# Theory Embed_Measure

theory Embed_Measure
imports Binary_Product_Measure
```(*  Title:      HOL/Analysis/Embed_Measure.thy
Author:     Manuel Eberl, TU München

Defines measure embeddings with injective functions, i.e. lifting a measure on some values
to a measure on "tagged" values (e.g. embed_measure M Inl lifts a measure on values 'a to a
measure on the left part of the sum type 'a + 'b)
*)

section ‹Embedding Measure Spaces with a Function›

theory Embed_Measure
imports Binary_Product_Measure
begin

text ‹
Given a measure space on some carrier set ‹Ω› and a function ‹f›, we can define a push-forward
measure on the carrier set ‹f(Ω)› whose ‹σ›-algebra is the one generated by mapping ‹f› over
the original sigma algebra.

This is useful e.\,g.\ when ‹f› is injective, i.\,e.\ it is some kind of ``tagging'' function.
For instance, suppose we have some algebraaic datatype of values with various constructors,
including a constructor ‹RealVal› for real numbers. Then ‹embed_measure› allows us to lift a
measure on real numbers to the appropriate subset of that algebraic datatype.
›
definition✐‹tag important› embed_measure :: "'a measure ⇒ ('a ⇒ 'b) ⇒ 'b measure" where
"embed_measure M f = measure_of (f ` space M) {f ` A |A. A ∈ sets M}
(λA. emeasure M (f -` A ∩ space M))"

lemma space_embed_measure: "space (embed_measure M f) = f ` space M"
unfolding embed_measure_def
by (subst space_measure_of) (auto dest: sets.sets_into_space)

lemma sets_embed_measure':
assumes inj: "inj_on f (space M)"
shows "sets (embed_measure M f) = {f ` A |A. A ∈ sets M}"
unfolding embed_measure_def
proof (intro sigma_algebra.sets_measure_of_eq sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI)
fix s assume "s ∈ {f ` A |A. A ∈ sets M}"
then obtain s' where s'_props: "s = f ` s'" "s' ∈ sets M" by auto
hence "f ` space M - s = f ` (space M - s')" using inj
by (auto dest: inj_onD sets.sets_into_space)
also have "... ∈ {f ` A |A. A ∈ sets M}" using s'_props by auto
finally show "f ` space M - s ∈ {f ` A |A. A ∈ sets M}" .
next
fix A :: "nat ⇒ _" assume "range A ⊆ {f ` A |A. A ∈ sets M}"
then obtain A' where A': "⋀i. A i = f ` A' i" "⋀i. A' i ∈ sets M"
by (auto simp: subset_eq choice_iff)
then have "(⋃x. f ` A' x) = f ` (⋃x. A' x)" by blast
with A' show "(⋃i. A i) ∈ {f ` A |A. A ∈ sets M}"
by simp blast
qed (auto dest: sets.sets_into_space)

lemma the_inv_into_vimage:
"inj_on f X ⟹ A ⊆ X ⟹ the_inv_into X f -` A ∩ (f`X) = f ` A"
by (auto simp: the_inv_into_f_f)

lemma sets_embed_eq_vimage_algebra:
assumes "inj_on f (space M)"
shows "sets (embed_measure M f) = sets (vimage_algebra (f`space M) (the_inv_into (space M) f) M)"
by (auto simp: sets_embed_measure'[OF assms] Pi_iff the_inv_into_f_f assms sets_vimage_algebra2 Setcompr_eq_image
dest: sets.sets_into_space
intro!: image_cong the_inv_into_vimage[symmetric])

lemma sets_embed_measure:
assumes inj: "inj f"
shows "sets (embed_measure M f) = {f ` A |A. A ∈ sets M}"
using assms by (subst sets_embed_measure') (auto intro!: inj_onI dest: injD)

lemma in_sets_embed_measure: "A ∈ sets M ⟹ f ` A ∈ sets (embed_measure M f)"
unfolding embed_measure_def
by (intro in_measure_of) (auto dest: sets.sets_into_space)

lemma measurable_embed_measure1:
assumes g: "(λx. g (f x)) ∈ measurable M N"
shows "g ∈ measurable (embed_measure M f) N"
unfolding measurable_def
proof safe
fix A assume "A ∈ sets N"
with g have "(λx. g (f x)) -` A ∩ space M ∈ sets M"
by (rule measurable_sets)
then have "f ` ((λx. g (f x)) -` A ∩ space M) ∈ sets (embed_measure M f)"
by (rule in_sets_embed_measure)
also have "f ` ((λx. g (f x)) -` A ∩ space M) = g -` A ∩ space (embed_measure M f)"
by (auto simp: space_embed_measure)
finally show "g -` A ∩ space (embed_measure M f) ∈ sets (embed_measure M f)" .
