--- a/src/HOL/Probability/Probability_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Probability_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -6,110 +6,219 @@
header {*Probability measure*}
theory Probability_Measure
-imports Lebesgue_Measure
+ imports Lebesgue_Measure Radon_Nikodym
begin
+lemma funset_eq_UN_fun_upd_I:
+ assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
+ and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
+ and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
+ shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
+proof safe
+ fix f assume f: "f \<in> F (insert a A)"
+ show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
+ proof (rule UN_I[of "f(a := d)"])
+ show "f(a := d) \<in> F A" using *[OF f] .
+ show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
+ proof (rule image_eqI[of _ _ "f a"])
+ show "f a \<in> G (f(a := d))" using **[OF f] .
+ qed simp
+ qed
+next
+ fix f x assume "f \<in> F A" "x \<in> G f"
+ from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
+qed
+
+lemma extensional_funcset_insert_eq[simp]:
+ assumes "a \<notin> A"
+ shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)"
+ apply (rule funset_eq_UN_fun_upd_I)
+ using assms
+ by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
+
+lemma finite_extensional_funcset[simp, intro]:
+ assumes "finite A" "finite B"
+ shows "finite (extensional A \<inter> (A \<rightarrow> B))"
+ using assms by induct auto
+
+lemma finite_PiE[simp, intro]:
+ assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
+ shows "finite (Pi\<^isub>E A B)"
+proof -
+ have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
+ show ?thesis
+ using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
+qed
+
+
+lemma countably_additiveI[case_names countably]:
+ assumes "\<And>A. \<lbrakk> range A \<subseteq> M ; disjoint_family A ; (\<Union>i. A i) \<in> M\<rbrakk> \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
+ shows "countably_additive M \<mu>"
+ using assms unfolding countably_additive_def by auto
+
+lemma convex_le_Inf_differential:
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex_on I f"
+ assumes "x \<in> interior I" "y \<in> I"
+ shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
+ (is "_ \<ge> _ + Inf (?F x) * (y - x)")
+proof -
+ show ?thesis
+ proof (cases rule: linorder_cases)
+ assume "x < y"
+ moreover
+ have "open (interior I)" by auto
+ from openE[OF this `x \<in> interior I`] guess e . note e = this
+ moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
+ ultimately have "x < t" "t < y" "t \<in> ball x e"
+ by (auto simp: dist_real_def field_simps split: split_min)
+ with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+ have "open (interior I)" by auto
+ from openE[OF this `x \<in> interior I`] guess e .
+ moreover def K \<equiv> "x - e / 2"
+ with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: dist_real_def)
+ ultimately have "K \<in> I" "K < x" "x \<in> I"
+ using interior_subset[of I] `x \<in> interior I` by auto
+
+ have "Inf (?F x) \<le> (f x - f y) / (x - y)"
+ proof (rule Inf_lower2)
+ show "(f x - f t) / (x - t) \<in> ?F x"
+ using `t \<in> I` `x < t` by auto
+ show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
+ using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
+ next
+ fix y assume "y \<in> ?F x"
+ with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
+ show "(f K - f x) / (K - x) \<le> y" by auto
+ qed
+ then show ?thesis
+ using `x < y` by (simp add: field_simps)
+ next
+ assume "y < x"
+ moreover
+ have "open (interior I)" by auto
+ from openE[OF this `x \<in> interior I`] guess e . note e = this
+ moreover def t \<equiv> "x + e / 2"
+ ultimately have "x < t" "t \<in> ball x e"
+ by (auto simp: dist_real_def field_simps)
+ with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+ have "(f x - f y) / (x - y) \<le> Inf (?F x)"
+ proof (rule Inf_greatest)
+ have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
+ using `y < x` by (auto simp: field_simps)
+ also
+ fix z assume "z \<in> ?F x"
+ with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
+ have "(f y - f x) / (y - x) \<le> z" by auto
+ finally show "(f x - f y) / (x - y) \<le> z" .
+ next
+ have "open (interior I)" by auto
+ from openE[OF this `x \<in> interior I`] guess e . note e = this
+ then have "x + e / 2 \<in> ball x e" by (auto simp: dist_real_def)
+ with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
+ then show "?F x \<noteq> {}" by blast
+ qed
+ then show ?thesis
+ using `y < x` by (simp add: field_simps)
+ qed simp
+qed
+
+lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
+ by (rule measure_eqI) (auto simp: emeasure_distr)
+
locale prob_space = finite_measure +
- assumes measure_space_1: "measure M (space M) = 1"
+ assumes emeasure_space_1: "emeasure M (space M) = 1"
lemma prob_spaceI[Pure.intro!]:
- assumes "measure_space M"
- assumes *: "measure M (space M) = 1"
+ assumes *: "emeasure M (space M) = 1"
shows "prob_space M"
proof -
interpret finite_measure M
proof
- show "measure_space M" by fact
- show "measure M (space M) \<noteq> \<infinity>" using * by simp
+ show "emeasure M (space M) \<noteq> \<infinity>" using * by simp
qed
show "prob_space M" by default fact
qed
abbreviation (in prob_space) "events \<equiv> sets M"
-abbreviation (in prob_space) "prob \<equiv> \<mu>'"
-abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'"
+abbreviation (in prob_space) "prob \<equiv> measure M"
+abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
-definition (in prob_space)
- "distribution X A = \<mu>' (X -` A \<inter> space M)"
-
-abbreviation (in prob_space)
- "joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
-
-lemma (in prob_space) prob_space_cong:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M"
- shows "prob_space N"
-proof
- show "measure_space N" by (intro measure_space_cong assms)
- show "measure N (space N) = 1"
- using measure_space_1 assms by simp
+lemma (in prob_space) prob_space_distr:
+ assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
+proof (rule prob_spaceI)
+ have "f -` space M' \<inter> 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_1)
qed
-lemma (in prob_space) distribution_cong:
- assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
- shows "distribution X = distribution Y"
- unfolding distribution_def fun_eq_iff
- using assms by (auto intro!: arg_cong[where f="\<mu>'"])
-
-lemma (in prob_space) joint_distribution_cong:
- assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
- assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
- shows "joint_distribution X Y = joint_distribution X' Y'"
- unfolding distribution_def fun_eq_iff
- using assms by (auto intro!: arg_cong[where f="\<mu>'"])
-
-lemma (in prob_space) distribution_id[simp]:
- "N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N"
- by (auto simp: distribution_def intro!: arg_cong[where f=prob])
-
lemma (in prob_space) prob_space: "prob (space M) = 1"
- using measure_space_1 unfolding \<mu>'_def by (simp add: one_ereal_def)
+ using emeasure_space_1 unfolding measure_def by (simp add: one_ereal_def)
lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
using bounded_measure[of A] by (simp add: prob_space)
-lemma (in prob_space) distribution_positive[simp, intro]:
- "0 \<le> distribution X A" unfolding distribution_def by auto
-
-lemma (in prob_space) not_zero_less_distribution[simp]:
- "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
- using distribution_positive[of X A] by arith
-
-lemma (in prob_space) joint_distribution_remove[simp]:
- "joint_distribution X X {(x, x)} = distribution X {x}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+lemma (in prob_space) not_empty: "space M \<noteq> {}"
+ using prob_space by auto
-lemma (in prob_space) distribution_unit[simp]: "distribution (\<lambda>x. ()) {()} = 1"
- unfolding distribution_def using prob_space by auto
-
-lemma (in prob_space) joint_distribution_unit[simp]: "distribution (\<lambda>x. (X x, ())) {(a, ())} = distribution X {a}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) not_empty: "space M \<noteq> {}"
- using prob_space empty_measure' by auto
-
-lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1"
