# HG changeset patch
# User hoelzl
# Date 1309160566 -7200
# Node ID 0d78c8d31d0d4c6269b79f45747c783228962560
# Parent  93c1fc6ac52791bdb5402b216d2dd2282055e710
move conditional expectation to its own theory file

diff -r 93c1fc6ac527 -r 0d78c8d31d0d src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Sun Jun 26 19:10:03 2011 +0200
+++ b/src/HOL/IsaMakefile	Mon Jun 27 09:42:46 2011 +0200
@@ -1203,6 +1203,7 @@
 $(OUT)/HOL-Probability: $(OUT)/HOL-Multivariate_Analysis		\
   Probability/Binary_Product_Measure.thy Probability/Borel_Space.thy	\
   Probability/Caratheodory.thy Probability/Complete_Measure.thy		\
+  Probability/Conditional_Probability.thy				\
   Probability/ex/Dining_Cryptographers.thy				\
   Probability/ex/Koepf_Duermuth_Countermeasure.thy			\
   Probability/Finite_Product_Measure.thy				\
diff -r 93c1fc6ac527 -r 0d78c8d31d0d src/HOL/Probability/Conditional_Probability.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Conditional_Probability.thy	Mon Jun 27 09:42:46 2011 +0200
@@ -0,0 +1,161 @@
+(*  Title:      HOL/Probability/Conditional_Probability.thy
+    Author:     Johannes Hölzl, TU München
+*)
+
+header {*Conditional probability*}
+
+theory Conditional_Probability
+imports Probability_Measure Radon_Nikodym
+begin
+
+section "Conditional Expectation and Probability"
+
+definition (in prob_space)
+  "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x)
+    \<and> (\<forall>C\<in>sets N. (\<integral>\<^isup>+x. Y x * indicator C x\<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x\<partial>M)))"
+
+lemma (in prob_space) conditional_expectation_exists:
+  fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
+  assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
+  and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
+  shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<and> (\<forall>C\<in>sets N.
+      (\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))"
+proof -
+  note N(4)[simp]
+  interpret P: prob_space N
+    using prob_space_subalgebra[OF N] .
+
+  let "?f A" = "\<lambda>x. X x * indicator A x"
+  let "?Q A" = "integral\<^isup>P M (?f A)"
+
+  from measure_space_density[OF borel]
+  have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)"
+    apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"])
+    using N by (auto intro!: P.sigma_algebra_cong)
+  then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" .
+
+  have "P.absolutely_continuous ?Q"
+    unfolding P.absolutely_continuous_def
+  proof safe
+    fix A assume "A \<in> sets N" "P.\<mu> A = 0"
+    then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A"
+      using borel N by (auto intro!: borel_measurable_indicator AE_not_in)
+    then show "?Q A = 0"
+      by (auto simp add: positive_integral_0_iff_AE)
+  qed
+  from P.Radon_Nikodym[OF Q this]
+  obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x"
+    "\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)"
+    by blast
+  with N(2) show ?thesis
+    by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)])
+qed
+
+lemma (in prob_space)
+  fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
+  assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
+  and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
+  shows borel_measurable_conditional_expectation:
+    "conditional_expectation N X \<in> borel_measurable N"
+  and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow>
+      (\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) =
+      (\<integral>\<^isup>+x. X x * indicator C x \<partial>M)"
+   (is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C")
+proof -
+  note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
+  then show "conditional_expectation N X \<in> borel_measurable N"
+    unfolding conditional_expectation_def by (rule someI2_ex) blast
+
+  from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C"
+    unfolding conditional_expectation_def by (rule someI2_ex) blast
+qed
+
+lemma (in sigma_algebra) factorize_measurable_function_pos:
+  fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
+  assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
+  assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)"
+  shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)"
+proof -
+  interpret M': sigma_algebra M' by fact
+  have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
+  from M'.sigma_algebra_vimage[OF this]
+  interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
+
+  from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this
+
+  have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
+  proof
+    fix i
+    from f(1)[of i] have "finite (f i`space M)" and B_ex:
+      "\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) -` {z} \<inter> space M = Y -` B \<inter> space M"
+      unfolding simple_function_def by auto
+    from B_ex[THEN bchoice] guess B .. note B = this
+
+    let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x"
+
+    show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
+    proof (intro exI[of _ ?g] conjI ballI)
+      show "simple_function M' ?g" using B by auto
+
+      fix x assume "x \<in> space M"
+      then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::extreal)"
+        unfolding indicator_def using B by auto
+      then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i]
+        by (subst va.simple_function_indicator_representation) auto
+    qed
+  qed
+  from choice[OF this] guess g .. note g = this
+
+  show ?thesis
+  proof (intro ballI bexI)
+    show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'"
+      using g by (auto intro: M'.borel_measurable_simple_function)
+    fix x assume "x \<in> space M"
+    have "max 0 (Z x) = (SUP i. f i x)" using f by simp
+    also have "\<dots> = (SUP i. g i (Y x))"
+      using g `x \<in> space M` by simp
+    finally show "max 0 (Z x) = (SUP i. g i (Y x))" .
+  qed
+qed
+
+lemma (in sigma_algebra) factorize_measurable_function:
+  fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
+  assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
+  shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)
+    \<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))"
+proof safe
+  interpret M': sigma_algebra M' by fact
+  have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
+  from M'.sigma_algebra_vimage[OF this]
+  interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
+
+  { fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'"
+    with M'.measurable_vimage_algebra[OF Y]
+    have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+      by (rule measurable_comp)
+    moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
+    then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
+       g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+       by (auto intro!: measurable_cong)
+    ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+      by simp }
+
+  assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+  with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M"
+    "(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
+    by auto
+  from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this
+  from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this
+  let "?g x" = "p x - n x"
+  show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
+  proof (intro bexI ballI)
+    show "?g \<in> borel_measurable M'" using p n by auto
+    fix x assume "x \<in> space M"
+    then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)"
+      using p n by auto
+    then show "Z x = ?g (Y x)"
+      by (auto split: split_max)
+  qed
+qed
+
+end
\ No newline at end of file
diff -r 93c1fc6ac527 -r 0d78c8d31d0d src/HOL/Probability/Information.thy
--- a/src/HOL/Probability/Information.thy	Sun Jun 26 19:10:03 2011 +0200
+++ b/src/HOL/Probability/Information.thy	Mon Jun 27 09:42:46 2011 +0200
@@ -8,6 +8,7 @@
 theory Information
 imports
   Independent_Family
+  Radon_Nikodym
   "~~/src/HOL/Library/Convex"
 begin
 