qed (insert measurable_space[OF assms], auto simp: space_embed_measure)

lemma measurable_embed_measure2':
assumes "inj_on f (space M)"
shows "f ∈ measurable M (embed_measure M f)"
proof-
{
fix A assume A: "A ∈ sets M"
also from A have "A = A ∩ space M" by auto
also have "... = f -` f ` A ∩ space M" using A assms
by (auto dest: inj_onD sets.sets_into_space)
finally have "f -` f ` A ∩ space M ∈ sets M" .
}
thus ?thesis using assms unfolding embed_measure_def
by (intro measurable_measure_of) (auto dest: sets.sets_into_space)
qed

lemma measurable_embed_measure2:
assumes [simp]: "inj f" shows "f ∈ measurable M (embed_measure M f)"
by (auto simp: inj_vimage_image_eq embed_measure_def
intro!: measurable_measure_of dest: sets.sets_into_space)

lemma embed_measure_eq_distr':
assumes "inj_on f (space M)"
shows "embed_measure M f = distr M (embed_measure M f) f"
proof-
have "distr M (embed_measure M f) f =
measure_of (f ` space M) {f ` A |A. A ∈ sets M}
(λA. emeasure M (f -` A ∩ space M))" unfolding distr_def
by (simp add: space_embed_measure sets_embed_measure'[OF assms])
also have "... = embed_measure M f" unfolding embed_measure_def ..
finally show ?thesis ..
qed

lemma embed_measure_eq_distr:
"inj f ⟹ embed_measure M f = distr M (embed_measure M f) f"
by (rule embed_measure_eq_distr') (auto intro!: inj_onI dest: injD)

lemma nn_integral_embed_measure':
"inj_on f (space M) ⟹ g ∈ borel_measurable (embed_measure M f) ⟹
nn_integral (embed_measure M f) g = nn_integral M (λx. g (f x))"
apply (subst embed_measure_eq_distr', simp)
apply (subst nn_integral_distr)
done

lemma nn_integral_embed_measure:
"inj f ⟹ g ∈ borel_measurable (embed_measure M f) ⟹
nn_integral (embed_measure M f) g = nn_integral M (λx. g (f x))"
by(erule nn_integral_embed_measure'[OF subset_inj_on]) simp

lemma emeasure_embed_measure':
assumes "inj_on f (space M)" "A ∈ sets (embed_measure M f)"
shows "emeasure (embed_measure M f) A = emeasure M (f -` A ∩ space M)"
by (subst embed_measure_eq_distr'[OF assms(1)])
(simp add: emeasure_distr[OF measurable_embed_measure2'[OF assms(1)] assms(2)])

lemma emeasure_embed_measure:
assumes "inj f" "A ∈ sets (embed_measure M f)"
shows "emeasure (embed_measure M f) A = emeasure M (f -` A ∩ space M)"
using assms by (intro emeasure_embed_measure') (auto intro!: inj_onI dest: injD)

lemma embed_measure_comp:
assumes [simp]: "inj f" "inj g"
shows "embed_measure (embed_measure M f) g = embed_measure M (g ∘ f)"
proof-
have [simp]: "inj (λx. g (f x))" by (subst o_def[symmetric]) (auto intro: inj_compose)
note measurable_embed_measure2[measurable]
have "embed_measure (embed_measure M f) g =
distr M (embed_measure (embed_measure M f) g) (g ∘ f)"
by (subst (1 2) embed_measure_eq_distr)
(simp_all add: distr_distr sets_embed_measure cong: distr_cong)
also have "... = embed_measure M (g ∘ f)"
by (subst (3) embed_measure_eq_distr, simp add: o_def, rule distr_cong)
(auto simp: sets_embed_measure o_def image_image[symmetric]
intro: inj_compose cong: distr_cong)
finally show ?thesis .
qed

lemma sigma_finite_embed_measure:
assumes "sigma_finite_measure M" and inj: "inj f"
shows "sigma_finite_measure (embed_measure M f)"
proof -
from assms(1) interpret sigma_finite_measure M .
