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Datei:
Giry_Monad.thy
Sprache: Isabelle
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(* Title: HOL/Probability/Giry_Monad.thy
Author: Johannes Hölzl, TU München
Author: Manuel Eberl, TU München
Defines the subprobability spaces, the subprobability functor and the Giry monad on subprobability
spaces.
*)
theory Giry_Monad
imports Probability_Measure "HOL-Library.Monad_Syntax"
begin
section \<open>Sub-probability spaces\<close>
locale subprob_space = finite_measure +
assumes emeasure_space_le_1: "emeasure M (space M) \ 1"
assumes subprob_not_empty: "space M \ {}"
lemma subprob_spaceI[Pure.intro!]:
assumes *: "emeasure M (space M) \ 1"
assumes "space M \ {}"
shows "subprob_space M"
proof -
interpret finite_measure M
proof
show "emeasure M (space M) \ \" using * by (auto simp: top_unique)
qed
show "subprob_space M" by standard fact+
qed
lemma (in subprob_space) emeasure_subprob_space_less_top: "emeasure M A \ top"
using emeasure_finite[of A] .
lemma prob_space_imp_subprob_space:
"prob_space M \ subprob_space M"
by (rule subprob_spaceI) (simp_all add: prob_space.emeasure_space_1 prob_space.not_empty)
lemma subprob_space_imp_sigma_finite: "subprob_space M \ sigma_finite_measure M"
unfolding subprob_space_def finite_measure_def by simp
sublocale prob_space \<subseteq> subprob_space
by (rule subprob_spaceI) (simp_all add: emeasure_space_1 not_empty)
lemma subprob_space_sigma [simp]: "\ \ {} \ subprob_space (sigma \ X)"
by(rule subprob_spaceI)(simp_all add: emeasure_sigma space_measure_of_conv)
lemma subprob_space_null_measure: "space M \ {} \ subprob_space (null_measure M)"
by(simp add: null_measure_def)
lemma (in subprob_space) subprob_space_distr:
assumes f: "f \ measurable M M'" and "space M' \ {}" shows "subprob_space (distr M M' f)"
proof (rule subprob_spaceI)
have "f -` space M' \ space M = space M" using f by (auto dest: measurable_space)
with f show "emeasure (distr M M' f) (space (distr M M' f)) \ 1"
by (auto simp: emeasure_distr emeasure_space_le_1)
show "space (distr M M' f) \ {}" by (simp add: assms)
qed
lemma (in subprob_space) subprob_emeasure_le_1: "emeasure M X \ 1"
by (rule order.trans[OF emeasure_space emeasure_space_le_1])
lemma (in subprob_space) subprob_measure_le_1: "measure M X \ 1"
using subprob_emeasure_le_1[of X] by (simp add: emeasure_eq_measure)
lemma (in subprob_space) nn_integral_le_const:
assumes "0 \ c" "AE x in M. f x \ c"
shows "(\\<^sup>+x. f x \M) \ c"
proof -
have "(\\<^sup>+ x. f x \M) \ (\\<^sup>+ x. c \M)"
by(rule nn_integral_mono_AE) fact
also have "\ \ c * emeasure M (space M)"
using \<open>0 \<le> c\<close> by simp
also have "\ \ c * 1" using emeasure_space_le_1 \0 \ c\ by(rule mult_left_mono)
finally show ?thesis by simp
qed
lemma emeasure_density_distr_interval:
fixes h :: "real \ real" and g :: "real \ real" and g' :: "real \ real"
assumes [simp]: "a \ b"
assumes Mf[measurable]: "f \ borel_measurable borel"
assumes Mg[measurable]: "g \ borel_measurable borel"
assumes Mg'[measurable]: "g' \<in> borel_measurable borel"
assumes Mh[measurable]: "h \ borel_measurable borel"
assumes prob: "subprob_space (density lborel f)"
assumes nonnegf: "\x. f x \ 0"
assumes derivg: "\x. x \ {a..b} \ (g has_real_derivative g' x) (at x)"
assumes contg': "continuous_on {a..b} g'"
assumes mono: "strict_mono_on g {a..b}" and inv: "\x. h x \ {a..b} \ g (h x) = x"
assumes range: "{a..b} \ range h"
shows "emeasure (distr (density lborel f) lborel h) {a..b} =
emeasure (density lborel (\<lambda>x. f (g x) * g' x)) {a..b}"
proof (cases "a < b")
assume "a < b"
from mono have inj: "inj_on g {a..b}" by (rule strict_mono_on_imp_inj_on)
from mono have mono': "mono_on g {a..b}" by (rule strict_mono_on_imp_mono_on)
from mono' derivg have "\x. x \ {a<.. g' x \ 0"
by (rule mono_on_imp_deriv_nonneg) auto
from contg' this have derivg_nonneg: "\x. x \ {a..b} \ g' x \ 0"
by (rule continuous_ge_on_Ioo) (simp_all add: \<open>a < b\<close>)
from derivg have contg: "continuous_on {a..b} g" by (rule has_real_derivative_imp_continuous_on)
have A: "h -` {a..b} = {g a..g b}"
proof (intro equalityI subsetI)
fix x assume x: "x \ h -` {a..b}"
hence "g (h x) \ {g a..g b}" by (auto intro: mono_onD[OF mono'])
with inv and x show "x \ {g a..g b}" by simp
next
fix y assume y: "y \ {g a..g b}"
with IVT'[OF _ _ _ contg, of y] obtain x where "x \ {a..b}" "y = g x" by auto
with range and inv show "y \ h -` {a..b}" by auto
qed
have prob': "subprob_space (distr (density lborel f) lborel h)"
by (rule subprob_space.subprob_space_distr[OF prob]) (simp_all add: Mh)
have B: "emeasure (distr (density lborel f) lborel h) {a..b} =
\<integral>\<^sup>+x. f x * indicator (h -` {a..b}) x \<partial>lborel"
by (subst emeasure_distr) (simp_all add: emeasure_density Mf Mh measurable_sets_borel[OF Mh])
also note A
also have "emeasure (distr (density lborel f) lborel h) {a..b} \ 1"
by (rule subprob_space.subprob_emeasure_le_1) (rule prob')
hence "emeasure (distr (density lborel f) lborel h) {a..b} \ \" by (auto simp: top_unique)
with assms have "(\\<^sup>+x. f x * indicator {g a..g b} x \lborel) =
(\<integral>\<^sup>+x. f (g x) * g' x * indicator {a..b} x \<partial>lborel)"
by (intro nn_integral_substitution_aux)
(auto simp: derivg_nonneg A B emeasure_density mult.commute \<open>a < b\<close>)
also have "... = emeasure (density lborel (\x. f (g x) * g' x)) {a..b}"
by (simp add: emeasure_density)
finally show ?thesis .
next
assume "\a < b"
with \<open>a \<le> b\<close> have [simp]: "b = a" by (simp add: not_less del: \<open>a \<le> b\<close>)
from inv and range have "h -` {a} = {g a}" by auto
thus ?thesis by (simp_all add: emeasure_distr emeasure_density measurable_sets_borel[OF Mh])
qed
locale pair_subprob_space =
pair_sigma_finite M1 M2 + M1: subprob_space M1 + M2: subprob_space M2 for M1 M2
sublocale pair_subprob_space \<subseteq> P?: subprob_space "M1 \<Otimes>\<^sub>M M2"
proof
from mult_le_one[OF M1.emeasure_space_le_1 _ M2.emeasure_space_le_1]
show "emeasure (M1 \\<^sub>M M2) (space (M1 \\<^sub>M M2)) \ 1"
by (simp add: M2.emeasure_pair_measure_Times space_pair_measure)
from M1.subprob_not_empty and M2.subprob_not_empty show "space (M1 \\<^sub>M M2) \ {}"
by (simp add: space_pair_measure)
qed
lemma subprob_space_null_measure_iff:
"subprob_space (null_measure M) \ space M \ {}"
by (auto intro!: subprob_spaceI dest: subprob_space.subprob_not_empty)
lemma subprob_space_restrict_space:
assumes M: "subprob_space M"
and A: "A \ space M \ sets M" "A \ space M \ {}"
shows "subprob_space (restrict_space M A)"
proof(rule subprob_spaceI)
have "emeasure (restrict_space M A) (space (restrict_space M A)) = emeasure M (A \ space M)"
using A by(simp add: emeasure_restrict_space space_restrict_space)
also have "\ \ 1" by(rule subprob_space.subprob_emeasure_le_1)(rule M)
finally show "emeasure (restrict_space M A) (space (restrict_space M A)) \ 1" .
