src/HOL/Probability/Probability_Measure.thy
 author hoelzl Fri Nov 02 14:23:40 2012 +0100 (2012-11-02) changeset 50002 ce0d316b5b44 parent 50001 382bd3173584 child 50003 8c213922ed49 permissions -rw-r--r--
add measurability prover; add support for Borel sets
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 lemma funset_eq_UN_fun_upd_I:
13   assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
14   and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
15   and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
16   shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
17 proof safe
18   fix f assume f: "f \<in> F (insert a A)"
19   show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
20   proof (rule UN_I[of "f(a := d)"])
21     show "f(a := d) \<in> F A" using *[OF f] .
22     show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
23     proof (rule image_eqI[of _ _ "f a"])
24       show "f a \<in> G (f(a := d))" using **[OF f] .
25     qed simp
26   qed
27 next
28   fix f x assume "f \<in> F A" "x \<in> G f"
29   from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
30 qed
32 lemma extensional_funcset_insert_eq[simp]:
33   assumes "a \<notin> A"
34   shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)"
35   apply (rule funset_eq_UN_fun_upd_I)
36   using assms
37   by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
39 lemma finite_extensional_funcset[simp, intro]:
40   assumes "finite A" "finite B"
41   shows "finite (extensional A \<inter> (A \<rightarrow> B))"
42   using assms by induct auto
44 lemma finite_PiE[simp, intro]:
45   assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
46   shows "finite (Pi\<^isub>E A B)"
47 proof -
48   have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
49   show ?thesis
50     using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
51 qed
54 lemma countably_additiveI[case_names countably]:
55   assumes "\<And>A. \<lbrakk> range A \<subseteq> M ; disjoint_family A ; (\<Union>i. A i) \<in> M\<rbrakk> \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
56   shows "countably_additive M \<mu>"
57   using assms unfolding countably_additive_def by auto
59 lemma convex_le_Inf_differential:
60   fixes f :: "real \<Rightarrow> real"
61   assumes "convex_on I f"
62   assumes "x \<in> interior I" "y \<in> I"
63   shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
64     (is "_ \<ge> _ + Inf (?F x) * (y - x)")
65 proof -
66   show ?thesis
67   proof (cases rule: linorder_cases)
68     assume "x < y"
69     moreover
70     have "open (interior I)" by auto
71     from openE[OF this `x \<in> interior I`] guess e . note e = this
72     moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
73     ultimately have "x < t" "t < y" "t \<in> ball x e"
74       by (auto simp: dist_real_def field_simps split: split_min)
75     with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
77     have "open (interior I)" by auto
78     from openE[OF this `x \<in> interior I`] guess e .
79     moreover def K \<equiv> "x - e / 2"
80     with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: dist_real_def)
81     ultimately have "K \<in> I" "K < x" "x \<in> I"
82       using interior_subset[of I] `x \<in> interior I` by auto
84     have "Inf (?F x) \<le> (f x - f y) / (x - y)"
85     proof (rule Inf_lower2)
86       show "(f x - f t) / (x - t) \<in> ?F x"
87         using `t \<in> I` `x < t` by auto
88       show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
89         using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
90     next
91       fix y assume "y \<in> ?F x"
92       with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
93       show "(f K - f x) / (K - x) \<le> y" by auto
94     qed
95     then show ?thesis
96       using `x < y` by (simp add: field_simps)
97   next
98     assume "y < x"
99     moreover
100     have "open (interior I)" by auto
101     from openE[OF this `x \<in> interior I`] guess e . note e = this
102     moreover def t \<equiv> "x + e / 2"
103     ultimately have "x < t" "t \<in> ball x e"
104       by (auto simp: dist_real_def field_simps)
105     with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
107     have "(f x - f y) / (x - y) \<le> Inf (?F x)"
108     proof (rule Inf_greatest)
109       have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
110         using `y < x` by (auto simp: field_simps)
111       also
112       fix z  assume "z \<in> ?F x"
113       with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
114       have "(f y - f x) / (y - x) \<le> z" by auto
115       finally show "(f x - f y) / (x - y) \<le> z" .
116     next
117       have "open (interior I)" by auto
118       from openE[OF this `x \<in> interior I`] guess e . note e = this
119       then have "x + e / 2 \<in> ball x e" by (auto simp: dist_real_def)
120       with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
121       then show "?F x \<noteq> {}" by blast
122     qed
123     then show ?thesis
124       using `y < x` by (simp add: field_simps)
125   qed simp
126 qed
128 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
129   by (rule measure_eqI) (auto simp: emeasure_distr)
131 locale prob_space = finite_measure +
132   assumes emeasure_space_1: "emeasure M (space M) = 1"
134 lemma prob_spaceI[Pure.intro!]:
135   assumes *: "emeasure M (space M) = 1"
136   shows "prob_space M"
137 proof -
138   interpret finite_measure M
139   proof
140     show "emeasure M (space M) \<noteq> \<infinity>" using * by simp
141   qed
142   show "prob_space M" by default fact
143 qed
145 abbreviation (in prob_space) "events \<equiv> sets M"
146 abbreviation (in prob_space) "prob \<equiv> measure M"
147 abbreviation (in prob_space) "random_variable M' X \<equiv> X \<in> measurable M M'"
148 abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
150 lemma (in prob_space) prob_space_distr:
151   assumes f: "f \<in> measurable M M'" shows "prob_space (distr M M' f)"
152 proof (rule prob_spaceI)
153   have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
154   with f show "emeasure (distr M M' f) (space (distr M M' f)) = 1"
155     by (auto simp: emeasure_distr emeasure_space_1)
156 qed
158 lemma (in prob_space) prob_space: "prob (space M) = 1"
159   using emeasure_space_1 unfolding measure_def by (simp add: one_ereal_def)
161 lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
162   using bounded_measure[of A] by (simp add: prob_space)