from sigma_finite_countable obtain A where
A_props: "countable A" "A ⊆ sets M" "⋃A = space M" "⋀X. X∈A ⟹ emeasure M X ≠ ∞" by blast
from A_props have "countable ((`) f`A)" by auto
moreover
from inj and A_props have "(`) f`A ⊆ sets (embed_measure M f)"
by (auto simp: sets_embed_measure)
moreover
from A_props and inj have "⋃((`) f`A) = space (embed_measure M f)"
by (auto simp: space_embed_measure intro!: imageI)
moreover
from A_props and inj have "∀a∈(`) f ` A. emeasure (embed_measure M f) a ≠ ∞"
by (intro ballI, subst emeasure_embed_measure)
(auto simp: inj_vimage_image_eq intro: in_sets_embed_measure)
ultimately show ?thesis by - (standard, blast)
qed

lemma embed_measure_count_space':
"inj_on f A ⟹ embed_measure (count_space A) f = count_space (f`A)"
apply (subst distr_bij_count_space[of f A "f`A", symmetric])
apply (subst embed_measure_eq_distr')
apply simp
apply(auto 4 3 intro!: measure_eqI imageI simp add: sets_embed_measure' subset_image_iff)
apply (subst (1 2) emeasure_distr)
apply (auto simp: space_embed_measure sets_embed_measure')
done

lemma embed_measure_count_space:
"inj f ⟹ embed_measure (count_space A) f = count_space (f`A)"
by(rule embed_measure_count_space')(erule subset_inj_on, simp)

lemma sets_embed_measure_alt:
"inj f ⟹ sets (embed_measure M f) = ((`) f) ` sets M"
by (auto simp: sets_embed_measure)

lemma emeasure_embed_measure_image':
assumes "inj_on f (space M)" "X ∈ sets M"
shows "emeasure (embed_measure M f) (f`X) = emeasure M X"
proof-
from assms have "emeasure (embed_measure M f) (f`X) = emeasure M (f -` f ` X ∩ space M)"
by (subst emeasure_embed_measure') (auto simp: sets_embed_measure')
also from assms have "f -` f ` X ∩ space M = X" by (auto dest: inj_onD sets.sets_into_space)
finally show ?thesis .
qed

lemma emeasure_embed_measure_image:
"inj f ⟹ X ∈ sets M ⟹ emeasure (embed_measure M f) (f`X) = emeasure M X"
by (simp_all add: emeasure_embed_measure in_sets_embed_measure inj_vimage_image_eq)

lemma embed_measure_eq_iff:
assumes "inj f"
shows "embed_measure A f = embed_measure B f ⟷ A = B" (is "?M = ?N ⟷ _")
proof
from assms have I: "inj ((`) f)" by (auto intro: injI dest: injD)
assume asm: "?M = ?N"
hence "sets (embed_measure A f) = sets (embed_measure B f)" by simp
with assms have "sets A = sets B" by (simp only: I inj_image_eq_iff sets_embed_measure_alt)
moreover {
fix X assume "X ∈ sets A"
from asm have "emeasure ?M (f`X) = emeasure ?N (f`X)" by simp
with ‹X ∈ sets A› and ‹sets A = sets B› and assms
have "emeasure A X = emeasure B X" by (simp add: emeasure_embed_measure_image)
}
ultimately show "A = B" by (rule measure_eqI)
qed simp

lemma the_inv_into_in_Pi: "inj_on f A ⟹ the_inv_into A f ∈ f ` A → A"
by (auto simp: the_inv_into_f_f)

lemma map_prod_image: "map_prod f g ` (A × B) = (f`A) × (g`B)"
using map_prod_surj_on[OF refl refl] .