next
show "space (restrict_space M A) \ {}"
using A by(simp add: space_restrict_space)
qed
definition subprob_algebra :: "'a measure \ 'a measure measure" where
"subprob_algebra K =
(SUP A \<in> sets K. vimage_algebra {M. subprob_space M \<and> sets M = sets K} (\<lambda>M. emeasure M A) borel)"
lemma space_subprob_algebra: "space (subprob_algebra A) = {M. subprob_space M \ sets M = sets A}"
by (auto simp add: subprob_algebra_def space_Sup_eq_UN)
lemma subprob_algebra_cong: "sets M = sets N \ subprob_algebra M = subprob_algebra N"
by (simp add: subprob_algebra_def)
lemma measurable_emeasure_subprob_algebra[measurable]:
"a \ sets A \ (\M. emeasure M a) \ borel_measurable (subprob_algebra A)"
by (auto intro!: measurable_Sup1 measurable_vimage_algebra1 simp: subprob_algebra_def)
lemma measurable_measure_subprob_algebra[measurable]:
"a \ sets A \ (\M. measure M a) \ borel_measurable (subprob_algebra A)"
unfolding measure_def by measurable
lemma subprob_measurableD:
assumes N: "N \ measurable M (subprob_algebra S)" and x: "x \ space M"
shows "space (N x) = space S"
and "sets (N x) = sets S"
and "measurable (N x) K = measurable S K"
and "measurable K (N x) = measurable K S"
using measurable_space[OF N x]
by (auto simp: space_subprob_algebra intro!: measurable_cong_sets dest: sets_eq_imp_space_eq)
ML \<open>
fun subprob_cong thm ctxt = (
let
val thm' = Thm.transfer' ctxt thm
val free = thm' |> Thm.concl_of |> HOLogic.dest_Trueprop |> dest_comb |> fst |>
dest_comb |> snd |> strip_abs_body |> head_of |> is_Free
in
if free then ([], Measurable.add_local_cong (thm' RS @{thm subprob_measurableD(2)}) ctxt)
else ([], ctxt)
end
handle THM _ => ([], ctxt) | TERM _ => ([], ctxt))
\<close>
setup \<open>
Context.theory_map (Measurable.add_preprocessor "subprob_cong" subprob_cong)
\<close>
context
fixes K M N assumes K: "K \ measurable M (subprob_algebra N)"
begin
lemma subprob_space_kernel: "a \ space M \ subprob_space (K a)"
using measurable_space[OF K] by (simp add: space_subprob_algebra)
lemma sets_kernel: "a \ space M \ sets (K a) = sets N"
using measurable_space[OF K] by (simp add: space_subprob_algebra)
lemma measurable_emeasure_kernel[measurable]:
"A \ sets N \ (\a. emeasure (K a) A) \ borel_measurable M"
using measurable_compose[OF K measurable_emeasure_subprob_algebra] .
end
lemma measurable_subprob_algebra:
"(\a. a \ space M \ subprob_space (K a)) \
(\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N) \<Longrightarrow>
(\<And>A. A \<in> sets N \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M) \<Longrightarrow>
K \<in> measurable M (subprob_algebra N)"
by (auto intro!: measurable_Sup2 measurable_vimage_algebra2 simp: subprob_algebra_def)
lemma measurable_submarkov:
"K \ measurable M (subprob_algebra M) \
(\<forall>x\<in>space M. subprob_space (K x) \<and> sets (K x) = sets M) \<and>
(\<forall>A\<in>sets M. (\<lambda>x. emeasure (K x) A) \<in> measurable M borel)"
proof
assume "(\x\space M. subprob_space (K x) \ sets (K x) = sets M) \
(\<forall>A\<in>sets M. (\<lambda>x. emeasure (K x) A) \<in> borel_measurable M)"
then show "K \ measurable M (subprob_algebra M)"
by (intro measurable_subprob_algebra) auto
next
assume "K \ measurable M (subprob_algebra M)"
then show "(\x\space M. subprob_space (K x) \ sets (K x) = sets M) \
(\<forall>A\<in>sets M. (\<lambda>x. emeasure (K x) A) \<in> borel_measurable M)"
by (auto dest: subprob_space_kernel sets_kernel)
qed
lemma measurable_subprob_algebra_generated:
assumes eq: "sets N = sigma_sets \ G" and "Int_stable G" "G \ Pow \"
assumes subsp: "\a. a \ space M \ subprob_space (K a)"
assumes sets: "\a. a \ space M \ sets (K a) = sets N"
assumes "\A. A \ G \ (\a. emeasure (K a) A) \ borel_measurable M"
assumes \<Omega>: "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M"
shows "K \ measurable M (subprob_algebra N)"
proof (rule measurable_subprob_algebra)
fix a assume "a \ space M" then show "subprob_space (K a)" "sets (K a) = sets N" by fact+
next
interpret G: sigma_algebra \<Omega> "sigma_sets \<Omega> G"
using \<open>G \<subseteq> Pow \<Omega>\<close> by (rule sigma_algebra_sigma_sets)
fix A assume "A \ sets N" with assms(2,3) show "(\a. emeasure (K a) A) \ borel_measurable M"
unfolding \<open>sets N = sigma_sets \<Omega> G\<close>
proof (induction rule: sigma_sets_induct_disjoint)
case (basic A) then show ?case by fact
next
case empty then show ?case by simp
next
case (compl A)
have "(\a. emeasure (K a) (\ - A)) \ borel_measurable M \
(\<lambda>a. emeasure (K a) \<Omega> - emeasure (K a) A) \<in> borel_measurable M"
using G.top G.sets_into_space sets eq compl subprob_space.emeasure_subprob_space_less_top[OF subsp]
by (intro measurable_cong emeasure_Diff) auto
with compl \<Omega> show ?case
by simp
next
case (union F)
moreover have "(\a. emeasure (K a) (\i. F i)) \ borel_measurable M \
(\<lambda>a. \<Sum>i. emeasure (K a) (F i)) \<in> borel_measurable M"
using sets union eq
by (intro measurable_cong suminf_emeasure[symmetric]) auto
ultimately show ?case
by auto
qed
qed
lemma space_subprob_algebra_empty_iff:
"space (subprob_algebra N) = {} \ space N = {}"
proof
have "\x. x \ space N \ density N (\_. 0) \ space (subprob_algebra N)"
by (auto simp: space_subprob_algebra emeasure_density intro!: subprob_spaceI)
then show "space (subprob_algebra N) = {} \ space N = {}"
by auto
next
assume "space N = {}"
hence "sets N = {{}}" by (simp add: space_empty_iff)
moreover have "\M. subprob_space M \ sets M \ {{}}"
by (simp add: subprob_space.subprob_not_empty space_empty_iff[symmetric])
ultimately show "space (subprob_algebra N) = {}" by (auto simp: space_subprob_algebra)
qed
lemma nn_integral_measurable_subprob_algebra[measurable]:
assumes f: "f \ borel_measurable N"
shows "(\M. integral\<^sup>N M f) \ borel_measurable (subprob_algebra N)" (is "_ \ ?B")
using f
proof induct
case (cong f g)
moreover have "(\M'. \\<^sup>+M''. f M'' \M') \ ?B \ (\M'. \\<^sup>+M''. g M'' \M') \ ?B"
by (intro measurable_cong nn_integral_cong cong)
(auto simp: space_subprob_algebra dest!: sets_eq_imp_space_eq)
ultimately show ?case by simp
next
case (set B)
then have "(\M'. \\<^sup>+M''. indicator B M'' \M') \ ?B \ (\M'. emeasure M' B) \ ?B"
by (intro measurable_cong nn_integral_indicator) (simp add: space_subprob_algebra)
with set show ?case
by (simp add: measurable_emeasure_subprob_algebra)
next
case (mult f c)
then have "(\M'. \\<^sup>+M''. c * f M'' \M') \ ?B \ (\M'. c * \\<^sup>+M''. f M'' \M') \ ?B"
by (intro measurable_cong nn_integral_cmult) (auto simp add: space_subprob_algebra)
with mult show ?case
by simp
next
case (add f g)
then have "(\M'. \\<^sup>+M''. f M'' + g M'' \M') \ ?B \ (\M'. (\\<^sup>+M''. f M'' \M') + (\\<^sup>+M''. g M'' \M')) \ ?B"
by (intro measurable_cong nn_integral_add) (auto simp add: space_subprob_algebra)
with add show ?case
by (simp add: ac_simps)
next
case (seq F)
then have "(\M'. \\<^sup>+M''. (SUP i. F i) M'' \M') \ ?B \ (\M'. SUP i. (\\<^sup>+M''. F i M'' \M')) \ ?B"
unfolding SUP_apply
by (intro measurable_cong nn_integral_monotone_convergence_SUP) (auto simp add: space_subprob_algebra)
with seq show ?case
by (simp add: ac_simps)
qed
lemma measurable_distr:
assumes [measurable]: "f \ measurable M N"
shows "(\M'. distr M' N f) \ measurable (subprob_algebra M) (subprob_algebra N)"
proof (cases "space N = {}")
assume not_empty: "space N \ {}"
show ?thesis
proof (rule measurable_subprob_algebra)
fix A assume A: "A \ sets N"
then have "(\M'. emeasure (distr M' N f) A) \ borel_measurable (subprob_algebra M) \
(\<lambda>M'. emeasure M' (f -` A \<inter> space M)) \<in> borel_measurable (subprob_algebra M)"
by (intro measurable_cong)
(auto simp: emeasure_distr space_subprob_algebra
intro!: arg_cong2[where f=emeasure] sets_eq_imp_space_eq arg_cong2[where f="(\)"])
also have "\"
using A by (intro measurable_emeasure_subprob_algebra) simp
finally show "(\M'. emeasure (distr M' N f) A) \ borel_measurable (subprob_algebra M)" .
