
also have "\<dots> \<le> (\<Sum>i. ennreal ((1/2)^Suc i))"
proof (intro suminf_le allI)
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 mult_right_mono) auto
also have "\<dots> = (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 "\<dots> \<le> 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 "\<dots> = ennreal (inverse 2 ^ Suc N)"
by (subst ennreal_power[symmetric], simp) (simp add: inverse_ennreal)
finally show "n N * emeasure M (A N) \<le> ennreal ((1/2)^Suc N)"
by simp
qed auto
also have "\<dots> < 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\<^sup>N M ?h \<noteq> \<infinity>"
by (auto simp: top_unique)
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 simp: less_top[symmetric]) }
note pos = this
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_continuousD:
"absolutely_continuous M N \<Longrightarrow> A \<in> sets M \<Longrightarrow> emeasure M A = 0 \<Longrightarrow> 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 \<open>AE x in M. P x\<close> obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
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 \<open>absolutely_continuous M M'\<close> 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_finite_measure:
assumes "finite_measure N" and sets_eq[simp]: "sets N = sets M"
assumes "absolutely_continuous M N"
shows "\<exists>f \<in> borel_measurable M. density M f = N"
proof -
interpret N: finite_measure N by fact
define G where "G = {g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A}"
have [measurable_dest]: "f \<in> G \<Longrightarrow> f \<in> borel_measurable M"
and G_D: "\<And>A. f \<in> G \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) \<le> N A" for f
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[measurable]: "f \<in> G" and g[measurable]: "g \<in> G"
have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
proof safe
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 [measurable]: "A \<in> sets M"
have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
using sets.sets_into_space[OF \<open>A \<in> sets M\<close>] 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>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) =
(\<integral>\<^sup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
(\<integral>\<^sup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
by (auto cong: nn_integral_cong intro!: nn_integral_add)
also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)"
using f g unfolding G_def by (auto intro!: add_mono)
also have "\<dots> = N A"
using union by (subst plus_emeasure) auto
finally show "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" .
qed auto }
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
next
fix A assume "A \<in> sets M"
have "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
(\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
by (intro nn_integral_cong) (simp split: split_indicator)
also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))"
using \<open>incseq f\<close> f \<open>A \<in> sets M\<close>
by (intro nn_integral_monotone_convergence_SUP)
(auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A"
using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_D)
qed }
note SUP_in_G = this
let ?y = "SUP g : G. integral\<^sup>N 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>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A"
from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)"
by (simp cong: nn_integral_cong)
qed
from ennreal_SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this
then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n"
proof safe
fix n assume "range ys \<subseteq> integral\<^sup>N M ` G"
hence "ys n \<in> integral\<^sup>N M ` G" by auto
thus "\<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" by auto
qed
from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^sup>N M (gs n) = ys n" by auto
hence y_eq: "?y = (SUP i. integral\<^sup>N M (gs i))" using ys by auto
let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})"
define f where [abs_def]: "f x = (SUP i. ?g i x)" for 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 \<open>gs (Suc i) \<in> G\<close> 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 [measurable]: "f \<in> borel_measurable M" unfolding G_def by auto
have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def
using g_in_G \<open>incseq ?g\<close> by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def)
also have "\<dots> = ?y"
proof (rule antisym)
show "(SUP i. integral\<^sup>N M (?g i)) \<le> ?y"
using g_in_G by (auto intro: SUP_mono)
show "?y \<le> (SUP i. integral\<^sup>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\<^sup>N M f = ?y" .
have upper_bound: "\<forall>A\<in>sets M. N A \<le> density M f A"
proof (rule ccontr)
assume "\<not> ?thesis"
then obtain A where A[measurable]: "A \<in> 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 \<open>absolutely_continuous M N\<close>[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 \<noteq> 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 = "\<lambda>x. f x + b"
have "nn_integral M f \<noteq> top"
using `f \<in> G`[THEN G_D, of "space M"] by (auto simp: top_unique cong: nn_integral_cong)
with \<open>b \<noteq> top\<close> 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 \<in> sets M" and [simp]: "Y \<subseteq> A"
and Y1: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> density M ?f C \<le> N C"
and Y2: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> A \<Longrightarrow> C \<inter> Y = {} \<Longrightarrow> N C \<le> density M ?f C"
by auto
let ?f' = "\<lambda>x. f x + b * indicator Y x"
have "M Y \<noteq> 0"
proof
assume "M Y = 0"
then have "N Y = 0"
using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, of Y] by auto
then have "N A = N (A - Y)"
by (subst emeasure_Diff) auto
also have "\<dots> \<le> density M ?f (A - Y)"
by (rule Y2) auto
also have "\<dots> \<le> density M ?f A - density M ?f Y"
by (subst emeasure_Diff) auto
also have "\<dots> \<le> 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 "\<dots> < 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 \<open>0 < b\<close> \<open>nn_integral M f \<noteq> top\<close>
by (simp add: nn_integral_add nn_integral_cmult_indicator ennreal_zero_less_mult_iff zero_less_iff_neq_zero)
moreover
have "?f' \<in> G"
unfolding G_def
proof safe
fix X assume [measurable]: "X \<in> sets M"
have "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) = density M f (X - Y) + density M ?f (X \<inter> Y)"
by (auto simp add: emeasure_density nn_integral_add[symmetric] split: split_indicator intro!: nn_integral_cong)
also have "\<dots> \<le> N (X - Y) + N (X \<inter> Y)"
using G_D[OF \<open>f \<in> G\<close>] by (intro add_mono Y1) (auto simp: emeasure_density)
also have "\<dots> = N X"
by (subst plus_emeasure) (auto intro!: arg_cong2[where f=emeasure])
finally show "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) \<le> N X" .
