src/HOL/Probability/Product_Measure.thy
 author hoelzl Tue Mar 16 16:27:28 2010 +0100 (2010-03-16) changeset 35833 7b7ae5aa396d child 35977 30d42bfd0174 permissions -rw-r--r--
Added product measure space
1 theory Product_Measure
2 imports "~~/src/HOL/Probability/Lebesgue"
3 begin
5 definition
6   "prod_measure M M' = (\<lambda>a. measure_space.integral M (\<lambda>s0. measure M' ((\<lambda>s1. (s0, s1)) -` a)))"
8 definition
9   "prod_measure_space M M' \<equiv>
10     \<lparr> space = space M \<times> space M',
11       sets = sets (sigma (space M \<times> space M') (prod_sets (sets M) (sets M'))),
12       measure = prod_measure M M' \<rparr>"
14 lemma prod_measure_times:
15   assumes "measure_space M" and "measure_space M'" and a: "a \<in> sets M"
16   shows "prod_measure M M' (a \<times> a') = measure M a * measure M' a'"
17 proof -
18   interpret M: measure_space M by fact
19   interpret M': measure_space M' by fact
21   { fix \<omega>
22     have "(\<lambda>\<omega>'. (\<omega>, \<omega>')) -` (a \<times> a') = (if \<omega> \<in> a then a' else {})"
23       by auto
24     hence "measure M' ((\<lambda>\<omega>'. (\<omega>, \<omega>')) -` (a \<times> a')) =
25       measure M' a' * indicator_fn a \<omega>"
26       unfolding indicator_fn_def by auto }
27   note vimage_eq_indicator = this
29   show ?thesis
30     unfolding prod_measure_def vimage_eq_indicator
31       M.integral_cmul_indicator(1)[OF `a \<in> sets M`]
32     by simp
33 qed
37 lemma measure_space_finite_prod_measure:
38   fixes M :: "('a, 'b) measure_space_scheme"
39     and M' :: "('c, 'd) measure_space_scheme"
40   assumes "measure_space M" and "measure_space M'"
41   and finM: "finite (space M)" "Pow (space M) = sets M"
42   and finM': "finite (space M')" "Pow (space M') = sets M'"
43   shows "measure_space (prod_measure_space M M')"
44 proof (rule finite_additivity_sufficient)
45   interpret M: measure_space M by fact
46   interpret M': measure_space M' by fact
48   have measure: "measure_space.measure (prod_measure_space M M') = prod_measure M M'"
49     unfolding prod_measure_space_def by simp
51   have prod_sets: "prod_sets (sets M) (sets M') \<subseteq> Pow (space M \<times> space M')"
52     using M.sets_into_space M'.sets_into_space unfolding prod_sets_def by auto
53   show sigma: "sigma_algebra (prod_measure_space M M')" unfolding prod_measure_space_def
54     by (rule sigma_algebra_sigma_sets[where a="prod_sets (sets M) (sets M')"])
55        (simp_all add: sigma_def prod_sets)
57   then interpret sa: sigma_algebra "prod_measure_space M M'" .