qed (auto intro!: subprob_space.subprob_space_distr simp: space_subprob_algebra not_empty cong: measurable_cong_sets)
qed (insert assms, auto simp: measurable_empty_iff space_subprob_algebra_empty_iff)
lemma emeasure_space_subprob_algebra[measurable]:
"(\a. emeasure a (space a)) \ borel_measurable (subprob_algebra N)"
proof-
have "(\a. emeasure a (space N)) \ borel_measurable (subprob_algebra N)" (is "?f \ ?M")
by (rule measurable_emeasure_subprob_algebra) simp
also have "?f \ ?M \ (\a. emeasure a (space a)) \ ?M"
by (rule measurable_cong) (auto simp: space_subprob_algebra dest: sets_eq_imp_space_eq)
finally show ?thesis .
qed
lemma integrable_measurable_subprob_algebra[measurable]:
fixes f :: "'a \ 'b::{banach, second_countable_topology}"
assumes [measurable]: "f \ borel_measurable N"
shows "Measurable.pred (subprob_algebra N) (\M. integrable M f)"
proof (rule measurable_cong[THEN iffD2])
show "M \ space (subprob_algebra N) \ integrable M f \ (\\<^sup>+x. norm (f x) \M) < \" for M
by (auto simp: space_subprob_algebra integrable_iff_bounded)
qed measurable
lemma integral_measurable_subprob_algebra[measurable]:
fixes f :: "'a \ 'b::{banach, second_countable_topology}"
assumes f [measurable]: "f \ borel_measurable N"
shows "(\M. integral\<^sup>L M f) \ subprob_algebra N \\<^sub>M borel"
proof -
from borel_measurable_implies_sequence_metric[OF f, of 0]
obtain F where F: "\i. simple_function N (F i)"
"\x. x \ space N \ (\i. F i x) \ f x"
"\i x. x \ space N \ norm (F i x) \ 2 * norm (f x)"
unfolding norm_conv_dist by blast
have [measurable]: "F i \ N \\<^sub>M count_space UNIV" for i
using F(1) by (rule measurable_simple_function)
define F' where [abs_def]:
"F' M i = (if integrable M f then integral\<^sup>L M (F i) else 0)" for M i
have "(\M. F' M i) \ subprob_algebra N \\<^sub>M borel" for i
proof (rule measurable_cong[THEN iffD2])
fix M assume "M \ space (subprob_algebra N)"
then have [simp]: "sets M = sets N" "space M = space N" "subprob_space M"
by (auto simp: space_subprob_algebra intro!: sets_eq_imp_space_eq)
interpret subprob_space M by fact
have "F' M i = (if integrable M f then Bochner_Integration.simple_bochner_integral M (F i) else 0)"
using F(1)
by (subst simple_bochner_integrable_eq_integral)
(auto simp: simple_bochner_integrable.simps simple_function_def F'_def)
then show "F' M i = (if integrable M f then \y\F i ` space N. measure M {x\space N. F i x = y} *\<^sub>R y else 0)"
unfolding simple_bochner_integral_def by simp
qed measurable
moreover
have "F' M \ integral\<^sup>L M f" if M: "M \ space (subprob_algebra N)" for M
proof cases
from M have [simp]: "sets M = sets N" "space M = space N"
by (auto simp: space_subprob_algebra intro!: sets_eq_imp_space_eq)
assume "integrable M f" then show ?thesis
unfolding F'_def using F(1)[THEN borel_measurable_simple_function] F
by (auto intro!: integral_dominated_convergence[where w="\x. 2 * norm (f x)"]
cong: measurable_cong_sets)
qed (auto simp: F'_def not_integrable_integral_eq)
ultimately show ?thesis
by (rule borel_measurable_LIMSEQ_metric)
qed
(* TODO: Rename. This name is too general -- Manuel *)
lemma measurable_pair_measure:
assumes f: "f \ measurable M (subprob_algebra N)"
assumes g: "g \ measurable M (subprob_algebra L)"
shows "(\x. f x \\<^sub>M g x) \ measurable M (subprob_algebra (N \\<^sub>M L))"
proof (rule measurable_subprob_algebra)
{ fix x assume "x \ space M"
with measurable_space[OF f] measurable_space[OF g]
have fx: "f x \ space (subprob_algebra N)" and gx: "g x \ space (subprob_algebra L)"
by auto
interpret F: subprob_space "f x"
using fx by (simp add: space_subprob_algebra)
interpret G: subprob_space "g x"
using gx by (simp add: space_subprob_algebra)
interpret pair_subprob_space "f x" "g x" ..
show "subprob_space (f x \\<^sub>M g x)" by unfold_locales
show sets_eq: "sets (f x \\<^sub>M g x) = sets (N \\<^sub>M L)"
using fx gx by (simp add: space_subprob_algebra)
have 1: "\A B. A \ sets N \ B \ sets L \ emeasure (f x \\<^sub>M g x) (A \ B) = emeasure (f x) A * emeasure (g x) B"
using fx gx by (intro G.emeasure_pair_measure_Times) (auto simp: space_subprob_algebra)
have "emeasure (f x \\<^sub>M g x) (space (f x \\<^sub>M g x)) =
emeasure (f x) (space (f x)) * emeasure (g x) (space (g x))"
by (subst G.emeasure_pair_measure_Times[symmetric]) (simp_all add: space_pair_measure)
hence 2: "\A. A \ sets (N \\<^sub>M L) \ emeasure (f x \\<^sub>M g x) (space N \ space L - A) =
... - emeasure (f x \<Otimes>\<^sub>M g x) A"
using emeasure_compl[simplified, OF _ P.emeasure_finite]
unfolding sets_eq
unfolding sets_eq_imp_space_eq[OF sets_eq]
by (simp add: space_pair_measure G.emeasure_pair_measure_Times)
note 1 2 sets_eq }
note Times = this(1) and Compl = this(2) and sets_eq = this(3)
fix A assume A: "A \ sets (N \\<^sub>M L)"
show "(\a. emeasure (f a \\<^sub>M g a) A) \ borel_measurable M"
using Int_stable_pair_measure_generator pair_measure_closed A
unfolding sets_pair_measure
proof (induct A rule: sigma_sets_induct_disjoint)
case (basic A) then show ?case
by (auto intro!: borel_measurable_times_ennreal simp: Times cong: measurable_cong)
(auto intro!: measurable_emeasure_kernel f g)
next
case (compl A)
then have A: "A \ sets (N \\<^sub>M L)"
by (auto simp: sets_pair_measure)
have "(\x. emeasure (f x) (space (f x)) * emeasure (g x) (space (g x)) -
emeasure (f x \<Otimes>\<^sub>M g x) A) \<in> borel_measurable M" (is "?f \<in> ?M")
using compl(2) f g by measurable
thus ?case by (simp add: Compl A cong: measurable_cong)
next
case (union A)
then have "range A \ sets (N \\<^sub>M L)" "disjoint_family A"
by (auto simp: sets_pair_measure)
then have "(\a. emeasure (f a \\<^sub>M g a) (\i. A i)) \ borel_measurable M \
(\<lambda>a. \<Sum>i. emeasure (f a \<Otimes>\<^sub>M g a) (A i)) \<in> borel_measurable M"
by (intro measurable_cong suminf_emeasure[symmetric])
(auto simp: sets_eq)
also have "\"
using union by auto
finally show ?case .
