src/HOL/Probability/Projective_Family.thy
 author immler@in.tum.de Fri Nov 09 14:31:26 2012 +0100 (2012-11-09) changeset 50042 6fe18351e9dd parent 50041 afe886a04198 child 50087 635d73673b5e permissions -rw-r--r--
1 (*  Title:      HOL/Probability/Projective_Family.thy
2     Author:     Fabian Immler, TU München
3     Author:     Johannes Hölzl, TU München
4 *)
8 theory Projective_Family
9 imports Finite_Product_Measure Probability_Measure
10 begin
12 definition
13   PiP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
14   "PiP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
15     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
16     (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
17     (\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))"
19 lemma space_PiP[simp]: "space (PiP I M P) = space (PiM I M)"
20   by (auto simp add: PiP_def space_PiM prod_emb_def intro!: space_extend_measure)
22 lemma sets_PiP[simp]: "sets (PiP I M P) = sets (PiM I M)"
23   by (auto simp add: PiP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure)
25 lemma measurable_PiP1[simp]: "measurable (PiP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'"
26   unfolding measurable_def by auto
28 lemma measurable_PiP2[simp]: "measurable M' (PiP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)"
29   unfolding measurable_def by auto
31 locale projective_family =
32   fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
33   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>
34      (P H) (prod_emb H M J X) = (P J) X"
35   assumes prob_space: "\<And>J. finite J \<Longrightarrow> prob_space (P J)"
36   assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
37   assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
38 begin
40 lemma emeasure_PiP:
41   assumes "finite J"
42   assumes "J \<subseteq> I"
43   assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
44   shows "emeasure (PiP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)"
45 proof -
46   have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
47   proof safe
48     fix x j assume "x \<in> Pi J (restrict A J)" "j \<in> J"
49     hence "x j \<in> restrict A J j" by (auto simp: Pi_def)
50     also have "\<dots> \<subseteq> space (M j)" using sets_into_space A `j \<in> J` by auto
51     finally show "x j \<in> space (M j)" .
52   qed
53   hence "emeasure (PiP J M P) (Pi\<^isub>E J A) =
54     emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
55     using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def)
56   also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
57   proof (rule emeasure_extend_measure_Pair[OF PiP_def])
58     show "positive (sets (PiP J M P)) (P J)" unfolding positive_def by auto
59     show "countably_additive (sets (PiP J M P)) (P J)" unfolding countably_additive_def
60       by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
61     show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
62       using assms by auto
63     fix K and X::"'i \<Rightarrow> 'a set"
64     show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))"
65       by (auto simp: prod_emb_def)
66     assume JX: "(K \<noteq> {} \<or> J = {}) \<and> finite K \<and> K \<subseteq> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))"
67     thus "emeasure (P J) (prod_emb J M K (Pi\<^isub>E K X)) = emeasure (P K) (Pi\<^isub>E K X)"
68       using assms
69       apply (cases "J = {}")
71       apply (fastforce simp add: intro!: projective sets_PiM_I_finite)
72       done
73   qed
74   finally show ?thesis .
75 qed
77 lemma PiP_finite:
78   assumes "finite J"
79   assumes "J \<subseteq> I"
80   shows "PiP J M P = P J" (is "?P = _")
81 proof (rule measure_eqI_generator_eq)
82   let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
83   let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
84   let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
85   show "Int_stable ?J"
86     by (rule Int_stable_PiE)
87   interpret prob_space "P J" using prob_space `finite J` by simp
88   show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_PiP)
89   show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space)
90   show "sets (PiP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J"
91     using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
92   fix X assume "X \<in> ?J"
93   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
94   with `finite J` have "X \<in> sets (PiP J M P)" by simp
95   have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E"
96     using E sets_into_space
97     by (auto intro!: prod_emb_PiE_same_index)
98   show "emeasure (PiP J M P) X = emeasure (P J) X"
99     unfolding X using E
100     by (intro emeasure_PiP assms) simp
101 qed (insert `finite J`, auto intro!: prod_algebraI_finite)
103 lemma emeasure_fun_emb[simp]:
104   assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)"
105   shows "emeasure (PiP L M P) (prod_emb L M J X) = emeasure (PiP J M P) X"
106   using assms
107   by (subst PiP_finite) (auto simp: PiP_finite finite_subset projective)
109 lemma prod_emb_injective:
110   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)"
111   assumes "prod_emb L M J X = prod_emb L M J Y"
112   shows "X = Y"
113 proof (rule injective_vimage_restrict)
114   show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
115     using sets[THEN sets_into_space] by (auto simp: space_PiM)
116   have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
117   proof
118     fix i assume "i \<in> L"
119     interpret prob_space "P {i}" using prob_space by simp
120     from not_empty show "\<exists>x. x \<in> space (M i)" by (auto simp add: proj_space space_PiM)
121   qed
122   from bchoice[OF this]
123   show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by auto
124   show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))"
125     using `prod_emb L M J X = prod_emb L M J Y` by (simp add: prod_emb_def)
126 qed fact
128 abbreviation
129   "emb L K X \<equiv> prod_emb L M K X"
131 definition generator :: "('i \<Rightarrow> 'a) set set" where
132   "generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M))"
134 lemma generatorI':
135   "J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> generator"
136   unfolding generator_def by auto
138 lemma algebra_generator:
139   assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G")
140   unfolding algebra_def algebra_axioms_def ring_of_sets_iff
141 proof (intro conjI ballI)
142   let ?G = generator
143   show "?G \<subseteq> Pow ?\<Omega>"