- unfolding measure_space_1[symmetric]
- using sets_into_space
- by (intro measure_mono) auto
+lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
+ using emeasure_space[of M X] by (simp add: emeasure_space_1)
lemma (in prob_space) AE_I_eq_1:
- assumes "\<mu> {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
- shows "AE x. P x"
+ assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
+ shows "AE x in M. P x"
proof (rule AE_I)
- show "\<mu> (space M - {x \<in> space M. P x}) = 0"
- using assms measure_space_1 by (simp add: measure_compl)
+ show "emeasure M (space M - {x \<in> space M. P x}) = 0"
+ using assms emeasure_space_1 by (simp add: emeasure_compl)
qed (insert assms, auto)
-lemma (in prob_space) distribution_1:
- "distribution X A \<le> 1"
- unfolding distribution_def by simp
-
lemma (in prob_space) prob_compl:
assumes A: "A \<in> events"
shows "prob (space M - A) = 1 - prob A"
using finite_measure_compl[OF A] by (simp add: prob_space)
+lemma (in prob_space) AE_in_set_eq_1:
+ assumes "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
+proof
+ assume ae: "AE x in M. x \<in> A"
+ have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
+ using `A \<in> events`[THEN sets_into_space] by auto
+ with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
+ by (simp add: emeasure_compl emeasure_space_1)
+ then show "prob A = 1"
+ using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
+next
+ assume prob: "prob A = 1"
+ show "AE x in M. x \<in> A"
+ proof (rule AE_I)
+ show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
+ show "emeasure M (space M - A) = 0"
+ using `A \<in> events` prob
+ by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
+ show "space M - A \<in> events"
+ using `A \<in> events` by auto
+ qed
+qed
+
+lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
+proof
+ assume "AE x in M. False"
+ then have "AE x in M. x \<in> {}" by simp
+ then show False
+ by (subst (asm) AE_in_set_eq_1) auto
+qed simp
+
+lemma (in prob_space) AE_prob_1:
+ assumes "prob A = 1" shows "AE x in M. x \<in> A"
+proof -
+ from `prob A = 1` have "A \<in> events"
+ by (metis measure_notin_sets zero_neq_one)
+ with AE_in_set_eq_1 assms show ?thesis by simp
+qed
+
lemma (in prob_space) prob_space_increasing: "increasing M prob"
by (auto intro!: finite_measure_mono simp: increasing_def)
@@ -164,9 +273,8 @@
shows "prob (\<Union> i :: nat. c i) = 0"
proof (rule antisym)
show "prob (\<Union> i :: nat. c i) \<le> 0"
- using finite_measure_countably_subadditive[OF assms(1)]
- by (simp add: assms(2) suminf_zero summable_zero)
-qed simp
+ using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))
+qed (simp add: measure_nonneg)
lemma (in prob_space) prob_equiprobable_finite_unions:
assumes "s \<in> events"
@@ -178,7 +286,7 @@
from someI_ex[OF this] assms
have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
have "prob s = (\<Sum> x \<in> s. prob {x})"
- using finite_measure_finite_singleton[OF s_finite] by simp
+ using finite_measure_eq_setsum_singleton[OF s_finite] by simp
also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
using setsum_constant assms by (simp add: real_eq_of_nat)
@@ -199,96 +307,20 @@
also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
proof (rule finite_measure_finite_Union)
show "finite s" by fact
- show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact
+ show "(\<lambda>i. e \<inter> f i)`s \<subseteq> events" using assms(2) by auto
show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
using disjoint by (auto simp: disjoint_family_on_def)
qed
finally show ?thesis .
qed
-lemma (in prob_space) prob_space_vimage:
- assumes S: "sigma_algebra S"
- assumes T: "T \<in> measure_preserving M S"
- shows "prob_space S"
-proof
- interpret S: measure_space S
- using S and T by (rule measure_space_vimage)
- show "measure_space S" ..
-
- from T[THEN measure_preservingD2]
- have "T -` space S \<inter> space M = space M"
- by (auto simp: measurable_def)
- with T[THEN measure_preservingD, of "space S", symmetric]
- show "measure S (space S) = 1"
- using measure_space_1 by simp
-qed
-
-lemma prob_space_unique_Int_stable:
- fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
- assumes E: "Int_stable E" "space E \<in> sets E"
- and M: "prob_space M" "space M = space E" "sets M = sets (sigma E)"
- and N: "prob_space N" "space N = space E" "sets N = sets (sigma E)"
- and eq: "\<And>X. X \<in> sets E \<Longrightarrow> finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
- assumes "X \<in> sets (sigma E)"
- shows "finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
-proof -
- interpret M!: prob_space M by fact
- interpret N!: prob_space N by fact
- have "measure M X = measure N X"
- proof (rule measure_unique_Int_stable[OF `Int_stable E`])
- show "range (\<lambda>i. space M) \<subseteq> sets E" "incseq (\<lambda>i. space M)" "(\<Union>i. space M) = space E"
- using E M N by auto
- show "\<And>i. M.\<mu> (space M) \<noteq> \<infinity>"
- using M.measure_space_1 by simp
- show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = M.\<mu>\<rparr>"
- using E M N by (auto intro!: M.measure_space_cong)
- show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = N.\<mu>\<rparr>"
- using E M N by (auto intro!: N.measure_space_cong)
- { fix X assume "X \<in> sets E"
- then have "X \<in> sets (sigma E)"
- by (auto simp: sets_sigma sigma_sets.Basic)
- with eq[OF `X \<in> sets E`] M N show "M.\<mu> X = N.\<mu> X"
- by (simp add: M.finite_measure_eq N.finite_measure_eq) }
- qed fact
- with `X \<in> sets (sigma E)` M N show ?thesis
- by (simp add: M.finite_measure_eq N.finite_measure_eq)
-qed
-
-lemma (in prob_space) distribution_prob_space:
- assumes X: "random_variable S X"
- shows "prob_space (S\<lparr>measure := ereal \<circ> distribution X\<rparr>)" (is "prob_space ?S")
-proof (rule prob_space_vimage)
- show "X \<in> measure_preserving M ?S"
- using X
- unfolding measure_preserving_def distribution_def [abs_def]
- by (auto simp: finite_measure_eq measurable_sets)
- show "sigma_algebra ?S" using X by simp
-qed
-
-lemma (in prob_space) AE_distribution:
- assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := ereal \<circ> distribution X\<rparr>. Q x"
- shows "AE x. Q (X x)"
-proof -
- interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space)
- obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N"
- using assms unfolding X.almost_everywhere_def by auto
- from X[unfolded measurable_def] N show "AE x. Q (X x)"
- by (intro AE_I'[where N="X -` N \<inter> space M"])
- (auto simp: finite_measure_eq distribution_def measurable_sets)
-qed
-
-lemma (in prob_space) distribution_eq_integral:
- "random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))"
- using finite_measure_eq[of "X -` A \<inter> space M"]
- by (auto simp: measurable_sets distribution_def)
-
lemma (in prob_space) expectation_less:
assumes [simp]: "integrable M X"
assumes gt: "\<forall>x\<in>space M. X x < b"
shows "expectation X < b"
proof -
have "expectation X < expectation (\<lambda>x. b)"
- using gt measure_space_1
+ using gt emeasure_space_1
by (intro integral_less_AE_space) auto
then show ?thesis using prob_space by simp
qed
@@ -299,80 +331,11 @@
shows "a < expectation X"
proof -
have "expectation (\<lambda>x. a) < expectation X"
- using gt measure_space_1
+ using gt emeasure_space_1
by (intro integral_less_AE_space) auto
then show ?thesis using prob_space by simp
qed
-lemma convex_le_Inf_differential:
- fixes f :: "real \<Rightarrow> real"
- assumes "convex_on I f"
- assumes "x \<in> interior I" "y \<in> I"
- shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
- (is "_ \<ge> _ + Inf (?F x) * (y - x)")
-proof -
- show ?thesis
- proof (cases rule: linorder_cases)
- assume "x < y"
- moreover
- have "open (interior I)" by auto
- from openE[OF this `x \<in> interior I`] guess e . note e = this
- moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
- ultimately have "x < t" "t < y" "t \<in> ball x e"
- by (auto simp: mem_ball dist_real_def field_simps split: split_min)
- with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
-
- have "open (interior I)" by auto
- from openE[OF this `x \<in> interior I`] guess e .