qed simp
qed
lemma restrict_space_measurable:
assumes X: "X \ {}" "X \ sets K"
assumes N: "N \ measurable M (subprob_algebra K)"
shows "(\x. restrict_space (N x) X) \ measurable M (subprob_algebra (restrict_space K X))"
proof (rule measurable_subprob_algebra)
fix a assume a: "a \ space M"
from N[THEN measurable_space, OF this]
have "subprob_space (N a)" and [simp]: "sets (N a) = sets K" "space (N a) = space K"
by (auto simp add: space_subprob_algebra dest: sets_eq_imp_space_eq)
then interpret subprob_space "N a"
by simp
show "subprob_space (restrict_space (N a) X)"
proof
show "space (restrict_space (N a) X) \ {}"
using X by (auto simp add: space_restrict_space)
show "emeasure (restrict_space (N a) X) (space (restrict_space (N a) X)) \ 1"
using X by (simp add: emeasure_restrict_space space_restrict_space subprob_emeasure_le_1)
qed
show "sets (restrict_space (N a) X) = sets (restrict_space K X)"
by (intro sets_restrict_space_cong) fact
next
fix A assume A: "A \ sets (restrict_space K X)"
show "(\a. emeasure (restrict_space (N a) X) A) \ borel_measurable M"
proof (subst measurable_cong)
fix a assume "a \ space M"
from N[THEN measurable_space, OF this]
have [simp]: "sets (N a) = sets K" "space (N a) = space K"
by (auto simp add: space_subprob_algebra dest: sets_eq_imp_space_eq)
show "emeasure (restrict_space (N a) X) A = emeasure (N a) (A \ X)"
using X A by (subst emeasure_restrict_space) (auto simp add: sets_restrict_space ac_simps)
next
show "(\w. emeasure (N w) (A \ X)) \ borel_measurable M"
using A X
by (intro measurable_compose[OF N measurable_emeasure_subprob_algebra])
(auto simp: sets_restrict_space)
qed
qed
section \<open>Properties of return\<close>
definition return :: "'a measure \ 'a \ 'a measure" where
"return R x = measure_of (space R) (sets R) (\A. indicator A x)"
lemma space_return[simp]: "space (return M x) = space M"
by (simp add: return_def)
lemma sets_return[simp]: "sets (return M x) = sets M"
by (simp add: return_def)
lemma measurable_return1[simp]: "measurable (return N x) L = measurable N L"
by (simp cong: measurable_cong_sets)
lemma measurable_return2[simp]: "measurable L (return N x) = measurable L N"
by (simp cong: measurable_cong_sets)
lemma return_sets_cong: "sets M = sets N \ return M = return N"
by (auto dest: sets_eq_imp_space_eq simp: fun_eq_iff return_def)
lemma return_cong: "sets A = sets B \ return A x = return B x"
by (auto simp add: return_def dest: sets_eq_imp_space_eq)
lemma emeasure_return[simp]:
assumes "A \ sets M"
shows "emeasure (return M x) A = indicator A x"
proof (rule emeasure_measure_of[OF return_def])
show "sets M \ Pow (space M)" by (rule sets.space_closed)
show "positive (sets (return M x)) (\A. indicator A x)" by (simp add: positive_def)
from assms show "A \ sets (return M x)" unfolding return_def by simp
show "countably_additive (sets (return M x)) (\A. indicator A x)"
by (auto intro!: countably_additiveI suminf_indicator)
qed
lemma prob_space_return: "x \ space M \ prob_space (return M x)"
by rule simp
lemma subprob_space_return: "x \ space M \ subprob_space (return M x)"
by (intro prob_space_return prob_space_imp_subprob_space)
lemma subprob_space_return_ne:
assumes "space M \ {}" shows "subprob_space (return M x)"
proof
show "emeasure (return M x) (space (return M x)) \ 1"
by (subst emeasure_return) (auto split: split_indicator)
qed (simp, fact)
lemma measure_return: assumes X: "X \ sets M" shows "measure (return M x) X = indicator X x"
unfolding measure_def emeasure_return[OF X, of x] by (simp split: split_indicator)
lemma AE_return:
assumes [simp]: "x \ space M" and [measurable]: "Measurable.pred M P"
shows "(AE y in return M x. P y) \ P x"
proof -
have "(AE y in return M x. y \ {x\space M. \ P x}) \ P x"
by (subst AE_iff_null_sets[symmetric]) (simp_all add: null_sets_def split: split_indicator)
also have "(AE y in return M x. y \ {x\space M. \ P x}) \ (AE y in return M x. P y)"
by (rule AE_cong) auto
finally show ?thesis .
qed
lemma nn_integral_return:
assumes "x \ space M" "g \ borel_measurable M"
shows "(\\<^sup>+ a. g a \return M x) = g x"
proof-
interpret prob_space "return M x" by (rule prob_space_return[OF \<open>x \<in> space M\<close>])
have "(\\<^sup>+ a. g a \return M x) = (\\<^sup>+ a. g x \return M x)" using assms
by (intro nn_integral_cong_AE) (auto simp: AE_return)
also have "... = g x"
using nn_integral_const[of "return M x"] emeasure_space_1 by simp
finally show ?thesis .
qed
lemma integral_return:
fixes g :: "_ \ 'a :: {banach, second_countable_topology}"
assumes "x \ space M" "g \ borel_measurable M"
shows "(\a. g a \return M x) = g x"
proof-
interpret prob_space "return M x" by (rule prob_space_return[OF \<open>x \<in> space M\<close>])
have "(\a. g a \return M x) = (\a. g x \return M x)" using assms
by (intro integral_cong_AE) (auto simp: AE_return)
then show ?thesis
using prob_space by simp
qed
lemma return_measurable[measurable]: "return N \ measurable N (subprob_algebra N)"
by (rule measurable_subprob_algebra) (auto simp: subprob_space_return)
lemma distr_return:
assumes "f \ measurable M N" and "x \ space M"
shows "distr (return M x) N f = return N (f x)"
using assms by (intro measure_eqI) (simp_all add: indicator_def emeasure_distr)
lemma return_restrict_space:
"\ \ sets M \ return (restrict_space M \) x = restrict_space (return M x) \"
by (auto intro!: measure_eqI simp: sets_restrict_space emeasure_restrict_space)
lemma measurable_distr2:
assumes f[measurable]: "case_prod f \ measurable (L \\<^sub>M M) N"
assumes g[measurable]: "g \ measurable L (subprob_algebra M)"
shows "(\x. distr (g x) N (f x)) \ measurable L (subprob_algebra N)"
proof -
have "(\x. distr (g x) N (f x)) \ measurable L (subprob_algebra N)
\<longleftrightarrow> (\<lambda>x. distr (return L x \<Otimes>\<^sub>M g x) N (case_prod f)) \<in> measurable L (subprob_algebra N)"
proof (rule measurable_cong)
fix x assume x: "x \ space L"
have gx: "g x \ space (subprob_algebra M)"
using measurable_space[OF g x] .
then have [simp]: "sets (g x) = sets M"
by (simp add: space_subprob_algebra)
then have [simp]: "space (g x) = space M"
by (rule sets_eq_imp_space_eq)
let ?R = "return L x"
from measurable_compose_Pair1[OF x f] have f_M': "f x \ measurable M N"
by simp
interpret subprob_space "g x"
using gx by (simp add: space_subprob_algebra)
have space_pair_M'[simp]: "\X. space (X \\<^sub>M g x) = space (X \\<^sub>M M)"
by (simp add: space_pair_measure)
show "distr (g x) N (f x) = distr (?R \\<^sub>M g x) N (case_prod f)" (is "?l = ?r")
proof (rule measure_eqI)
show "sets ?l = sets ?r"
by simp
next
fix A assume "A \ sets ?l"
then have A[measurable]: "A \ sets N"
by simp
then have "emeasure ?r A = emeasure (?R \\<^sub>M g x) ((\(x, y). f x y) -` A \ space (?R \\<^sub>M g x))"
by (auto simp add: emeasure_distr f_M' cong: measurable_cong_sets)
also have "\ = (\\<^sup>+M''. emeasure (g x) (f M'' -` A \ space M) \?R)"
apply (subst emeasure_pair_measure_alt)
apply (rule measurable_sets[OF _ A])
apply (auto simp add: f_M' cong: measurable_cong_sets)
apply (intro nn_integral_cong arg_cong[where f="emeasure (g x)"])
apply (auto simp: space_subprob_algebra space_pair_measure)
done
also have "\ = emeasure (g x) (f x -` A \ space M)"
by (subst nn_integral_return)
(auto simp: x intro!: measurable_emeasure)
also have "\ = emeasure ?l A"
by (simp add: emeasure_distr f_M' cong: measurable_cong_sets)
finally show "emeasure ?l A = emeasure ?r A" ..
