src/HOL/Probability/Probability_Measure.thy
 author haftmann Sun Jun 23 21:16:07 2013 +0200 (2013-06-23) changeset 52435 6646bb548c6b parent 51683 baefa3b461c2 child 53015 a1119cf551e8 permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
1 (*  Title:      HOL/Probability/Probability_Measure.thy
2     Author:     Johannes Hölzl, TU München
3     Author:     Armin Heller, TU München
4 *)
6 header {*Probability measure*}
8 theory Probability_Measure
9   imports Lebesgue_Measure Radon_Nikodym
10 begin
12 locale prob_space = finite_measure +
13   assumes emeasure_space_1: "emeasure M (space M) = 1"
15 lemma prob_spaceI[Pure.intro!]:
16   assumes *: "emeasure M (space M) = 1"
17   shows "prob_space M"
18 proof -
19   interpret finite_measure M
20   proof
21     show "emeasure M (space M) \<noteq> \<infinity>" using * by simp
22   qed
23   show "prob_space M" by default fact
24 qed
26 abbreviation (in prob_space) "events \<equiv> sets M"
27 abbreviation (in prob_space) "prob \<equiv> measure M"
28 abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
29 abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
31 lemma (in prob_space) prob_space_distr:
32   assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
33 proof (rule prob_spaceI)
34   have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
35   with f show "emeasure (distr M M' f) (space (distr M M' f)) = 1"
36     by (auto simp: emeasure_distr emeasure_space_1)
37 qed
39 lemma (in prob_space) prob_space: "prob (space M) = 1"
40   using emeasure_space_1 unfolding measure_def by (simp add: one_ereal_def)
42 lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
43   using bounded_measure[of A] by (simp add: prob_space)
45 lemma (in prob_space) not_empty: "space M \<noteq> {}"
46   using prob_space by auto
48 lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
49   using emeasure_space[of M X] by (simp add: emeasure_space_1)
51 lemma (in prob_space) AE_I_eq_1:
52   assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
53   shows "AE x in M. P x"
54 proof (rule AE_I)
55   show "emeasure M (space M - {x \<in> space M. P x}) = 0"
56     using assms emeasure_space_1 by (simp add: emeasure_compl)
57 qed (insert assms, auto)
59 lemma (in prob_space) prob_compl:
60   assumes A: "A \<in> events"
61   shows "prob (space M - A) = 1 - prob A"
62   using finite_measure_compl[OF A] by (simp add: prob_space)
64 lemma (in prob_space) AE_in_set_eq_1:
65   assumes "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
66 proof
67   assume ae: "AE x in M. x \<in> A"
68   have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
69     using `A \<in> events`[THEN sets.sets_into_space] by auto
70   with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
71     by (simp add: emeasure_compl emeasure_space_1)
72   then show "prob A = 1"
73     using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
74 next
75   assume prob: "prob A = 1"
76   show "AE x in M. x \<in> A"
77   proof (rule AE_I)
78     show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
79     show "emeasure M (space M - A) = 0"
80       using `A \<in> events` prob
81       by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
82     show "space M - A \<in> events"
83       using `A \<in> events` by auto
84   qed
85 qed
87 lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
88 proof
89   assume "AE x in M. False"
90   then have "AE x in M. x \<in> {}" by simp
91   then show False
92     by (subst (asm) AE_in_set_eq_1) auto
93 qed simp
95 lemma (in prob_space) AE_prob_1:
96   assumes "prob A = 1" shows "AE x in M. x \<in> A"
97 proof -
98   from `prob A = 1` have "A \<in> events"
99     by (metis measure_notin_sets zero_neq_one)
100   with AE_in_set_eq_1 assms show ?thesis by simp
101 qed
103 lemma (in prob_space) AE_const[simp]: "(AE x in M. P) \<longleftrightarrow> P"
104   by (cases P) (auto simp: AE_False)
106 lemma (in prob_space) AE_contr:
107   assumes ae: "AE \<omega> in M. P \<omega>" "AE \<omega> in M. \<not> P \<omega>"
108   shows False
109 proof -
110   from ae have "AE \<omega> in M. False" by eventually_elim auto
111   then show False by auto
112 qed
114 lemma (in prob_space) expectation_less:
115   assumes [simp]: "integrable M X"
116   assumes gt: "AE x in M. X x < b"
117   shows "expectation X < b"
118 proof -
119   have "expectation X < expectation (\<lambda>x. b)"
120     using gt emeasure_space_1
121     by (intro integral_less_AE_space) auto
122   then show ?thesis using prob_space by simp
123 qed
125 lemma (in prob_space) expectation_greater:
126   assumes [simp]: "integrable M X"
127   assumes gt: "AE x in M. a < X x"
128   shows "a < expectation X"
129 proof -
130   have "expectation (\<lambda>x. a) < expectation X"
131     using gt emeasure_space_1
132     by (intro integral_less_AE_space) auto
133   then show ?thesis using prob_space by simp
134 qed
136 lemma (in prob_space) jensens_inequality:
137   fixes a b :: real
138   assumes X: "integrable M X" "AE x in M. X x \<in> I"
139   assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
140   assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
141   shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
142 proof -
143   let ?F = "\<lambda>x. Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))"
144   from X(2) AE_False have "I \<noteq> {}" by auto
146   from I have "open I" by auto
148   note I
149   moreover
150   { assume "I \<subseteq> {a <..}"