- moreover def K \<equiv> "x - e / 2"
- with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: mem_ball dist_real_def)
- ultimately have "K \<in> I" "K < x" "x \<in> I"
- using interior_subset[of I] `x \<in> interior I` by auto
-
- have "Inf (?F x) \<le> (f x - f y) / (x - y)"
- proof (rule Inf_lower2)
- show "(f x - f t) / (x - t) \<in> ?F x"
- using `t \<in> I` `x < t` by auto
- show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
- using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
- next
- fix y assume "y \<in> ?F x"
- with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
- show "(f K - f x) / (K - x) \<le> y" by auto
- qed
- then show ?thesis
- using `x < y` by (simp add: field_simps)
- next
- assume "y < x"
- moreover
- have "open (interior I)" by auto
- from openE[OF this `x \<in> interior I`] guess e . note e = this
- moreover def t \<equiv> "x + e / 2"
- ultimately have "x < t" "t \<in> ball x e"
- by (auto simp: mem_ball dist_real_def field_simps)
- with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
-
- have "(f x - f y) / (x - y) \<le> Inf (?F x)"
- proof (rule Inf_greatest)
- have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
- using `y < x` by (auto simp: field_simps)
- also
- fix z assume "z \<in> ?F x"
- with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
- have "(f y - f x) / (y - x) \<le> z" by auto
- finally show "(f x - f y) / (x - y) \<le> z" .
- next
- have "open (interior I)" by auto
- from openE[OF this `x \<in> interior I`] guess e . note e = this
- then have "x + e / 2 \<in> ball x e" by (auto simp: mem_ball dist_real_def)
- with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
- then show "?F x \<noteq> {}" by blast
- qed
- then show ?thesis
- using `y < x` by (simp add: field_simps)
- qed simp
-qed
-
lemma (in prob_space) jensens_inequality:
fixes a b :: real
assumes X: "integrable M X" "X ` space M \<subseteq> I"
@@ -410,8 +373,7 @@
fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
then guess x .. note x = this
have "q x + ?F x * (expectation X - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
- using prob_space
- by (simp add: integral_add integral_cmult integral_diff lebesgue_integral_const X)
+ using prob_space by (simp add: X)
also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
using `x \<in> I` `open I` X(2)
by (intro integral_mono integral_add integral_cmult integral_diff
@@ -422,31 +384,6 @@
finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
qed
-lemma (in prob_space) distribution_eq_translated_integral:
- assumes "random_variable S X" "A \<in> sets S"
- shows "distribution X A = integral\<^isup>P (S\<lparr>measure := ereal \<circ> distribution X\<rparr>) (indicator A)"
-proof -
- interpret S: prob_space "S\<lparr>measure := ereal \<circ> distribution X\<rparr>"
- using assms(1) by (rule distribution_prob_space)
- show ?thesis
- using S.positive_integral_indicator(1)[of A] assms by simp
-qed
-
-lemma (in prob_space) finite_expectation1:
- assumes f: "finite (X`space M)" and rv: "random_variable borel X"
- shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r")
-proof (subst integral_on_finite)
- show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto
- show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r"
- "\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>"
- using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto
-qed
-
-lemma (in prob_space) finite_expectation:
- assumes "finite (X`space M)" "random_variable borel X"
- shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})"
- using assms unfolding distribution_def using finite_expectation1 by auto
-
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0:
assumes "{x} \<in> events"
assumes "prob {x} = 1"
@@ -455,119 +392,25 @@
shows "prob {y} = 0"
using prob_one_inter[of "{y}" "{x}"] assms by auto
-lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0"
- unfolding distribution_def by simp
-
-lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1"
-proof -
- have "X -` X ` space M \<inter> space M = space M" by auto
- thus ?thesis unfolding distribution_def by (simp add: prob_space)
-qed
-
-lemma (in prob_space) distribution_one:
- assumes "random_variable M' X" and "A \<in> sets M'"
- shows "distribution X A \<le> 1"
-proof -
- have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def
- using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono)
- thus ?thesis by (simp add: prob_space)
-qed
-
-lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0:
- assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X"
- (is "random_variable ?S X")
- assumes "distribution X {x} = 1"
- assumes "y \<noteq> x"
- shows "distribution X {y} = 0"
-proof cases
- { fix x have "X -` {x} \<inter> space M \<in> sets M"
- proof cases
- assume "x \<in> X`space M" with X show ?thesis
- by (auto simp: measurable_def image_iff)
- next
- assume "x \<notin> X`space M" then have "X -` {x} \<inter> space M = {}" by auto
- then show ?thesis by auto
- qed } note single = this
- have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> space M"
- "X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> space M) = {}"
- using `y \<noteq> x` by auto
- with finite_measure_inter_full_set[OF single single, of x y] assms(2)
- show ?thesis by (auto simp: distribution_def prob_space)
-next
- assume "{y} \<notin> sets ?S"
- then have "X -` {y} \<inter> space M = {}" by auto
- thus "distribution X {y} = 0" unfolding distribution_def by auto
-qed
-
lemma (in prob_space) joint_distribution_Times_le_fst:
- assumes X: "random_variable MX X" and Y: "random_variable MY Y"
- and A: "A \<in> sets MX" and B: "B \<in> sets MY"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
- unfolding distribution_def
-proof (intro finite_measure_mono)
- show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
- show "X -` A \<inter> space M \<in> events"
- using X A unfolding measurable_def by simp
- have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
- (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-qed
-
-lemma (in prob_space) joint_distribution_commute:
- "joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+ "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
+ \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MX X) A"
+ by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
lemma (in prob_space) joint_distribution_Times_le_snd:
- assumes X: "random_variable MX X" and Y: "random_variable MY Y"
- and A: "A \<in> sets MX" and B: "B \<in> sets MY"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
- using assms
- by (subst joint_distribution_commute)
- (simp add: swap_product joint_distribution_Times_le_fst)
-
-lemma (in prob_space) random_variable_pairI:
- assumes "random_variable MX X"
- assumes "random_variable MY Y"
- shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
-proof
- interpret MX: sigma_algebra MX using assms by simp
- interpret MY: sigma_algebra MY using assms by simp
- interpret P: pair_sigma_algebra MX MY by default
- show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
- have sa: "sigma_algebra M" by default
- show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
- unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
-qed
-
-lemma (in prob_space) joint_distribution_commute_singleton:
- "joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
-
-lemma (in prob_space) joint_distribution_assoc_singleton:
- "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} =
- joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}"
- unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
+ "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
+ \<Longrightarrow> emeasure (distr M (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))) (A \<times> B) \<le> emeasure (distr M MY Y) B"
+ by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
-sublocale pair_prob_space \<subseteq> P: prob_space P
+sublocale pair_prob_space \<subseteq> P: prob_space "M1 \<Otimes>\<^isub>M M2"
proof
- show "measure_space P" ..
- show "measure P (space P) = 1"
- by (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure)
+ show "emeasure (M1 \<Otimes>\<^isub>M M2) (space (M1 \<Otimes>\<^isub>M M2)) = 1"
+ by (simp add: emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
qed
-lemma countably_additiveI[case_names countably]:
- assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow>
- (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
- shows "countably_additive M \<mu>"
- using assms unfolding countably_additive_def by auto
-
-lemma (in prob_space) joint_distribution_prob_space:
- assumes "random_variable MX X" "random_variable MY Y"
- shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)"
- using random_variable_pairI[OF assms] by (rule distribution_prob_space)
-
-locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +
+locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
fixes I :: "'i set"
assumes prob_space: "\<And>i. prob_space (M i)"
@@ -578,648 +421,401 @@
sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
proof
- show "measure_space P" ..
