```(*  Title:      HOL/Probability/Giry_Monad.thy
Author:     Johannes Hölzl, TU München
Author:     Manuel Eberl, TU München

Defines subprobability spaces, the subprobability functor and the Giry monad on subprobability
spaces.
*)

begin

subsection ‹Sub-probability spaces›

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"
by simp

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 ⊆ subprob_space
by (rule subprob_spaceI) (simp_all add: emeasure_space_1 not_empty)

lemma subprob_space_sigma [simp]: "Ω ≠ {} ⟹ subprob_space (sigma Ω X)"

lemma subprob_space_null_measure: "space M ≠ {} ⟹ subprob_space (null_measure M)"

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 "(∫⇧+x. f x ∂M) ≤ c"
proof -
have "(∫⇧+ x. f x ∂M) ≤ (∫⇧+ x. c ∂M)"
by(rule nn_integral_mono_AE) fact
also have "… ≤ c * emeasure M (space M)"
using ‹0 ≤ c› 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' ∈ 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 {a..b} g" 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 (λx. 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 {a..b} g" by (rule strict_mono_on_imp_mono_on)
from mono' derivg have "⋀x. x ∈ {a<..<b} ⟹ 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: ‹a < b›)

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} =
∫⇧+x. f x * indicator (h -` {a..b}) x ∂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 "(∫⇧+x. f x * indicator {g a..g b} x ∂lborel) =
(∫⇧+x. f (g x) * g' x * indicator {a..b} x ∂lborel)"
by (intro nn_integral_substitution_aux)
(auto simp: derivg_nonneg A B emeasure_density mult.commute ‹a < b›)
also have "... = emeasure (density lborel (λx. f (g x) * g' x)) {a..b}"
finally show ?thesis .
next
assume "¬a < b"
with ‹a ≤ b› have [simp]: "b = a" by (simp add: not_less del: ‹a ≤ b›)
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 ⊆ P?: subprob_space "M1 ⨂⇩M M2"
proof
from mult_le_one[OF M1.emeasure_space_le_1 _ M2.emeasure_space_le_1]
show "emeasure (M1 ⨂⇩M M2) (space (M1 ⨂⇩M M2)) ≤ 1"
from M1.subprob_not_empty and M2.subprob_not_empty show "space (M1 ⨂⇩M M2) ≠ {}"
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) ≠ {}"
qed

definition subprob_algebra :: "'a measure ⇒ 'a measure measure" where
"subprob_algebra K =
(SUP A ∈ sets K. vimage_algebra {M. subprob_space M ∧ sets M = sets K} (λM. 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"

lemma measurable_emeasure_subprob_algebra[measurable]:
"a ∈ sets A ⟹ (λM. 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 ⟹ (λM. 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 ‹

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))

›

setup ‹
›

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 ⟹ (λa. 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)) ⟹
(⋀a. a ∈ space M ⟹ sets (K a) = sets N) ⟹
(⋀A. A ∈ sets N ⟹ (λa. emeasure (K a) A) ∈ borel_measurable M) ⟹
K ∈ 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) ⟷
(∀x∈space M. subprob_space (K x) ∧ sets (K x) = sets M) ∧
(∀A∈sets M. (λx. emeasure (K x) A) ∈ measurable M borel)"
proof
assume "(∀x∈space M. subprob_space (K x) ∧ sets (K x) = sets M) ∧
(∀A∈sets M. (λx. emeasure (K x) A) ∈ 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) ∧
(∀A∈sets M. (λx. emeasure (K x) A) ∈ borel_measurable M)"
by (auto dest: subprob_space_kernel sets_kernel)
qed

lemma measurable_subprob_algebra_generated:
assumes eq: "sets N = sigma_sets Ω G" and "Int_stable G" "G ⊆ Pow Ω"
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 ⟹ (λa. emeasure (K a) A) ∈ borel_measurable M"
assumes Ω: "(λa. emeasure (K a) Ω) ∈ 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 Ω "sigma_sets Ω G"
using ‹G ⊆ Pow Ω› by (rule sigma_algebra_sigma_sets)
fix A assume "A ∈ sets N" with assms(2,3) show "(λa. emeasure (K a) A) ∈ borel_measurable M"
unfolding ‹sets N = sigma_sets Ω G›
