src/HOL/Analysis/Embed_Measure.thy

+(*  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 \<open>Embed Measure Spaces with a Function\<close>
+
+theory Embed_Measure
+imports Binary_Product_Measure
+begin
+
+definition embed_measure :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
+  "embed_measure M f = measure_of (f ` space M) {f ` A |A. A \<in> sets M}
+                           (\<lambda>A. emeasure M (f -` A \<inter> 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 \<in> 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 \<in> {f ` A |A. A \<in> sets M}"
+  then obtain s' where s'_props: "s = f ` s'" "s' \<in> 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 "... \<in> {f ` A |A. A \<in> sets M}" using s'_props by auto
+  finally show "f ` space M - s \<in> {f ` A |A. A \<in> sets M}" .
+next
+  fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> {f ` A |A. A \<in> sets M}"
+  then obtain A' where A': "\<And>i. A i = f ` A' i" "\<And>i. A' i \<in> sets M"
+    by (auto simp: subset_eq choice_iff)
+  then have "(\<Union>x. f ` A' x) = f ` (\<Union>x. A' x)" by blast
+  with A' show "(\<Union>i. A i) \<in> {f ` A |A. A \<in> sets M}"
+    by simp blast
+qed (auto dest: sets.sets_into_space)
+
+lemma the_inv_into_vimage:
+  "inj_on f X \<Longrightarrow> A \<subseteq> X \<Longrightarrow> the_inv_into X f -` A \<inter> (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 simple_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 \<in> sets M}"
+  using assms by (subst sets_embed_measure') (auto intro!: inj_onI dest: injD)
+
+lemma in_sets_embed_measure: "A \<in> sets M \<Longrightarrow> f ` A \<in> 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: "(\<lambda>x. g (f x)) \<in> measurable M N"
+  shows "g \<in> measurable (embed_measure M f) N"
+  unfolding measurable_def
+proof safe
+  fix A assume "A \<in> sets N"
+  with g have "(\<lambda>x. g (f x)) -` A \<inter> space M \<in> sets M"
+    by (rule measurable_sets)
+  then have "f ` ((\<lambda>x. g (f x)) -` A \<inter> space M) \<in> sets (embed_measure M f)"
+    by (rule in_sets_embed_measure)
+  also have "f ` ((\<lambda>x. g (f x)) -` A \<inter> space M) = g -` A \<inter> space (embed_measure M f)"
+    by (auto simp: space_embed_measure)
+  finally show "g -` A \<inter> space (embed_measure M f) \<in> 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 \<in> measurable M (embed_measure M f)"
+proof-
+  {
+    fix A assume A: "A \<in> sets M"
+    also from A have "A = A \<inter> space M" by auto
+    also have "... = f -` f ` A \<inter> space M" using A assms
+      by (auto dest: inj_onD sets.sets_into_space)
+    finally have "f -` f ` A \<inter> space M \<in> 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 \<in> 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 \<in> sets M}
+                       (\<lambda>A. emeasure M (f -` A \<inter> 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 \<Longrightarrow> 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) \<Longrightarrow> g \<in> borel_measurable (embed_measure M f) \<Longrightarrow>
+  nn_integral (embed_measure M f) g = nn_integral M (\<lambda>x. g (f x))"
+  apply (subst embed_measure_eq_distr', simp)
+  apply (subst nn_integral_distr)
+  apply (simp_all add: measurable_embed_measure2')
+  done
+
+lemma nn_integral_embed_measure:
+  "inj f \<Longrightarrow> g \<in> borel_measurable (embed_measure M f) \<Longrightarrow>
+  nn_integral (embed_measure M f) g = nn_integral M (\<lambda>x. 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 \<in> sets (embed_measure M f)"
+    shows "emeasure (embed_measure M f) A = emeasure M (f -` A \<inter> 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 \<in> sets (embed_measure M f)"
+    shows "emeasure (embed_measure M f) A = emeasure M (f -` A \<inter> 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 \<circ> f)"
+proof-
+  have [simp]: "inj (\<lambda>x. g (f x))" by (subst o_def[symmetric]) (auto intro: inj_comp)
+  note measurable_embed_measure2[measurable]
+  have "embed_measure (embed_measure M f) g =
+            distr M (embed_measure (embed_measure M f) g) (g \<circ> 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 \<circ> 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_comp 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 \<subseteq> sets M" "\<Union>A = space M" "\<And>X. X\<in>A \<Longrightarrow> emeasure M X \<noteq> \<infinity>" by blast
+  from A_props have "countable (op ` f`A)" by auto
+  moreover
+  from inj and A_props have "op ` f`A \<subseteq> sets (embed_measure M f)"
+    by (auto simp: sets_embed_measure)
+  moreover
+  from A_props and inj have "\<Union>(op ` f`A) = space (embed_measure M f)"
+    by (auto simp: space_embed_measure intro!: imageI)
+  moreover
+  from A_props and inj have "\<forall>a\<in>op ` f ` A. emeasure (embed_measure M f) a \<noteq> \<infinity>"
+    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 \<Longrightarrow> embed_measure (count_space A) f = count_space (f`A)"
+  apply (subst distr_bij_count_space[of f A "f`A", symmetric])
+  apply (simp add: inj_on_def bij_betw_def)
+  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 \<Longrightarrow> 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 \<Longrightarrow> sets (embed_measure M f) = (op`f) ` sets M"
+  by (auto simp: sets_embed_measure)
+
+lemma emeasure_embed_measure_image':
+  assumes "inj_on f (space M)" "X \<in> 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 \<inter> space M)"
+    by (subst emeasure_embed_measure') (auto simp: sets_embed_measure')
+  also from assms have "f -` f ` X \<inter> space M = X" by (auto dest: inj_onD sets.sets_into_space)
+  finally show ?thesis .