- show "measure P (space P) = 1"
- by (simp add: measure_times M.measure_space_1 setprod_1)
+ show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (space (\<Pi>\<^isub>M i\<in>I. M i)) = 1"
+ by (simp add: measure_times M.emeasure_space_1 setprod_1 space_PiM)
qed
lemma (in finite_product_prob_space) prob_times:
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
proof -
- have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
- using X by (intro finite_measure_eq[symmetric] in_P) auto
- also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
+ have "ereal (measure (\<Pi>\<^isub>M i\<in>I. M i) (\<Pi>\<^isub>E i\<in>I. X i)) = emeasure (\<Pi>\<^isub>M i\<in>I. M i) (\<Pi>\<^isub>E i\<in>I. X i)"
+ using X by (simp add: emeasure_eq_measure)
+ also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
using measure_times X by simp
- also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
- using X by (simp add: M.finite_measure_eq setprod_ereal)
- finally show ?thesis by simp
-qed
-
-lemma (in prob_space) random_variable_restrict:
- assumes I: "finite I"
- assumes X: "\<And>i. i \<in> I \<Longrightarrow> random_variable (N i) (X i)"
- shows "random_variable (Pi\<^isub>M I N) (\<lambda>x. \<lambda>i\<in>I. X i x)"
-proof
- { fix i assume "i \<in> I"
- with X interpret N: sigma_algebra "N i" by simp
- have "sets (N i) \<subseteq> Pow (space (N i))" by (rule N.space_closed) }
- note N_closed = this
- then show "sigma_algebra (Pi\<^isub>M I N)"
- by (simp add: product_algebra_def)
- (intro sigma_algebra_sigma product_algebra_generator_sets_into_space)
- show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
- using X by (intro measurable_restrict[OF I N_closed]) auto
-qed
-
-section "Probability spaces on finite sets"
-
-locale finite_prob_space = prob_space + finite_measure_space
-
-abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'"
-
-lemma (in prob_space) finite_random_variableD:
- assumes "finite_random_variable M' X" shows "random_variable M' X"
-proof -
- interpret M': finite_sigma_algebra M' using assms by simp
- show "random_variable M' X" using assms by simp default
-qed
-
-lemma (in prob_space) distribution_finite_prob_space:
- assumes "finite_random_variable MX X"
- shows "finite_prob_space (MX\<lparr>measure := ereal \<circ> distribution X\<rparr>)"
-proof -
- interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>"
- using assms[THEN finite_random_variableD] by (rule distribution_prob_space)
- interpret MX: finite_sigma_algebra MX
- using assms by auto
- show ?thesis by default (simp_all add: MX.finite_space)
-qed
-
-lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]:
- assumes "simple_function M X"
- shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X"
- (is "finite_random_variable ?X _")
-proof (intro conjI)
- have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp
- interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow)
- show "finite_sigma_algebra ?X"
- by default auto
- show "X \<in> measurable M ?X"
- proof (unfold measurable_def, clarsimp)
- fix A assume A: "A \<subseteq> X`space M"
- then have "finite A" by (rule finite_subset) simp
- then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events"
- unfolding vimage_UN UN_extend_simps
- apply (rule finite_UN)
- using A assms unfolding simple_function_def by auto
- then show "X -` A \<inter> space M \<in> events" by simp
- qed
-qed
-
-lemma (in prob_space) simple_function_imp_random_variable[simp, intro]:
- assumes "simple_function M X"
- shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X"
- using simple_function_imp_finite_random_variable[OF assms, of ext]
- by (auto dest!: finite_random_variableD)
-
-lemma (in prob_space) sum_over_space_real_distribution:
- "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def prob_space[symmetric]
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def simple_function_def
- intro!: arg_cong[where f=prob])
-
-lemma (in prob_space) finite_random_variable_pairI:
- assumes "finite_random_variable MX X"
- assumes "finite_random_variable MY Y"
- shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
-proof
- interpret MX: finite_sigma_algebra MX using assms by simp
- interpret MY: finite_sigma_algebra MY using assms by simp
- interpret P: pair_finite_sigma_algebra MX MY by default
- show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" ..
- have sa: "sigma_algebra M" by default
- show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
- unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
-qed
-
-lemma (in prob_space) finite_random_variable_imp_sets:
- "finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX"
- unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
-
-lemma (in prob_space) finite_random_variable_measurable:
- assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events"
-proof -
- interpret X: finite_sigma_algebra MX using X by simp
- from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and
- "X \<in> space M \<rightarrow> space MX"
- by (auto simp: measurable_def)
- then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M"
- by auto
- show "X -` A \<inter> space M \<in> events"
- unfolding * by (intro vimage) auto
-qed
-
-lemma (in prob_space) joint_distribution_finite_Times_le_fst:
- assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
- unfolding distribution_def
-proof (intro finite_measure_mono)
- show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
- show "X -` A \<inter> space M \<in> events"
- using finite_random_variable_measurable[OF X] .
- have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
- (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
-qed
-
-lemma (in prob_space) joint_distribution_finite_Times_le_snd:
- assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
- shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
- using assms
- by (subst joint_distribution_commute)
- (simp add: swap_product joint_distribution_finite_Times_le_fst)
-
-lemma (in prob_space) finite_distribution_order:
- fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
- assumes "finite_random_variable MX X" "finite_random_variable MY Y"
- shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
- and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
- and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"]
- using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"]
- by (auto intro: antisym)
-
-lemma (in prob_space) setsum_joint_distribution:
- assumes X: "finite_random_variable MX X"
- assumes Y: "random_variable MY Y" "B \<in> sets MY"
- shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B"
- unfolding distribution_def
-proof (subst finite_measure_finite_Union[symmetric])
- interpret MX: finite_sigma_algebra MX using X by auto
- show "finite (space MX)" using MX.finite_space .
- let ?d = "\<lambda>i. (\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M"
- { fix i assume "i \<in> space MX"
- moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
- ultimately show "?d i \<in> events"
- using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y
- using MX.sets_eq_Pow by auto }
- show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def)
- show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)"
- using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>'])
-qed
-
-lemma (in prob_space) setsum_joint_distribution_singleton:
- assumes X: "finite_random_variable MX X"
- assumes Y: "finite_random_variable MY Y" "b \<in> space MY"
- shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}"
- using setsum_joint_distribution[OF X
- finite_random_variableD[OF Y(1)]
- finite_random_variable_imp_sets[OF Y]] by simp
-
-lemma (in prob_space) setsum_distribution:
- assumes X: "finite_random_variable MX X" shows "(\<Sum>a\<in>space MX. distribution X {a}) = 1"
- using setsum_joint_distribution[OF assms, of "\<lparr> space = UNIV, sets = Pow UNIV \<rparr>" "\<lambda>x. ()" "{()}"]
- using sigma_algebra_Pow[of "UNIV::unit set" "()"] by simp
-
-locale pair_finite_prob_space = pair_prob_space M1 M2 + pair_finite_space M1 M2 + M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2
-
-sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default
-
-lemma funset_eq_UN_fun_upd_I:
- assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
- and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
- and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
- shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
-proof safe
- fix f assume f: "f \<in> F (insert a A)"
- show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
- proof (rule UN_I[of "f(a := d)"])
- show "f(a := d) \<in> F A" using *[OF f] .
- show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
- proof (rule image_eqI[of _ _ "f a"])
- show "f a \<in> G (f(a := d))" using **[OF f] .
- qed simp
- qed
-next
- fix f x assume "f \<in> F A" "x \<in> G f"
- from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
-qed
-
-lemma extensional_funcset_insert_eq[simp]:
- assumes "a \<notin> A"
- shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)"
- apply (rule funset_eq_UN_fun_upd_I)
- using assms
- by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
-
-lemma finite_extensional_funcset[simp, intro]:
- assumes "finite A" "finite B"
- shows "finite (extensional A \<inter> (A \<rightarrow> B))"
- using assms by induct (auto simp: extensional_funcset_insert_eq)
-
-lemma finite_PiE[simp, intro]:
- assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
- shows "finite (Pi\<^isub>E A B)"
-proof -
- have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
- show ?thesis
- using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
-qed
-
-locale finite_product_finite_prob_space = finite_product_prob_space M I for M I +
- assumes finite_space: "\<And>i. finite_prob_space (M i)"
-
-sublocale finite_product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space .
-
-lemma (in finite_product_finite_prob_space) singleton_eq_product:
- assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})"
-proof (safe intro!: ext[of _ x])
- fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I"
- with x show "y i = x i"
- by (cases "i \<in> I") (auto simp: extensional_def)
-qed (insert x, auto)
-
-sublocale finite_product_finite_prob_space \<subseteq> finite_prob_space "Pi\<^isub>M I M"
-proof
- show "finite (space P)"
- using finite_index M.finite_space by auto
-
- { fix x assume "x \<in> space P"
- with this[THEN singleton_eq_product]
- have "{x} \<in> sets P"
- by (auto intro!: in_P) }
- note x_in_P = this
-
- have "Pow (space P) \<subseteq> sets P"
- proof
- fix X assume "X \<in> Pow (space P)"
- moreover then have "finite X"
- using `finite (space P)` by (blast intro: finite_subset)
- ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P"
- by (intro finite_UN x_in_P) auto
- then show "X \<in> sets P" by simp
- qed
- with space_closed show [simp]: "sets P = Pow (space P)" ..