164 lemma (in prob_space) not_empty: "space M \<noteq> {}"
165   using prob_space by auto
167 lemma (in prob_space) measure_le_1: "emeasure M X \<le> 1"
168   using emeasure_space[of M X] by (simp add: emeasure_space_1)
170 lemma (in prob_space) AE_I_eq_1:
171   assumes "emeasure M {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
172   shows "AE x in M. P x"
173 proof (rule AE_I)
174   show "emeasure M (space M - {x \<in> space M. P x}) = 0"
175     using assms emeasure_space_1 by (simp add: emeasure_compl)
176 qed (insert assms, auto)
178 lemma (in prob_space) prob_compl:
179   assumes A: "A \<in> events"
180   shows "prob (space M - A) = 1 - prob A"
181   using finite_measure_compl[OF A] by (simp add: prob_space)
183 lemma (in prob_space) AE_in_set_eq_1:
184   assumes "A \<in> events" shows "(AE x in M. x \<in> A) \<longleftrightarrow> prob A = 1"
185 proof
186   assume ae: "AE x in M. x \<in> A"
187   have "{x \<in> space M. x \<in> A} = A" "{x \<in> space M. x \<notin> A} = space M - A"
188     using `A \<in> events`[THEN sets_into_space] by auto
189   with AE_E2[OF ae] `A \<in> events` have "1 - emeasure M A = 0"
190     by (simp add: emeasure_compl emeasure_space_1)
191   then show "prob A = 1"
192     using `A \<in> events` by (simp add: emeasure_eq_measure one_ereal_def)
193 next
194   assume prob: "prob A = 1"
195   show "AE x in M. x \<in> A"
196   proof (rule AE_I)
197     show "{x \<in> space M. x \<notin> A} \<subseteq> space M - A" by auto
198     show "emeasure M (space M - A) = 0"
199       using `A \<in> events` prob
200       by (simp add: prob_compl emeasure_space_1 emeasure_eq_measure one_ereal_def)
201     show "space M - A \<in> events"
202       using `A \<in> events` by auto
203   qed
204 qed
206 lemma (in prob_space) AE_False: "(AE x in M. False) \<longleftrightarrow> False"
207 proof
208   assume "AE x in M. False"
209   then have "AE x in M. x \<in> {}" by simp
210   then show False
211     by (subst (asm) AE_in_set_eq_1) auto
212 qed simp
214 lemma (in prob_space) AE_prob_1:
215   assumes "prob A = 1" shows "AE x in M. x \<in> A"
216 proof -
217   from `prob A = 1` have "A \<in> events"
218     by (metis measure_notin_sets zero_neq_one)
219   with AE_in_set_eq_1 assms show ?thesis by simp
220 qed
222 lemma (in finite_measure) prob_space_increasing: "increasing M (measure M)"
223   by (auto intro!: finite_measure_mono simp: increasing_def)
225 lemma (in finite_measure) prob_zero_union:
226   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"
227   shows "measure M (s \<union> t) = measure M s"
228 using assms
229 proof -
230   have "measure M (s \<union> t) \<le> measure M s"
231     using finite_measure_subadditive[of s t] assms by auto
232   moreover have "measure M (s \<union> t) \<ge> measure M s"
233     using assms by (blast intro: finite_measure_mono)
234   ultimately show ?thesis by simp
235 qed
237 lemma (in finite_measure) prob_eq_compl:
238   assumes "s \<in> sets M" "t \<in> sets M"
239   assumes "measure M (space M - s) = measure M (space M - t)"
240   shows "measure M s = measure M t"
241   using assms finite_measure_compl by auto
243 lemma (in prob_space) prob_one_inter:
244   assumes events:"s \<in> events" "t \<in> events"
245   assumes "prob t = 1"
246   shows "prob (s \<inter> t) = prob s"
247 proof -
248   have "prob ((space M - s) \<union> (space M - t)) = prob (space M - s)"
249     using events assms  prob_compl[of "t"] by (auto intro!: prob_zero_union)
250   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
251     by blast
252   finally show "prob (s \<inter> t) = prob s"
253     using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s])
254 qed
256 lemma (in finite_measure) prob_eq_bigunion_image:
257   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"
258   assumes "disjoint_family f" "disjoint_family g"
259   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"
260   shows "measure M (\<Union> i. f i) = measure M (\<Union> i. g i)"
261 using assms
262 proof -
263   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union> i. f i))"
264     by (rule finite_measure_UNION[OF assms(1,3)])
265   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union> i. g i))"
266     by (rule finite_measure_UNION[OF assms(2,4)])
267   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
268 qed
270 lemma (in finite_measure) prob_countably_zero:
271   assumes "range c \<subseteq> sets M"
272   assumes "\<And> i. measure M (c i) = 0"
273   shows "measure M (\<Union> i :: nat. c i) = 0"
274 proof (rule antisym)
275   show "measure M (\<Union> i :: nat. c i) \<le> 0"
276     using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))
277 qed (simp add: measure_nonneg)
279 lemma (in prob_space) prob_equiprobable_finite_unions:
280   assumes "s \<in> events"
281   assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events"
282   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})"
283   shows "prob s = real (card s) * prob {SOME x. x \<in> s}"
284 proof (cases "s = {}")
285   case False hence "\<exists> x. x \<in> s" by blast
286   from someI_ex[OF this] assms
287   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
288   have "prob s = (\<Sum> x \<in> s. prob {x})"
289     using finite_measure_eq_setsum_singleton[OF s_finite] by simp
290   also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
291   also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
292     using setsum_constant assms by (simp add: real_eq_of_nat)
293   finally show ?thesis by simp
294 qed simp
296 lemma (in prob_space) prob_real_sum_image_fn:
297   assumes "e \<in> events"
298   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events"
299   assumes "finite s"
300   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
301   assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)"
302   shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))"
303 proof -
304   have e: "e = (\<Union> i \<in> s. e \<inter> f i)"
305     using `e \<in> events` sets_into_space upper by blast
306   hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp
307   also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
308   proof (rule finite_measure_finite_Union)
309     show "finite s" by fact
310     show "(\<lambda>i. e \<inter> f i)`s \<subseteq> events" using assms(2) by auto
311     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
312       using disjoint by (auto simp: disjoint_family_on_def)
313   qed
314   finally show ?thesis .
