55 shows "emeasure (Pi\<^isub>M L M) (prod_emb L M J X) = emeasure (Pi\<^isub>M J M) X" |
55 shows "emeasure (Pi\<^isub>M L M) (prod_emb L M J X) = emeasure (Pi\<^isub>M J M) X" |
56 by (subst distr_restrict[OF L]) |
56 by (subst distr_restrict[OF L]) |
57 (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X) |
57 (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X) |
58 |
58 |
59 definition |
59 definition |
60 PiP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where |
60 limP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where |
61 "PiP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i)) |
61 "limP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i)) |
62 {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} |
62 {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} |
63 (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) |
63 (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) |
64 (\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))" |
64 (\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))" |
65 |
65 |
66 lemma space_PiP[simp]: "space (PiP I M P) = space (PiM I M)" |
66 abbreviation "lim\<^isub>P \<equiv> limP" |
67 by (auto simp add: PiP_def space_PiM prod_emb_def intro!: space_extend_measure) |
67 |
68 |
68 lemma space_limP[simp]: "space (limP I M P) = space (PiM I M)" |
69 lemma sets_PiP[simp]: "sets (PiP I M P) = sets (PiM I M)" |
69 by (auto simp add: limP_def space_PiM prod_emb_def intro!: space_extend_measure) |
70 by (auto simp add: PiP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure) |
70 |
71 |
71 lemma sets_limP[simp]: "sets (limP I M P) = sets (PiM I M)" |
72 lemma measurable_PiP1[simp]: "measurable (PiP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'" |
72 by (auto simp add: limP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure) |
|
73 |
|
74 lemma measurable_limP1[simp]: "measurable (limP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'" |
73 unfolding measurable_def by auto |
75 unfolding measurable_def by auto |
74 |
76 |
75 lemma measurable_PiP2[simp]: "measurable M' (PiP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)" |
77 lemma measurable_limP2[simp]: "measurable M' (limP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)" |
76 unfolding measurable_def by auto |
78 unfolding measurable_def by auto |
77 |
79 |
78 locale projective_family = |
80 locale projective_family = |
79 fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)" |
81 fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)" |
80 assumes projective: "\<And>J H X. J \<noteq> {} \<Longrightarrow> J \<subseteq> H \<Longrightarrow> H \<subseteq> I \<Longrightarrow> finite H \<Longrightarrow> X \<in> sets (PiM J M) \<Longrightarrow> |
82 assumes projective: "\<And>J H X. J \<noteq> {} \<Longrightarrow> J \<subseteq> H \<Longrightarrow> H \<subseteq> I \<Longrightarrow> finite H \<Longrightarrow> X \<in> sets (PiM J M) \<Longrightarrow> |
82 assumes prob_space: "\<And>J. finite J \<Longrightarrow> prob_space (P J)" |
84 assumes prob_space: "\<And>J. finite J \<Longrightarrow> prob_space (P J)" |
83 assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)" |
85 assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)" |
84 assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)" |
86 assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)" |
85 begin |
87 begin |
86 |
88 |
87 lemma emeasure_PiP: |
89 lemma emeasure_limP: |
88 assumes "finite J" |
90 assumes "finite J" |
89 assumes "J \<subseteq> I" |
91 assumes "J \<subseteq> I" |
90 assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)" |
92 assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)" |
91 shows "emeasure (PiP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)" |
93 shows "emeasure (limP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)" |
92 proof - |
94 proof - |
93 have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" |
95 have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" |
94 proof safe |
96 proof safe |
95 fix x j assume "x \<in> Pi J (restrict A J)" "j \<in> J" |
97 fix x j assume "x \<in> Pi J (restrict A J)" "j \<in> J" |
96 hence "x j \<in> restrict A J j" by (auto simp: Pi_def) |
98 hence "x j \<in> restrict A J j" by (auto simp: Pi_def) |
97 also have "\<dots> \<subseteq> space (M j)" using sets_into_space A `j \<in> J` by auto |
99 also have "\<dots> \<subseteq> space (M j)" using sets_into_space A `j \<in> J` by auto |
98 finally show "x j \<in> space (M j)" . |
100 finally show "x j \<in> space (M j)" . |
99 qed |
101 qed |
100 hence "emeasure (PiP J M P) (Pi\<^isub>E J A) = |
102 hence "emeasure (limP J M P) (Pi\<^isub>E J A) = |
101 emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))" |
103 emeasure (limP J M P) (prod_emb J M J (Pi\<^isub>E J A))" |
102 using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def) |
104 using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def) |
103 also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)" |
105 also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)" |
104 proof (rule emeasure_extend_measure_Pair[OF PiP_def]) |
106 proof (rule emeasure_extend_measure_Pair[OF limP_def]) |
105 show "positive (sets (PiP J M P)) (P J)" unfolding positive_def by auto |
107 show "positive (sets (limP J M P)) (P J)" unfolding positive_def by auto |
106 show "countably_additive (sets (PiP J M P)) (P J)" unfolding countably_additive_def |