lemma map_prod_vimage: "map_prod f g -` (A × B) = (f-`A) × (g-`B)"
by auto

lemma embed_measure_prod:
assumes f: "inj f" and g: "inj g" and [simp]: "sigma_finite_measure M" "sigma_finite_measure N"
shows "embed_measure M f ⨂⇩M embed_measure N g = embed_measure (M ⨂⇩M N) (λ(x, y). (f x, g y))"
(is "?L = _")
unfolding map_prod_def[symmetric]
proof (rule pair_measure_eqI)
have fg[simp]: "⋀A. inj_on (map_prod f g) A" "⋀A. inj_on f A" "⋀A. inj_on g A"
using f g by (auto simp: inj_on_def)

note complete_lattice_class.Sup_insert[simp del] ccSup_insert[simp del]
ccSUP_insert[simp del]
show sets: "sets ?L = sets (embed_measure (M ⨂⇩M N) (map_prod f g))"
unfolding map_prod_def[symmetric]
cong: vimage_algebra_cong)
apply (subst sets_vimage_Sup_eq[where Y="space (M ⨂⇩M N)"])
simp del: map_prod_simp
del: prod_fun_imageE) []
apply auto []
apply (subst (1 2 3 4 ) vimage_algebra_vimage_algebra_eq)
apply (simp_all add: the_inv_into_in_Pi Pi_iff[of snd] Pi_iff[of fst] space_pair_measure)
apply (simp_all add: Pi_iff[of snd] Pi_iff[of fst] the_inv_into_in_Pi vimage_algebra_vimage_algebra_eq
space_pair_measure[symmetric] map_prod_image[symmetric])
apply (intro arg_cong[where f=sets] arg_cong[where f=Sup] arg_cong2[where f=insert] vimage_algebra_cong)
apply (auto simp: map_prod_image the_inv_into_f_f
simp del: map_prod_simp del: prod_fun_imageE)
done

note measurable_embed_measure2[measurable]
fix A B assume AB: "A ∈ sets (embed_measure M f)" "B ∈ sets (embed_measure N g)"
moreover have "f -` A × g -` B ∩ space (M ⨂⇩M N) = (f -` A ∩ space M) × (g -` B ∩ space N)"
by (auto simp: space_pair_measure)
ultimately show "emeasure (embed_measure M f) A * emeasure (embed_measure N g) B =
emeasure (embed_measure (M ⨂⇩M N) (map_prod f g)) (A × B)"
by (simp add: map_prod_vimage sets[symmetric] emeasure_embed_measure
sigma_finite_measure.emeasure_pair_measure_Times)
qed (insert assms, simp_all add: sigma_finite_embed_measure)

lemma mono_embed_measure:
"space M = space M' ⟹ sets M ⊆ sets M' ⟹ sets (embed_measure M f) ⊆ sets (embed_measure M' f)"
unfolding embed_measure_def
apply (subst (1 2) sets_measure_of)
apply (blast dest: sets.sets_into_space)
apply (blast dest: sets.sets_into_space)
apply simp
apply (intro sigma_sets_mono')
apply safe
apply metis
done

lemma density_embed_measure:
assumes inj: "inj f" and Mg[measurable]: "g ∈ borel_measurable (embed_measure M f)"
shows "density (embed_measure M f) g = embed_measure (density M (g ∘ f)) f" (is "?M1 = ?M2")
proof (rule measure_eqI)
fix X assume X: "X ∈ sets ?M1"
from inj have Mf[measurable]: "f ∈ measurable M (embed_measure M f)"
by (rule measurable_embed_measure2)
from Mg and X have "emeasure ?M1 X = ∫⇧+ x. g x * indicator X x ∂embed_measure M f"
by (subst emeasure_density) simp_all
also from X have "... = ∫⇧+ x. g (f x) * indicator X (f x) ∂M"
by (subst embed_measure_eq_distr[OF inj], subst nn_integral_distr) auto
also have "... = ∫⇧+ x. g (f x) * indicator (f -` X ∩ space M) x ∂M"
by (intro nn_integral_cong) (auto split: split_indicator)
also from X have "... = emeasure (density M (g ∘ f)) (f -` X ∩ space M)"
by (subst emeasure_density) (simp_all add: measurable_comp[OF Mf Mg] measurable_sets[OF Mf])
also from X and inj have "... = emeasure ?M2 X"
by (subst emeasure_embed_measure) (simp_all add: sets_embed_measure)
finally show "emeasure ?M1 X = emeasure ?M2 X" .

lemma density_embed_measure':
assumes inj: "inj f" and inv: "⋀x. f' (f x) = x" and Mg[measurable]: "g ∈ borel_measurable M"
shows "density (embed_measure M f) (g ∘ f') = embed_measure (density M g) f"
proof-
have "density (embed_measure M f) (g ∘ f') = embed_measure (density M (g ∘ f' ∘ f)) f"
by (rule density_embed_measure[OF inj])
(rule measurable_comp, rule measurable_embed_measure1, subst measurable_cong,
rule inv, rule measurable_ident_sets, simp, rule Mg)
also have "density M (g ∘ f' ∘ f) = density M g"
by (intro density_cong) (subst measurable_cong, simp add: o_def inv, simp_all add: Mg inv)
finally show ?thesis .