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 "(λa. emeasure (K a) (Ω - A)) ∈ borel_measurable M ⟷
(λa. emeasure (K a) Ω - emeasure (K a) A) ∈ 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 Ω show ?case
by simp
next
case (union F)
moreover have "(λa. emeasure (K a) (⋃i. F i)) ∈ borel_measurable M ⟷
(λa. ∑i. emeasure (K a) (F i)) ∈ 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 (λ_. 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 ≠ {{}}"
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 "(λM. integral⇧N M f) ∈ borel_measurable (subprob_algebra N)" (is "_ ∈ ?B")
using f
proof induct
case (cong f g)
moreover have "(λM'. ∫⇧+M''. f M'' ∂M') ∈ ?B ⟷ (λM'. ∫⇧+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 "(λM'. ∫⇧+M''. indicator B M'' ∂M') ∈ ?B ⟷ (λM'. emeasure M' B) ∈ ?B"
by (intro measurable_cong nn_integral_indicator) (simp add: space_subprob_algebra)
with set show ?case
next
case (mult f c)
then have "(λM'. ∫⇧+M''. c * f M'' ∂M') ∈ ?B ⟷ (λM'. c * ∫⇧+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
then have "(λM'. ∫⇧+M''. f M'' + g M'' ∂M') ∈ ?B ⟷ (λM'. (∫⇧+M''. f M'' ∂M') + (∫⇧+M''. g M'' ∂M')) ∈ ?B"
next
case (seq F)
then have "(λM'. ∫⇧+M''. (SUP i. F i) M'' ∂M') ∈ ?B ⟷ (λM'. SUP i. (∫⇧+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
qed

lemma measurable_distr:
assumes [measurable]: "f ∈ measurable M N"
shows "(λM'. distr M' N f) ∈ measurable (subprob_algebra M) (subprob_algebra N)"
proof (cases "space N = {}")
case False
show ?thesis
proof (rule measurable_subprob_algebra)
fix A assume A: "A ∈ sets N"
then have "(λM'. emeasure (distr M' N f) A) ∈ borel_measurable (subprob_algebra M) ⟷
(λM'. emeasure M' (f -` A ∩ space M)) ∈ 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 "(λM'. emeasure (distr M' N f) A) ∈ borel_measurable (subprob_algebra M)" .
qed (auto intro!: subprob_space.subprob_space_distr simp: space_subprob_algebra False cong: measurable_cong_sets)
qed (use assms in ‹auto simp: measurable_empty_iff space_subprob_algebra_empty_iff›)

lemma emeasure_space_subprob_algebra[measurable]:
"(λa. emeasure a (space a)) ∈ borel_measurable (subprob_algebra N)"
proof-
have "(λa. emeasure a (space N)) ∈ borel_measurable (subprob_algebra N)" (is "?f ∈ ?M")
by (rule measurable_emeasure_subprob_algebra) simp
also have "?f ∈ ?M ⟷ (λa. 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) (λM. integrable M f)"
proof (rule measurable_cong[THEN iffD2])
show "M ∈ space (subprob_algebra N) ⟹ integrable M f ⟷ (∫⇧+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 "(λM. integral⇧L M f) ∈ subprob_algebra N →⇩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 ⟹ (λi. 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 →⇩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⇧L M (F i) else 0)" for M i

have "(λM. F' M i) ∈ subprob_algebra N →⇩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} *⇩R y else 0)"
unfolding simple_bochner_integral_def by simp
qed measurable
moreover
have "F' M ⇢ integral⇧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="λx. 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 "(λx. f x ⨂⇩M g x) ∈ measurable M (subprob_algebra (N ⨂⇩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 ⨂⇩M g x)" by unfold_locales
show sets_eq: "sets (f x ⨂⇩M g x) = sets (N ⨂⇩M L)"
using fx gx by (simp add: space_subprob_algebra)

have 1: "⋀A B. A ∈ sets N ⟹ B ∈ sets L ⟹ emeasure (f x ⨂⇩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 ⨂⇩M g x) (space (f x ⨂⇩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 ⨂⇩M L) ⟹ emeasure (f x ⨂⇩M g x) (space N × space L - A) =
... - emeasure (f x ⨂⇩M g x) A"
using emeasure_compl[simplified, OF _ P.emeasure_finite]
unfolding sets_eq
unfolding sets_eq_imp_space_eq[OF sets_eq]
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 ⨂⇩M L)"