108 show "countably_additive (sets (limP J M P)) (P J)" unfolding countably_additive_def |
107 by (auto simp: suminf_emeasure proj_sets[OF `finite J`]) |
109 by (auto simp: suminf_emeasure proj_sets[OF `finite J`]) |
108 show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))" |
110 show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))" |
109 using assms by auto |
111 using assms by auto |
110 fix K and X::"'i \<Rightarrow> 'a set" |
112 fix K and X::"'i \<Rightarrow> 'a set" |
111 show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))" |
113 show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))" |
119 done |
121 done |
120 qed |
122 qed |
121 finally show ?thesis . |
123 finally show ?thesis . |
122 qed |
124 qed |
123 |
125 |
124 lemma PiP_finite: |
126 lemma limP_finite: |
125 assumes "finite J" |
127 assumes "finite J" |
126 assumes "J \<subseteq> I" |
128 assumes "J \<subseteq> I" |
127 shows "PiP J M P = P J" (is "?P = _") |
129 shows "limP J M P = P J" (is "?P = _") |
128 proof (rule measure_eqI_generator_eq) |
130 proof (rule measure_eqI_generator_eq) |
129 let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}" |
131 let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}" |
130 let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)" |
132 let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)" |
131 let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))" |
133 let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))" |
132 show "Int_stable ?J" |
134 show "Int_stable ?J" |
133 by (rule Int_stable_PiE) |
135 by (rule Int_stable_PiE) |
134 interpret prob_space "P J" using prob_space `finite J` by simp |
136 interpret prob_space "P J" using prob_space `finite J` by simp |
135 show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_PiP) |
137 show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_limP) |
136 show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space) |
138 show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space) |
137 show "sets (PiP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J" |
139 show "sets (limP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J" |
138 using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff) |
140 using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff) |
139 fix X assume "X \<in> ?J" |
141 fix X assume "X \<in> ?J" |
140 then obtain E where X: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto |
142 then obtain E where X: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto |
141 with `finite J` have "X \<in> sets (PiP J M P)" by simp |
143 with `finite J` have "X \<in> sets (limP J M P)" by simp |
142 have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E" |
144 have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E" |
143 using E sets_into_space |
145 using E sets_into_space |
144 by (auto intro!: prod_emb_PiE_same_index) |
146 by (auto intro!: prod_emb_PiE_same_index) |
145 show "emeasure (PiP J M P) X = emeasure (P J) X" |
147 show "emeasure (limP J M P) X = emeasure (P J) X" |
146 unfolding X using E |
148 unfolding X using E |
147 by (intro emeasure_PiP assms) simp |
149 by (intro emeasure_limP assms) simp |
148 qed (insert `finite J`, auto intro!: prod_algebraI_finite) |
150 qed (insert `finite J`, auto intro!: prod_algebraI_finite) |
149 |
151 |
150 lemma emeasure_fun_emb[simp]: |
152 lemma emeasure_fun_emb[simp]: |
151 assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)" |
153 assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)" |
152 shows "emeasure (PiP L M P) (prod_emb L M J X) = emeasure (PiP J M P) X" |
154 shows "emeasure (limP L M P) (prod_emb L M J X) = emeasure (limP J M P) X" |
153 using assms |
155 using assms |
154 by (subst PiP_finite) (auto simp: PiP_finite finite_subset projective) |
156 by (subst limP_finite) (auto simp: limP_finite finite_subset projective) |
155 |
157 |
156 lemma prod_emb_injective: |
158 lemma prod_emb_injective: |
157 assumes "J \<noteq> {}" "J \<subseteq> L" "finite J" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)" |
159 assumes "J \<noteq> {}" "J \<subseteq> L" "finite J" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)" |
158 assumes "prod_emb L M J X = prod_emb L M J Y" |
160 assumes "prod_emb L M J X = prod_emb L M J Y" |
159 shows "X = Y" |
161 shows "X = Y" |
233 "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> generator" |
235 "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> generator" |
234 unfolding generator_def by auto |
236 unfolding generator_def by auto |
235 |
237 |
236 definition |
238 definition |
237 "\<mu>G A = |
239 "\<mu>G A = |
238 (THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = emeasure (PiP J M P) X))" |
240 (THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = emeasure (limP J M P) X))" |
239 |
241 |
240 lemma \<mu>G_spec: |
242 lemma \<mu>G_spec: |
241 assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" |
243 assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" |
242 shows "\<mu>G A = emeasure (PiP J M P) X" |
244 shows "\<mu>G A = emeasure (limP J M P) X" |
243 unfolding \<mu>G_def |
245 unfolding \<mu>G_def |
244 proof (intro the_equality allI impI ballI) |
246 proof (intro the_equality allI impI ballI) |
245 fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^isub>M K M)" |
247 fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^isub>M K M)" |