151     with X have "a < expectation X"
152       by (intro expectation_greater) auto }
153   moreover
154   { assume "I \<subseteq> {..< b}"
155     with X have "expectation X < b"
156       by (intro expectation_less) auto }
157   ultimately have "expectation X \<in> I"
158     by (elim disjE)  (auto simp: subset_eq)
159   moreover
160   { fix y assume y: "y \<in> I"
161     with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
162       by (auto intro!: cSup_eq_maximum convex_le_Inf_differential image_eqI[OF _ y] simp: interior_open) }
163   ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
164     by simp
165   also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
166   proof (rule cSup_least)
167     show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
168       using `I \<noteq> {}` by auto
169   next
170     fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
171     then guess x .. note x = this
172     have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
173       using prob_space by (simp add: X)
174     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
175       using `x \<in> I` `open I` X(2)
176       apply (intro integral_mono_AE integral_add integral_cmult integral_diff
177                 lebesgue_integral_const X q)
178       apply (elim eventually_elim1)
179       apply (intro convex_le_Inf_differential)
180       apply (auto simp: interior_open q)
181       done
182     finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
183   qed
184   finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
185 qed
187 subsection  {* Introduce binder for probability *}
189 syntax
190   "_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _'))")
192 translations
193   "\<P>(x in M. P)" => "CONST measure M {x \<in> CONST space M. P}"
195 definition
196   "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
198 syntax
199   "_conditional_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _ \<bar>/ _'))")
201 translations
202   "\<P>(x in M. P \<bar> Q)" => "CONST cond_prob M (\<lambda>x. P) (\<lambda>x. Q)"
204 lemma (in prob_space) AE_E_prob:
205   assumes ae: "AE x in M. P x"
206   obtains S where "S \<subseteq> {x \<in> space M. P x}" "S \<in> events" "prob S = 1"
207 proof -
208   from ae[THEN AE_E] guess N .
209   then show thesis
210     by (intro that[of "space M - N"])
211        (auto simp: prob_compl prob_space emeasure_eq_measure)
212 qed
214 lemma (in prob_space) prob_neg: "{x\<in>space M. P x} \<in> events \<Longrightarrow> \<P>(x in M. \<not> P x) = 1 - \<P>(x in M. P x)"
215   by (auto intro!: arg_cong[where f=prob] simp add: prob_compl[symmetric])
217 lemma (in prob_space) prob_eq_AE:
218   "(AE x in M. P x \<longleftrightarrow> Q x) \<Longrightarrow> {x\<in>space M. P x} \<in> events \<Longrightarrow> {x\<in>space M. Q x} \<in> events \<Longrightarrow> \<P>(x in M. P x) = \<P>(x in M. Q x)"
219   by (rule finite_measure_eq_AE) auto
221 lemma (in prob_space) prob_eq_0_AE:
222   assumes not: "AE x in M. \<not> P x" shows "\<P>(x in M. P x) = 0"
223 proof cases
224   assume "{x\<in>space M. P x} \<in> events"
225   with not have "\<P>(x in M. P x) = \<P>(x in M. False)"
226     by (intro prob_eq_AE) auto
227   then show ?thesis by simp
228 qed (simp add: measure_notin_sets)
230 lemma (in prob_space) prob_Collect_eq_0:
231   "{x \<in> space M. P x} \<in> sets M \<Longrightarrow> \<P>(x in M. P x) = 0 \<longleftrightarrow> (AE x in M. \<not> P x)"
232   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"] by (simp add: emeasure_eq_measure)
234 lemma (in prob_space) prob_Collect_eq_1:
235   "{x \<in> space M. P x} \<in> sets M \<Longrightarrow> \<P>(x in M. P x) = 1 \<longleftrightarrow> (AE x in M. P x)"
236   using AE_in_set_eq_1[of "{x\<in>space M. P x}"] by simp
238 lemma (in prob_space) prob_eq_0:
239   "A \<in> sets M \<Longrightarrow> prob A = 0 \<longleftrightarrow> (AE x in M. x \<notin> A)"
240   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. x \<notin> A"]
241   by (auto simp add: emeasure_eq_measure Int_def[symmetric])
243 lemma (in prob_space) prob_eq_1:
244   "A \<in> sets M \<Longrightarrow> prob A = 1 \<longleftrightarrow> (AE x in M. x \<in> A)"
245   using AE_in_set_eq_1[of A] by simp
247 lemma (in prob_space) prob_sums:
248   assumes P: "\<And>n. {x\<in>space M. P n x} \<in> events"
249   assumes Q: "{x\<in>space M. Q x} \<in> events"
250   assumes ae: "AE x in M. (\<forall>n. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. P n x))"
251   shows "(\<lambda>n. \<P>(x in M. P n x)) sums \<P>(x in M. Q x)"
252 proof -
253   from ae[THEN AE_E_prob] guess S . note S = this
254   then have disj: "disjoint_family (\<lambda>n. {x\<in>space M. P n x} \<inter> S)"
255     by (auto simp: disjoint_family_on_def)
256   from S have ae_S:
257     "AE x in M. x \<in> {x\<in>space M. Q x} \<longleftrightarrow> x \<in> (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
258     "\<And>n. AE x in M. x \<in> {x\<in>space M. P n x} \<longleftrightarrow> x \<in> {x\<in>space M. P n x} \<inter> S"
259     using ae by (auto dest!: AE_prob_1)
260   from ae_S have *:
261     "\<P>(x in M. Q x) = prob (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
262     using P Q S by (intro finite_measure_eq_AE) auto
263   from ae_S have **:
264     "\<And>n. \<P>(x in M. P n x) = prob ({x\<in>space M. P n x} \<inter> S)"
265     using P Q S by (intro finite_measure_eq_AE) auto
266   show ?thesis