qed simp
then have "nn_integral M ?f' \<le> ?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 \<in> sets (density M f)" then show "emeasure (density M f) A \<le> emeasure N A"
by (auto simp: emeasure_density intro!: G_D[OF \<open>f \<in> G\<close>])
next
fix A assume A: "A \<in> sets (density M f)" then show "emeasure N A \<le> 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 "\<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> (\<forall>i. N (B i) \<noteq> \<infinity>) \<and>
(\<forall>A\<in>sets M. A \<inter> (\<Union>i. B i) = {} \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>))"
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 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'' \<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[measurable]: "\<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: emeasure_subadditive_finite)
also have "\<dots> < \<infinity>" using Q' by (simp add: less_top)
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"
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
have "disjointed Q' i \<in> 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') \<subseteq> sets M"
using Q'_sets by (intro sets.range_disjointed_sets) auto
{ fix A assume A: "A \<in> sets M" "A \<inter> (\<Union>i. disjointed Q' i) = {}"
then have A1: "A \<inter> (\<Union>i. Q' i) = {}"
unfolding UN_disjointed_eq by auto
show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
proof (rule disjCI, simp)
assume *: "emeasure M A = 0 \<or> N A \<noteq> top"
show "emeasure M A = 0 \<and> 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 \<noteq> \<infinity>" by auto
with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)"
using Q' A1 by (auto intro!: plus_emeasure simp: set_eq_iff)
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 \<open>N A \<noteq> \<infinity>\<close> 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
with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>"
using \<open>N A \<noteq> \<infinity>\<close> by (auto simp: top_unique)
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 \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto
qed
qed }
{ fix i
have "N (disjointed Q' i) \<le> N (Q' i)"
by (auto intro!: emeasure_mono simp: disjointed_def)
then show "N (disjointed Q' i) \<noteq> \<infinity>"
using Q'(2)[of i] by (auto simp: top_unique) }
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. density M f = N"
proof -
from split_space_into_finite_sets_and_rest[OF assms]
obtain Q :: "nat \<Rightarrow> 'a set"
where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
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). 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]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq)
show "absolutely_continuous (?M i) (?N i)"
using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close>
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 force
{ fix A i assume A: "A \<in> sets M"
with Q borel have "(\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A"
by (auto simp add: emeasure_density nn_integral_density subset_eq
intro!: nn_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>\<^sup>+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 (space M - (\<Union>i. Q i)) x"
show ?thesis
proof (safe intro!: bexI[of _ ?f])
show "?f \<in> 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 \<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 \<open>A \<in> sets M\<close> by auto
have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator ((space M - (\<Union>i. Q i)) \<inter> A) x \<partial>M)"
using borel by (intro nn_integral_cong) (auto simp: indicator_def)
also have "\<dots> = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)"
using borel Qi \<open>A \<in> sets M\<close>
by (subst nn_integral_add)
(auto simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le)
also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)"
by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto
finally have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" .
moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)"
using Q Q_sets \<open>A \<in> sets M\<close>
by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
moreover
have "(space M - (\<Union>x. Q x)) \<inter> A \<inter> (\<Union>x. Q x) = {}"
by auto
then have "\<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A) = N ((space M - (\<Union>i. Q i)) \<inter> A)"
using in_Q0[of "(space M - (\<Union>i. Q i)) \<inter> A"] \<open>A \<in> sets M\<close> Q by (auto simp: ennreal_top_mult)
moreover have "(space M - (\<Union>i. Q i)) \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
using Q_sets \<open>A \<in> sets M\<close> by auto
moreover have "((\<Union>i. Q i) \<inter> A) \<union> ((space M - (\<Union>i. Q i)) \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> ((space M - (\<Union>i. Q i)) \<inter> A) = {}"
using \<open>A \<in> sets M\<close> sets.sets_into_space by auto
ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)"
using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "(space M - (\<Union>i. Q i)) \<inter> A"] by (simp add: sets_eq)
with \<open>A \<in> sets M\<close> borel Q 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. density M f = N"
proof -
from Ex_finite_integrable_function
obtain h where finite: "integral\<^sup>N 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>\<^sup>+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: 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 \<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_mono) (auto simp: not_le[symmetric])
then have "A \<in> null_sets M"
using \<open>A \<in> sets M\<close> 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" "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
subsection \<open>Uniqueness of densities\<close>
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\<^sup>N 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>\<^sup>+ 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\<^sup>N M g \<noteq> \<infinity>" by (simp cong: nn_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\<^sup>N 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\<^sup>N M g" using pos
by (intro nn_integral_mono_AE) (auto split: split_indicator)
then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by (auto simp: top_unique)
have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^sup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
by (auto intro!: nn_integral_cong simp: indicator_def)
also have "\<dots> = ?P f ?N - ?P g ?N"
proof (rule nn_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"
using pos by (auto split: split_indicator)
qed fact
also have "\<dots> = 0"
unfolding eq[THEN bspec, OF N] using Pg_fin by auto
finally have "AE x in M. f x \<le> 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 \<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>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+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 Q :: "nat \<Rightarrow> 'a set"
where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>"
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 \<inter> (\<Union>i. Q i) = {} \<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[measurable]: "\<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::ennreal. 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 (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x"
proof (rule AE_I')
{ fix f :: "'a \<Rightarrow> ennreal" assume borel: "f \<in> borel_measurable M"
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
let ?A = "\<lambda>i. (space M - (\<Union>i. Q i)) \<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 by auto
have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ennreal) * indicator (?A i) x \<partial>M)"
unfolding eq[OF \<open>?A i \<in> sets M\<close>]
by (auto intro!: nn_integral_mono simp: indicator_def)
also have "\<dots> = i * emeasure M (?A i)"
using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator)
also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by (auto simp: ennreal_mult_less_top of_nat_less_top)
finally have "?N (?A i) \<noteq> \<infinity>" by simp
then show "?A i \<in> null_sets M" using in_Q0[OF \<open>?A i \<in> sets M\<close>] \<open>?A i \<in> sets M\<close> by auto
qed
also have "(\<Union>i. ?A i) = (space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
by (auto simp: ennreal_Ex_less_of_nat less_top[symmetric])
finally have "(space M - (\<Union>i. Q i)) \<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 "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all
then show "((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un)
show "{x \<in> space M. ?f (space M - (\<Union>i. Q i)) x \<noteq> ?f' (space M - (\<Union>i. Q i)) x} \<subseteq>
((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def)
qed
moreover have "AE x in M. (?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
?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
lemma (in sigma_finite_measure) density_unique:
assumes f: "f \<in> borel_measurable M"
assumes f': "f' \<in> 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 \<in> borel_measurable M"
and fin: "integral\<^sup>N M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x"
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: nn_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>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto }
note h_null_sets = this
{ fix A assume "A \<in> sets M"
have "(\<integral>\<^sup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?fM)"
using \<open>A \<in> sets M\<close> h_borel h_nn f f'
by (intro nn_integral_density[symmetric]) auto
also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)"
by (simp_all add: density_eq)
also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)"
using \<open>A \<in> sets M\<close> h_borel h_nn f f'
by (intro nn_integral_density) auto
finally have "(\<integral>\<^sup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * (f' x * indicator A x) \<partial>M)"
by (simp add: ac_simps)
then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)"
using \<open>A \<in> sets M\<close> 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 \<in> borel_measurable M" and f': "f' \<in> borel_measurable M"
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"
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>\<^sup>+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 cover(1)
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"
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\<^sup>N ?N h \<noteq> \<infinity>" 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\<^sup>N ?N h = (\<integral>\<^sup>+x. f x * h x \<partial>M)"
using f h by (auto intro!: nn_integral_density)
