src/HOL/Probability/Probability_Measure.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 43920 cedb5cb948fd child 44890 22f665a2e91c permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     1 (*  Title:      HOL/Probability/Probability_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Armin Heller, TU München
```
```     4 *)
```
```     5
```
```     6 header {*Probability measure*}
```
```     7
```
```     8 theory Probability_Measure
```
```     9 imports Lebesgue_Measure
```
```    10 begin
```
```    11
```
```    12 locale prob_space = measure_space +
```
```    13   assumes measure_space_1: "measure M (space M) = 1"
```
```    14
```
```    15 sublocale prob_space < finite_measure
```
```    16 proof
```
```    17   from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp
```
```    18 qed
```
```    19
```
```    20 abbreviation (in prob_space) "events \<equiv> sets M"
```
```    21 abbreviation (in prob_space) "prob \<equiv> \<mu>'"
```
```    22 abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'"
```
```    23 abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
```
```    24
```
```    25 definition (in prob_space)
```
```    26   "distribution X A = \<mu>' (X -` A \<inter> space M)"
```
```    27
```
```    28 abbreviation (in prob_space)
```
```    29   "joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
```
```    30
```
```    31 lemma (in prob_space) prob_space_cong:
```
```    32   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M"
```
```    33   shows "prob_space N"
```
```    34 proof -
```
```    35   interpret N: measure_space N by (intro measure_space_cong assms)
```
```    36   show ?thesis by default (insert assms measure_space_1, simp)
```
```    37 qed
```
```    38
```
```    39 lemma (in prob_space) distribution_cong:
```
```    40   assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
```
```    41   shows "distribution X = distribution Y"
```
```    42   unfolding distribution_def fun_eq_iff
```
```    43   using assms by (auto intro!: arg_cong[where f="\<mu>'"])
```
```    44
```
```    45 lemma (in prob_space) joint_distribution_cong:
```
```    46   assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
```
```    47   assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
```
```    48   shows "joint_distribution X Y = joint_distribution X' Y'"
```
```    49   unfolding distribution_def fun_eq_iff
```
```    50   using assms by (auto intro!: arg_cong[where f="\<mu>'"])
```
```    51
```
```    52 lemma (in prob_space) distribution_id[simp]:
```
```    53   "N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N"
```
```    54   by (auto simp: distribution_def intro!: arg_cong[where f=prob])
```
```    55
```
```    56 lemma (in prob_space) prob_space: "prob (space M) = 1"
```
```    57   using measure_space_1 unfolding \<mu>'_def by (simp add: one_ereal_def)
```
```    58
```
```    59 lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
```
```    60   using bounded_measure[of A] by (simp add: prob_space)
```
```    61
```
```    62 lemma (in prob_space) distribution_positive[simp, intro]:
```
```    63   "0 \<le> distribution X A" unfolding distribution_def by auto
```
```    64
```
```    65 lemma (in prob_space) not_zero_less_distribution[simp]:
```
```    66   "(\<not> 0 < distribution X A) \<longleftrightarrow> distribution X A = 0"
```
```    67   using distribution_positive[of X A] by arith
```
```    68
```
```    69 lemma (in prob_space) joint_distribution_remove[simp]:
```
```    70     "joint_distribution X X {(x, x)} = distribution X {x}"
```
```    71   unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
```
```    72
```
```    73 lemma (in prob_space) not_empty: "space M \<noteq> {}"
```
```    74   using prob_space empty_measure' by auto
```
```    75
```
```    76 lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1"
```
```    77   unfolding measure_space_1[symmetric]
```
```    78   using sets_into_space
```
```    79   by (intro measure_mono) auto
```
```    80
```
```    81 lemma (in prob_space) AE_I_eq_1:
```