59   { fix x y assume "y \<in> sets (prod_measure_space M M')" and "x \<in> space M"
60     hence "y \<subseteq> space M \<times> space M'"
61       using sa.sets_into_space unfolding prod_measure_space_def by simp
62     hence "Pair x -` y \<in> sets M'"
63       using `x \<in> space M` unfolding finM'(2)[symmetric] by auto }
64   note Pair_in_sets = this
66   show "additive (prod_measure_space M M') (measure (prod_measure_space M M'))"
67     unfolding measure additive_def
68   proof safe
69     fix x y assume x: "x \<in> sets (prod_measure_space M M')" and y: "y \<in> sets (prod_measure_space M M')"
70       and disj_x_y: "x \<inter> y = {}"
71     { fix z have "Pair z -` x \<inter> Pair z -` y = {}" using disj_x_y by auto }
72     note Pair_disj = this
74     from M'.measure_additive[OF Pair_in_sets[OF x] Pair_in_sets[OF y] Pair_disj, symmetric]
75     show "prod_measure M M' (x \<union> y) = prod_measure M M' x + prod_measure M M' y"
76       unfolding prod_measure_def
77       apply (subst (1 2 3) M.integral_finite_singleton[OF finM])
79   qed
81   show "finite (space (prod_measure_space M M'))"
82     unfolding prod_measure_space_def using finM finM' by simp
84   have singletonM: "\<And>x. x \<in> space M \<Longrightarrow> {x} \<in> sets M"
85     unfolding finM(2)[symmetric] by simp
87   show "positive (prod_measure_space M M') (measure (prod_measure_space M M'))"
88     unfolding positive_def
89   proof (safe, simp add: M.integral_zero prod_measure_space_def prod_measure_def)
90     fix Q assume "Q \<in> sets (prod_measure_space M M')"
91     from Pair_in_sets[OF this]
92     show "0 \<le> measure (prod_measure_space M M') Q"
93       unfolding prod_measure_space_def prod_measure_def
94       apply (subst M.integral_finite_singleton[OF finM])
95       using M.positive M'.positive singletonM
96       by (auto intro!: setsum_nonneg mult_nonneg_nonneg)
97   qed
98 qed
100 lemma measure_space_finite_prod_measure_alterantive:
101   assumes "measure_space M" and "measure_space M'"
102   and finM: "finite (space M)" "Pow (space M) = sets M"
103   and finM': "finite (space M')" "Pow (space M') = sets M'"
104   shows "measure_space \<lparr> space = space M \<times> space M',
105                          sets = Pow (space M \<times> space M'),
106 		         measure = prod_measure M M' \<rparr>"
107   (is "measure_space ?space")
108 proof (rule finite_additivity_sufficient)
109   interpret M: measure_space M by fact
110   interpret M': measure_space M' by fact
112   show "sigma_algebra ?space"
113     using sigma_algebra.sigma_algebra_extend[where M="\<lparr> space = space M \<times> space M', sets = Pow (space M \<times> space M') \<rparr>"]
114     by (auto intro!: sigma_algebra_Pow)
115   then interpret sa: sigma_algebra ?space .
117   have measure: "measure_space.measure (prod_measure_space M M') = prod_measure M M'"
118     unfolding prod_measure_space_def by simp
120   { fix x y assume "y \<in> sets ?space" and "x \<in> space M"
121     hence "y \<subseteq> space M \<times> space M'"
122       using sa.sets_into_space by simp
123     hence "Pair x -` y \<in> sets M'"
124       using `x \<in> space M` unfolding finM'(2)[symmetric] by auto }
125   note Pair_in_sets = this
127   show "additive ?space (measure ?space)"
128     unfolding measure additive_def
129   proof safe
130     fix x y assume x: "x \<in> sets ?space" and y: "y \<in> sets ?space"
131       and disj_x_y: "x \<inter> y = {}"
132     { fix z have "Pair z -` x \<inter> Pair z -` y = {}" using disj_x_y by auto }
133     note Pair_disj = this
135     from M'.measure_additive[OF Pair_in_sets[OF x] Pair_in_sets[OF y] Pair_disj, symmetric]
136     show "measure ?space (x \<union> y) = measure ?space x + measure ?space y"
137       apply (simp add: prod_measure_def)
138       apply (subst (1 2 3) M.integral_finite_singleton[OF finM])
140   qed
142   show "finite (space ?space)" using finM finM' by simp
144   have singletonM: "\<And>x. x \<in> space M \<Longrightarrow> {x} \<in> sets M"
145     unfolding finM(2)[symmetric] by simp
147   show "positive ?space (measure ?space)"
148     unfolding positive_def
149   proof (safe, simp add: M.integral_zero prod_measure_def)
150     fix Q assume "Q \<in> sets ?space"
151     from Pair_in_sets[OF this]
152     show "0 \<le> measure ?space Q"
153       unfolding prod_measure_space_def prod_measure_def
154       apply (subst M.integral_finite_singleton[OF finM])
155       using M.positive M'.positive singletonM
156       by (auto intro!: setsum_nonneg mult_nonneg_nonneg)
157   qed
158 qed
160 end