+qed
+
+lemma emeasure_embed_measure_image:
+    "inj f \<Longrightarrow> X \<in> sets M \<Longrightarrow> 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 \<longleftrightarrow> A = B" (is "?M = ?N \<longleftrightarrow> _")
+proof
+  from assms have I: "inj (op` 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 \<in> sets A"
+    from asm have "emeasure ?M (f`X) = emeasure ?N (f`X)" by simp
+    with \<open>X \<in> sets A\<close> and \<open>sets A = sets B\<close> 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 \<Longrightarrow> the_inv_into A f \<in> f ` A \<rightarrow> A"
+  by (auto simp: the_inv_into_f_f)
+
+lemma map_prod_image: "map_prod f g ` (A \<times> B) = (f`A) \<times> (g`B)"
+  using map_prod_surj_on[OF refl refl] .
+
+lemma map_prod_vimage: "map_prod f g -` (A \<times> B) = (f-`A) \<times> (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 \<Otimes>\<^sub>M embed_measure N g = embed_measure (M \<Otimes>\<^sub>M N) (\<lambda>(x, y). (f x, g y))"
+    (is "?L = _")
+  unfolding map_prod_def[symmetric]
+proof (rule pair_measure_eqI)
+  have fg[simp]: "\<And>A. inj_on (map_prod f g) A" "\<And>A. inj_on f A" "\<And>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] SUP_insert[simp del]
+     ccSUP_insert[simp del]
+  show sets: "sets ?L = sets (embed_measure (M \<Otimes>\<^sub>M N) (map_prod f g))"
+    unfolding map_prod_def[symmetric]
+    apply (simp add: sets_pair_eq_sets_fst_snd sets_embed_eq_vimage_algebra
+      cong: vimage_algebra_cong)
+    apply (subst sets_vimage_Sup_eq[where Y="space (M \<Otimes>\<^sub>M N)"])
+    apply (simp_all add: space_pair_measure[symmetric])
+    apply (auto simp add: the_inv_into_f_f
+                simp del: map_prod_simp
+                del: prod_fun_imageE) []
+    apply auto []
+    apply (subst image_insert)
+    apply simp
+    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)
+    apply (simp_all add: the_inv_into_f_f space_pair_measure)
+    done
+
+  note measurable_embed_measure2[measurable]
+  fix A B assume AB: "A \<in> sets (embed_measure M f)" "B \<in> sets (embed_measure N g)"
+  moreover have "f -` A \<times> g -` B \<inter> space (M \<Otimes>\<^sub>M N) = (f -` A \<inter> space M) \<times> (g -` B \<inter> 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 \<Otimes>\<^sub>M N) (map_prod f g)) (A \<times> 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' \<Longrightarrow> sets M \<subseteq> sets M' \<Longrightarrow> sets (embed_measure M f) \<subseteq> 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 (simp add: subset_eq)
+  apply metis
+  done
+
+lemma density_embed_measure:
+  assumes inj: "inj f" and Mg[measurable]: "g \<in> borel_measurable (embed_measure M f)"
+  shows "density (embed_measure M f) g = embed_measure (density M (g \<circ> f)) f" (is "?M1 = ?M2")
+proof (rule measure_eqI)
+  fix X assume X: "X \<in> sets ?M1"
+  from inj have Mf[measurable]: "f \<in> measurable M (embed_measure M f)"
+    by (rule measurable_embed_measure2)
+  from Mg and X have "emeasure ?M1 X = \<integral>\<^sup>+ x. g x * indicator X x \<partial>embed_measure M f"
+    by (subst emeasure_density) simp_all
+  also from X have "... = \<integral>\<^sup>+ x. g (f x) * indicator X (f x) \<partial>M"
+    by (subst embed_measure_eq_distr[OF inj], subst nn_integral_distr) auto
+  also have "... = \<integral>\<^sup>+ x. g (f x) * indicator (f -` X \<inter> space M) x \<partial>M"
+    by (intro nn_integral_cong) (auto split: split_indicator)
+  also from X have "... = emeasure (density M (g \<circ> f)) (f -` X \<inter> 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" .