```    82   assumes "\<mu> {x\<in>space M. P x} = 1" "{x\<in>space M. P x} \<in> sets M"
```
```    83   shows "AE x. P x"
```
```    84 proof (rule AE_I)
```
```    85   show "\<mu> (space M - {x \<in> space M. P x}) = 0"
```
```    86     using assms measure_space_1 by (simp add: measure_compl)
```
```    87 qed (insert assms, auto)
```
```    88
```
```    89 lemma (in prob_space) distribution_1:
```
```    90   "distribution X A \<le> 1"
```
```    91   unfolding distribution_def by simp
```
```    92
```
```    93 lemma (in prob_space) prob_compl:
```
```    94   assumes A: "A \<in> events"
```
```    95   shows "prob (space M - A) = 1 - prob A"
```
```    96   using finite_measure_compl[OF A] by (simp add: prob_space)
```
```    97
```
```    98 lemma (in prob_space) prob_space_increasing: "increasing M prob"
```
```    99   by (auto intro!: finite_measure_mono simp: increasing_def)
```
```   100
```
```   101 lemma (in prob_space) prob_zero_union:
```
```   102   assumes "s \<in> events" "t \<in> events" "prob t = 0"
```
```   103   shows "prob (s \<union> t) = prob s"
```
```   104 using assms
```
```   105 proof -
```
```   106   have "prob (s \<union> t) \<le> prob s"
```
```   107     using finite_measure_subadditive[of s t] assms by auto
```
```   108   moreover have "prob (s \<union> t) \<ge> prob s"
```
```   109     using assms by (blast intro: finite_measure_mono)
```
```   110   ultimately show ?thesis by simp
```
```   111 qed
```
```   112
```
```   113 lemma (in prob_space) prob_eq_compl:
```
```   114   assumes "s \<in> events" "t \<in> events"
```
```   115   assumes "prob (space M - s) = prob (space M - t)"
```
```   116   shows "prob s = prob t"
```
```   117   using assms prob_compl by auto
```
```   118
```
```   119 lemma (in prob_space) prob_one_inter:
```
```   120   assumes events:"s \<in> events" "t \<in> events"
```
```   121   assumes "prob t = 1"
```
```   122   shows "prob (s \<inter> t) = prob s"
```
```   123 proof -
```
```   124   have "prob ((space M - s) \<union> (space M - t)) = prob (space M - s)"
```
```   125     using events assms  prob_compl[of "t"] by (auto intro!: prob_zero_union)
```
```   126   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
```
```   127     by blast
```
```   128   finally show "prob (s \<inter> t) = prob s"
```
```   129     using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s])
```
```   130 qed
```
```   131
```
```   132 lemma (in prob_space) prob_eq_bigunion_image:
```
```   133   assumes "range f \<subseteq> events" "range g \<subseteq> events"
```
```   134   assumes "disjoint_family f" "disjoint_family g"
```
```   135   assumes "\<And> n :: nat. prob (f n) = prob (g n)"
```
```   136   shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))"
```
```   137 using assms
```
```   138 proof -
```
```   139   have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))"
```
```   140     by (rule finite_measure_UNION[OF assms(1,3)])
```
```   141   have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))"
```
```   142     by (rule finite_measure_UNION[OF assms(2,4)])
```
```   143   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
```
```   144 qed
```
```   145
```
```   146 lemma (in prob_space) prob_countably_zero:
```
```   147   assumes "range c \<subseteq> events"
```
```   148   assumes "\<And> i. prob (c i) = 0"
```
```   149   shows "prob (\<Union> i :: nat. c i) = 0"
```
```   150 proof (rule antisym)
```
```   151   show "prob (\<Union> i :: nat. c i) \<le> 0"
```
```   152     using finite_measure_countably_subadditive[OF assms(1)]
```
```   153     by (simp add: assms(2) suminf_zero summable_zero)
```
```   154 qed simp
```
```   155
```
```   156 lemma (in prob_space) prob_equiprobable_finite_unions:
```
```   157   assumes "s \<in> events"
```
```   158   assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events"
```
```   159   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})"
```
```   160   shows "prob s = real (card s) * prob {SOME x. x \<in> s}"
```
```   161 proof (cases "s = {}")
```