315 qed
317 lemma (in prob_space) expectation_less:
318   assumes [simp]: "integrable M X"
319   assumes gt: "AE x in M. X x < b"
320   shows "expectation X < b"
321 proof -
322   have "expectation X < expectation (\<lambda>x. b)"
323     using gt emeasure_space_1
324     by (intro integral_less_AE_space) auto
325   then show ?thesis using prob_space by simp
326 qed
328 lemma (in prob_space) expectation_greater:
329   assumes [simp]: "integrable M X"
330   assumes gt: "AE x in M. a < X x"
331   shows "a < expectation X"
332 proof -
333   have "expectation (\<lambda>x. a) < expectation X"
334     using gt emeasure_space_1
335     by (intro integral_less_AE_space) auto
336   then show ?thesis using prob_space by simp
337 qed
339 lemma (in prob_space) jensens_inequality:
340   fixes a b :: real
341   assumes X: "integrable M X" "AE x in M. X x \<in> I"
342   assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
343   assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
344   shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
345 proof -
346   let ?F = "\<lambda>x. Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))"
347   from X(2) AE_False have "I \<noteq> {}" by auto
349   from I have "open I" by auto
351   note I
352   moreover
353   { assume "I \<subseteq> {a <..}"
354     with X have "a < expectation X"
355       by (intro expectation_greater) auto }
356   moreover
357   { assume "I \<subseteq> {..< b}"
358     with X have "expectation X < b"
359       by (intro expectation_less) auto }
360   ultimately have "expectation X \<in> I"
361     by (elim disjE)  (auto simp: subset_eq)
362   moreover
363   { fix y assume y: "y \<in> I"
364     with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
365       by (auto intro!: Sup_eq_maximum convex_le_Inf_differential image_eqI[OF _ y] simp: interior_open) }
366   ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
367     by simp
368   also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
369   proof (rule Sup_least)
370     show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
371       using `I \<noteq> {}` by auto
372   next
373     fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
374     then guess x .. note x = this
375     have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
376       using prob_space by (simp add: X)
377     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
378       using `x \<in> I` `open I` X(2)
379       apply (intro integral_mono_AE integral_add integral_cmult integral_diff
380                 lebesgue_integral_const X q)
381       apply (elim eventually_elim1)
382       apply (intro convex_le_Inf_differential)
383       apply (auto simp: interior_open q)
384       done
385     finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
386   qed
387   finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
388 qed
390 lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0:
391   assumes "{x} \<in> events"
392   assumes "prob {x} = 1"
393   assumes "{y} \<in> events"
394   assumes "y \<noteq> x"
395   shows "prob {y} = 0"
396   using prob_one_inter[of "{y}" "{x}"] assms by auto
398 subsection  {* Introduce binder for probability *}
400 syntax
401   "_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _'))")
403 translations
404   "\<P>(x in M. P)" => "CONST measure M {x \<in> CONST space M. P}"
406 definition
407   "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
409 syntax
410   "_conditional_prob" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("('\<P>'(_ in _. _ \<bar>/ _'))")
412 translations
413   "\<P>(x in M. P \<bar> Q)" => "CONST cond_prob M (\<lambda>x. P) (\<lambda>x. Q)"
415 lemma (in prob_space) AE_E_prob:
416   assumes ae: "AE x in M. P x"
417   obtains S where "S \<subseteq> {x \<in> space M. P x}" "S \<in> events" "prob S = 1"
418 proof -
419   from ae[THEN AE_E] guess N .
420   then show thesis
421     by (intro that[of "space M - N"])
422        (auto simp: prob_compl prob_space emeasure_eq_measure)
423 qed
425 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)"
426   by (auto intro!: arg_cong[where f=prob] simp add: prob_compl[symmetric])
428 lemma (in prob_space) prob_eq_AE:
429   "(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)"
430   by (rule finite_measure_eq_AE) auto
432 lemma (in prob_space) prob_eq_0_AE:
433   assumes not: "AE x in M. \<not> P x" shows "\<P>(x in M. P x) = 0"
434 proof cases
435   assume "{x\<in>space M. P x} \<in> events"
436   with not have "\<P>(x in M. P x) = \<P>(x in M. False)"
437     by (intro prob_eq_AE) auto
438   then show ?thesis by simp
439 qed (simp add: measure_notin_sets)
441 lemma (in prob_space) prob_sums:
442   assumes P: "\<And>n. {x\<in>space M. P n x} \<in> events"
443   assumes Q: "{x\<in>space M. Q x} \<in> events"
444   assumes ae: "AE x in M. (\<forall>n. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. P n x))"
445   shows "(\<lambda>n. \<P>(x in M. P n x)) sums \<P>(x in M. Q x)"
446 proof -
447   from ae[THEN AE_E_prob] guess S . note S = this
448   then have disj: "disjoint_family (\<lambda>n. {x\<in>space M. P n x} \<inter> S)"
449     by (auto simp: disjoint_family_on_def)
450   from S have ae_S:
451     "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)"
452     "\<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"
453     using ae by (auto dest!: AE_prob_1)
454   from ae_S have *:
455     "\<P>(x in M. Q x) = prob (\<Union>n. {x\<in>space M. P n x} \<inter> S)"
456     using P Q S by (intro finite_measure_eq_AE) auto
457   from ae_S have **:
458     "\<And>n. \<P>(x in M. P n x) = prob ({x\<in>space M. P n x} \<inter> S)"
459     using P Q S by (intro finite_measure_eq_AE) auto
460   show ?thesis
461     unfolding * ** using S P disj
462     by (intro finite_measure_UNION) auto
463 qed
465 lemma (in prob_space) cond_prob_eq_AE:
466   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"
467   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"
468   shows "cond_prob M P Q = cond_prob M P' Q'"
469   using P Q
470   by (auto simp: cond_prob_def intro!: arg_cong2[where f="op /"] prob_eq_AE sets_Collect_conj)
473 lemma (in prob_space) joint_distribution_Times_le_fst:
474   "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
475     \<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"
476   by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
478 lemma (in prob_space) joint_distribution_Times_le_snd:
479   "random_variable MX X \<Longrightarrow> random_variable MY Y \<Longrightarrow> A \<in> sets MX \<Longrightarrow> B \<in> sets MY
480     \<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"
481   by (auto simp: emeasure_distr measurable_pair_iff comp_def intro!: emeasure_mono measurable_sets)