246 have "emeasure (PiP K M P) Y = emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) K Y)" |
248 have "emeasure (limP K M P) Y = emeasure (limP (K \<union> J) M P) (emb (K \<union> J) K Y)" |
247 using K J by simp |
249 using K J by simp |
248 also have "emb (K \<union> J) K Y = emb (K \<union> J) J X" |
250 also have "emb (K \<union> J) K Y = emb (K \<union> J) J X" |
249 using K J by (simp add: prod_emb_injective[of "K \<union> J" I]) |
251 using K J by (simp add: prod_emb_injective[of "K \<union> J" I]) |
250 also have "emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (PiP J M P) X" |
252 also have "emeasure (limP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (limP J M P) X" |
251 using K J by simp |
253 using K J by simp |
252 finally show "emeasure (PiP J M P) X = emeasure (PiP K M P) Y" .. |
254 finally show "emeasure (limP J M P) X = emeasure (limP K M P) Y" .. |
253 qed (insert J, force) |
255 qed (insert J, force) |
254 |
256 |
255 lemma \<mu>G_eq: |
257 lemma \<mu>G_eq: |
256 "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = emeasure (PiP J M P) X" |
258 "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = emeasure (limP J M P) X" |
257 by (intro \<mu>G_spec) auto |
259 by (intro \<mu>G_spec) auto |
258 |
260 |
259 lemma generator_Ex: |
261 lemma generator_Ex: |
260 assumes *: "A \<in> generator" |
262 assumes *: "A \<in> generator" |
261 shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = emeasure (PiP J M P) X" |
263 shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = emeasure (limP J M P) X" |
262 proof - |
264 proof - |
263 from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" |
265 from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)" |
264 unfolding generator_def by auto |
266 unfolding generator_def by auto |
265 with \<mu>G_spec[OF this] show ?thesis by auto |
267 with \<mu>G_spec[OF this] show ?thesis by auto |
266 qed |
268 qed |
267 |
269 |
268 lemma generatorE: |
270 lemma generatorE: |
269 assumes A: "A \<in> generator" |
271 assumes A: "A \<in> generator" |
270 obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = emeasure (PiP J M P) X" |
272 obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = emeasure (limP J M P) X" |
271 proof - |
273 proof - |
272 from generator_Ex[OF A] obtain X J where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" |
274 from generator_Ex[OF A] obtain X J where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" |
273 "\<mu>G A = emeasure (PiP J M P) X" by auto |
275 "\<mu>G A = emeasure (limP J M P) X" by auto |
274 then show thesis by (intro that) auto |
276 then show thesis by (intro that) auto |
275 qed |
277 qed |
276 |
278 |
277 lemma merge_sets: |
279 lemma merge_sets: |
278 "J \<inter> K = {} \<Longrightarrow> A \<in> sets (Pi\<^isub>M (J \<union> K) M) \<Longrightarrow> x \<in> space (Pi\<^isub>M J M) \<Longrightarrow> (\<lambda>y. merge J K (x,y)) -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)" |
280 "J \<inter> K = {} \<Longrightarrow> A \<in> sets (Pi\<^isub>M (J \<union> K) M) \<Longrightarrow> x \<in> space (Pi\<^isub>M J M) \<Longrightarrow> (\<lambda>y. merge J K (x,y)) -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)" |
332 done |
334 done |
333 have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)" |
335 have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)" |
334 using J K by simp_all |
336 using J K by simp_all |
335 then have "\<mu>G (A \<union> B) = \<mu>G (emb I (J \<union> K) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y))" |
337 then have "\<mu>G (A \<union> B) = \<mu>G (emb I (J \<union> K) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y))" |
336 by simp |
338 by simp |
337 also have "\<dots> = emeasure (PiP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)" |
339 also have "\<dots> = emeasure (limP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)" |
338 using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un) |
340 using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un) |
339 also have "\<dots> = \<mu>G A + \<mu>G B" |
341 also have "\<dots> = \<mu>G A + \<mu>G B" |
340 using J K JK_disj by (simp add: plus_emeasure[symmetric]) |
342 using J K JK_disj by (simp add: plus_emeasure[symmetric]) |
341 finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" . |
343 finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" . |
342 qed |
344 qed |
354 (\<forall>i. emeasure (Pi\<^isub>M J M) (A i) \<noteq> \<infinity>)" using sigma_finite[OF `finite J`] |
356 (\<forall>i. emeasure (Pi\<^isub>M J M) (A i) \<noteq> \<infinity>)" using sigma_finite[OF `finite J`] |
355 by (auto simp add: sigma_finite_measure_def) |
357 by (auto simp add: sigma_finite_measure_def) |
356 show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) = 1" by (rule f.emeasure_space_1) |
358 show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) = 1" by (rule f.emeasure_space_1) |
357 qed simp_all |
359 qed simp_all |
358 |
360 |
359 lemma (in product_prob_space) PiP_PiM_finite[simp]: |
361 lemma (in product_prob_space) limP_PiM_finite[simp]: |
360 assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "PiP J M (\<lambda>J. PiM J M) = PiM J M" |
362 assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "limP J M (\<lambda>J. PiM J M) = PiM J M" |
361 using assms by (simp add: PiP_finite) |
363 using assms by (simp add: limP_finite) |
362 |
364 |
363 end |
365 end |