qed
qed
also have "\"
apply (intro measurable_compose[OF measurable_pair_measure measurable_distr])
apply (rule return_measurable)
apply measurable
done
finally show ?thesis .
qed
lemma nn_integral_measurable_subprob_algebra2:
assumes f[measurable]: "(\(x, y). f x y) \ borel_measurable (M \\<^sub>M N)"
assumes N[measurable]: "L \ measurable M (subprob_algebra N)"
shows "(\x. integral\<^sup>N (L x) (f x)) \ borel_measurable M"
proof -
note nn_integral_measurable_subprob_algebra[measurable]
note measurable_distr2[measurable]
have "(\x. integral\<^sup>N (distr (L x) (M \\<^sub>M N) (\y. (x, y))) (\(x, y). f x y)) \ borel_measurable M"
by measurable
then show "(\x. integral\<^sup>N (L x) (f x)) \ borel_measurable M"
by (rule measurable_cong[THEN iffD1, rotated])
(simp add: nn_integral_distr)
qed
lemma emeasure_measurable_subprob_algebra2:
assumes A[measurable]: "(SIGMA x:space M. A x) \ sets (M \\<^sub>M N)"
assumes L[measurable]: "L \ measurable M (subprob_algebra N)"
shows "(\x. emeasure (L x) (A x)) \ borel_measurable M"
proof -
{ fix x assume "x \ space M"
then have "Pair x -` Sigma (space M) A = A x"
by auto
with sets_Pair1[OF A, of x] have "A x \ sets N"
by auto }
note ** = this
have *: "\x. fst x \ space M \ snd x \ A (fst x) \ x \ (SIGMA x:space M. A x)"
by (auto simp: fun_eq_iff)
have "(\(x, y). indicator (A x) y::ennreal) \ borel_measurable (M \\<^sub>M N)"
apply measurable
apply (subst measurable_cong)
apply (rule *)
apply (auto simp: space_pair_measure)
done
then have "(\x. integral\<^sup>N (L x) (indicator (A x))) \ borel_measurable M"
by (intro nn_integral_measurable_subprob_algebra2[where N=N] L)
then show "(\x. emeasure (L x) (A x)) \ borel_measurable M"
apply (rule measurable_cong[THEN iffD1, rotated])
apply (rule nn_integral_indicator)
apply (simp add: subprob_measurableD[OF L] **)
done
qed
lemma measure_measurable_subprob_algebra2:
assumes A[measurable]: "(SIGMA x:space M. A x) \ sets (M \\<^sub>M N)"
assumes L[measurable]: "L \ measurable M (subprob_algebra N)"
shows "(\x. measure (L x) (A x)) \ borel_measurable M"
unfolding measure_def
by (intro borel_measurable_enn2real emeasure_measurable_subprob_algebra2[OF assms])
definition "select_sets M = (SOME N. sets M = sets (subprob_algebra N))"
lemma select_sets1:
"sets M = sets (subprob_algebra N) \ sets M = sets (subprob_algebra (select_sets M))"
unfolding select_sets_def by (rule someI)
lemma sets_select_sets[simp]:
assumes sets: "sets M = sets (subprob_algebra N)"
shows "sets (select_sets M) = sets N"
unfolding select_sets_def
proof (rule someI2)
show "sets M = sets (subprob_algebra N)"
by fact
next
fix L assume "sets M = sets (subprob_algebra L)"
with sets have eq: "space (subprob_algebra N) = space (subprob_algebra L)"
by (intro sets_eq_imp_space_eq) simp
show "sets L = sets N"
proof cases
assume "space (subprob_algebra N) = {}"
with space_subprob_algebra_empty_iff[of N] space_subprob_algebra_empty_iff[of L]
show ?thesis
by (simp add: eq space_empty_iff)
next
assume "space (subprob_algebra N) \ {}"
with eq show ?thesis
by (fastforce simp add: space_subprob_algebra)
qed
qed
lemma space_select_sets[simp]:
"sets M = sets (subprob_algebra N) \ space (select_sets M) = space N"
by (intro sets_eq_imp_space_eq sets_select_sets)
section \<open>Join\<close>
definition join :: "'a measure measure \ 'a measure" where
"join M = measure_of (space (select_sets M)) (sets (select_sets M)) (\B. \\<^sup>+ M'. emeasure M' B \M)"
lemma
shows space_join[simp]: "space (join M) = space (select_sets M)"
and sets_join[simp]: "sets (join M) = sets (select_sets M)"
by (simp_all add: join_def)
lemma emeasure_join:
assumes M[simp, measurable_cong]: "sets M = sets (subprob_algebra N)" and A: "A \ sets N"
shows "emeasure (join M) A = (\\<^sup>+ M'. emeasure M' A \M)"
proof (rule emeasure_measure_of[OF join_def])
show "countably_additive (sets (join M)) (\B. \\<^sup>+ M'. emeasure M' B \M)"
proof (rule countably_additiveI)
fix A :: "nat \ 'a set" assume A: "range A \ sets (join M)" "disjoint_family A"
have "(\i. \\<^sup>+ M'. emeasure M' (A i) \M) = (\\<^sup>+M'. (\i. emeasure M' (A i)) \M)"
using A by (subst nn_integral_suminf) (auto simp: measurable_emeasure_subprob_algebra)
also have "\ = (\\<^sup>+M'. emeasure M' (\i. A i) \M)"
proof (rule nn_integral_cong)
fix M' assume "M' \<in> space M"
then show "(\i. emeasure M' (A i)) = emeasure M' (\i. A i)"
using A sets_eq_imp_space_eq[OF M] by (simp add: suminf_emeasure space_subprob_algebra)
qed
finally show "(\i. \\<^sup>+M'. emeasure M' (A i) \M) = (\\<^sup>+M'. emeasure M' (\i. A i) \M)" .
qed
qed (auto simp: A sets.space_closed positive_def)
lemma measurable_join:
"join \ measurable (subprob_algebra (subprob_algebra N)) (subprob_algebra N)"
proof (cases "space N \ {}", rule measurable_subprob_algebra)
fix A assume "A \ sets N"
let ?B = "borel_measurable (subprob_algebra (subprob_algebra N))"
have "(\M'. emeasure (join M') A) \ ?B \ (\M'. (\\<^sup>+ M''. emeasure M'' A \M')) \ ?B"
proof (rule measurable_cong)
fix M' assume "M' \<in> space (subprob_algebra (subprob_algebra N))"
then show "emeasure (join M') A = (\\<^sup>+ M''. emeasure M'' A \M')"
by (intro emeasure_join) (auto simp: space_subprob_algebra \<open>A\<in>sets N\<close>)
qed
also have "(\M'. \\<^sup>+M''. emeasure M'' A \M') \ ?B"
using measurable_emeasure_subprob_algebra[OF \<open>A\<in>sets N\<close>]
by (rule nn_integral_measurable_subprob_algebra)
finally show "(\M'. emeasure (join M') A) \ borel_measurable (subprob_algebra (subprob_algebra N))" .