483 locale pair_prob_space = pair_sigma_finite M1 M2 + M1: prob_space M1 + M2: prob_space M2 for M1 M2
485 sublocale pair_prob_space \<subseteq> P: prob_space "M1 \<Otimes>\<^isub>M M2"
486 proof
487   show "emeasure (M1 \<Otimes>\<^isub>M M2) (space (M1 \<Otimes>\<^isub>M M2)) = 1"
488     by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_space_1 M2.emeasure_space_1 space_pair_measure)
489 qed
491 locale product_prob_space = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" +
492   fixes I :: "'i set"
493   assumes prob_space: "\<And>i. prob_space (M i)"
495 sublocale product_prob_space \<subseteq> M: prob_space "M i" for i
496   by (rule prob_space)
498 locale finite_product_prob_space = finite_product_sigma_finite M I + product_prob_space M I for M I
500 sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
501 proof
502   show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (space (\<Pi>\<^isub>M i\<in>I. M i)) = 1"
503     by (simp add: measure_times M.emeasure_space_1 setprod_1 space_PiM)
504 qed
506 lemma (in finite_product_prob_space) prob_times:
507   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
508   shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
509 proof -
510   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)"
511     using X by (simp add: emeasure_eq_measure)
512   also have "\<dots> = (\<Prod>i\<in>I. emeasure (M i) (X i))"
513     using measure_times X by simp
514   also have "\<dots> = ereal (\<Prod>i\<in>I. measure (M i) (X i))"
515     using X by (simp add: M.emeasure_eq_measure setprod_ereal)
516   finally show ?thesis by simp
517 qed
519 section {* Distributions *}
521 definition "distributed M N X f \<longleftrightarrow> distr M N X = density N f \<and>
522   f \<in> borel_measurable N \<and> (AE x in N. 0 \<le> f x) \<and> X \<in> measurable M N"
524 lemma
525   shows distributed_distr_eq_density: "distributed M N X f \<Longrightarrow> distr M N X = density N f"
526     and distributed_measurable: "distributed M N X f \<Longrightarrow> X \<in> measurable M N"
527     and distributed_borel_measurable: "distributed M N X f \<Longrightarrow> f \<in> borel_measurable N"
528     and distributed_AE: "distributed M N X f \<Longrightarrow> (AE x in N. 0 \<le> f x)"
529   by (simp_all add: distributed_def)
531 lemma
532   shows distributed_real_measurable: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> f \<in> borel_measurable N"
533     and distributed_real_AE: "distributed M N X (\<lambda>x. ereal (f x)) \<Longrightarrow> (AE x in N. 0 \<le> f x)"
534   by (simp_all add: distributed_def borel_measurable_ereal_iff)
536 lemma distributed_count_space:
537   assumes X: "distributed M (count_space A) X P" and a: "a \<in> A" and A: "finite A"
538   shows "P a = emeasure M (X -` {a} \<inter> space M)"
539 proof -
540   have "emeasure M (X -` {a} \<inter> space M) = emeasure (distr M (count_space A) X) {a}"
541     using X a A by (simp add: distributed_measurable emeasure_distr)
542   also have "\<dots> = emeasure (density (count_space A) P) {a}"
543     using X by (simp add: distributed_distr_eq_density)
544   also have "\<dots> = (\<integral>\<^isup>+x. P a * indicator {a} x \<partial>count_space A)"
545     using X a by (auto simp add: emeasure_density distributed_def indicator_def intro!: positive_integral_cong)
546   also have "\<dots> = P a"
547     using X a by (subst positive_integral_cmult_indicator) (auto simp: distributed_def one_ereal_def[symmetric] AE_count_space)
548   finally show ?thesis ..
549 qed
551 lemma distributed_cong_density:
552   "(AE x in N. f x = g x) \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable N \<Longrightarrow>
553     distributed M N X f \<longleftrightarrow> distributed M N X g"
554   by (auto simp: distributed_def intro!: density_cong)
556 lemma subdensity:
557   assumes T: "T \<in> measurable P Q"
558   assumes f: "distributed M P X f"
559   assumes g: "distributed M Q Y g"
560   assumes Y: "Y = T \<circ> X"
561   shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
562 proof -
563   have "{x\<in>space Q. g x = 0} \<in> null_sets (distr M Q (T \<circ> X))"
564     using g Y by (auto simp: null_sets_density_iff distributed_def)
565   also have "distr M Q (T \<circ> X) = distr (distr M P X) Q T"
566     using T f[THEN distributed_measurable] by (rule distr_distr[symmetric])
567   finally have "T -` {x\<in>space Q. g x = 0} \<inter> space P \<in> null_sets (distr M P X)"
568     using T by (subst (asm) null_sets_distr_iff) auto
569   also have "T -` {x\<in>space Q. g x = 0} \<inter> space P = {x\<in>space P. g (T x) = 0}"
570     using T by (auto dest: measurable_space)
571   finally show ?thesis
572     using f g by (auto simp add: null_sets_density_iff distributed_def)
573 qed
575 lemma subdensity_real:
576   fixes g :: "'a \<Rightarrow> real" and f :: "'b \<Rightarrow> real"
577   assumes T: "T \<in> measurable P Q"
578   assumes f: "distributed M P X f"
579   assumes g: "distributed M Q Y g"
580   assumes Y: "Y = T \<circ> X"
581   shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
582   using subdensity[OF T, of M X "\<lambda>x. ereal (f x)" Y "\<lambda>x. ereal (g x)"] assms by auto
584 lemma distributed_emeasure:
585   "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)"
586   by (auto simp: distributed_measurable distributed_AE distributed_borel_measurable
587                  distributed_distr_eq_density[symmetric] emeasure_density[symmetric] emeasure_distr)
589 lemma distributed_positive_integral:
590   "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)"
591   by (auto simp: distributed_measurable distributed_AE distributed_borel_measurable
592                  distributed_distr_eq_density[symmetric] positive_integral_density[symmetric] positive_integral_distr)
594 lemma distributed_integral:
595   "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)"
596   by (auto simp: distributed_real_measurable distributed_real_AE distributed_measurable
597                  distributed_distr_eq_density[symmetric] integral_density[symmetric] integral_distr)
599 lemma distributed_transform_integral:
600   assumes Px: "distributed M N X Px"
601   assumes "distributed M P Y Py"
602   assumes Y: "Y = T \<circ> X" and T: "T \<in> measurable N P" and f: "f \<in> borel_measurable P"
603   shows "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. Px x * f (T x) \<partial>N)"
604 proof -
605   have "(\<integral>x. Py x * f x \<partial>P) = (\<integral>x. f (Y x) \<partial>M)"
606     by (rule distributed_integral) fact+
607   also have "\<dots> = (\<integral>x. f (T (X x)) \<partial>M)"
608     using Y by simp
609   also have "\<dots> = (\<integral>x. Px x * f (T x) \<partial>N)"
610     using measurable_comp[OF T f] Px by (intro distributed_integral[symmetric]) (auto simp: comp_def)
611   finally show ?thesis .