diff -r 93c1fc6ac527 -r 0d78c8d31d0d src/HOL/Probability/Measure.thy
--- a/src/HOL/Probability/Measure.thy	Sun Jun 26 19:10:03 2011 +0200
+++ b/src/HOL/Probability/Measure.thy	Mon Jun 27 09:42:46 2011 +0200
@@ -1205,6 +1205,10 @@
 lemma (in finite_measure) positive_measure'[simp, intro]: "0 \<le> \<mu>' A"
   unfolding \<mu>'_def by (auto simp: real_of_extreal_pos)
 
+lemma (in finite_measure) real_measure:
+  assumes A: "A \<in> sets M" shows "\<exists>r. 0 \<le> r \<and> \<mu> A = extreal r"
+  using finite_measure[OF A] positive_measure[OF A] by (cases "\<mu> A") auto
+
 lemma (in finite_measure) bounded_measure: "\<mu>' A \<le> \<mu>' (space M)"
 proof cases
   assume "A \<in> sets M"
diff -r 93c1fc6ac527 -r 0d78c8d31d0d src/HOL/Probability/Probability.thy
--- a/src/HOL/Probability/Probability.thy	Sun Jun 26 19:10:03 2011 +0200
+++ b/src/HOL/Probability/Probability.thy	Mon Jun 27 09:42:46 2011 +0200
@@ -4,6 +4,7 @@
   Probability_Measure
   Infinite_Product_Measure
   Independent_Family
+  Conditional_Probability
   Information
   "ex/Dining_Cryptographers"
   "ex/Koepf_Duermuth_Countermeasure"
diff -r 93c1fc6ac527 -r 0d78c8d31d0d src/HOL/Probability/Probability_Measure.thy
--- a/src/HOL/Probability/Probability_Measure.thy	Sun Jun 26 19:10:03 2011 +0200
+++ b/src/HOL/Probability/Probability_Measure.thy	Mon Jun 27 09:42:46 2011 +0200
@@ -6,7 +6,7 @@
 header {*Probability measure*}
 
 theory Probability_Measure
-imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure Lebesgue_Measure
+imports Lebesgue_Measure
 begin
 
 locale prob_space = measure_space +
@@ -1107,159 +1107,6 @@
     by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
 qed
 