267     unfolding * ** using S P disj
268     by (intro finite_measure_UNION) auto
269 qed
271 lemma (in prob_space) cond_prob_eq_AE:
272   assumes P: "AE x in M. Q x \<longrightarrow> P x \<longleftrightarrow> P' x" "{x\<in>space M. P x} \<in> events" "{x\<in>space M. P' x} \<in> events"
273   assumes Q: "AE x in M. Q x \<longleftrightarrow> Q' x" "{x\<in>space M. Q x} \<in> events" "{x\<in>space M. Q' x} \<in> events"
274   shows "cond_prob M P Q = cond_prob M P' Q'"
275   using P Q
276   by (auto simp: cond_prob_def intro!: arg_cong2[where f="op /"] prob_eq_AE sets.sets_Collect_conj)
279 lemma (in prob_space) joint_distribution_Times_le_fst:
280   "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
281     \<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"
282   by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
284 lemma (in prob_space) joint_distribution_Times_le_snd:
285   "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
286     \<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"
287   by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
289 locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
291 sublocale pair_prob_space \<subseteq> P: prob_space "M1 \<Otimes>\<^isub>M M2"
292 proof
293   show "emeasure (M1 \<Otimes>\<^isub>M M2) (space (M1 \<Otimes>\<^isub>M M2)) = 1"
294     by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
295 qed
297 locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
298   fixes I :: "'i set"
299   assumes prob_space: "\<And>i. prob_space (M i)"
301 sublocale product_prob_space \<subseteq> M: prob_space "M i" for i
302   by (rule prob_space)
304 locale finite_product_prob_space = finite_product_sigma_finite M I + product_prob_space M I for M I
306 sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
307 proof
308   show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (space (\<Pi>\<^isub>M i\<in>I. M i)) = 1"
309     by (simp add: measure_times M.emeasure_space_1 setprod_1 space_PiM)
310 qed
312 lemma (in finite_product_prob_space) prob_times:
313   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
314   shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
315 proof -
316   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)"
317     using X by (simp add: emeasure_eq_measure)
318   also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
319     using measure_times X by simp
320   also have "\<dots> = ereal (\<Prod>i\<in>I. measure (M i) (X i))"
321     using X by (simp add: M.emeasure_eq_measure setprod_ereal)
322   finally show ?thesis by simp
323 qed
325 section {* Distributions *}
327 definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and>
328   f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
330 lemma
331   assumes "distributed M N X f"
332   shows distributed_distr_eq_density: "distr M N X = density N f"
333     and distributed_measurable: "X \<in> measurable M N"
334     and distributed_borel_measurable: "f \<in> borel_measurable N"
335     and distributed_AE: "(AE x in N. 0 \<le> f x)"
336   using assms by (simp_all add: distributed_def)
338 lemma
339   assumes D: "distributed M N X f"
340   shows distributed_measurable'[measurable_dest]:
341       "g \<in> measurable L M \<Longrightarrow> (\<lambda>x. X (g x)) \<in> measurable L N"
342     and distributed_borel_measurable'[measurable_dest]:
343       "h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
344   using distributed_measurable[OF D] distributed_borel_measurable[OF D]
345   by simp_all
347 lemma
348   shows distributed_real_measurable: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
349     and distributed_real_AE: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> (AE x in N. 0 \<le> f x)"
350   by (simp_all add: distributed_def borel_measurable_ereal_iff)
352 lemma
353   assumes D: "distributed M N X (\<lambda>x. ereal (f x))"
354   shows distributed_real_measurable'[measurable_dest]:
355       "h \<in> measurable L N \<Longrightarrow> (\<lambda>x. f (h x)) \<in> borel_measurable L"
356   using distributed_real_measurable[OF D]
357   by simp_all
359 lemma
360   assumes D: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) f"
361   shows joint_distributed_measurable1[measurable_dest]:
362       "h1 \<in> measurable N M \<Longrightarrow> (\<lambda>x. X (h1 x)) \<in> measurable N S"
363     and joint_distributed_measurable2[measurable_dest]:
364       "h2 \<in> measurable N M \<Longrightarrow> (\<lambda>x. Y (h2 x)) \<in> measurable N T"
365   using measurable_compose[OF distributed_measurable[OF D] measurable_fst]
366   using measurable_compose[OF distributed_measurable[OF D] measurable_snd]
367   by auto
369 lemma distributed_count_space:
370   assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
371   shows "P a = emeasure M (X -` {a} \<inter> space M)"
372 proof -
373   have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
374     using X a A by (simp add: emeasure_distr)
375   also have "\<dots> = emeasure (density (count_space A) P) {a}"
376     using X by (simp add: distributed_distr_eq_density)
377   also have "\<dots> = (\<integral>\<^isup>+x. P a * indicator {a} x \<partial>count_space A)"
378     using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: positive_integral_cong)
379   also have "\<dots> = P a"
380     using X a by (subst positive_integral_cmult_indicator) (auto simp: distributed_def one_ereal_def[symmetric] AE_count_space)
381   finally show ?thesis ..