show "(λa. emeasure (f a ⨂⇩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 ⨂⇩M L)"
by (auto simp: sets_pair_measure)
have "(λx. emeasure (f x) (space (f x)) * emeasure (g x) (space (g x)) -
emeasure (f x ⨂⇩M g x) A) ∈ borel_measurable M" (is "?f ∈ ?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 ⨂⇩M L)" "disjoint_family A"
by (auto simp: sets_pair_measure)
then have "(λa. emeasure (f a ⨂⇩M g a) (⋃i. A i)) ∈ borel_measurable M ⟷
(λa. ∑i. emeasure (f a ⨂⇩M g a) (A i)) ∈ 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 "(λx. 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 "(λa. 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 "(λw. 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

subsection ‹Properties of ``return''›

definition return :: "'a measure ⇒ 'a ⇒ 'a measure" where
"return R x = measure_of (space R) (sets R) (λA. indicator A x)"

lemma space_return[simp]: "space (return M x) = space M"

lemma sets_return[simp]: "sets (return M x) = sets M"

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)) (λA. 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)) (λA. indicator A x)"
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)"
by (metis assms emeasure_return indicator_simps(2) sets.top space_return subprob_spaceI subprob_space_return zero_le)

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 "(∫⇧+ a. g a ∂return M x) = g x"
proof-
interpret prob_space "return M x" by (rule prob_space_return[OF ‹x ∈ space M›])
have "(∫⇧+ a. g a ∂return M x) = (∫⇧+ 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 ‹x ∈ space M›])
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:
"Ω ∈ sets M ⟹ return (restrict_space M Ω) x = restrict_space (return M x) Ω"
by (auto intro!: measure_eqI simp: sets_restrict_space emeasure_restrict_space)

lemma measurable_distr2:
assumes f[measurable]: "case_prod f ∈ measurable (L ⨂⇩M M) N"
assumes g[measurable]: "g ∈ measurable L (subprob_algebra M)"
shows "(λx. distr (g x) N (f x)) ∈ measurable L (subprob_algebra N)"
proof -
have "(λx. distr (g x) N (f x)) ∈ measurable L (subprob_algebra N)
⟷ (λx. distr (return L x ⨂⇩M g x) N (case_prod f)) ∈ 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"
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 ⨂⇩M g x) = space (X ⨂⇩M M)"
show "distr (g x) N (f x) = distr (?R ⨂⇩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 ⨂⇩M g x) ((λ(x, y). f x y) -` A ∩ space (?R ⨂⇩M g x))"
by (auto simp add: emeasure_distr f_M' cong: measurable_cong_sets)
also have "… = (∫⇧+M''. emeasure (g x) (f M'' -` A ∩ space M) ∂?R)"
apply (subst emeasure_pair_measure_alt)
apply (force simp add: f_M' cong: measurable_cong_sets intro!: measurable_sets[OF _ A])
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 "…"
proof (intro measurable_compose[OF measurable_pair_measure measurable_distr])
show "return L ∈ L →⇩M subprob_algebra L"
by (rule return_measurable)
qed measurable
finally show ?thesis .
qed

lemma nn_integral_measurable_subprob_algebra2:
assumes f[measurable]: "(λ(x, y). f x y) ∈ borel_measurable (M ⨂⇩M N)"
assumes N[measurable]: "L ∈ measurable M (subprob_algebra N)"
shows "(λx. integral⇧N (L x) (f x)) ∈ borel_measurable M"
proof -
note nn_integral_measurable_subprob_algebra[measurable]
note measurable_distr2[measurable]
have "(λx. integral⇧N (distr (L x) (M ⨂⇩M N) (λy. (x, y))) (λ(x, y). f x y)) ∈ borel_measurable M"
by measurable
then show "(λx. integral⇧N (L x) (f x)) ∈ borel_measurable M"
by (rule measurable_cong[THEN iffD1, rotated])
qed

lemma emeasure_measurable_subprob_algebra2:
assumes A[measurable]: "(SIGMA x:space M. A x) ∈ sets (M ⨂⇩M N)"
assumes L[measurable]: "L ∈ measurable M (subprob_algebra N)"
shows "(λx. 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 MN: "Measurable.pred (M ⨂⇩M N) (λw. w ∈ Sigma (space M) A)"