-qed
-
-lemma (in finite_product_finite_prob_space) measure_finite_times:
- "(\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)) \<Longrightarrow> \<mu> (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.\<mu> i (X i))"
- by (rule measure_times) simp
-
-lemma (in finite_product_finite_prob_space) measure_singleton_times:
- assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})"
- unfolding singleton_eq_product[OF x] using x
- by (intro measure_finite_times) auto
-
-lemma (in finite_product_finite_prob_space) prob_finite_times:
- assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)"
- shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
-proof -
- have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
- using X by (intro finite_measure_eq[symmetric] in_P) auto
- also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
- using measure_finite_times X by simp
- also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
- using X by (simp add: M.finite_measure_eq setprod_ereal)
+ also have "\<dots> = ereal (\<Prod>i\<in>I. measure (M i) (X i))"
+ using X by (simp add: M.emeasure_eq_measure setprod_ereal)
finally show ?thesis by simp
qed
-lemma (in finite_product_finite_prob_space) prob_singleton_times:
- assumes x: "x \<in> space P"
- shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})"
- unfolding singleton_eq_product[OF x] using x
- by (intro prob_finite_times) auto
-
-lemma (in finite_product_finite_prob_space) prob_finite_product:
- "A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})"
- by (auto simp add: finite_measure_singleton prob_singleton_times
- simp del: space_product_algebra
- intro!: setsum_cong prob_singleton_times)
+section {* Distributions *}
-lemma (in prob_space) joint_distribution_finite_prob_space:
- assumes X: "finite_random_variable MX X"
- assumes Y: "finite_random_variable MY Y"
- shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)"
- by (intro distribution_finite_prob_space finite_random_variable_pairI X Y)
-
-lemma finite_prob_space_eq:
- "finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1"
- unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def
- by auto
-
-lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1"
- using measure_space_1 sum_over_space by simp
+definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and>
+ f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
-lemma (in finite_prob_space) joint_distribution_restriction_fst:
- "joint_distribution X Y A \<le> distribution X (fst ` A)"
- unfolding distribution_def
-proof (safe intro!: finite_measure_mono)
- fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
- show "x \<in> X -` fst ` A"
- by (auto intro!: image_eqI[OF _ *])
-qed (simp_all add: sets_eq_Pow)
-
-lemma (in finite_prob_space) joint_distribution_restriction_snd:
- "joint_distribution X Y A \<le> distribution Y (snd ` A)"
- unfolding distribution_def
-proof (safe intro!: finite_measure_mono)
- fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
- show "x \<in> Y -` snd ` A"
- by (auto intro!: image_eqI[OF _ *])
-qed (simp_all add: sets_eq_Pow)
-
-lemma (in finite_prob_space) distribution_order:
- shows "0 \<le> distribution X x'"
- and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')"
- and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
- and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
- and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
- and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
- using
- joint_distribution_restriction_fst[of X Y "{(x, y)}"]
- joint_distribution_restriction_snd[of X Y "{(x, y)}"]
- by (auto intro: antisym)
-
-lemma (in finite_prob_space) distribution_mono:
- assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
- shows "distribution X x \<le> distribution Y y"
- unfolding distribution_def
- using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono)
+lemma
+ shows distributed_distr_eq_density: "distributed M N X f \<Longrightarrow> distr M N X = density N f"
+ and distributed_measurable: "distributed M N X f \<Longrightarrow> X \<in> measurable M N"
+ and distributed_borel_measurable: "distributed M N X f \<Longrightarrow> f \<in> borel_measurable N"
+ and distributed_AE: "distributed M N X f \<Longrightarrow> (AE x in N. 0 \<le> f x)"
+ by (simp_all add: distributed_def)
-lemma (in finite_prob_space) distribution_mono_gt_0:
- assumes gt_0: "0 < distribution X x"
- assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
- shows "0 < distribution Y y"
- by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
-
-lemma (in finite_prob_space) sum_over_space_distrib:
- "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
- unfolding distribution_def prob_space[symmetric] using finite_space
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow
- intro!: arg_cong[where f=\<mu>'])
-
-lemma (in finite_prob_space) finite_sum_over_space_eq_1:
- "(\<Sum>x\<in>space M. prob {x}) = 1"
- using prob_space finite_space
- by (subst (asm) finite_measure_finite_singleton) auto
-
-lemma (in prob_space) distribution_remove_const:
- shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
- and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
- and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
- and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
- and "distribution (\<lambda>x. ()) {()} = 1"
- by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric])
+lemma
+ shows distributed_real_measurable: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
+ and distributed_real_AE: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> (AE x in N. 0 \<le> f x)"
+ by (simp_all add: distributed_def borel_measurable_ereal_iff)
-lemma (in finite_prob_space) setsum_distribution_gen:
- assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
- and "inj_on f (X`space M)"
- shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
- unfolding distribution_def assms
- using finite_space assms
- by (subst finite_measure_finite_Union[symmetric])
- (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
- intro!: arg_cong[where f=prob])
-
-lemma (in finite_prob_space) setsum_distribution_cut:
- "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
- "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
- "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
- "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
- "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
- by (auto intro!: inj_onI setsum_distribution_gen)
-
-lemma (in finite_prob_space) uniform_prob:
- assumes "x \<in> space M"
- assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
- shows "prob {x} = 1 / card (space M)"
+lemma distributed_count_space:
+ assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
+ shows "P a = emeasure M (X -` {a} \<inter> space M)"
proof -
- have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
- using assms(2)[OF _ `x \<in> space M`] by blast
- have "1 = prob (space M)"
- using prob_space by auto
- also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
- using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
- sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
- finite_space unfolding disjoint_family_on_def prob_space[symmetric]
- by (auto simp add:setsum_restrict_set)
- also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
- using prob_x by auto
- also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
- finally have one: "1 = real (card (space M)) * prob {x}"
- using real_eq_of_nat by auto
- hence two: "real (card (space M)) \<noteq> 0" by fastforce
- from one have three: "prob {x} \<noteq> 0" by fastforce
- thus ?thesis using one two three divide_cancel_right
- by (auto simp:field_simps)
+ have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
+ using X a A by (simp add: distributed_measurable emeasure_distr)
+ also have "\<dots> = emeasure (density (count_space A) P) {a}"
+ using X by (simp add: distributed_distr_eq_density)
+ also have "\<dots> = (\<integral>\<^isup>+x. P a * indicator {a} x \<partial>count_space A)"
+ using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: positive_integral_cong)
+ also have "\<dots> = P a"
+ using X a by (subst positive_integral_cmult_indicator) (auto simp: distributed_def one_ereal_def[symmetric] AE_count_space)
+ finally show ?thesis ..
qed
-lemma (in prob_space) prob_space_subalgebra:
- assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M"
- and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
- shows "prob_space N"
-proof
- interpret N: measure_space N
- by (rule measure_space_subalgebra[OF assms])
- show "measure_space N" ..