-section "Conditional Expectation and Probability"
-
-lemma (in prob_space) conditional_expectation_exists:
-  fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
-  assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
-  and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
-  shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<and> (\<forall>C\<in>sets N.
-      (\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))"
-proof -
-  note N(4)[simp]
-  interpret P: prob_space N
-    using prob_space_subalgebra[OF N] .
-
-  let "?f A" = "\<lambda>x. X x * indicator A x"
-  let "?Q A" = "integral\<^isup>P M (?f A)"
-
-  from measure_space_density[OF borel]
-  have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)"
-    apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"])
-    using N by (auto intro!: P.sigma_algebra_cong)
-  then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" .
-
-  have "P.absolutely_continuous ?Q"
-    unfolding P.absolutely_continuous_def
-  proof safe
-    fix A assume "A \<in> sets N" "P.\<mu> A = 0"
-    then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A"
-      using borel N by (auto intro!: borel_measurable_indicator AE_not_in)
-    then show "?Q A = 0"
-      by (auto simp add: positive_integral_0_iff_AE)
-  qed
-  from P.Radon_Nikodym[OF Q this]
-  obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y x"
-    "\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)"
-    by blast
-  with N(2) show ?thesis
-    by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)])
-qed
-
-definition (in prob_space)
-  "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y x)
-    \<and> (\<forall>C\<in>sets N. (\<integral>\<^isup>+x. Y x * indicator C x\<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x\<partial>M)))"
-
-abbreviation (in prob_space)
-  "conditional_prob N A \<equiv> conditional_expectation N (indicator A)"
-
-lemma (in prob_space)
-  fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
-  assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X x"
-  and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
-  shows borel_measurable_conditional_expectation:
-    "conditional_expectation N X \<in> borel_measurable N"
-  and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow>
-      (\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) =
-      (\<integral>\<^isup>+x. X x * indicator C x \<partial>M)"
-   (is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C")
-proof -
-  note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
-  then show "conditional_expectation N X \<in> borel_measurable N"
-    unfolding conditional_expectation_def by (rule someI2_ex) blast
-
-  from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C"
-    unfolding conditional_expectation_def by (rule someI2_ex) blast
-qed
-
-lemma (in sigma_algebra) factorize_measurable_function_pos:
-  fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
-  assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
-  assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)"
-  shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)"
-proof -
-  interpret M': sigma_algebra M' by fact
-  have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
-  from M'.sigma_algebra_vimage[OF this]
-  interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
-
-  from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this
-
-  have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
-  proof
-    fix i
-    from f(1)[of i] have "finite (f i`space M)" and B_ex:
-      "\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) -` {z} \<inter> space M = Y -` B \<inter> space M"
-      unfolding simple_function_def by auto
-    from B_ex[THEN bchoice] guess B .. note B = this
-
-    let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x"
-
-    show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
-    proof (intro exI[of _ ?g] conjI ballI)
-      show "simple_function M' ?g" using B by auto
-
-      fix x assume "x \<in> space M"
-      then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::extreal)"
-        unfolding indicator_def using B by auto
-      then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i]
-        by (subst va.simple_function_indicator_representation) auto
-    qed
-  qed
-  from choice[OF this] guess g .. note g = this
-
-  show ?thesis
-  proof (intro ballI bexI)
-    show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'"
-      using g by (auto intro: M'.borel_measurable_simple_function)
-    fix x assume "x \<in> space M"
-    have "max 0 (Z x) = (SUP i. f i x)" using f by simp
-    also have "\<dots> = (SUP i. g i (Y x))"
-      using g `x \<in> space M` by simp
-    finally show "max 0 (Z x) = (SUP i. g i (Y x))" .
-  qed
-qed
-
-lemma (in sigma_algebra) factorize_measurable_function:
-  fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
-  assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
-  shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)
-    \<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))"
-proof safe
-  interpret M': sigma_algebra M' by fact
-  have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
-  from M'.sigma_algebra_vimage[OF this]
-  interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
-
-  { fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'"
-    with M'.measurable_vimage_algebra[OF Y]
-    have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
-      by (rule measurable_comp)
-    moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
-    then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
-       g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
-       by (auto intro!: measurable_cong)
-    ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
-      by simp }
-
-  assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
-  with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M"
-    "(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
-    by auto
-  from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this
-  from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this
-  let "?g x" = "p x - n x"
-  show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
-  proof (intro bexI ballI)
-    show "?g \<in> borel_measurable M'" using p n by auto
-    fix x assume "x \<in> space M"
-    then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)"
-      using p n by auto
-    then show "Z x = ?g (Y x)"
-      by (auto split: split_max)
-  qed
-qed
-
 subsection "Borel Measure on {0 .. 1}"
 
 definition pborel :: "real measure_space" where
diff -r 93c1fc6ac527 -r 0d78c8d31d0d src/HOL/Probability/Radon_Nikodym.thy
--- a/src/HOL/Probability/Radon_Nikodym.thy	Sun Jun 26 19:10:03 2011 +0200
+++ b/src/HOL/Probability/Radon_Nikodym.thy	Mon Jun 27 09:42:46 2011 +0200
@@ -314,10 +314,6 @@
   qed
 qed
 
-lemma (in finite_measure) real_measure:
-  assumes A: "A \<in> sets M" shows "\<exists>r. 0 \<le> r \<and> \<mu> A = extreal r"
-  using finite_measure[OF A] positive_measure[OF A] by (cases "\<mu> A") auto
-
 lemma (in finite_measure) Radon_Nikodym_finite_measure:
   assumes "finite_measure (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
   assumes "absolutely_continuous \<nu>"