by auto
then have "(λ(x, y). indicator (A x) y::ennreal) ∈ borel_measurable (M ⨂⇩M N)"
apply measurable
by (smt (verit, best) MN measurable_cong mem_Sigma_iff prod.collapse space_pair_measure)
then have "(λx. integral⇧N (L x) (indicator (A x))) ∈ borel_measurable M"
by (intro nn_integral_measurable_subprob_algebra2[where N=N] L)
then show "(λx. emeasure (L x) (A x)) ∈ borel_measurable M"
by (smt (verit) "**" L measurable_cong_simp nn_integral_indicator sets_kernel)
qed

lemma measure_measurable_subprob_algebra2:
assumes A[measurable]: "(SIGMA x:space M. A x) ∈ sets (M ⨂⇩M N)"
assumes L[measurable]: "L ∈ measurable M (subprob_algebra N)"
shows "(λx. 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
next
assume "space (subprob_algebra N) ≠ {}"
with eq show ?thesis
by (smt (verit) equals0I mem_Collect_eq 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)

subsection ‹Join›

definition join :: "'a measure measure ⇒ 'a measure" where
"join M = measure_of (space (select_sets M)) (sets (select_sets M)) (λB. ∫⇧+ 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)"

lemma emeasure_join:
assumes M[simp, measurable_cong]: "sets M = sets (subprob_algebra N)" and A: "A ∈ sets N"
shows "emeasure (join M) A = (∫⇧+ M'. emeasure M' A ∂M)"
proof (rule emeasure_measure_of[OF join_def])
show "countably_additive (sets (join M)) (λB. ∫⇧+ M'. emeasure M' B ∂M)"
fix A :: "nat ⇒ 'a set" assume A: "range A ⊆ sets (join M)" "disjoint_family A"
have "(∑i. ∫⇧+ M'. emeasure M' (A i) ∂M) = (∫⇧+M'. (∑i. emeasure M' (A i)) ∂M)"
using A by (subst nn_integral_suminf) (auto simp: measurable_emeasure_subprob_algebra)
also have "… = (∫⇧+M'. emeasure M' (⋃i. A i) ∂M)"
proof (rule nn_integral_cong)
fix M' assume "M' ∈ 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. ∫⇧+M'. emeasure M' (A i) ∂M) = (∫⇧+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 "(λM'. emeasure (join M') A) ∈ ?B ⟷ (λM'. (∫⇧+ M''. emeasure M'' A ∂M')) ∈ ?B"
proof (rule measurable_cong)
fix M' assume "M' ∈ space (subprob_algebra (subprob_algebra N))"
then show "emeasure (join M') A = (∫⇧+ M''. emeasure M'' A ∂M')"
by (intro emeasure_join) (auto simp: space_subprob_algebra ‹A∈sets N›)
qed
also have "(λM'. ∫⇧+M''. emeasure M'' A ∂M') ∈ ?B"
using measurable_emeasure_subprob_algebra[OF ‹A∈sets N›]
by (rule nn_integral_measurable_subprob_algebra)
finally show "(λM'. 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 "(∫⇧+M'. emeasure M' (space N) ∂M) ≤ (∫⇧+M'. 1 ∂M)"
proof (intro nn_integral_mono)
show "⋀x. ⟦M ∈ space (subprob_algebra (subprob_algebra N)); x ∈ space M⟧
⟹ emeasure x (space N) ≤ 1"
by (smt (verit) mem_Collect_eq sets_eq_imp_space_eq space_subprob_algebra subprob_space.subprob_emeasure_le_1)
qed
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 "(∫⇧+x. f x ∂join M) = (∫⇧+M'. ∫⇧+x. f x ∂M' ∂M)"
using f
proof induct
case (cong f g)
moreover have "integral⇧N (join M) f = integral⇧N (join M) g"
by (intro nn_integral_cong cong) (simp add: M)
moreover from M have "(∫⇧+ M'. integral⇧N M' f ∂M) = (∫⇧+ M'. integral⇧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 "(∫⇧+ M'. integral⇧N M' (indicator A) ∂M) = (∫⇧+ 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 "(∫⇧+ M'. ∫⇧+ x. c * f x ∂M' ∂M) = (∫⇧+ M'. c * ∫⇧+ 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 * (∫⇧+ M'. ∫⇧+ 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⇧N (join M) f)"
also have "… = (∫⇧+ 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
have "(∫⇧+ M'. ∫⇧+ x. f x + g x ∂M' ∂M) = (∫⇧+ M'. (∫⇧+ x. f x ∂M') + (∫⇧+ x. g x ∂M') ∂M)"
also have "… = (∫⇧+ M'. ∫⇧+ x. f x ∂M' ∂M) + (∫⇧+ M'. ∫⇧+ x. g x ∂M' ∂M)"
also have "… = (integral⇧N (join M) f) + (integral⇧N (join M) g)"
also have "… = (∫⇧+ x. f x + g x ∂join M)"
finally show ?case by (simp add: ac_simps)
next
case (seq F)