- show "measure N (space N) = 1"
- using assms(4)[OF N.top] by (simp add: assms measure_space_1)
+lemma distributed_cong_density:
+ "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
+ distributed M N X f \<longleftrightarrow> distributed M N X g"
+ by (auto simp: distributed_def intro!: density_cong)
+
+lemma subdensity:
+ assumes T: "T \<in> measurable P Q"
+ assumes f: "distributed M P X f"
+ assumes g: "distributed M Q Y g"
+ assumes Y: "Y = T \<circ> X"
+ shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
+proof -
+ have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
+ using g Y by (auto simp: null_sets_density_iff distributed_def)
+ also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
+ using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
+ finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
+ using T by (subst (asm) null_sets_distr_iff) auto
+ also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
+ using T by (auto dest: measurable_space)
+ finally show ?thesis
+ using f g by (auto simp add: null_sets_density_iff distributed_def)
qed
-lemma (in prob_space) prob_space_of_restricted_space:
- assumes "\<mu> A \<noteq> 0" "A \<in> sets M"
- shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)"
- (is "prob_space ?P")
+lemma subdensity_real:
+ fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
+ assumes T: "T \<in> measurable P Q"
+ assumes f: "distributed M P X f"
+ assumes g: "distributed M Q Y g"
+ assumes Y: "Y = T \<circ> X"
+ shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
+ using subdensity[OF T, of M X "\<lambda>x. ereal (f x)" Y "\<lambda>x. ereal (g x)"] assms by auto
+
+lemma distributed_integral:
+ "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>x. f x * g x \<partial>N) = (\<integral>x. g (X x) \<partial>M)"
+ by (auto simp: distributed_real_measurable distributed_real_AE distributed_measurable
+ distributed_distr_eq_density[symmetric] integral_density[symmetric] integral_distr)
+
+lemma distributed_transform_integral:
+ assumes Px: "distributed M N X Px"
+ assumes "distributed M P Y Py"
+ assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
+ shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
proof -
- interpret A: measure_space "restricted_space A"
- using `A \<in> sets M` by (rule restricted_measure_space)
- interpret A': sigma_algebra ?P
- by (rule A.sigma_algebra_cong) auto
- show "prob_space ?P"
- proof
- show "measure_space ?P"
- proof
- show "positive ?P (measure ?P)"
- proof (simp add: positive_def, safe)
- fix B assume "B \<in> events"
- with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M`
- show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int)
- qed
- show "countably_additive ?P (measure ?P)"
- proof (simp add: countably_additive_def, safe)
- fix B and F :: "nat \<Rightarrow> 'a set"
- assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F"
- { fix i
- from F have "F i \<in> op \<inter> A ` events" by auto
- with `A \<in> events` have "F i \<in> events" by auto }
- moreover then have "range F \<subseteq> events" by auto
- moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S"
- by (simp add: mult_commute divide_ereal_def)
- moreover have "0 \<le> inverse (\<mu> A)"
- using real_measure[OF `A \<in> events`] by auto
- ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A"
- using measure_countably_additive[of F] F
- by (auto simp: suminf_cmult_ereal)
- qed
- qed
- show "measure ?P (space ?P) = 1"
- using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto
- qed
-qed
-
-lemma finite_prob_spaceI:
- assumes "finite (space M)" "sets M = Pow(space M)"
- and 1: "measure M (space M) = 1" and "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> measure M {x}"
- and add: "\<And>A B. A \<subseteq> space M \<Longrightarrow> measure M A = (\<Sum>x\<in>A. measure M {x})"
- shows "finite_prob_space M"
-proof -
- interpret finite_measure_space M
- proof
- show "measure M (space M) \<noteq> \<infinity>" using 1 by simp
- qed fact+
- show ?thesis by default fact
-qed
-
-lemma (in finite_prob_space) distribution_eq_setsum:
- "distribution X A = (\<Sum>x\<in>A \<inter> X ` space M. distribution X {x})"
-proof -
- have *: "X -` A \<inter> space M = (\<Union>x\<in>A \<inter> X ` space M. X -` {x} \<inter> space M)"
- by auto
- then show "distribution X A = (\<Sum>x\<in>A \<inter> X ` space M. distribution X {x})"
- using finite_space unfolding distribution_def *
- by (intro finite_measure_finite_Union)
- (auto simp: disjoint_family_on_def)
-qed
-
-lemma (in finite_prob_space) distribution_eq_setsum_finite:
- assumes "finite A"
- shows "distribution X A = (\<Sum>x\<in>A. distribution X {x})"
-proof -
- note distribution_eq_setsum[of X A]
- also have "(\<Sum>x\<in>A \<inter> X ` space M. distribution X {x}) = (\<Sum>x\<in>A. distribution X {x})"
- proof (intro setsum_mono_zero_cong_left ballI)
- fix i assume "i\<in>A - A \<inter> X ` space M"
- then have "X -` {i} \<inter> space M = {}" by auto
- then show "distribution X {i} = 0"
- by (simp add: distribution_def)
- next
- show "finite A" by fact
- qed simp_all
+ have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
+ by (rule distributed_integral) fact+
+ also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
+ using Y by simp
+ also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
+ using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
finally show ?thesis .
qed
-lemma (in finite_prob_space) finite_measure_space:
- fixes X :: "'a \<Rightarrow> 'x"
- shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X\<rparr>"
- (is "finite_measure_space ?S")
-proof (rule finite_measure_spaceI, simp_all)
- show "finite (X ` space M)" using finite_space by simp
-next
- fix A assume "A \<subseteq> X ` space M"
- then show "distribution X A = (\<Sum>x\<in>A. distribution X {x})"
- by (subst distribution_eq_setsum) (simp add: Int_absorb2)
+lemma distributed_marginal_eq_joint:
+ assumes T: "sigma_finite_measure T"
+ assumes S: "sigma_finite_measure S"
+ assumes Px: "distributed M S X Px"
+ assumes Py: "distributed M T Y Py"
+ assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+ shows "AE y in T. Py y = (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S)"
+proof (rule sigma_finite_measure.density_unique[OF T])
+ interpret ST: pair_sigma_finite S T using S T unfolding pair_sigma_finite_def by simp
+ show "Py \<in> borel_measurable T" "AE y in T. 0 \<le> Py y"
+ "(\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S) \<in> borel_measurable T" "AE y in T. 0 \<le> \<integral>\<^isup>+ x. Pxy (x, y) \<partial>S"
+ using Pxy[THEN distributed_borel_measurable]
+ by (auto intro!: Py[THEN distributed_borel_measurable] Py[THEN distributed_AE]
+ ST.positive_integral_snd_measurable' positive_integral_positive)
+
+ show "density T Py = density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)"
+ proof (rule measure_eqI)
+ fix A assume A: "A \<in> sets (density T Py)"
+ have *: "\<And>x y. x \<in> space S \<Longrightarrow> indicator (space S \<times> A) (x, y) = indicator A y"
+ by (auto simp: indicator_def)
+ have "emeasure (density T Py) A = emeasure (distr M T Y) A"
+ unfolding Py[THEN distributed_distr_eq_density] ..
+ also have "\<dots> = emeasure (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (space S \<times> A)"
+ using A Px Py Pxy
+ by (subst (1 2) emeasure_distr)
+ (auto dest: measurable_space distributed_measurable intro!: arg_cong[where f="emeasure M"])
+ also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (space S \<times> A)"
+ unfolding Pxy[THEN distributed_distr_eq_density] ..
+ also have "\<dots> = (\<integral>\<^isup>+ x. Pxy x * indicator (space S \<times> A) x \<partial>(S \<Otimes>\<^isub>M T))"
+ using A Pxy by (simp add: emeasure_density distributed_borel_measurable)
+ also have "\<dots> = (\<integral>\<^isup>+y. \<integral>\<^isup>+x. Pxy (x, y) * indicator (space S \<times> A) (x, y) \<partial>S \<partial>T)"
+ using A Pxy
+ by (subst ST.positive_integral_snd_measurable) (simp_all add: emeasure_density distributed_borel_measurable)
+ also have "\<dots> = (\<integral>\<^isup>+y. (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S) * indicator A y \<partial>T)"
+ using measurable_comp[OF measurable_Pair1[OF measurable_identity] distributed_borel_measurable[OF Pxy]]
+ using distributed_borel_measurable[OF Pxy] distributed_AE[OF Pxy, THEN ST.AE_pair]
+ by (subst (asm) ST.AE_commute) (auto intro!: positive_integral_cong_AE positive_integral_multc cong: positive_integral_cong simp: * comp_def)
+ also have "\<dots> = emeasure (density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)) A"
+ using A by (intro emeasure_density[symmetric]) (auto intro!: ST.positive_integral_snd_measurable' Pxy[THEN distributed_borel_measurable])
+ finally show "emeasure (density T Py) A = emeasure (density T (\<lambda>x. \<integral>\<^isup>+ xa. Pxy (xa, x) \<partial>S)) A" .