next
assume [simp]: "space N \ {}"
fix M assume M: "M \ space (subprob_algebra (subprob_algebra N))"
then have "(\\<^sup>+M'. emeasure M' (space N) \M) \ (\\<^sup>+M'. 1 \M)"
apply (intro nn_integral_mono)
apply (auto simp: space_subprob_algebra
dest!: sets_eq_imp_space_eq subprob_space.emeasure_space_le_1)
done
with M show "subprob_space (join M)"
by (intro subprob_spaceI)
(auto simp: emeasure_join space_subprob_algebra M dest: subprob_space.emeasure_space_le_1)
next
assume "\(space N \ {})"
thus ?thesis by (simp add: measurable_empty_iff space_subprob_algebra_empty_iff)
qed (auto simp: space_subprob_algebra)
lemma nn_integral_join:
assumes f: "f \ borel_measurable N"
and M[measurable_cong]: "sets M = sets (subprob_algebra N)"
shows "(\\<^sup>+x. f x \join M) = (\\<^sup>+M'. \\<^sup>+x. f x \M' \M)"
using f
proof induct
case (cong f g)
moreover have "integral\<^sup>N (join M) f = integral\<^sup>N (join M) g"
by (intro nn_integral_cong cong) (simp add: M)
moreover from M have "(\\<^sup>+ M'. integral\<^sup>N M' f \M) = (\\<^sup>+ M'. integral\<^sup>N M' g \M)"
by (intro nn_integral_cong cong)
(auto simp add: space_subprob_algebra dest!: sets_eq_imp_space_eq)
ultimately show ?case
by simp
next
case (set A)
with M have "(\\<^sup>+ M'. integral\<^sup>N M' (indicator A) \M) = (\\<^sup>+ M'. emeasure M' A \M)"
by (intro nn_integral_cong nn_integral_indicator)
(auto simp: space_subprob_algebra dest!: sets_eq_imp_space_eq)
with set show ?case
using M by (simp add: emeasure_join)
next
case (mult f c)
have "(\\<^sup>+ M'. \\<^sup>+ x. c * f x \M' \M) = (\\<^sup>+ M'. c * \\<^sup>+ x. f x \M' \M)"
using mult M M[THEN sets_eq_imp_space_eq]
by (intro nn_integral_cong nn_integral_cmult) (auto simp add: space_subprob_algebra)
also have "\ = c * (\\<^sup>+ M'. \\<^sup>+ x. f x \M' \M)"
using nn_integral_measurable_subprob_algebra[OF mult(2)]
by (intro nn_integral_cmult mult) (simp add: M)
also have "\ = c * (integral\<^sup>N (join M) f)"
by (simp add: mult)
also have "\ = (\\<^sup>+ x. c * f x \join M)"
using mult(2,3) by (intro nn_integral_cmult[symmetric] mult) (simp add: M cong: measurable_cong_sets)
finally show ?case by simp
next
case (add f g)
have "(\\<^sup>+ M'. \\<^sup>+ x. f x + g x \M' \M) = (\\<^sup>+ M'. (\\<^sup>+ x. f x \M') + (\\<^sup>+ x. g x \M') \M)"
using add M M[THEN sets_eq_imp_space_eq]
by (intro nn_integral_cong nn_integral_add) (auto simp add: space_subprob_algebra)
also have "\ = (\\<^sup>+ M'. \\<^sup>+ x. f x \M' \M) + (\\<^sup>+ M'. \\<^sup>+ x. g x \M' \M)"
using nn_integral_measurable_subprob_algebra[OF add(1)]
using nn_integral_measurable_subprob_algebra[OF add(4)]
by (intro nn_integral_add add) (simp_all add: M)
also have "\ = (integral\<^sup>N (join M) f) + (integral\<^sup>N (join M) g)"
by (simp add: add)
also have "\ = (\\<^sup>+ x. f x + g x \join M)"
using add by (intro nn_integral_add[symmetric] add) (simp_all add: M cong: measurable_cong_sets)
finally show ?case by (simp add: ac_simps)
next
case (seq F)
have "(\\<^sup>+ M'. \\<^sup>+ x. (SUP i. F i) x \M' \M) = (\\<^sup>+ M'. (SUP i. \\<^sup>+ x. F i x \M') \M)"
using seq M M[THEN sets_eq_imp_space_eq] unfolding SUP_apply
by (intro nn_integral_cong nn_integral_monotone_convergence_SUP)
(auto simp add: space_subprob_algebra)
also have "\ = (SUP i. \\<^sup>+ M'. \\<^sup>+ x. F i x \M' \M)"
using nn_integral_measurable_subprob_algebra[OF seq(1)] seq
by (intro nn_integral_monotone_convergence_SUP)
(simp_all add: M incseq_nn_integral incseq_def le_fun_def nn_integral_mono )
also have "\ = (SUP i. integral\<^sup>N (join M) (F i))"
by (simp add: seq)
also have "\ = (\\<^sup>+ x. (SUP i. F i x) \join M)"
using seq by (intro nn_integral_monotone_convergence_SUP[symmetric] seq)
(simp_all add: M cong: measurable_cong_sets)
finally show ?case by (simp add: ac_simps image_comp)
qed
lemma measurable_join1:
"\ f \ measurable N K; sets M = sets (subprob_algebra N) \
\<Longrightarrow> f \<in> measurable (join M) K"
by(simp add: measurable_def)
lemma
fixes f :: "_ \ real"
assumes f_measurable [measurable]: "f \ borel_measurable N"
and f_bounded: "\x. x \ space N \ \f x\ \ B"
and M [measurable_cong]: "sets M = sets (subprob_algebra N)"
and fin: "finite_measure M"
and M_bounded: "AE M' in M. emeasure M' (space M') \ ennreal B'"
shows integrable_join: "integrable (join M) f" (is ?integrable)
and integral_join: "integral\<^sup>L (join M) f = \ M'. integral\<^sup>L M' f \M" (is ?integral)
proof(case_tac [!] "space N = {}")
assume *: "space N = {}"
show ?integrable
using M * by(simp add: real_integrable_def measurable_def nn_integral_empty)
have "(\ M'. integral\<^sup>L M' f \M) = (\ M'. 0 \M)"
proof(rule Bochner_Integration.integral_cong)
fix M'
assume "M' \ space M"
with sets_eq_imp_space_eq[OF M] have "space M' = space N"
by(auto simp add: space_subprob_algebra dest: sets_eq_imp_space_eq)
with * show "(\ x. f x \M') = 0" by(simp add: Bochner_Integration.integral_empty)
qed simp
then show ?integral
using M * by(simp add: Bochner_Integration.integral_empty)
next
assume *: "space N \ {}"
from * have B [simp]: "0 \ B" by(auto dest: f_bounded)
have [measurable]: "f \ borel_measurable (join M)" using f_measurable M
by(rule measurable_join1)
{ fix f M'
assume [measurable]: "f \ borel_measurable N"
and f_bounded: "\x. x \ space N \ f x \ B"
and "M' \ space M" "emeasure M' (space M') \ ennreal B'"
have "AE x in M'. ennreal (f x) \ ennreal B"
proof(rule AE_I2)
fix x
assume "x \ space M'"
with \<open>M' \<in> space M\<close> sets_eq_imp_space_eq[OF M]
have "x \ space N" by(auto simp add: space_subprob_algebra dest: sets_eq_imp_space_eq)
from f_bounded[OF this] show "ennreal (f x) \ ennreal B" by simp
qed
then have "(\\<^sup>+ x. ennreal (f x) \M') \ (\\<^sup>+ x. ennreal B \M')"
by(rule nn_integral_mono_AE)
also have "\ = ennreal B * emeasure M' (space M')" by(simp)
also have "\ \ ennreal B * ennreal B'" by(rule mult_left_mono)(fact, simp)
also have "\ \ ennreal B * ennreal \B'\" by(rule mult_left_mono)(simp_all)
finally have "(\\<^sup>+ x. ennreal (f x) \M') \ ennreal (B * \B'\)" by (simp add: ennreal_mult) }
note bounded1 = this
have bounded:
"\f. \ f \ borel_measurable N; \x. x \ space N \ f x \ B \
\<Longrightarrow> (\<integral>\<^sup>+ x. ennreal (f x) \<partial>join M) \<noteq> top"
proof -
fix f
assume [measurable]: "f \ borel_measurable N"
and f_bounded: "\x. x \ space N \ f x \ B"
have "(\\<^sup>+ x. ennreal (f x) \join M) = (\\<^sup>+ M'. \\<^sup>+ x. ennreal (f x) \M' \M)"
by(rule nn_integral_join[OF _ M]) simp
also have "\ \ \\<^sup>+ M'. B * \B'\ \M"
using bounded1[OF \<open>f \<in> borel_measurable N\<close> f_bounded]
by(rule nn_integral_mono_AE[OF AE_mp[OF M_bounded AE_I2], rule_format])