612 qed
614 lemma (in prob_space) distributed_unique:
615   assumes Px: "distributed M S X Px"
616   assumes Py: "distributed M S X Py"
617   shows "AE x in S. Px x = Py x"
618 proof -
619   interpret X: prob_space "distr M S X"
620     using distributed_measurable[OF Px] by (rule prob_space_distr)
621   have "sigma_finite_measure (distr M S X)" ..
622   with sigma_finite_density_unique[of Px S Py ] Px Py
623   show ?thesis
624     by (auto simp: distributed_def)
625 qed
627 lemma (in prob_space) distributed_jointI:
628   assumes "sigma_finite_measure S" "sigma_finite_measure T"
629   assumes X[simp]: "X \<in> measurable M S" and Y[simp]: "Y \<in> measurable M T"
630   assumes f[simp]: "f \<in> borel_measurable (S \<Otimes>\<^isub>M T)" "AE x in S \<Otimes>\<^isub>M T. 0 \<le> f x"
631   assumes eq: "\<And>A B. A \<in> sets S \<Longrightarrow> B \<in> sets T \<Longrightarrow>
632     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)"
633   shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) f"
634   unfolding distributed_def
635 proof safe
636   interpret S: sigma_finite_measure S by fact
637   interpret T: sigma_finite_measure T by fact
638   interpret ST: pair_sigma_finite S T by default
640   from ST.sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('b \<times> 'c) set" .. note F = this
641   let ?E = "{a \<times> b |a b. a \<in> sets S \<and> b \<in> sets T}"
642   let ?P = "S \<Otimes>\<^isub>M T"
643   show "distr M ?P (\<lambda>x. (X x, Y x)) = density ?P f" (is "?L = ?R")
644   proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of S T]])
645     show "?E \<subseteq> Pow (space ?P)"
646       using space_closed[of S] space_closed[of T] by (auto simp: space_pair_measure)
647     show "sets ?L = sigma_sets (space ?P) ?E"
648       by (simp add: sets_pair_measure space_pair_measure)
649     then show "sets ?R = sigma_sets (space ?P) ?E"
650       by simp
651   next
652     interpret L: prob_space ?L
653       by (rule prob_space_distr) (auto intro!: measurable_Pair)
654     show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?L (F i) \<noteq> \<infinity>"
655       using F by (auto simp: space_pair_measure)
656   next
657     fix E assume "E \<in> ?E"
658     then obtain A B where E[simp]: "E = A \<times> B" and A[simp]: "A \<in> sets S" and B[simp]: "B \<in> sets T" by auto
659     have "emeasure ?L E = emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B}"
660       by (auto intro!: arg_cong[where f="emeasure M"] simp add: emeasure_distr measurable_Pair)
661     also have "\<dots> = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. (f (x, y) * indicator B y) * indicator A x \<partial>T) \<partial>S)"
662       by (auto simp add: eq measurable_Pair measurable_compose[OF _ f(1)] positive_integral_multc
663                intro!: positive_integral_cong)
664     also have "\<dots> = emeasure ?R E"
665       by (auto simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric]
666                intro!: positive_integral_cong split: split_indicator)
667     finally show "emeasure ?L E = emeasure ?R E" .
668   qed
669 qed (auto intro!: measurable_Pair)
671 lemma (in prob_space) distributed_swap:
672   assumes "sigma_finite_measure S" "sigma_finite_measure T"
673   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
674   shows "distributed M (T \<Otimes>\<^isub>M S) (\<lambda>x. (Y x, X x)) (\<lambda>(x, y). Pxy (y, x))"
675 proof -
676   interpret S: sigma_finite_measure S by fact
677   interpret T: sigma_finite_measure T by fact
678   interpret ST: pair_sigma_finite S T by default
679   interpret TS: pair_sigma_finite T S by default
681   note measurable_Pxy = measurable_compose[OF _ distributed_borel_measurable[OF Pxy]]
682   show ?thesis
683     apply (subst TS.distr_pair_swap)
684     unfolding distributed_def
685   proof safe
686     let ?D = "distr (S \<Otimes>\<^isub>M T) (T \<Otimes>\<^isub>M S) (\<lambda>(x, y). (y, x))"
687     show 1: "(\<lambda>(x, y). Pxy (y, x)) \<in> borel_measurable ?D"
688       by (auto simp: measurable_split_conv intro!: measurable_Pair measurable_Pxy)
689     with Pxy
690     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))"
691       by (subst AE_distr_iff)
692          (auto dest!: distributed_AE
693                simp: measurable_split_conv split_beta
694                intro!: measurable_Pair borel_measurable_ereal_le)
695     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))"
696       using measurable_compose[OF distributed_measurable[OF Pxy] measurable_fst]
697       using measurable_compose[OF distributed_measurable[OF Pxy] measurable_snd]
698       by (auto intro!: measurable_Pair)
699     { fix A assume A: "A \<in> sets (T \<Otimes>\<^isub>M S)"
700       let ?B = "(\<lambda>(x, y). (y, x)) -` A \<inter> space (S \<Otimes>\<^isub>M T)"
701       from sets_into_space[OF A]
702       have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
703         emeasure M ((\<lambda>x. (X x, Y x)) -` ?B \<inter> space M)"
704         by (auto intro!: arg_cong2[where f=emeasure] simp: space_pair_measure)
705       also have "\<dots> = (\<integral>\<^isup>+ x. Pxy x * indicator ?B x \<partial>(S \<Otimes>\<^isub>M T))"
706         using Pxy A by (intro distributed_emeasure measurable_sets) (auto simp: measurable_split_conv measurable_Pair)
707       finally have "emeasure M ((\<lambda>x. (Y x, X x)) -` A \<inter> space M) =
708         (\<integral>\<^isup>+ x. Pxy x * indicator A (snd x, fst x) \<partial>(S \<Otimes>\<^isub>M T))"
709         by (auto intro!: positive_integral_cong split: split_indicator) }
710     note * = this
711     show "distr M ?D (\<lambda>x. (Y x, X x)) = density ?D (\<lambda>(x, y). Pxy (y, x))"