qed

lemma inj_on_image_subset_iff:
assumes "inj_on f C" "A ⊆ C"  "B ⊆ C"
shows "f ` A ⊆ f ` B ⟷ A ⊆ B"
proof (intro iffI subsetI)
fix x assume A: "f ` A ⊆ f ` B" and B: "x ∈ A"
from B have "f x ∈ f ` A" by blast
with A have "f x ∈ f ` B" by blast
then obtain y where "f x = f y" and "y ∈ B" by blast
with assms and B have "x = y" by (auto dest: inj_onD)
with ‹y ∈ B› show "x ∈ B" by simp
qed auto

lemma AE_embed_measure':
assumes inj: "inj_on f (space M)"
shows "(AE x in embed_measure M f. P x) ⟷ (AE x in M. P (f x))"
proof
let ?M = "embed_measure M f"
assume "AE x in ?M. P x"
then obtain A where A_props: "A ∈ sets ?M" "emeasure ?M A = 0" "{x∈space ?M. ¬P x} ⊆ A"
by (force elim: AE_E)
then obtain A' where A'_props: "A = f ` A'" "A' ∈ sets M" by (auto simp: sets_embed_measure' inj)
moreover have B: "{x∈space ?M. ¬P x} = f ` {x∈space M. ¬P (f x)}"
by (auto simp: inj space_embed_measure)
from A_props(3) have "{x∈space M. ¬P (f x)} ⊆ A'"
by (subst (asm) B, subst (asm) A'_props, subst (asm) inj_on_image_subset_iff[OF inj])
(insert A'_props, auto dest: sets.sets_into_space)
moreover from A_props A'_props have "emeasure M A' = 0"
ultimately show "AE x in M. P (f x)" by (intro AE_I)
next
let ?M = "embed_measure M f"
assume "AE x in M. P (f x)"
then obtain A where A_props: "A ∈ sets M" "emeasure M A = 0" "{x∈space M. ¬P (f x)} ⊆ A"
by (force elim: AE_E)
hence "f`A ∈ sets ?M" "emeasure ?M (f`A) = 0" "{x∈space ?M. ¬P x} ⊆ f`A"
by (auto simp: space_embed_measure emeasure_embed_measure_image' sets_embed_measure' inj)
thus "AE x in ?M. P x" by (intro AE_I)
qed

lemma AE_embed_measure:
assumes inj: "inj f"
shows "(AE x in embed_measure M f. P x) ⟷ (AE x in M. P (f x))"
using assms by (intro AE_embed_measure') (auto intro!: inj_onI dest: injD)

lemma nn_integral_monotone_convergence_SUP_countable:
fixes f :: "'a ⇒ 'b ⇒ ennreal"
assumes nonempty: "Y ≠ {}"
and chain: "Complete_Partial_Order.chain (≤) (f ` Y)"
and countable: "countable B"
shows "(∫⇧+ x. (SUP i∈Y. f i x) ∂count_space B) = (SUP i∈Y. (∫⇧+ x. f i x ∂count_space B))"
(is "?lhs = ?rhs")
proof -
let ?f = "(λi x. f i (from_nat_into B x) * indicator (to_nat_on B ` B) x)"
have "?lhs = ∫⇧+ x. (SUP i∈Y. f i (from_nat_into B (to_nat_on B x))) ∂count_space B"
also have "… = ∫⇧+ x. (SUP i∈Y. f i (from_nat_into B x)) ∂count_space (to_nat_on B ` B)"
by(simp add: embed_measure_count_space'[symmetric] inj_on_to_nat_on countable nn_integral_embed_measure' measurable_embed_measure1)
also have "… = ∫⇧+ x. (SUP i∈Y. ?f i x) ∂count_space UNIV"
by(simp add: nn_integral_count_space_indicator ennreal_indicator[symmetric] SUP_mult_right_ennreal nonempty)
also have "… = (SUP i∈Y. ∫⇧+ x. ?f i x ∂count_space UNIV)"
proof(rule nn_integral_monotone_convergence_SUP_nat)
show "Complete_Partial_Order.chain (≤) (?f ` Y)"
by(rule chain_imageI[OF chain, unfolded image_image])(auto intro!: le_funI split: split_indicator dest: le_funD)
qed fact
also have "… = (SUP i∈Y. ∫⇧+ x. f i (from_nat_into B x) ∂count_space (to_nat_on B ` B))"