also have "\ = B * \B'\ * emeasure M (space M)" by simp
also have "\ < \"
using finite_measure.finite_emeasure_space[OF fin]
by(simp add: ennreal_mult_less_top less_top)
finally show "?thesis f" by simp
qed
have f_pos: "(\\<^sup>+ x. ennreal (f x) \join M) \ \"
and f_neg: "(\\<^sup>+ x. ennreal (- f x) \join M) \ \"
using f_bounded by(auto del: notI intro!: bounded simp add: abs_le_iff)
show ?integrable using f_pos f_neg by(simp add: real_integrable_def)
note [measurable] = nn_integral_measurable_subprob_algebra
have int_f: "(\\<^sup>+ x. f x \join M) = \\<^sup>+ M'. \\<^sup>+ x. f x \M' \M"
by(simp add: nn_integral_join[OF _ M])
have int_mf: "(\\<^sup>+ x. - f x \join M) = (\\<^sup>+ M'. \\<^sup>+ x. - f x \M' \M)"
by(simp add: nn_integral_join[OF _ M])
have pos_finite: "AE M' in M. (\\<^sup>+ x. f x \M') \ \"
using AE_space M_bounded
proof eventually_elim
fix M' assume "M' \<in> space M" "emeasure M' (space M') \<le> ennreal B'"
then have "(\\<^sup>+ x. ennreal (f x) \M') \ ennreal (B * \B'\)"
using f_measurable by(auto intro!: bounded1 dest: f_bounded)
then show "(\\<^sup>+ x. ennreal (f x) \M') \ \"
by (auto simp: top_unique)
qed
hence [simp]: "(\\<^sup>+ M'. ennreal (enn2real (\\<^sup>+ x. f x \M')) \M) = (\\<^sup>+ M'. \\<^sup>+ x. f x \M' \M)"
by (rule nn_integral_cong_AE[OF AE_mp]) (simp add: less_top)
from f_pos have [simp]: "integrable M (\M'. enn2real (\\<^sup>+ x. f x \M'))"
by(simp add: int_f real_integrable_def nn_integral_0_iff_AE[THEN iffD2] ennreal_neg enn2real_nonneg)
have neg_finite: "AE M' in M. (\\<^sup>+ x. - f x \M') \ \"
using AE_space M_bounded
proof eventually_elim
fix M' assume "M' \<in> space M" "emeasure M' (space M') \<le> ennreal B'"
then have "(\\<^sup>+ x. ennreal (- f x) \M') \ ennreal (B * \B'\)"
using f_measurable by(auto intro!: bounded1 dest: f_bounded)
then show "(\\<^sup>+ x. ennreal (- f x) \M') \ \"
by (auto simp: top_unique)
qed
hence [simp]: "(\\<^sup>+ M'. ennreal (enn2real (\\<^sup>+ x. - f x \M')) \M) = (\\<^sup>+ M'. \\<^sup>+ x. - f x \M' \M)"
by (rule nn_integral_cong_AE[OF AE_mp]) (simp add: less_top)
from f_neg have [simp]: "integrable M (\M'. enn2real (\\<^sup>+ x. - f x \M'))"
by(simp add: int_mf real_integrable_def nn_integral_0_iff_AE[THEN iffD2] ennreal_neg enn2real_nonneg)
have "(\ x. f x \join M) = enn2real (\\<^sup>+ N. \\<^sup>+x. f x \N \M) - enn2real (\\<^sup>+ N. \\<^sup>+x. - f x \N \M)"
unfolding real_lebesgue_integral_def[OF \<open>?integrable\<close>] by (simp add: nn_integral_join[OF _ M])
also have "\ = (\N. enn2real (\\<^sup>+x. f x \N) \M) - (\N. enn2real (\\<^sup>+x. - f x \N) \M)"
using pos_finite neg_finite by (subst (1 2) integral_eq_nn_integral) (auto simp: enn2real_nonneg)
also have "\ = (\N. enn2real (\\<^sup>+x. f x \N) - enn2real (\\<^sup>+x. - f x \N) \M)"
by simp
also have "\ = \M'. \ x. f x \M' \M"
proof (rule integral_cong_AE)
show "AE x in M.
enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>x) - enn2real (\<integral>\<^sup>+ x. ennreal (- f x) \<partial>x) = integral\<^sup>L x f"
using AE_space M_bounded
proof eventually_elim
fix M' assume "M' \<in> space M" "emeasure M' (space M') \<le> B'"
then interpret subprob_space M'
by (auto simp: M[THEN sets_eq_imp_space_eq] space_subprob_algebra)
from \<open>M' \<in> space M\<close> sets_eq_imp_space_eq[OF M]
have [measurable_cong]: "sets M' = sets N" by(simp add: space_subprob_algebra)
hence [simp]: "space M' = space N" by(rule sets_eq_imp_space_eq)
have "integrable M' f"
by(rule integrable_const_bound[where B=B])(auto simp add: f_bounded)
then show "enn2real (\\<^sup>+ x. f x \M') - enn2real (\\<^sup>+ x. - f x \M') = \ x. f x \M'"
by(simp add: real_lebesgue_integral_def)
qed
qed simp_all
finally show ?integral by simp
qed
lemma join_assoc:
assumes M[measurable_cong]: "sets M = sets (subprob_algebra (subprob_algebra N))"
shows "join (distr M (subprob_algebra N) join) = join (join M)"
proof (rule measure_eqI)
fix A assume "A \ sets (join (distr M (subprob_algebra N) join))"
then have A: "A \ sets N" by simp
show "emeasure (join (distr M (subprob_algebra N) join)) A = emeasure (join (join M)) A"
using measurable_join[of N]
by (auto simp: M A nn_integral_distr emeasure_join measurable_emeasure_subprob_algebra
sets_eq_imp_space_eq[OF M] space_subprob_algebra nn_integral_join[OF _ M]
intro!: nn_integral_cong emeasure_join)
qed (simp add: M)
lemma join_return:
assumes "sets M = sets N" and "subprob_space M"
shows "join (return (subprob_algebra N) M) = M"
by (rule measure_eqI)
(simp_all add: emeasure_join space_subprob_algebra
measurable_emeasure_subprob_algebra nn_integral_return assms)
lemma join_return':
assumes "sets N = sets M"
shows "join (distr M (subprob_algebra N) (return N)) = M"
apply (rule measure_eqI)
apply (simp add: assms)
apply (subgoal_tac "return N \ measurable M (subprob_algebra N)")
apply (simp add: emeasure_join nn_integral_distr measurable_emeasure_subprob_algebra assms)
apply (subst measurable_cong_sets, rule assms[symmetric], rule refl, rule return_measurable)
done
lemma join_distr_distr:
fixes f :: "'a \ 'b" and M :: "'a measure measure" and N :: "'b measure"
assumes "sets M = sets (subprob_algebra R)" and "f \ measurable R N"
shows "join (distr M (subprob_algebra N) (\M. distr M N f)) = distr (join M) N f" (is "?r = ?l")
proof (rule measure_eqI)
fix A assume "A \ sets ?r"
hence A_in_N: "A \ sets N" by simp
from assms have "f \ measurable (join M) N"
by (simp cong: measurable_cong_sets)
moreover from assms and A_in_N have "f-`A \ space R \ sets R"
by (intro measurable_sets) simp_all
ultimately have "emeasure (distr (join M) N f) A = \\<^sup>+M'. emeasure M' (f-`A \ space R) \M"
by (simp_all add: A_in_N emeasure_distr emeasure_join assms)
also have "... = \\<^sup>+ x. emeasure (distr x N f) A \M" using A_in_N
proof (intro nn_integral_cong, subst emeasure_distr)
fix M' assume "M' \<in> space M"
from assms have "space M = space (subprob_algebra R)"
using sets_eq_imp_space_eq by blast
with \<open>M' \<in> space M\<close> have [simp]: "sets M' = sets R" using space_subprob_algebra by blast
show "f \ measurable M' N" by (simp cong: measurable_cong_sets add: assms)
have "space M' = space R" by (rule sets_eq_imp_space_eq) simp
thus "emeasure M' (f -` A \ space R) = emeasure M' (f -` A \ space M')" by simp
qed
also have "(\M. distr M N f) \ measurable M (subprob_algebra N)"
by (simp cong: measurable_cong_sets add: assms measurable_distr)
hence "(\\<^sup>+ x. emeasure (distr x N f) A \M) =
emeasure (join (distr M (subprob_algebra N) (\<lambda>M. distr M N f))) A"
by (simp_all add: emeasure_join assms A_in_N nn_integral_distr measurable_emeasure_subprob_algebra)
finally show "emeasure ?r A = emeasure ?l A" ..