382 qed
384 lemma distributed_cong_density:
385   "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
386     distributed M N X f \<longleftrightarrow> distributed M N X g"
387   by (auto simp: distributed_def intro!: density_cong)
389 lemma subdensity:
390   assumes T: "T \<in> measurable P Q"
391   assumes f: "distributed M P X f"
392   assumes g: "distributed M Q Y g"
393   assumes Y: "Y = T \<circ> X"
394   shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
395 proof -
396   have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
397     using g Y by (auto simp: null_sets_density_iff distributed_def)
398   also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
399     using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
400   finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
401     using T by (subst (asm) null_sets_distr_iff) auto
402   also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
403     using T by (auto dest: measurable_space)
404   finally show ?thesis
405     using f g by (auto simp add: null_sets_density_iff distributed_def)
406 qed
408 lemma subdensity_real:
409   fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
410   assumes T: "T \<in> measurable P Q"
411   assumes f: "distributed M P X f"
412   assumes g: "distributed M Q Y g"
413   assumes Y: "Y = T \<circ> X"
414   shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
415   using subdensity[OF T, of M X "\<lambda>x. ereal (f x)" Y "\<lambda>x. ereal (g x)"] assms by auto
417 lemma distributed_emeasure:
418   "distributed M N X f \<Longrightarrow> A \<in> sets N \<Longrightarrow> emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+x. f x * indicator A x \<partial>N)"
419   by (auto simp: distributed_AE
420                  distributed_distr_eq_density[symmetric] emeasure_density[symmetric] emeasure_distr)
422 lemma distributed_positive_integral:
423   "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>\<^isup>+x. f x * g x \<partial>N) = (\<integral>\<^isup>+x. g (X x) \<partial>M)"
424   by (auto simp: distributed_AE
425                  distributed_distr_eq_density[symmetric] positive_integral_density[symmetric] positive_integral_distr)
427 lemma distributed_integral:
428   "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)"
429   by (auto simp: distributed_real_AE
430                  distributed_distr_eq_density[symmetric] integral_density[symmetric] integral_distr)
432 lemma distributed_transform_integral:
433   assumes Px: "distributed M N X Px"
434   assumes "distributed M P Y Py"
435   assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
436   shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
437 proof -
438   have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
439     by (rule distributed_integral) fact+
440   also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
441     using Y by simp
442   also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
443     using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
444   finally show ?thesis .
445 qed
447 lemma (in prob_space) distributed_unique:
448   assumes Px: "distributed M S X Px"
449   assumes Py: "distributed M S X Py"
450   shows "AE x in S. Px x = Py x"
451 proof -
452   interpret X: prob_space "distr M S X"
453     using Px by (intro prob_space_distr) simp
454   have "sigma_finite_measure (distr M S X)" ..
455   with sigma_finite_density_unique[of Px S Py ] Px Py
456   show ?thesis
457     by (auto simp: distributed_def)
458 qed
460 lemma (in prob_space) distributed_jointI:
461   assumes "sigma_finite_measure S" "sigma_finite_measure T"
462   assumes X[measurable]: "X \<in> measurable M S" and Y[measurable]: "Y \<in> measurable M T"
463   assumes [measurable]: "f \<in> borel_measurable (S \<Otimes>\<^isub>M T)" and f: "AE x in S \<Otimes>\<^isub>M T. 0 \<le> f x"
464   assumes eq: "\<And>A B. A \<in> sets S \<Longrightarrow> B \<in> sets T \<Longrightarrow>
465     emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B} = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. f (x, y) * indicator B y \<partial>T) * indicator A x \<partial>S)"
466   shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) f"
467   unfolding distributed_def
468 proof safe
469   interpret S: sigma_finite_measure S by fact
470   interpret T: sigma_finite_measure T by fact
471   interpret ST: pair_sigma_finite S T by default
473   from ST.sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('b \<times> 'c) set" .. note F = this
474   let ?E = "{a \<times> b |a b. a \<in> sets S \<and> b \<in> sets T}"
475   let ?P = "S \<Otimes>\<^isub>M T"
476   show "distr M ?P (\<lambda>x. (X x, Y x)) = density ?P f" (is "?L = ?R")
477   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of S T]])
478     show "?E \<subseteq> Pow (space ?P)"
479       using sets.space_closed[of S] sets.space_closed[of T] by (auto simp: space_pair_measure)
480     show "sets ?L = sigma_sets (space ?P) ?E"
481       by (simp add: sets_pair_measure space_pair_measure)
482     then show "sets ?R = sigma_sets (space ?P) ?E"
483       by simp
484   next
485     interpret L: prob_space ?L
486       by (rule prob_space_distr) (auto intro!: measurable_Pair)
487     show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?L (F i) \<noteq> \<infinity>"
488       using F by (auto simp: space_pair_measure)
489   next
490     fix E assume "E \<in> ?E"
491     then obtain A B where E[simp]: "E = A \<times> B"
492       and A[measurable]: "A \<in> sets S" and B[measurable]: "B \<in> sets T" by auto
493     have "emeasure ?L E = emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B}"
494       by (auto intro!: arg_cong[where f="emeasure M"] simp add: emeasure_distr measurable_Pair)
495     also have "\<dots> = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. (f (x, y) * indicator B y) * indicator A x \<partial>T) \<partial>S)"
496       using f by (auto simp add: eq positive_integral_multc intro!: positive_integral_cong)
497     also have "\<dots> = emeasure ?R E"
498       by (auto simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric]
499                intro!: positive_integral_cong split: split_indicator)
500     finally show "emeasure ?L E = emeasure ?R E" .