```   162   case False hence "\<exists> x. x \<in> s" by blast
```
```   163   from someI_ex[OF this] assms
```
```   164   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
```
```   165   have "prob s = (\<Sum> x \<in> s. prob {x})"
```
```   166     using finite_measure_finite_singleton[OF s_finite] by simp
```
```   167   also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
```
```   168   also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
```
```   169     using setsum_constant assms by (simp add: real_eq_of_nat)
```
```   170   finally show ?thesis by simp
```
```   171 qed simp
```
```   172
```
```   173 lemma (in prob_space) prob_real_sum_image_fn:
```
```   174   assumes "e \<in> events"
```
```   175   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events"
```
```   176   assumes "finite s"
```
```   177   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
```
```   178   assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)"
```
```   179   shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))"
```
```   180 proof -
```
```   181   have e: "e = (\<Union> i \<in> s. e \<inter> f i)"
```
```   182     using `e \<in> events` sets_into_space upper by blast
```
```   183   hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp
```
```   184   also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
```
```   185   proof (rule finite_measure_finite_Union)
```
```   186     show "finite s" by fact
```
```   187     show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact
```
```   188     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
```
```   189       using disjoint by (auto simp: disjoint_family_on_def)
```
```   190   qed
```
```   191   finally show ?thesis .
```
```   192 qed
```
```   193
```
```   194 lemma (in prob_space) prob_space_vimage:
```
```   195   assumes S: "sigma_algebra S"
```
```   196   assumes T: "T \<in> measure_preserving M S"
```
```   197   shows "prob_space S"
```
```   198 proof -
```
```   199   interpret S: measure_space S
```
```   200     using S and T by (rule measure_space_vimage)
```
```   201   show ?thesis
```
```   202   proof
```
```   203     from T[THEN measure_preservingD2]
```
```   204     have "T -` space S \<inter> space M = space M"
```
```   205       by (auto simp: measurable_def)
```
```   206     with T[THEN measure_preservingD, of "space S", symmetric]
```
```   207     show  "measure S (space S) = 1"
```
```   208       using measure_space_1 by simp
```
```   209   qed
```
```   210 qed
```
```   211
```
```   212 lemma prob_space_unique_Int_stable:
```
```   213   fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
```
```   214   assumes E: "Int_stable E" "space E \<in> sets E"
```
```   215   and M: "prob_space M" "space M = space E" "sets M = sets (sigma E)"
```
```   216   and N: "prob_space N" "space N = space E" "sets N = sets (sigma E)"
```
```   217   and eq: "\<And>X. X \<in> sets E \<Longrightarrow> finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
```
```   218   assumes "X \<in> sets (sigma E)"
```
```   219   shows "finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
```
```   220 proof -
```
```   221   interpret M!: prob_space M by fact
```
```   222   interpret N!: prob_space N by fact
```
```   223   have "measure M X = measure N X"
```
```   224   proof (rule measure_unique_Int_stable[OF `Int_stable E`])
```
```   225     show "range (\<lambda>i. space M) \<subseteq> sets E" "incseq (\<lambda>i. space M)" "(\<Union>i. space M) = space E"
```
```   226       using E M N by auto
```
```   227     show "\<And>i. M.\<mu> (space M) \<noteq> \<infinity>"
```
```   228       using M.measure_space_1 by simp
```
```   229     show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = M.\<mu>\<rparr>"
```
```   230       using E M N by (auto intro!: M.measure_space_cong)
```
```   231     show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = N.\<mu>\<rparr>"
```
```   232       using E M N by (auto intro!: N.measure_space_cong)
```
```   233     { fix X assume "X \<in> sets E"
```
```   234       then have "X \<in> sets (sigma E)"