qed simp
definition bind :: "'a measure \ ('a \ 'b measure) \ 'b measure" where
"bind M f = (if space M = {} then count_space {} else
join (distr M (subprob_algebra (f (SOME x. x \<in> space M))) f))"
adhoc_overloading Monad_Syntax.bind bind
lemma bind_empty:
"space M = {} \ bind M f = count_space {}"
by (simp add: bind_def)
lemma bind_nonempty:
"space M \ {} \ bind M f = join (distr M (subprob_algebra (f (SOME x. x \ space M))) f)"
by (simp add: bind_def)
lemma sets_bind_empty: "sets M = {} \ sets (bind M f) = {{}}"
by (auto simp: bind_def)
lemma space_bind_empty: "space M = {} \ space (bind M f) = {}"
by (simp add: bind_def)
lemma sets_bind[simp, measurable_cong]:
assumes f: "\x. x \ space M \ sets (f x) = sets N" and M: "space M \ {}"
shows "sets (bind M f) = sets N"
using f [of "SOME x. x \ space M"] by (simp add: bind_nonempty M some_in_eq)
lemma space_bind[simp]:
assumes "\x. x \ space M \ sets (f x) = sets N" and "space M \ {}"
shows "space (bind M f) = space N"
using assms by (intro sets_eq_imp_space_eq sets_bind)
lemma bind_cong_All:
assumes "\x \ space M. f x = g x"
shows "bind M f = bind M g"
proof (cases "space M = {}")
assume "space M \ {}"
hence "(SOME x. x \ space M) \ space M" by (rule_tac someI_ex) blast
with assms have "f (SOME x. x \ space M) = g (SOME x. x \ space M)" by blast
with \<open>space M \<noteq> {}\<close> and assms show ?thesis by (simp add: bind_nonempty cong: distr_cong)
qed (simp add: bind_empty)
lemma bind_cong:
"M = N \ (\x. x \ space M \ f x = g x) \ bind M f = bind N g"
using bind_cong_All[of M f g] by auto
lemma bind_nonempty':
assumes "f \ measurable M (subprob_algebra N)" "x \ space M"
shows "bind M f = join (distr M (subprob_algebra N) f)"
using assms
apply (subst bind_nonempty, blast)
apply (subst subprob_algebra_cong[OF sets_kernel[OF assms(1) someI_ex]], blast)
apply (simp add: subprob_algebra_cong[OF sets_kernel[OF assms]])
done
lemma bind_nonempty'':
assumes "f \ measurable M (subprob_algebra N)" "space M \ {}"
shows "bind M f = join (distr M (subprob_algebra N) f)"
using assms by (auto intro: bind_nonempty')
lemma emeasure_bind:
"\space M \ {}; f \ measurable M (subprob_algebra N);X \ sets N\
\<Longrightarrow> emeasure (M \<bind> f) X = \<integral>\<^sup>+x. emeasure (f x) X \<partial>M"
by (simp add: bind_nonempty'' emeasure_join nn_integral_distr measurable_emeasure_subprob_algebra)
lemma nn_integral_bind:
assumes f: "f \ borel_measurable B"
assumes N: "N \ measurable M (subprob_algebra B)"
shows "(\\<^sup>+x. f x \(M \ N)) = (\\<^sup>+x. \\<^sup>+y. f y \N x \M)"
proof cases
assume M: "space M \ {}" show ?thesis
unfolding bind_nonempty''[OF N M] nn_integral_join[OF f sets_distr]
by (rule nn_integral_distr[OF N])
(simp add: f nn_integral_measurable_subprob_algebra)
qed (simp add: bind_empty space_empty[of M] nn_integral_count_space)
lemma AE_bind:
assumes N[measurable]: "N \ measurable M (subprob_algebra B)"
assumes P[measurable]: "Measurable.pred B P"
shows "(AE x in M \ N. P x) \ (AE x in M. AE y in N x. P y)"
proof cases
assume M: "space M = {}" show ?thesis
unfolding bind_empty[OF M] unfolding space_empty[OF M] by (simp add: AE_count_space)
next
assume M: "space M \ {}"
note sets_kernel[OF N, simp]
have *: "(\\<^sup>+x. indicator {x. \ P x} x \(M \ N)) = (\\<^sup>+x. indicator {x\space B. \ P x} x \(M \ N))"
by (intro nn_integral_cong) (simp add: space_bind[OF _ M] split: split_indicator)
have "(AE x in M \ N. P x) \ (\\<^sup>+ x. integral\<^sup>N (N x) (indicator {x \ space B. \ P x}) \M) = 0"
by (simp add: AE_iff_nn_integral sets_bind[OF _ M] space_bind[OF _ M] * nn_integral_bind[where B=B]
del: nn_integral_indicator)
also have "\ = (AE x in M. AE y in N x. P y)"
apply (subst nn_integral_0_iff_AE)
apply (rule measurable_compose[OF N nn_integral_measurable_subprob_algebra])
apply measurable
apply (intro eventually_subst AE_I2)
apply (auto simp add: subprob_measurableD(1)[OF N]
intro!: AE_iff_measurable[symmetric])
done
finally show ?thesis .
qed
lemma measurable_bind':
assumes M1: "f \ measurable M (subprob_algebra N)" and
M2: "case_prod g \ measurable (M \\<^sub>M N) (subprob_algebra R)"
shows "(\x. bind (f x) (g x)) \ measurable M (subprob_algebra R)"
proof (subst measurable_cong)
fix x assume x_in_M: "x \ space M"
with assms have "space (f x) \ {}"
by (blast dest: subprob_space_kernel subprob_space.subprob_not_empty)
moreover from M2 x_in_M have "g x \ measurable (f x) (subprob_algebra R)"
by (subst measurable_cong_sets[OF sets_kernel[OF M1 x_in_M] refl])
(auto dest: measurable_Pair2)
ultimately show "bind (f x) (g x) = join (distr (f x) (subprob_algebra R) (g x))"
by (simp_all add: bind_nonempty'')
next
show "(\w. join (distr (f w) (subprob_algebra R) (g w))) \ measurable M (subprob_algebra R)"
apply (rule measurable_compose[OF _ measurable_join])
apply (rule measurable_distr2[OF M2 M1])
done
qed
lemma measurable_bind[measurable (raw)]:
assumes M1: "f \ measurable M (subprob_algebra N)" and
M2: "(\x. g (fst x) (snd x)) \ measurable (M \\<^sub>M N) (subprob_algebra R)"
shows "(\x. bind (f x) (g x)) \ measurable M (subprob_algebra R)"
using assms by (auto intro: measurable_bind' simp: measurable_split_conv)
lemma measurable_bind2:
assumes "f \ measurable M (subprob_algebra N)" and "g \ measurable N (subprob_algebra R)"
shows "(\x. bind (f x) g) \ measurable M (subprob_algebra R)"
using assms by (intro measurable_bind' measurable_const) auto
lemma subprob_space_bind:
assumes "subprob_space M" "f \ measurable M (subprob_algebra N)"
shows "subprob_space (M \ f)"
proof (rule subprob_space_kernel[of "\x. x \ f"])
show "(\x. x \ f) \ measurable (subprob_algebra M) (subprob_algebra N)"
by (rule measurable_bind, rule measurable_ident_sets, rule refl,
rule measurable_compose[OF measurable_snd assms(2)])
from assms(1) show "M \ space (subprob_algebra M)"
by (simp add: space_subprob_algebra)
qed
lemma
fixes f :: "_ \ real"
assumes f_measurable [measurable]: "f \ borel_measurable K"
and f_bounded: "\x. x \ space K \ \f x\ \ B"
and N [measurable]: "N \ measurable M (subprob_algebra K)"
and fin: "finite_measure M"
and M_bounded: "AE x in M. emeasure (N x) (space (N x)) \ ennreal B'"
shows integrable_bind: "integrable (bind M N) f" (is ?integrable)
and integral_bind: "integral\<^sup>L (bind M N) f = \ x. integral\<^sup>L (N x) f \M" (is ?integral)
proof(case_tac [!] "space M = {}")
assume [simp]: "space M \ {}"
interpret finite_measure M by(rule fin)
have "integrable (join (distr M (subprob_algebra K) N)) f"
using f_measurable f_bounded
by(rule integrable_join[where B'=B'])(simp_all add: finite_measure_distr AE_distr_iff M_bounded)
then show ?integrable by(simp add: bind_nonempty''[where N=K])
have "integral\<^sup>L (join (distr M (subprob_algebra K) N)) f = \ M'. integral\<^sup>L M' f \distr M (subprob_algebra K) N"
using f_measurable f_bounded
by(rule integral_join[where B'=B'])(simp_all add: finite_measure_distr AE_distr_iff M_bounded)
also have "\ = \ x. integral\<^sup>L (N x) f \M"
by(rule integral_distr)(simp_all add: integral_measurable_subprob_algebra[OF _])
finally show ?integral by(simp add: bind_nonempty''[where N=K])
qed(simp_all add: bind_def integrable_count_space lebesgue_integral_count_space_finite Bochner_Integration.integral_empty)
lemma (in prob_space) prob_space_bind:
assumes ae: "AE x in M. prob_space (N x)"
and N[measurable]: "N \ measurable M (subprob_algebra S)"
shows "prob_space (M \ N)"
proof
have "emeasure (M \ N) (space (M \ N)) = (\\<^sup>+x. emeasure (N x) (space (N x)) \M)"
by (subst emeasure_bind[where N=S])
(auto simp: not_empty space_bind[OF sets_kernel] subprob_measurableD[OF N] intro!: nn_integral_cong)
also have "\ = (\\<^sup>+x. 1 \M)"
using ae by (intro nn_integral_cong_AE, eventually_elim) (rule prob_space.emeasure_space_1)
finally show "emeasure (M \ N) (space (M \ N)) = 1"
by (simp add: emeasure_space_1)
qed
lemma (in subprob_space) bind_in_space:
"A \ measurable M (subprob_algebra N) \ (M \ A) \ space (subprob_algebra N)"
--> --------------------
--> maximum size reached
--> --------------------
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