501   qed
502 qed (auto simp: f)
504 lemma (in prob_space) distributed_swap:
505   assumes "sigma_finite_measure S" "sigma_finite_measure T"
506   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
507   shows "distributed M (T \<Otimes>\<^isub>M S) (\<lambda>x. (Y x, X x)) (\<lambda>(x, y). Pxy (y, x))"
508 proof -
509   interpret S: sigma_finite_measure S by fact
510   interpret T: sigma_finite_measure T by fact
511   interpret ST: pair_sigma_finite S T by default
512   interpret TS: pair_sigma_finite T S by default
514   note Pxy[measurable]
515   show ?thesis
516     apply (subst TS.distr_pair_swap)
517     unfolding distributed_def
518   proof safe
519     let ?D = "distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x))"
520     show 1: "(\<lambda>(x, y). Pxy (y, x)) \<in> borel_measurable ?D"
521       by auto
522     with Pxy
523     show "AE x in distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x)). 0 \<le> (case x of (x, y) \<Rightarrow> Pxy (y, x))"
524       by (subst AE_distr_iff)
525          (auto dest!: distributed_AE
526                simp: measurable_split_conv split_beta
527                intro!: measurable_Pair)
528     show 2: "random_variable (distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x))) (\<lambda>x. (Y x, X x))"
529       using Pxy by auto
530     { fix A assume A: "A \<in> sets (T \<Otimes>\<^isub>M S)"
531       let ?B = "(\<lambda>(x, y). (y, x)) -` A \<inter> space (S \<Otimes>\<^isub>M T)"
532       from sets.sets_into_space[OF A]
533       have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
534         emeasure M ((\<lambda>x. (X x, Y x)) -` ?B \<inter> space M)"
535         by (auto intro!: arg_cong2[where f=emeasure] simp: space_pair_measure)
536       also have "\<dots> = (\<integral>\<^isup>+ x. Pxy x * indicator ?B x \<partial>(S \<Otimes>\<^isub>M T))"
537         using Pxy A by (intro distributed_emeasure) auto
538       finally have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
539         (\<integral>\<^isup>+ x. Pxy x * indicator A (snd x, fst x) \<partial>(S \<Otimes>\<^isub>M T))"
540         by (auto intro!: positive_integral_cong split: split_indicator) }
541     note * = this
542     show "distr M ?D (\<lambda>x. (Y x, X x)) = density ?D (\<lambda>(x, y). Pxy (y, x))"
543       apply (intro measure_eqI)
544       apply (simp_all add: emeasure_distr[OF 2] emeasure_density[OF 1])
545       apply (subst positive_integral_distr)
546       apply (auto intro!: * simp: comp_def split_beta)
547       done
548   qed
549 qed
551 lemma (in prob_space) distr_marginal1:
552   assumes "sigma_finite_measure S" "sigma_finite_measure T"
553   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
554   defines "Px \<equiv> \<lambda>x. (\<integral>\<^isup>+z. Pxy (x, z) \<partial>T)"
555   shows "distributed M S X Px"
556   unfolding distributed_def
557 proof safe
558   interpret S: sigma_finite_measure S by fact
559   interpret T: sigma_finite_measure T by fact
560   interpret ST: pair_sigma_finite S T by default
562   note Pxy[measurable]
563   show X: "X \<in> measurable M S" by simp
565   show borel: "Px \<in> borel_measurable S"
566     by (auto intro!: T.positive_integral_fst_measurable simp: Px_def)
568   interpret Pxy: prob_space "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
569     by (intro prob_space_distr) simp
570   have "(\<integral>\<^isup>+ x. max 0 (- Pxy x) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>\<^isup>+ x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
571     using Pxy
572     by (intro positive_integral_cong_AE) (auto simp: max_def dest: distributed_AE)
574   show "distr M S X = density S Px"
575   proof (rule measure_eqI)
576     fix A assume A: "A \<in> sets (distr M S X)"
577     with X measurable_space[of Y M T]
578     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)"
579       by (auto simp add: emeasure_distr intro!: arg_cong[where f="emeasure M"])
580     also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (A \<times> space T)"
581       using Pxy by (simp add: distributed_def)
582     also have "\<dots> = \<integral>\<^isup>+ x. \<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
583       using A borel Pxy
584       by (simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric])
585     also have "\<dots> = \<integral>\<^isup>+ x. Px x * indicator A x \<partial>S"
586       apply (rule positive_integral_cong_AE)
587       using Pxy[THEN distributed_AE, THEN ST.AE_pair] AE_space
588     proof eventually_elim
589       fix x assume "x \<in> space S" "AE y in T. 0 \<le> Pxy (x, y)"
590       moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
591         by (auto simp: indicator_def)
592       ultimately have "(\<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = (\<integral>\<^isup>+ y. Pxy (x, y) \<partial>T) * indicator A x"
593         by (simp add: eq positive_integral_multc cong: positive_integral_cong)
594       also have "(\<integral>\<^isup>+ y. Pxy (x, y) \<partial>T) = Px x"
595         by (simp add: Px_def ereal_real positive_integral_positive)
596       finally show "(\<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = Px x * indicator A x" .