have "(∫⇧+ M'. ∫⇧+ x. (SUP i. F i) x ∂M' ∂M) = (∫⇧+ M'. (SUP i. ∫⇧+ 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)
also have "… = (SUP i. ∫⇧+ M'. ∫⇧+ 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⇧N (join M) (F i))"
also have "… = (∫⇧+ x. (SUP i. F i x) ∂join M)"
using seq by (intro nn_integral_monotone_convergence_SUP[symmetric] seq)
finally show ?case by (simp add: ac_simps image_comp)
qed

lemma measurable_join1:
"⟦ f ∈ measurable N K; sets M = sets (subprob_algebra N) ⟧
⟹ f ∈ measurable (join M) K"

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⇧L (join M) f = ∫ M'. integral⇧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⇧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 ‹M' ∈ space M› 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 "(∫⇧+ x. ennreal (f x) ∂M') ≤ (∫⇧+ 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 "(∫⇧+ 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 ⟧
⟹ (∫⇧+ x. ennreal (f x) ∂join M) ≠ top"
proof -
fix f
assume [measurable]: "f ∈ borel_measurable N"
and f_bounded: "⋀x. x ∈ space N ⟹ f x ≤ B"
have "(∫⇧+ x. ennreal (f x) ∂join M) = (∫⇧+ M'. ∫⇧+ x. ennreal (f x) ∂M' ∂M)"
by(rule nn_integral_join[OF _ M]) simp
also have "… ≤ ∫⇧+ M'. B * ¦B'¦ ∂M"
using bounded1[OF ‹f ∈ borel_measurable N› 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]
finally show "?thesis f" by simp
qed
have f_pos: "(∫⇧+ x. ennreal (f x) ∂join M) ≠ ∞"
and f_neg: "(∫⇧+ 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: "(∫⇧+ x. f x ∂join M) = ∫⇧+ M'. ∫⇧+ x. f x ∂M' ∂M"
have int_mf: "(∫⇧+ x. - f x ∂join M) = (∫⇧+ M'. ∫⇧+ x. - f x ∂M' ∂M)"

have pos_finite: "AE M' in M. (∫⇧+ x. f x ∂M') ≠ ∞"
using AE_space M_bounded
proof eventually_elim
fix M' assume "M' ∈ space M" "emeasure M' (space M') ≤ ennreal B'"
then have "(∫⇧+ x. ennreal (f x) ∂M') ≤ ennreal (B * ¦B'¦)"
using f_measurable by(auto intro!: bounded1 dest: f_bounded)
then show "(∫⇧+ x. ennreal (f x) ∂M') ≠ ∞"
by (auto simp: top_unique)
qed
hence [simp]: "(∫⇧+ M'. ennreal (enn2real (∫⇧+ x. f x ∂M')) ∂M) = (∫⇧+ M'. ∫⇧+ 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 (λM'. enn2real (∫⇧+ 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. (∫⇧+ x. - f x ∂M') ≠ ∞"
using AE_space M_bounded
proof eventually_elim
fix M' assume "M' ∈ space M" "emeasure M' (space M') ≤ ennreal B'"
then have "(∫⇧+ x. ennreal (- f x) ∂M') ≤ ennreal (B * ¦B'¦)"
using f_measurable by(auto intro!: bounded1 dest: f_bounded)
then show "(∫⇧+ x. ennreal (- f x) ∂M') ≠ ∞"
by (auto simp: top_unique)
qed
hence [simp]: "(∫⇧+ M'. ennreal (enn2real (∫⇧+ x. - f x ∂M')) ∂M) = (∫⇧+ M'. ∫⇧+ 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 (λM'. enn2real (∫⇧+ 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 (∫⇧+ N. ∫⇧+x. f x ∂N ∂M) - enn2real (∫⇧+ N. ∫⇧+x. - f x ∂N ∂M)"
unfolding real_lebesgue_integral_def[OF ‹?integrable›] by (simp add: nn_integral_join[OF _ M])
also have "… = (∫N. enn2real (∫⇧+x. f x ∂N) ∂M) - (∫N. enn2real (∫⇧+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 (∫⇧+x. f x ∂N) - enn2real (∫⇧+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 (∫⇧+ x. ennreal (f x) ∂x) - enn2real (∫⇧+ x. ennreal (- f x) ∂x) = integral⇧L x f"
using AE_space M_bounded
proof eventually_elim
fix M' assume "M' ∈ space M" "emeasure M' (space M') ≤ B'"
then interpret subprob_space M'
by (auto simp: M[THEN sets_eq_imp_space_eq] space_subprob_algebra)

from ‹M' ∈ space M› 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 (∫⇧+ x. f x ∂M') - enn2real (∫⇧+ x. - f x ∂M') = ∫ x. f x ∂M'"
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)

lemma join_return:
assumes "sets M = sets N" and "subprob_space M"
shows "join (return (subprob_algebra N) M) = M"
by (rule measure_eqI)
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"
proof (rule measure_eqI)
fix A
have "return N ∈ measurable M (subprob_algebra N)"