712       apply (intro measure_eqI)
713       apply (simp_all add: emeasure_distr[OF 2] emeasure_density[OF 1])
714       apply (subst positive_integral_distr)
715       apply (auto intro!: measurable_pair measurable_Pxy * simp: comp_def split_beta)
716       done
717   qed
718 qed
720 lemma (in prob_space) distr_marginal1:
721   assumes "sigma_finite_measure S" "sigma_finite_measure T"
722   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
723   defines "Px \<equiv> \<lambda>x. (\<integral>\<^isup>+z. Pxy (x, z) \<partial>T)"
724   shows "distributed M S X Px"
725   unfolding distributed_def
726 proof safe
727   interpret S: sigma_finite_measure S by fact
728   interpret T: sigma_finite_measure T by fact
729   interpret ST: pair_sigma_finite S T by default
731   have XY: "(\<lambda>x. (X x, Y x)) \<in> measurable M (S \<Otimes>\<^isub>M T)"
732     using Pxy by (rule distributed_measurable)
733   then show X: "X \<in> measurable M S"
734     unfolding measurable_pair_iff by (simp add: comp_def)
735   from XY have Y: "Y \<in> measurable M T"
736     unfolding measurable_pair_iff by (simp add: comp_def)
738   from Pxy show borel: "Px \<in> borel_measurable S"
739     by (auto intro!: T.positive_integral_fst_measurable dest!: distributed_borel_measurable simp: Px_def)
741   interpret Pxy: prob_space "distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
742     using XY by (rule prob_space_distr)
743   have "(\<integral>\<^isup>+ x. max 0 (- Pxy x) \<partial>(S \<Otimes>\<^isub>M T)) = (\<integral>\<^isup>+ x. 0 \<partial>(S \<Otimes>\<^isub>M T))"
744     using Pxy
745     by (intro positive_integral_cong_AE) (auto simp: max_def dest: distributed_borel_measurable distributed_AE)
747   show "distr M S X = density S Px"
748   proof (rule measure_eqI)
749     fix A assume A: "A \<in> sets (distr M S X)"
750     with X Y XY have "emeasure (distr M S X) A = emeasure (distr M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))) (A \<times> space T)"
751       by (auto simp add: emeasure_distr
752                intro!: arg_cong[where f="emeasure M"] dest: measurable_space)
753     also have "\<dots> = emeasure (density (S \<Otimes>\<^isub>M T) Pxy) (A \<times> space T)"
754       using Pxy by (simp add: distributed_def)
755     also have "\<dots> = \<integral>\<^isup>+ x. \<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
756       using A borel Pxy
757       by (simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric] distributed_def)
758     also have "\<dots> = \<integral>\<^isup>+ x. Px x * indicator A x \<partial>S"
759       apply (rule positive_integral_cong_AE)
760       using Pxy[THEN distributed_AE, THEN ST.AE_pair] AE_space
761     proof eventually_elim
762       fix x assume "x \<in> space S" "AE y in T. 0 \<le> Pxy (x, y)"
763       moreover have eq: "\<And>y. y \<in> space T \<Longrightarrow> indicator (A \<times> space T) (x, y) = indicator A x"
764         by (auto simp: indicator_def)
765       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"
766         using Pxy[THEN distributed_borel_measurable] by (simp add: eq positive_integral_multc measurable_Pair2 cong: positive_integral_cong)
767       also have "(\<integral>\<^isup>+ y. Pxy (x, y) \<partial>T) = Px x"
768         by (simp add: Px_def ereal_real positive_integral_positive)
769       finally show "(\<integral>\<^isup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T) = Px x * indicator A x" .
770     qed
771     finally show "emeasure (distr M S X) A = emeasure (density S Px) A"
772       using A borel Pxy by (simp add: emeasure_density)
773   qed simp
775   show "AE x in S. 0 \<le> Px x"
776     by (simp add: Px_def positive_integral_positive real_of_ereal_pos)
777 qed
779 lemma (in prob_space) distr_marginal2:
780   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
781   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
782   shows "distributed M T Y (\<lambda>y. (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S))"
783   using distr_marginal1[OF T S distributed_swap[OF S T]] Pxy by simp
785 lemma (in prob_space) distributed_marginal_eq_joint1:
786   assumes T: "sigma_finite_measure T"
787   assumes S: "sigma_finite_measure S"
788   assumes Px: "distributed M S X Px"
789   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
790   shows "AE x in S. Px x = (\<integral>\<^isup>+y. Pxy (x, y) \<partial>T)"
791   using Px distr_marginal1[OF S T Pxy] by (rule distributed_unique)
793 lemma (in prob_space) distributed_marginal_eq_joint2:
794   assumes T: "sigma_finite_measure T"
795   assumes S: "sigma_finite_measure S"
796   assumes Py: "distributed M T Y Py"
797   assumes Pxy: "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) Pxy"
798   shows "AE y in T. Py y = (\<integral>\<^isup>+x. Pxy (x, y) \<partial>S)"
799   using Py distr_marginal2[OF S T Pxy] by (rule distributed_unique)
801 lemma (in prob_space) distributed_joint_indep':
802   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
803   assumes X: "distributed M S X Px" and Y: "distributed M T Y Py"
804   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))"
805   shows "distributed M (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
806   unfolding distributed_def
807 proof safe
808   interpret S: sigma_finite_measure S by fact
809   interpret T: sigma_finite_measure T by fact
810   interpret ST: pair_sigma_finite S T by default
812   interpret X: prob_space "density S Px"
813     unfolding distributed_distr_eq_density[OF X, symmetric]
814     using distributed_measurable[OF X]
815     by (rule prob_space_distr)
816   have sf_X: "sigma_finite_measure (density S Px)" ..