597     qed
598     finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
599       using A borel Pxy by (simp add: emeasure_density)
600   qed simp
602   show "AE x in S. 0 \<le> Px x"
603     by (simp add: Px_def positive_integral_positive real_of_ereal_pos)
604 qed
606 lemma (in prob_space) distr_marginal2:
607   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
608   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
609   shows "distributed M T Y (\<lambda>y. (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S))"
610   using distr_marginal1[OF T S distributed_swap[OF S T]] Pxy by simp
612 lemma (in prob_space) distributed_marginal_eq_joint1:
613   assumes T: "sigma_finite_measure T"
614   assumes S: "sigma_finite_measure S"
615   assumes Px: "distributed M S X Px"
616   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
617   shows "AE x in S. Px x = (\<integral>\<^isup>+y. Pxy (x, y) \<partial>T)"
618   using Px distr_marginal1[OF S T Pxy] by (rule distributed_unique)
620 lemma (in prob_space) distributed_marginal_eq_joint2:
621   assumes T: "sigma_finite_measure T"
622   assumes S: "sigma_finite_measure S"
623   assumes Py: "distributed M T Y Py"
624   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
625   shows "AE y in T. Py y = (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S)"
626   using Py distr_marginal2[OF S T Pxy] by (rule distributed_unique)
628 lemma (in prob_space) distributed_joint_indep':
629   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
630   assumes X[measurable]: "distributed M S X Px" and Y[measurable]: "distributed M T Y Py"
631   assumes indep: "distr M S X \<Otimes>\<^isub>M distr M T Y = distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
632   shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
633   unfolding distributed_def
634 proof safe
635   interpret S: sigma_finite_measure S by fact
636   interpret T: sigma_finite_measure T by fact
637   interpret ST: pair_sigma_finite S T by default
639   interpret X: prob_space "density S Px"
640     unfolding distributed_distr_eq_density[OF X, symmetric]
641     by (rule prob_space_distr) simp
642   have sf_X: "sigma_finite_measure (density S Px)" ..
644   interpret Y: prob_space "density T Py"
645     unfolding distributed_distr_eq_density[OF Y, symmetric]
646     by (rule prob_space_distr) simp
647   have sf_Y: "sigma_finite_measure (density T Py)" ..
649   show "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) = density (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). Px x * Py y)"
650     unfolding indep[symmetric] distributed_distr_eq_density[OF X] distributed_distr_eq_density[OF Y]
651     using distributed_borel_measurable[OF X] distributed_AE[OF X]
652     using distributed_borel_measurable[OF Y] distributed_AE[OF Y]
653     by (rule pair_measure_density[OF _ _ _ _ T sf_Y])
655   show "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))" by auto
657   show Pxy: "(\<lambda>(x, y). Px x * Py y) \<in> borel_measurable (S \<Otimes>\<^isub>M T)" by auto
659   show "AE x in S \<Otimes>\<^isub>M T. 0 \<le> (case x of (x, y) \<Rightarrow> Px x * Py y)"
660     apply (intro ST.AE_pair_measure borel_measurable_le Pxy borel_measurable_const)
661     using distributed_AE[OF X]
662     apply eventually_elim
663     using distributed_AE[OF Y]
664     apply eventually_elim
665     apply auto
666     done
667 qed
669 definition
670   "simple_distributed M X f \<longleftrightarrow> distributed M (count_space (X`space M)) X (\<lambda>x. ereal (f x)) \<and>
671     finite (X`space M)"
673 lemma simple_distributed:
674   "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
675   unfolding simple_distributed_def by auto
677 lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
678   by (simp add: simple_distributed_def)
680 lemma (in prob_space) distributed_simple_function_superset:
681   assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
682   assumes A: "X`space M \<subseteq> A" "finite A"
683   defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
684   shows "distributed M S X P'"
685   unfolding distributed_def
686 proof safe
687   show "(\<lambda>x. ereal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
688   show "AE x in S. 0 \<le> ereal (P' x)"
689     using X by (auto simp: S_def P'_def simple_distributed_def intro!: measure_nonneg)
690   show "distr M S X = density S P'"
691   proof (rule measure_eqI_finite)
692     show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
693       using A unfolding S_def by auto
694     show "finite A" by fact
695     fix a assume a: "a \<in> A"
696     then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
697     with A a X have "emeasure (distr M S X) {a} = P' a"
698       by (subst emeasure_distr)
699          (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure measurable_count_space_eq2
700                intro!: arg_cong[where f=prob])
701     also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' a) * indicator {a} x \<partial>S)"
702       using A X a
703       by (subst positive_integral_cmult_indicator)
704          (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
705     also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' x) * indicator {a} x \<partial>S)"
706       by (auto simp: indicator_def intro!: positive_integral_cong)
707     also have "\<dots> = emeasure (density S P') {a}"
708       using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
709     finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
710   qed
711   show "random_variable S X"
712     using X(1) A by (auto simp: measurable_def simple_functionD S_def)
713 qed
715 lemma (in prob_space) simple_distributedI:
716   assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
717   shows "simple_distributed M X P"
718   unfolding simple_distributed_def
719 proof
720   have "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then P x else 0))"
721     (is "?A")
722     using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X]) auto
723   also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (P x))"
724     by (rule distributed_cong_density) auto
725   finally show "\<dots>" .