then have "(\<integral>\<^sup>+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!: nn_integral_PInf[unfolded infinity_ennreal_def] borel_measurable_vimage)
qed auto
then show "AE x in M. f x \<noteq> \<infinity>"
using fin by (auto elim!: AE_Ball_mp simp: less_top ennreal_mult_less_top)
next
assume AE: "AE x in M. f x \<noteq> \<infinity>"
from sigma_finite guess Q . note Q = this
define A where "A i =
f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ennreal(of_nat (Suc n))}) \<inter> space M" for i
{ 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
show "sigma_finite_measure ?N"
proof (standard, intro exI conjI ballI)
show "countable (range (\<lambda>(i, j). A i \<inter> Q j))"
by auto
show "range (\<lambda>(i, j). A i \<inter> Q j) \<subseteq> sets (density M f)"
using A_in_sets by auto
next
have "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = (\<Union>i j. A i \<inter> Q j)"
by 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" 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 "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = space ?N" by simp
next
fix X assume "X \<in> range (\<lambda>(i, j). A i \<inter> Q j)"
then obtain i j where [simp]:"X = A i \<inter> Q j" by auto
have "(\<integral>\<^sup>+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 \<open>i = 0\<close>)
from nn_integral_cong_AE[OF this] show ?thesis by simp
next
case (Suc n)
then have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
(\<integral>\<^sup>+x. (Suc n :: ennreal) * indicator (Q j) x \<partial>M)"
by (auto intro!: nn_integral_mono simp: indicator_def A_def ennreal_of_nat_eq_real_of_nat)
also have "\<dots> = Suc n * emeasure M (Q j)"
using Q by (auto intro!: nn_integral_cmult_indicator)
also have "\<dots> < \<infinity>"
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 \<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>)"
by (subst sigma_finite_iff_density_finite')
(auto simp: max_def intro!: measurable_If)
subsection \<open>Radon-Nikodym derivative\<close>
definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ennreal" where
"RN_deriv M N =
(if \<exists>f. f \<in> borel_measurable M \<and> density M f = N
then SOME f. f \<in> borel_measurable M \<and> density M f = N
else (\<lambda>_. 0))"
lemma RN_derivI:
assumes "f \<in> borel_measurable M" "density M f = N"
shows "density M (RN_deriv M N) = N"
proof -
have "\<exists>f. f \<in> borel_measurable M \<and> density M f = N"
using assms by auto
moreover then have "density M (SOME f. f \<in> borel_measurable M \<and> density M f = N) = N"
by (rule someI2_ex) auto
ultimately show ?thesis
by (auto simp: RN_deriv_def)
qed
lemma borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M"
proof -
{ assume ex: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N"
have 1: "(SOME f. f \<in> borel_measurable M \<and> density M f = N) \<in> 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 \<in> 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 \<Longrightarrow> sets N = sets M \<Longrightarrow> 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 \<in> borel_measurable M"
shows "integral\<^sup>N N f = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
proof -
have "integral\<^sup>N N f = integral\<^sup>N (density M (RN_deriv M N)) f"
using N by (simp add: density_RN_deriv)
also have "\<dots> = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
using f by (simp add: nn_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"
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 \<in> 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 \<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_countable guess F .. note F = this
show "\<exists>A. countable A \<and> A \<subseteq> sets (distr M M' T) \<and> \<Union>A = space (distr M M' T) \<and> (\<forall>a\<in>A. emeasure (distr M M' T) a \<noteq> \<infinity>)"
proof (intro exI conjI ballI)
show *: "(\<lambda>A. T' -` A \<inter> space ?M') ` F \<subseteq> sets ?M'"
using F T' by (auto simp: measurable_def)
show "\<Union>((\<lambda>A. T' -` A \<inter> space ?M')`F) = space ?M'"
using F T'[THEN measurable_space] by (auto simp: set_eq_iff)
next
fix X assume "X \<in> (\<lambda>A. T' -` A \<inter> space ?M')`F"
then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto
have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto
moreover
have Fi: "A \<in> sets M" using F \<open>A\<in>F\<close> by auto
ultimately show "emeasure ?M' X \<noteq> \<infinity>"
using F T T' \<open>A\<in>F\<close> by (simp add: emeasure_distr)
qed (insert F, auto)
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)
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)"
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 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 \<in> borel_measurable M"
shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
integrable M (\<lambda>x. enn2real (RN_deriv M N x) * f x)" (is ?integrable)
and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. enn2real (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 eq: "density M (RN_deriv M N) = density M (\<lambda>x. 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
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 = ennreal (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 = borel_measurable_RN_deriv density_RN_deriv[OF ac]
let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
show "(\<lambda>x. enn2real (RN_deriv M N x)) \<in> borel_measurable M"
using RN by auto
have "N (?RN \<infinity>) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)"
using RN(1) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
also have "\<dots> = (\<integral>\<^sup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
by (intro nn_integral_cong) (auto simp: indicator_def)
also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)"
using RN by (intro nn_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"
then have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}"
by (auto simp: zero_less_iff_neq_zero)
with eq have "N (?RN \<infinity>) = \<infinity>" by (simp add: ennreal_mult_eq_top_iff)
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 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) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)"
by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
also have "\<dots> = (\<integral>\<^sup>+ x. 0 \<partial>M)"
by (intro nn_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 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} \<in> sets M"
shows "N {x} = RN_deriv M N x * emeasure M {x}"
proof -
from \<open>{x} \<in> sets M\<close>
have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong)
with x density_RN_deriv[OF ac] show ?thesis
by (auto simp: max_def)
qed
end