+ qed simp
qed
-lemma (in finite_prob_space) finite_prob_space_of_images:
- "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X \<rparr>"
- by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_ereal_def)
+lemma (in prob_space) distr_marginal1:
+ fixes Pxy :: "('b \<times> 'c) \<Rightarrow> real"
+ assumes "sigma_finite_measure S" "sigma_finite_measure T"
+ assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+ defines "Px \<equiv> \<lambda>x. real (\<integral>\<^isup>+z. Pxy (x, z) \<partial>T)"
+ shows "distributed M S X Px"
+ unfolding distributed_def
+proof safe
+ interpret S: sigma_finite_measure S by fact
+ interpret T: sigma_finite_measure T by fact
+ interpret ST: pair_sigma_finite S T by default
+
+ have XY: "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
+ using Pxy by (rule distributed_measurable)
+ then show X: "X \<in> measurable M S"
+ unfolding measurable_pair_iff by (simp add: comp_def)
+ from XY have Y: "Y \<in> measurable M T"
+ unfolding measurable_pair_iff by (simp add: comp_def)
+
+ from Pxy show borel: "(\<lambda>x. ereal (Px x)) \<in> borel_measurable S"
+ by (auto intro!: ST.positive_integral_fst_measurable borel_measurable_real_of_ereal dest!: distributed_real_measurable simp: Px_def)
-lemma (in finite_prob_space) finite_product_measure_space:
- fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
- assumes "finite s1" "finite s2"
- shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = ereal \<circ> joint_distribution X Y\<rparr>"
- (is "finite_measure_space ?M")
-proof (rule finite_measure_spaceI, simp_all)
- show "finite (s1 \<times> s2)"
- using assms by auto
-next
- fix A assume "A \<subseteq> (s1 \<times> s2)"
- with assms show "joint_distribution X Y A = (\<Sum>x\<in>A. joint_distribution X Y {x})"
- by (intro distribution_eq_setsum_finite) (auto dest: finite_subset)
-qed
-
-lemma (in finite_prob_space) finite_product_measure_space_of_images:
- shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M,
- sets = Pow (X ` space M \<times> Y ` space M),
- measure = ereal \<circ> joint_distribution X Y \<rparr>"
- using finite_space by (auto intro!: finite_product_measure_space)
-
-lemma (in finite_prob_space) finite_product_prob_space_of_images:
- "finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M),
- measure = ereal \<circ> joint_distribution X Y \<rparr>"
- (is "finite_prob_space ?S")
-proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_ereal_def)
- have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto
- thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1"
- by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
+ interpret Pxy: prob_space "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
+ using XY by (rule prob_space_distr)
+ have "(\<integral>\<^isup>+ x. max 0 (ereal (- Pxy x)) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>\<^isup>+ x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
+ using Pxy
+ by (intro positive_integral_cong_AE) (auto simp: max_def dest: distributed_real_measurable distributed_real_AE)
+ then have Pxy_integrable: "integrable (S \<Otimes>\<^isub>M T) Pxy"
+ using Pxy Pxy.emeasure_space_1
+ by (simp add: integrable_def emeasure_density positive_integral_max_0 distributed_def borel_measurable_ereal_iff cong: positive_integral_cong)
+
+ show "distr M S X = density S Px"
+ proof (rule measure_eqI)
+ fix A assume A: "A \<in> sets (distr M S X)"
+ with X Y XY have "emeasure (distr M S X) A = emeasure (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (A \<times> space T)"
+ by (auto simp add: emeasure_distr
+ intro!: arg_cong[where f="emeasure M"] dest: measurable_space)
+ also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (A \<times> space T)"
+ using Pxy by (simp add: distributed_def)
+ also have "\<dots> = \<integral>\<^isup>+ x. \<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
+ using A borel Pxy
+ by (simp add: emeasure_density ST.positive_integral_fst_measurable(2)[symmetric] distributed_def)
+ also have "\<dots> = \<integral>\<^isup>+ x. ereal (Px x) * indicator A x \<partial>S"
+ apply (rule positive_integral_cong_AE)
+ using Pxy[THEN distributed_real_AE, THEN ST.AE_pair] ST.integrable_fst_measurable(1)[OF Pxy_integrable] AE_space
+ proof eventually_elim
+ fix x assume "x \<in> space S" "AE y in T. 0 \<le> Pxy (x, y)" and i: "integrable T (\<lambda>y. Pxy (x, y))"
+ moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
+ by (auto simp: indicator_def)
+ ultimately have "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T) =
+ (\<integral>\<^isup>+ y. ereal (Pxy (x, y)) \<partial>T) * indicator A x"
+ using Pxy[THEN distributed_real_measurable] by (simp add: eq positive_integral_multc measurable_Pair2 cong: positive_integral_cong)
+ also have "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) \<partial>T) = Px x"
+ using i by (simp add: Px_def ereal_real integrable_def positive_integral_positive)
+ finally show "(\<integral>\<^isup>+ y. ereal (Pxy (x, y)) * indicator (A \<times> space T) (x, y) \<partial>T) = ereal (Px x) * indicator A x" .
+ qed
+ finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
+ using A borel Pxy by (simp add: emeasure_density)
+ qed simp
+
+ show "AE x in S. 0 \<le> ereal (Px x)"
+ by (simp add: Px_def positive_integral_positive real_of_ereal_pos)
qed
-subsection "Borel Measure on {0 ..< 1}"
-
-definition pborel :: "real measure_space" where
- "pborel = lborel.restricted_space {0 ..< 1}"
-
-lemma space_pborel[simp]:
- "space pborel = {0 ..< 1}"
- unfolding pborel_def by auto
-
-lemma sets_pborel:
- "A \<in> sets pborel \<longleftrightarrow> A \<in> sets borel \<and> A \<subseteq> { 0 ..< 1}"
- unfolding pborel_def by auto
+definition
+ "simple_distributed M X f \<longleftrightarrow> distributed M (count_space (X`space M)) X (\<lambda>x. ereal (f x)) \<and>
+ finite (X`space M)"
-lemma in_pborel[intro, simp]:
- "A \<subseteq> {0 ..< 1} \<Longrightarrow> A \<in> sets borel \<Longrightarrow> A \<in> sets pborel"
- unfolding pborel_def by auto
-
-interpretation pborel: measure_space pborel
- using lborel.restricted_measure_space[of "{0 ..< 1}"]
- by (simp add: pborel_def)
+lemma simple_distributed:
+ "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
+ unfolding simple_distributed_def by auto
-interpretation pborel: prob_space pborel
-proof
- show "measure pborel (space pborel) = 1"
- by (simp add: one_ereal_def pborel_def)
-qed default
-
-lemma pborel_prob: "pborel.prob A = (if A \<in> sets borel \<and> A \<subseteq> {0 ..< 1} then real (lborel.\<mu> A) else 0)"
- unfolding pborel.\<mu>'_def by (auto simp: pborel_def)
+lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
+ by (simp add: simple_distributed_def)
-lemma pborel_singelton[simp]: "pborel.prob {a} = 0"
- by (auto simp: pborel_prob)
-
-lemma
- shows pborel_atLeastAtMost[simp]: "pborel.\<mu>' {a .. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)"
- and pborel_atLeastLessThan[simp]: "pborel.\<mu>' {a ..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
- and pborel_greaterThanAtMost[simp]: "pborel.\<mu>' {a <.. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)"
- and pborel_greaterThanLessThan[simp]: "pborel.\<mu>' {a <..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
- unfolding pborel_prob
- by (auto simp: atLeastAtMost_subseteq_atLeastLessThan_iff
- greaterThanAtMost_subseteq_atLeastLessThan_iff greaterThanLessThan_subseteq_atLeastLessThan_iff)
-
-lemma pborel_lebesgue_measure:
- "A \<in> sets pborel \<Longrightarrow> pborel.prob A = real (measure lebesgue A)"
- by (simp add: sets_pborel pborel_prob)
+lemma (in prob_space) distributed_simple_function_superset:
+ assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
+ assumes A: "X`space M \<subseteq> A" "finite A"
+ defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
+ shows "distributed M S X P'"
+ unfolding distributed_def
+proof safe
+ show "(\<lambda>x. ereal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
+ show "AE x in S. 0 \<le> ereal (P' x)"
+ using X by (auto simp: S_def P'_def simple_distributed_def intro!: measure_nonneg)
+ show "distr M S X = density S P'"
+ proof (rule measure_eqI_finite)
+ show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
+ using A unfolding S_def by auto
+ show "finite A" by fact
+ fix a assume a: "a \<in> A"
+ then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
+ with A a X have "emeasure (distr M S X) {a} = P' a"
+ by (subst emeasure_distr)
+ (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure
+ intro!: arg_cong[where f=prob])
+ also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' a) * indicator {a} x \<partial>S)"
+ using A X a
+ by (subst positive_integral_cmult_indicator)
+ (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
+ also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' x) * indicator {a} x \<partial>S)"
+ by (auto simp: indicator_def intro!: positive_integral_cong)
+ also have "\<dots> = emeasure (density S P') {a}"
+ using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
+ finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
+ qed
+ show "random_variable S X"
+ using X(1) A by (auto simp: measurable_def simple_functionD S_def)
+qed
-lemma pborel_alt:
- "pborel = sigma \<lparr>
- space = {0..<1},
- sets = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1}),
- measure = measure lborel \<rparr>" (is "_ = ?R")
+lemma (in prob_space) simple_distributedI:
+ assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
+ shows "simple_distributed M X P"
+ unfolding simple_distributed_def
+proof
+ have "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then P x else 0))"
+ (is "?A")
+ using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X]) auto
+ also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (P x))"
+ by (rule distributed_cong_density) auto
+ finally show "\<dots>" .