726 qed (rule simple_functionD[OF X(1)])
728 lemma simple_distributed_joint_finite:
729   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
730   shows "finite (X ` space M)" "finite (Y ` space M)"
731 proof -
732   have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
733     using X by (auto simp: simple_distributed_def simple_functionD)
734   then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
735     by auto
736   then show fin: "finite (X ` space M)" "finite (Y ` space M)"
737     by (auto simp: image_image)
738 qed
740 lemma simple_distributed_joint2_finite:
741   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
742   shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
743 proof -
744   have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
745     using X by (auto simp: simple_distributed_def simple_functionD)
746   then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
747     "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
748     "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
749     by auto
750   then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
751     by (auto simp: image_image)
752 qed
754 lemma simple_distributed_simple_function:
755   "simple_distributed M X Px \<Longrightarrow> simple_function M X"
756   unfolding simple_distributed_def distributed_def
757   by (auto simp: simple_function_def measurable_count_space_eq2)
759 lemma simple_distributed_measure:
760   "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
761   using distributed_count_space[of M "X`space M" X P a, symmetric]
762   by (auto simp: simple_distributed_def measure_def)
764 lemma simple_distributed_nonneg: "simple_distributed M X f \<Longrightarrow> x \<in> space M \<Longrightarrow> 0 \<le> f (X x)"
765   by (auto simp: simple_distributed_measure measure_nonneg)
767 lemma (in prob_space) simple_distributed_joint:
768   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
769   defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)"
770   defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
771   shows "distributed M S (\<lambda>x. (X x, Y x)) P"
772 proof -
773   from simple_distributed_joint_finite[OF X, simp]
774   have S_eq: "S = count_space (X`space M \<times> Y`space M)"
775     by (simp add: S_def pair_measure_count_space)
776   show ?thesis
777     unfolding S_eq P_def
778   proof (rule distributed_simple_function_superset)
779     show "simple_function M (\<lambda>x. (X x, Y x))"
780       using X by (rule simple_distributed_simple_function)
781     fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
782     from simple_distributed_measure[OF X this]
783     show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
784   qed auto
785 qed
787 lemma (in prob_space) simple_distributed_joint2:
788   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
789   defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M) \<Otimes>\<^isub>M count_space (Z`space M)"
790   defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
791   shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
792 proof -
793   from simple_distributed_joint2_finite[OF X, simp]
794   have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
795     by (simp add: S_def pair_measure_count_space)
796   show ?thesis
797     unfolding S_eq P_def
798   proof (rule distributed_simple_function_superset)
799     show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
800       using X by (rule simple_distributed_simple_function)
801     fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
802     from simple_distributed_measure[OF X this]
803     show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
804   qed auto
805 qed
807 lemma (in prob_space) simple_distributed_setsum_space:
808   assumes X: "simple_distributed M X f"
809   shows "setsum f (X`space M) = 1"
810 proof -
811   from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
812     by (subst finite_measure_finite_Union)
813        (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
814              intro!: setsum_cong arg_cong[where f="prob"])
815   also have "\<dots> = prob (space M)"
816     by (auto intro!: arg_cong[where f=prob])
817   finally show ?thesis
818     using emeasure_space_1 by (simp add: emeasure_eq_measure one_ereal_def)
819 qed
821 lemma (in prob_space) distributed_marginal_eq_joint_simple:
822   assumes Px: "simple_function M X"
823   assumes Py: "simple_distributed M Y Py"
824   assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
825   assumes y: "y \<in> Y`space M"
826   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)"
827 proof -
828   note Px = simple_distributedI[OF Px refl]
829   have *: "\<And>f A. setsum (\<lambda>x. max 0 (ereal (f x))) A = ereal (setsum (\<lambda>x. max 0 (f x)) A)"
830     by (simp add: setsum_ereal[symmetric] zero_ereal_def)
831   from distributed_marginal_eq_joint2[OF