+qed (simp_all add: sets_embed_measure inj)
+
+lemma density_embed_measure':
+  assumes inj: "inj f" and inv: "\<And>x. f' (f x) = x" and Mg[measurable]: "g \<in> borel_measurable M"
+  shows "density (embed_measure M f) (g \<circ> f') = embed_measure (density M g) f"
+proof-
+  have "density (embed_measure M f) (g \<circ> f') = embed_measure (density M (g \<circ> f' \<circ> 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 \<circ> f' \<circ> 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 \<subseteq> C"  "B \<subseteq> C"
+  shows "f ` A \<subseteq> f ` B \<longleftrightarrow> A \<subseteq> B"
+proof (intro iffI subsetI)
+  fix x assume A: "f ` A \<subseteq> f ` B" and B: "x \<in> A"
+  from B have "f x \<in> f ` A" by blast
+  with A have "f x \<in> f ` B" by blast
+  then obtain y where "f x = f y" and "y \<in> B" by blast
+  with assms and B have "x = y" by (auto dest: inj_onD)
+  with \<open>y \<in> B\<close> show "x \<in> 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) \<longleftrightarrow> (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 \<in> sets ?M" "emeasure ?M A = 0" "{x\<in>space ?M. \<not>P x} \<subseteq> A"
+    by (force elim: AE_E)
+  then obtain A' where A'_props: "A = f ` A'" "A' \<in> sets M" by (auto simp: sets_embed_measure' inj)
+  moreover have B: "{x\<in>space ?M. \<not>P x} = f ` {x\<in>space M. \<not>P (f x)}"
+    by (auto simp: inj space_embed_measure)
+  from A_props(3) have "{x\<in>space M. \<not>P (f x)} \<subseteq> 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"
+    by (simp add: emeasure_embed_measure_image' inj)
+  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 \<in> sets M" "emeasure M A = 0" "{x\<in>space M. \<not>P (f x)} \<subseteq> A"
+    by (force elim: AE_E)
+  hence "f`A \<in> sets ?M" "emeasure ?M (f`A) = 0" "{x\<in>space ?M. \<not>P x} \<subseteq> 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) \<longleftrightarrow> (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 \<Rightarrow> 'b \<Rightarrow> ennreal"
+  assumes nonempty: "Y \<noteq> {}"
+  and chain: "Complete_Partial_Order.chain op \<le> (f ` Y)"
+  and countable: "countable B"
+  shows "(\<integral>\<^sup>+ x. (SUP i:Y. f i x) \<partial>count_space B) = (SUP i:Y. (\<integral>\<^sup>+ x. f i x \<partial>count_space B))"
+  (is "?lhs = ?rhs")
+proof -
+  let ?f = "(\<lambda>i x. f i (from_nat_into B x) * indicator (to_nat_on B ` B) x)"
+  have "?lhs = \<integral>\<^sup>+ x. (SUP i:Y. f i (from_nat_into B (to_nat_on B x))) \<partial>count_space B"
+    by(rule nn_integral_cong)(simp add: countable)
+  also have "\<dots> = \<integral>\<^sup>+ x. (SUP i:Y. f i (from_nat_into B x)) \<partial>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 "\<dots> = \<integral>\<^sup>+ x. (SUP i:Y. ?f i x) \<partial>count_space UNIV"
+    by(simp add: nn_integral_count_space_indicator ennreal_indicator[symmetric] SUP_mult_right_ennreal nonempty)
+  also have "\<dots> = (SUP i:Y. \<integral>\<^sup>+ x. ?f i x \<partial>count_space UNIV)"
+  proof(rule nn_integral_monotone_convergence_SUP_nat)
+    show "Complete_Partial_Order.chain op \<le> (?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 "\<dots> = (SUP i:Y. \<integral>\<^sup>+ x. f i (from_nat_into B x) \<partial>count_space (to_nat_on B ` B))"
+    by(simp add: nn_integral_count_space_indicator)
+  also have "\<dots> = (SUP i:Y. \<integral>\<^sup>+ x. f i (from_nat_into B (to_nat_on B x)) \<partial>count_space B)"
+    by(simp add: embed_measure_count_space'[symmetric] inj_on_to_nat_on countable nn_integral_embed_measure' measurable_embed_measure1)
+  also have "\<dots> = ?rhs"
+    by(intro arg_cong2[where f="SUPREMUM"] ext nn_integral_cong_AE)(simp_all add: AE_count_space countable)
+  finally show ?thesis .
+qed
+
+end