144     by (auto simp: generator_def prod_emb_def)
145   from `I \<noteq> {}` obtain i where "i \<in> I" by auto
146   then show "{} \<in> ?G"
147     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
148              simp: sigma_sets.Empty generator_def prod_emb_def)
149   from `i \<in> I` show "?\<Omega> \<in> ?G"
150     by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
151              simp: generator_def prod_emb_def)
152   fix A assume "A \<in> ?G"
153   then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA"
154     by (auto simp: generator_def)
155   fix B assume "B \<in> ?G"
156   then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB"
157     by (auto simp: generator_def)
158   let ?RA = "emb (JA \<union> JB) JA XA"
159   let ?RB = "emb (JA \<union> JB) JB XB"
160   have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
161     using XA A XB B by auto
162   show "A - B \<in> ?G" "A \<union> B \<in> ?G"
163     unfolding * using XA XB by (safe intro!: generatorI') auto
164 qed
166 lemma sets_PiM_generator:
167   "sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
168 proof cases
169   assume "I = {}" then show ?thesis
170     unfolding generator_def
171     by (auto simp: sets_PiM_empty sigma_sets_empty_eq cong: conj_cong)
172 next
173   assume "I \<noteq> {}"
174   show ?thesis
175   proof
176     show "sets (Pi\<^isub>M I M) \<subseteq> sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
177       unfolding sets_PiM
178     proof (safe intro!: sigma_sets_subseteq)
179       fix A assume "A \<in> prod_algebra I M" with `I \<noteq> {}` show "A \<in> generator"
180         by (auto intro!: generatorI' sets_PiM_I_finite elim!: prod_algebraE)
181     qed
182   qed (auto simp: generator_def space_PiM[symmetric] intro!: sigma_sets_subset)
183 qed
185 lemma generatorI:
186   "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"
187   unfolding generator_def by auto
189 definition
190   "\<mu>G A =
191     (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))"
193 lemma \<mu>G_spec:
194   assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
195   shows "\<mu>G A = emeasure (PiP J M P) X"
196   unfolding \<mu>G_def
197 proof (intro the_equality allI impI ballI)
198   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)"
199   have "emeasure (PiP K M P) Y = emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) K Y)"
200     using K J by simp
201   also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
202     using K J by (simp add: prod_emb_injective[of "K \<union> J" I])
203   also have "emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (PiP J M P) X"
204     using K J by simp
205   finally show "emeasure (PiP J M P) X = emeasure (PiP K M P) Y" ..
206 qed (insert J, force)
208 lemma \<mu>G_eq:
209   "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"
210   by (intro \<mu>G_spec) auto
212 lemma generator_Ex:
213   assumes *: "A \<in> generator"
214   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"
215 proof -
216   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)"
217     unfolding generator_def by auto
218   with \<mu>G_spec[OF this] show ?thesis by auto
219 qed
221 lemma generatorE:
222   assumes A: "A \<in> generator"
223   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"
224 proof -
225   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"
226     "\<mu>G A = emeasure (PiP J M P) X" by auto
227   then show thesis by (intro that) auto
228 qed
230 lemma merge_sets:
231   "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)"
232   by simp
234 lemma merge_emb:
235   assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
236   shows "((\<lambda>x. merge J (I - J) (y, x)) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
237     emb I (K - J) ((\<lambda>x. merge J (K - J) (y, x)) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
238 proof -
239   have [simp]: "\<And>x J K L. merge J K (y, restrict x L) = merge J (K \<inter> L) (y, x)"
240     by (auto simp: restrict_def merge_def)
241   have [simp]: "\<And>x J K L. restrict (merge J K (y, x)) L = merge (J \<inter> L) (K \<inter> L) (y, x)"
242     by (auto simp: restrict_def merge_def)
243   have [simp]: "(I - J) \<inter> K = K - J" using `K \<subseteq> I` `J \<subseteq> I` by auto
244   have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
245   have [simp]: "(K - J) \<inter> K = K - J" by auto
246   from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
247     by (simp split: split_merge add: prod_emb_def Pi_iff extensional_merge_sub set_eq_iff space_PiM)
248        auto
249 qed
251 lemma positive_\<mu>G:
252   assumes "I \<noteq> {}"
253   shows "positive generator \<mu>G"
254 proof -
255   interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
256   show ?thesis
257   proof (intro positive_def[THEN iffD2] conjI ballI)
258     from generatorE[OF G.empty_sets] guess J X . note this[simp]
259     have "X = {}"
260       by (rule prod_emb_injective[of J I]) simp_all
261     then show "\<mu>G {} = 0" by simp
262   next
263     fix A assume "A \<in> generator"
264     from generatorE[OF this] guess J X . note this[simp]
265     show "0 \<le> \<mu>G A" by (simp add: emeasure_nonneg)
266   qed
267 qed
270   assumes "I \<noteq> {}"
272 proof -
273   interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
274   show ?thesis
275   proof (intro additive_def[THEN iffD2] ballI impI)
276     fix A assume "A \<in> generator" with generatorE guess J X . note J = this
277     fix B assume "B \<in> generator" with generatorE guess K Y . note K = this
278     assume "A \<inter> B = {}"
279     have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
280       using J K by auto
281     have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
282       apply (rule prod_emb_injective[of "J \<union> K" I])
283       apply (insert `A \<inter> B = {}` JK J K)
284       apply (simp_all add: Int prod_emb_Int)
285       done
286     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)"
287       using J K by simp_all
288     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))"
289       by simp
290     also have "\<dots> = emeasure (PiP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
291       using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un)
292     also have "\<dots> = \<mu>G A + \<mu>G B"
293       using J K JK_disj by (simp add: plus_emeasure[symmetric])
294     finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
295   qed
296 qed
298 end
300 end