832     sigma_finite_measure_count_space_finite
833     sigma_finite_measure_count_space_finite
834     simple_distributed[OF Py] simple_distributed_joint[OF Pxy],
835     OF Py[THEN simple_distributed_finite] Px[THEN simple_distributed_finite]]
836     y
837     Px[THEN simple_distributed_finite]
838     Py[THEN simple_distributed_finite]
839     Pxy[THEN simple_distributed, THEN distributed_real_AE]
840   show ?thesis
841     unfolding AE_count_space
842     apply (auto simp add: positive_integral_count_space_finite * intro!: setsum_cong split: split_max)
843     done
844 qed
846 lemma distributedI_real:
847   fixes f :: "'a \<Rightarrow> real"
848   assumes gen: "sets M1 = sigma_sets (space M1) E" and "Int_stable E"
849     and A: "range A \<subseteq> E" "(\<Union>i::nat. A i) = space M1" "\<And>i. emeasure (distr M M1 X) (A i) \<noteq> \<infinity>"
850     and X: "X \<in> measurable M M1"
851     and f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x"
852     and eq: "\<And>A. A \<in> E \<Longrightarrow> emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M1)"
853   shows "distributed M M1 X f"
854   unfolding distributed_def
855 proof (intro conjI)
856   show "distr M M1 X = density M1 f"
857   proof (rule measure_eqI_generator_eq[where A=A])
858     { fix A assume A: "A \<in> E"
859       then have "A \<in> sigma_sets (space M1) E" by auto
860       then have "A \<in> sets M1"
861         using gen by simp
862       with f A eq[of A] X show "emeasure (distr M M1 X) A = emeasure (density M1 f) A"
863         by (simp add: emeasure_distr emeasure_density borel_measurable_ereal
864                       times_ereal.simps[symmetric] ereal_indicator
865                  del: times_ereal.simps) }
866     note eq_E = this
867     show "Int_stable E" by fact
868     { fix e assume "e \<in> E"
869       then have "e \<in> sigma_sets (space M1) E" by auto
870       then have "e \<in> sets M1" unfolding gen .
871       then have "e \<subseteq> space M1" by (rule sets.sets_into_space) }
872     then show "E \<subseteq> Pow (space M1)" by auto
873     show "sets (distr M M1 X) = sigma_sets (space M1) E"
874       "sets (density M1 (\<lambda>x. ereal (f x))) = sigma_sets (space M1) E"
875       unfolding gen[symmetric] by auto
876   qed fact+
877 qed (insert X f, auto)
879 lemma distributedI_borel_atMost:
880   fixes f :: "real \<Rightarrow> real"
881   assumes [measurable]: "X \<in> borel_measurable M"
882     and [measurable]: "f \<in> borel_measurable borel" and f[simp]: "AE x in lborel. 0 \<le> f x"
883     and g_eq: "\<And>a. (\<integral>\<^isup>+x. f x * indicator {..a} x \<partial>lborel)  = ereal (g a)"
884     and M_eq: "\<And>a. emeasure M {x\<in>space M. X x \<le> a} = ereal (g a)"
885   shows "distributed M lborel X f"
886 proof (rule distributedI_real)
887   show "sets lborel = sigma_sets (space lborel) (range atMost)"
888     by (simp add: borel_eq_atMost)
889   show "Int_stable (range atMost :: real set set)"
890     by (auto simp: Int_stable_def)
891   have vimage_eq: "\<And>a. (X -` {..a} \<inter> space M) = {x\<in>space M. X x \<le> a}" by auto
892   def A \<equiv> "\<lambda>i::nat. {.. real i}"
893   then show "range A \<subseteq> range atMost" "(\<Union>i. A i) = space lborel"
894     "\<And>i. emeasure (distr M lborel X) (A i) \<noteq> \<infinity>"
895     by (auto simp: real_arch_simple emeasure_distr vimage_eq M_eq)
897   fix A :: "real set" assume "A \<in> range atMost"
898   then obtain a where A: "A = {..a}" by auto
899   show "emeasure M (X -` A \<inter> space M) = (\<integral>\<^isup>+x. f x * indicator A x \<partial>lborel)"
900     unfolding vimage_eq A M_eq g_eq ..
901 qed auto
903 lemma (in prob_space) uniform_distributed_params:
904   assumes X: "distributed M MX X (\<lambda>x. indicator A x / measure MX A)"
905   shows "A \<in> sets MX" "measure MX A \<noteq> 0"
906 proof -
907   interpret X: prob_space "distr M MX X"
908     using distributed_measurable[OF X] by (rule prob_space_distr)
910   show "measure MX A \<noteq> 0"
911   proof
912     assume "measure MX A = 0"
913     with X.emeasure_space_1 X.prob_space distributed_distr_eq_density[OF X]
914     show False
915       by (simp add: emeasure_density zero_ereal_def[symmetric])
916   qed
917   with measure_notin_sets[of A MX] show "A \<in> sets MX"
918     by blast
919 qed
921 lemma prob_space_uniform_measure:
922   assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
923   shows "prob_space (uniform_measure M A)"
924 proof
925   show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
926     using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
927     using sets.sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
928     by (simp add: Int_absorb2 emeasure_nonneg)
929 qed
931 lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
932   by default (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ereal_def)
934 end