+qed (rule simple_functionD[OF X(1)])
+
+lemma simple_distributed_joint_finite:
+ assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
+ shows "finite (X ` space M)" "finite (Y ` space M)"
proof -
- have *: "{0..<1::real} \<in> sets borel" by auto
- have **: "op \<inter> {0..<1::real} ` range (\<lambda>(x, y). {x..<y}) = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1})"
- unfolding image_image by (intro arg_cong[where f=range]) auto
- have "pborel = algebra.restricted_space (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a ..< b :: real}),
- measure = measure pborel\<rparr>) {0 ..< 1}"
- by (simp add: sigma_def lebesgue_def pborel_def borel_eq_atLeastLessThan lborel_def)
- also have "\<dots> = ?R"
- by (subst restricted_sigma)
- (simp_all add: sets_sigma sigma_sets.Basic ** pborel_def image_eqI[of _ _ "(0,1)"])
- finally show ?thesis .
+ have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
+ using X by (auto simp: simple_distributed_def simple_functionD)
+ then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
+ by auto
+ then show fin: "finite (X ` space M)" "finite (Y ` space M)"
+ by (auto simp: image_image)
+qed
+
+lemma simple_distributed_joint2_finite:
+ assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
+ shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
+proof -
+ have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
+ using X by (auto simp: simple_distributed_def simple_functionD)
+ then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+ "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+ "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
+ by auto
+ then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
+ by (auto simp: image_image)
qed
-subsection "Bernoulli space"
+lemma simple_distributed_simple_function:
+ "simple_distributed M X Px \<Longrightarrow> simple_function M X"
+ unfolding simple_distributed_def distributed_def
+ by (auto simp: simple_function_def)
+
+lemma simple_distributed_measure:
+ "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
+ using distributed_count_space[of M "X`space M" X P a, symmetric]
+ by (auto simp: simple_distributed_def measure_def)
+
+lemma simple_distributed_nonneg: "simple_distributed M X f \<Longrightarrow> x \<in> space M \<Longrightarrow> 0 \<le> f (X x)"
+ by (auto simp: simple_distributed_measure measure_nonneg)
-definition "bernoulli_space p = \<lparr> space = UNIV, sets = UNIV,
- measure = ereal \<circ> setsum (\<lambda>b. if b then min 1 (max 0 p) else 1 - min 1 (max 0 p)) \<rparr>"
+lemma (in prob_space) simple_distributed_joint:
+ assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
+ defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)"
+ defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
+ shows "distributed M S (\<lambda>x. (X x, Y x)) P"
+proof -
+ from simple_distributed_joint_finite[OF X, simp]
+ have S_eq: "S = count_space (X`space M \<times> Y`space M)"
+ by (simp add: S_def pair_measure_count_space)
+ show ?thesis
+ unfolding S_eq P_def
+ proof (rule distributed_simple_function_superset)
+ show "simple_function M (\<lambda>x. (X x, Y x))"
+ using X by (rule simple_distributed_simple_function)
+ fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
+ from simple_distributed_measure[OF X this]
+ show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
+ qed auto
+qed
-interpretation bernoulli: finite_prob_space "bernoulli_space p" for p
- by (rule finite_prob_spaceI)
- (auto simp: bernoulli_space_def UNIV_bool one_ereal_def setsum_Un_disjoint intro!: setsum_nonneg)
+lemma (in prob_space) simple_distributed_joint2:
+ assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
+ defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M) \<Otimes>\<^isub>M count_space (Z`space M)"
+ defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
+ shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
+proof -
+ from simple_distributed_joint2_finite[OF X, simp]
+ have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
+ by (simp add: S_def pair_measure_count_space)
+ show ?thesis
+ unfolding S_eq P_def
+ proof (rule distributed_simple_function_superset)
+ show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
+ using X by (rule simple_distributed_simple_function)
+ fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
+ from simple_distributed_measure[OF X this]
+ show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
+ qed auto
+qed
+
+lemma (in prob_space) simple_distributed_setsum_space:
+ assumes X: "simple_distributed M X f"
+ shows "setsum f (X`space M) = 1"
+proof -
+ from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
+ by (subst finite_measure_finite_Union)
+ (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
+ intro!: setsum_cong arg_cong[where f="prob"])
+ also have "\<dots> = prob (space M)"
+ by (auto intro!: arg_cong[where f=prob])
+ finally show ?thesis
+ using emeasure_space_1 by (simp add: emeasure_eq_measure one_ereal_def)
+qed
-lemma bernoulli_measure:
- "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p B = (\<Sum>b\<in>B. if b then p else 1 - p)"
- unfolding bernoulli.\<mu>'_def unfolding bernoulli_space_def by (auto intro!: setsum_cong)
+lemma (in prob_space) distributed_marginal_eq_joint_simple:
+ assumes Px: "simple_function M X"
+ assumes Py: "simple_distributed M Y Py"
+ assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
+ assumes y: "y \<in> Y`space M"
+ shows "Py y = (\<Sum>x\<in>X`space M. if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0)"
+proof -
+ note Px = simple_distributedI[OF Px refl]
+ have *: "\<And>f A. setsum (\<lambda>x. max 0 (ereal (f x))) A = ereal (setsum (\<lambda>x. max 0 (f x)) A)"
+ by (simp add: setsum_ereal[symmetric] zero_ereal_def)
+ from distributed_marginal_eq_joint[OF sigma_finite_measure_count_space_finite sigma_finite_measure_count_space_finite
+ simple_distributed[OF Px] simple_distributed[OF Py] simple_distributed_joint[OF Pxy],
+ OF Py[THEN simple_distributed_finite] Px[THEN simple_distributed_finite]]
+ y Px[THEN simple_distributed_finite] Py[THEN simple_distributed_finite]
+ Pxy[THEN simple_distributed, THEN distributed_real_AE]
+ show ?thesis
+ unfolding AE_count_space
+ apply (elim ballE[where x=y])
+ apply (auto simp add: positive_integral_count_space_finite * intro!: setsum_cong split: split_max)
+ done
+qed
-lemma bernoulli_measure_True: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {True} = p"
- and bernoulli_measure_False: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {False} = 1 - p"
- unfolding bernoulli_measure by simp_all
+
+lemma prob_space_uniform_measure:
+ assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
+ shows "prob_space (uniform_measure M A)"
+proof
+ show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
+ using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
+ using sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
+ by (simp add: Int_absorb2 emeasure_nonneg)
+qed
+
+lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
+ by default (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ereal_def)
end