using assms by auto
moreover
assume "A ∈ sets (join (distr M (subprob_algebra N) (return N)))"
ultimately show "emeasure (join (distr M (subprob_algebra N) (return N))) A = emeasure M A"
by (simp add: emeasure_join nn_integral_distr measurable_emeasure_subprob_algebra assms)

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) (λM. 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 = ∫⇧+M'. emeasure M' (f-`A ∩ space R) ∂M"
by (simp_all add: A_in_N emeasure_distr emeasure_join assms)

also have "... = ∫⇧+ x. emeasure (distr x N f) A ∂M" using A_in_N
proof (intro nn_integral_cong, subst emeasure_distr)
fix M' assume "M' ∈ space M"
from assms have "space M = space (subprob_algebra R)"
using sets_eq_imp_space_eq by blast
with ‹M' ∈ space M› 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 "(λM. distr M N f) ∈ measurable M (subprob_algebra N)"
by (simp cong: measurable_cong_sets add: assms measurable_distr)
hence "(∫⇧+ x. emeasure (distr x N f) A ∂M) =
emeasure (join (distr M (subprob_algebra N) (λM. 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 ∈ space M))) f))"

lemma bind_empty:
"space M = {} ⟹ bind M f = count_space {}"

lemma bind_nonempty:
"space M ≠ {} ⟹ bind M f = join (distr M (subprob_algebra (f (SOME x. x ∈ space M))) f)"

lemma sets_bind_empty: "sets M = {} ⟹ sets (bind M f) = {{}}"
by auto

lemma space_bind_empty: "space M = {} ⟹ space (bind M f) = {}"

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 ‹space M ≠ {}› and assms show ?thesis by (simp add: bind_nonempty cong: distr_cong)

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)"
proof -
have "join (distr M (subprob_algebra (f (SOME x. x ∈ space M))) f) = join (distr M (subprob_algebra N) f)"
by (metis assms someI_ex subprob_algebra_cong subprob_measurableD(2))
with assms show ?thesis
by (metis bind_nonempty empty_iff)
qed

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⟧
⟹ emeasure (M ⤜ f) X = ∫⇧+x. emeasure (f x) X ∂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 "(∫⇧+x. f x ∂(M ⤜ N)) = (∫⇧+x. ∫⇧+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])
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 *: "(∫⇧+x. indicator {x. ¬ P x} x ∂(M ⤜ N)) = (∫⇧+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) ⟷ (∫⇧+ x. integral⇧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. integral⇧N (N x) (indicator {x ∈ space B. ¬ P x}) = 0)"
proof (rule nn_integral_0_iff_AE)
show "(λx. integral⇧N (N x) (indicator {x ∈ space B. ¬ P x})) ∈ borel_measurable M"
apply (rule measurable_compose[OF N nn_integral_measurable_subprob_algebra])
by measurable
qed
also have "… = (AE x in M. AE y in N x. P y)"
apply (intro eventually_subst AE_I2)
by (auto simp add: subprob_measurableD(1)[OF N] intro!: AE_iff_measurable[symmetric])
finally show ?thesis .
qed

lemma measurable_bind':
assumes M1: "f ∈ measurable M (subprob_algebra N)" and
M2: "case_prod g ∈ measurable (M ⨂⇩M N) (subprob_algebra R)"
shows "(λx. 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))"
next
show "(λw. 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: "(λx. g (fst x) (snd x)) ∈ measurable (M ⨂⇩M N) (subprob_algebra R)"
shows "(λx. 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 "(λx. 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 "λx. x ⤜ f"])
show "(λx. 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)"
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⇧L (bind M N) f = ∫ x. integral⇧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⇧L (join (distr M (subprob_algebra K) N)) f = ∫ M'. integral⇧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⇧L (N x) f ∂M"
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)) = (∫⇧+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 "… = (∫⇧+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"