818   interpret Y: prob_space "density T Py"
819     unfolding distributed_distr_eq_density[OF Y, symmetric]
820     using distributed_measurable[OF Y]
821     by (rule prob_space_distr)
822   have sf_Y: "sigma_finite_measure (density T Py)" ..
824   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)"
825     unfolding indep[symmetric] distributed_distr_eq_density[OF X] distributed_distr_eq_density[OF Y]
826     using distributed_borel_measurable[OF X] distributed_AE[OF X]
827     using distributed_borel_measurable[OF Y] distributed_AE[OF Y]
828     by (rule pair_measure_density[OF _ _ _ _ S T sf_X sf_Y])
830   show "random_variable (S \<Otimes>\<^isub>M T) (\<lambda>x. (X x, Y x))"
831     using distributed_measurable[OF X] distributed_measurable[OF Y]
832     by (auto intro: measurable_Pair)
834   show Pxy: "(\<lambda>(x, y). Px x * Py y) \<in> borel_measurable (S \<Otimes>\<^isub>M T)"
835     by (auto simp: split_beta'
836              intro!: measurable_compose[OF _ distributed_borel_measurable[OF X]]
837                      measurable_compose[OF _ distributed_borel_measurable[OF Y]])
839   show "AE x in S \<Otimes>\<^isub>M T. 0 \<le> (case x of (x, y) \<Rightarrow> Px x * Py y)"
840     apply (intro ST.AE_pair_measure borel_measurable_ereal_le Pxy borel_measurable_const)
841     using distributed_AE[OF X]
842     apply eventually_elim
843     using distributed_AE[OF Y]
844     apply eventually_elim
845     apply auto
846     done
847 qed
849 definition
850   "simple_distributed M X f \<longleftrightarrow> distributed M (count_space (X`space M)) X (\<lambda>x. ereal (f x)) \<and>
851     finite (X`space M)"
853 lemma simple_distributed:
854   "simple_distributed M X Px \<Longrightarrow> distributed M (count_space (X`space M)) X Px"
855   unfolding simple_distributed_def by auto
857 lemma simple_distributed_finite[dest]: "simple_distributed M X P \<Longrightarrow> finite (X`space M)"
858   by (simp add: simple_distributed_def)
860 lemma (in prob_space) distributed_simple_function_superset:
861   assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
862   assumes A: "X`space M \<subseteq> A" "finite A"
863   defines "S \<equiv> count_space A" and "P' \<equiv> (\<lambda>x. if x \<in> X`space M then P x else 0)"
864   shows "distributed M S X P'"
865   unfolding distributed_def
866 proof safe
867   show "(\<lambda>x. ereal (P' x)) \<in> borel_measurable S" unfolding S_def by simp
868   show "AE x in S. 0 \<le> ereal (P' x)"
869     using X by (auto simp: S_def P'_def simple_distributed_def intro!: measure_nonneg)
870   show "distr M S X = density S P'"
871   proof (rule measure_eqI_finite)
872     show "sets (distr M S X) = Pow A" "sets (density S P') = Pow A"
873       using A unfolding S_def by auto
874     show "finite A" by fact
875     fix a assume a: "a \<in> A"
876     then have "a \<notin> X`space M \<Longrightarrow> X -` {a} \<inter> space M = {}" by auto
877     with A a X have "emeasure (distr M S X) {a} = P' a"
878       by (subst emeasure_distr)
879          (auto simp add: S_def P'_def simple_functionD emeasure_eq_measure measurable_count_space_eq2
880                intro!: arg_cong[where f=prob])
881     also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' a) * indicator {a} x \<partial>S)"
882       using A X a
883       by (subst positive_integral_cmult_indicator)
884          (auto simp: S_def P'_def simple_distributed_def simple_functionD measure_nonneg)
885     also have "\<dots> = (\<integral>\<^isup>+x. ereal (P' x) * indicator {a} x \<partial>S)"
886       by (auto simp: indicator_def intro!: positive_integral_cong)
887     also have "\<dots> = emeasure (density S P') {a}"
888       using a A by (intro emeasure_density[symmetric]) (auto simp: S_def)
889     finally show "emeasure (distr M S X) {a} = emeasure (density S P') {a}" .
890   qed
891   show "random_variable S X"
892     using X(1) A by (auto simp: measurable_def simple_functionD S_def)
893 qed
895 lemma (in prob_space) simple_distributedI:
896   assumes X: "simple_function M X" "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = measure M (X -` {x} \<inter> space M)"
897   shows "simple_distributed M X P"
898   unfolding simple_distributed_def
899 proof
900   have "distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (if x \<in> X`space M then P x else 0))"
901     (is "?A")
902     using simple_functionD[OF X(1)] by (intro distributed_simple_function_superset[OF X]) auto
903   also have "?A \<longleftrightarrow> distributed M (count_space (X ` space M)) X (\<lambda>x. ereal (P x))"
904     by (rule distributed_cong_density) auto
905   finally show "\<dots>" .
906 qed (rule simple_functionD[OF X(1)])
908 lemma simple_distributed_joint_finite:
909   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
910   shows "finite (X ` space M)" "finite (Y ` space M)"
911 proof -
912   have "finite ((\<lambda>x. (X x, Y x)) ` space M)"
913     using X by (auto simp: simple_distributed_def simple_functionD)
914   then have "finite (fst ` (\<lambda>x. (X x, Y x)) ` space M)" "finite (snd ` (\<lambda>x. (X x, Y x)) ` space M)"
915     by auto
916   then show fin: "finite (X ` space M)" "finite (Y ` space M)"
917     by (auto simp: image_image)
918 qed
920 lemma simple_distributed_joint2_finite:
921   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
922   shows "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
923 proof -
924   have "finite ((\<lambda>x. (X x, Y x, Z x)) ` space M)"
925     using X by (auto simp: simple_distributed_def simple_functionD)
926   then have "finite (fst ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
927     "finite ((fst \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
928     "finite ((snd \<circ> snd) ` (\<lambda>x. (X x, Y x, Z x)) ` space M)"
929     by auto
930   then show fin: "finite (X ` space M)" "finite (Y ` space M)" "finite (Z ` space M)"
931     by (auto simp: image_image)
932 qed
934 lemma simple_distributed_simple_function:
935   "simple_distributed M X Px \<Longrightarrow> simple_function M X"
936   unfolding simple_distributed_def distributed_def
937   by (auto simp: simple_function_def measurable_count_space_eq2)
939 lemma simple_distributed_measure:
940   "simple_distributed M X P \<Longrightarrow> a \<in> X`space M \<Longrightarrow> P a = measure M (X -` {a} \<inter> space M)"
941   using distributed_count_space[of M "X`space M" X P a, symmetric]
942   by (auto simp: simple_distributed_def measure_def)
944 lemma simple_distributed_nonneg: "simple_distributed M X f \<Longrightarrow> x \<in> space M \<Longrightarrow> 0 \<le> f (X x)"
945   by (auto simp: simple_distributed_measure measure_nonneg)
947 lemma (in prob_space) simple_distributed_joint:
948   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
949   defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M)"
950   defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x))`space M then Px x else 0)"
951   shows "distributed M S (\<lambda>x. (X x, Y x)) P"
952 proof -
953   from simple_distributed_joint_finite[OF X, simp]
954   have S_eq: "S = count_space (X`space M \<times> Y`space M)"
955     by (simp add: S_def pair_measure_count_space)
956   show ?thesis
957     unfolding S_eq P_def
958   proof (rule distributed_simple_function_superset)
959     show "simple_function M (\<lambda>x. (X x, Y x))"
960       using X by (rule simple_distributed_simple_function)
961     fix x assume "x \<in> (\<lambda>x. (X x, Y x)) ` space M"
962     from simple_distributed_measure[OF X this]
963     show "Px x = prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M)" .
964   qed auto
965 qed
967 lemma (in prob_space) simple_distributed_joint2:
968   assumes X: "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
969   defines "S \<equiv> count_space (X`space M) \<Otimes>\<^isub>M count_space (Y`space M) \<Otimes>\<^isub>M count_space (Z`space M)"
970   defines "P \<equiv> (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x))`space M then Px x else 0)"
971   shows "distributed M S (\<lambda>x. (X x, Y x, Z x)) P"
972 proof -
973   from simple_distributed_joint2_finite[OF X, simp]
974   have S_eq: "S = count_space (X`space M \<times> Y`space M \<times> Z`space M)"
975     by (simp add: S_def pair_measure_count_space)
976   show ?thesis
977     unfolding S_eq P_def
978   proof (rule distributed_simple_function_superset)
979     show "simple_function M (\<lambda>x. (X x, Y x, Z x))"
980       using X by (rule simple_distributed_simple_function)
981     fix x assume "x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M"
982     from simple_distributed_measure[OF X this]
983     show "Px x = prob ((\<lambda>x. (X x, Y x, Z x)) -` {x} \<inter> space M)" .
984   qed auto
985 qed
987 lemma (in prob_space) simple_distributed_setsum_space:
988   assumes X: "simple_distributed M X f"
989   shows "setsum f (X`space M) = 1"
990 proof -
991   from X have "setsum f (X`space M) = prob (\<Union>i\<in>X`space M. X -` {i} \<inter> space M)"
992     by (subst finite_measure_finite_Union)
993        (auto simp add: disjoint_family_on_def simple_distributed_measure simple_distributed_simple_function simple_functionD
994              intro!: setsum_cong arg_cong[where f="prob"])
995   also have "\<dots> = prob (space M)"
996     by (auto intro!: arg_cong[where f=prob])
997   finally show ?thesis
998     using emeasure_space_1 by (simp add: emeasure_eq_measure one_ereal_def)
999 qed
1001 lemma (in prob_space) distributed_marginal_eq_joint_simple:
1002   assumes Px: "simple_function M X"
1003   assumes Py: "simple_distributed M Y Py"
1004   assumes Pxy: "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
1005   assumes y: "y \<in> Y`space M"
1006   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)"
1007 proof -
1008   note Px = simple_distributedI[OF Px refl]
1009   have *: "\<And>f A. setsum (\<lambda>x. max 0 (ereal (f x))) A = ereal (setsum (\<lambda>x. max 0 (f x)) A)"
1010     by (simp add: setsum_ereal[symmetric] zero_ereal_def)
1011   from distributed_marginal_eq_joint2[OF
1012     sigma_finite_measure_count_space_finite
1013     sigma_finite_measure_count_space_finite
1014     simple_distributed[OF Py] simple_distributed_joint[OF Pxy],
1015     OF Py[THEN simple_distributed_finite] Px[THEN simple_distributed_finite]]
1016     y
1017     Px[THEN simple_distributed_finite]
1018     Py[THEN simple_distributed_finite]
1019     Pxy[THEN simple_distributed, THEN distributed_real_AE]
1020   show ?thesis
1021     unfolding AE_count_space
1022     apply (auto simp add: positive_integral_count_space_finite * intro!: setsum_cong split: split_max)
1023     done
1024 qed
1026 lemma prob_space_uniform_measure:
1027   assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>"
1028   shows "prob_space (uniform_measure M A)"
1029 proof
1030   show "emeasure (uniform_measure M A) (space (uniform_measure M A)) = 1"
1031     using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)], of "space M"]
1032     using sets_into_space[OF emeasure_neq_0_sets[OF A(1)]] A
1033     by (simp add: Int_absorb2 emeasure_nonneg)
1034 qed
1036 lemma prob_space_uniform_count_measure: "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> prob_space (uniform_count_measure A)"
1037   by default (auto simp: emeasure_uniform_count_measure space_uniform_count_measure one_ereal_def)
1039 end