```
```   235         by (auto simp: sets_sigma sigma_sets.Basic)
```
```   236       with eq[OF `X \<in> sets E`] M N show "M.\<mu> X = N.\<mu> X"
```
```   237         by (simp add: M.finite_measure_eq N.finite_measure_eq) }
```
```   238   qed fact
```
```   239   with `X \<in> sets (sigma E)` M N show ?thesis
```
```   240     by (simp add: M.finite_measure_eq N.finite_measure_eq)
```
```   241 qed
```
```   242
```
```   243 lemma (in prob_space) distribution_prob_space:
```
```   244   assumes X: "random_variable S X"
```
```   245   shows "prob_space (S\<lparr>measure := ereal \<circ> distribution X\<rparr>)" (is "prob_space ?S")
```
```   246 proof (rule prob_space_vimage)
```
```   247   show "X \<in> measure_preserving M ?S"
```
```   248     using X
```
```   249     unfolding measure_preserving_def distribution_def_raw
```
```   250     by (auto simp: finite_measure_eq measurable_sets)
```
```   251   show "sigma_algebra ?S" using X by simp
```
```   252 qed
```
```   253
```
```   254 lemma (in prob_space) AE_distribution:
```
```   255   assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := ereal \<circ> distribution X\<rparr>. Q x"
```
```   256   shows "AE x. Q (X x)"
```
```   257 proof -
```
```   258   interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space)
```
```   259   obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N"
```
```   260     using assms unfolding X.almost_everywhere_def by auto
```
```   261   from X[unfolded measurable_def] N show "AE x. Q (X x)"
```
```   262     by (intro AE_I'[where N="X -` N \<inter> space M"])
```
```   263        (auto simp: finite_measure_eq distribution_def measurable_sets)
```
```   264 qed
```
```   265
```
```   266 lemma (in prob_space) distribution_eq_integral:
```
```   267   "random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))"
```
```   268   using finite_measure_eq[of "X -` A \<inter> space M"]
```
```   269   by (auto simp: measurable_sets distribution_def)
```
```   270
```
```   271 lemma (in prob_space) expectation_less:
```
```   272   assumes [simp]: "integrable M X"
```
```   273   assumes gt: "\<forall>x\<in>space M. X x < b"
```
```   274   shows "expectation X < b"
```
```   275 proof -
```
```   276   have "expectation X < expectation (\<lambda>x. b)"
```
```   277     using gt measure_space_1
```
```   278     by (intro integral_less_AE_space) auto
```
```   279   then show ?thesis using prob_space by simp
```
```   280 qed
```
```   281
```
```   282 lemma (in prob_space) expectation_greater:
```
```   283   assumes [simp]: "integrable M X"
```
```   284   assumes gt: "\<forall>x\<in>space M. a < X x"
```
```   285   shows "a < expectation X"
```
```   286 proof -
```
```   287   have "expectation (\<lambda>x. a) < expectation X"
```
```   288     using gt measure_space_1
```
```   289     by (intro integral_less_AE_space) auto
```
```   290   then show ?thesis using prob_space by simp
```
```   291 qed
```
```   292
```
```   293 lemma convex_le_Inf_differential:
```
```   294   fixes f :: "real \<Rightarrow> real"
```
```   295   assumes "convex_on I f"
```
```   296   assumes "x \<in> interior I" "y \<in> I"
```
```   297   shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
```
```   298     (is "_ \<ge> _ + Inf (?F x) * (y - x)")
```
```   299 proof -
```
```   300   show ?thesis
```
```   301   proof (cases rule: linorder_cases)
```
```   302     assume "x < y"
```
```   303     moreover
```
```   304     have "open (interior I)" by auto
```
```   305     from openE[OF this `x \<in> interior I`] guess e . note e = this
```
```   306     moreover def t \<equiv> "min (x + e / 2) ((x + y) / 2)"
```
```   307     ultimately have "x < t" "t < y" "t \<in> ball x e"
```
```   308       by (auto simp: mem_ball dist_real_def field_simps split: split_min)
```
```   309     with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
```
```   310
```
```   311     have "open (interior I)" by auto
```
```   312     from openE[OF this `x \<in> interior I`] guess e .
```
```   313     moreover def K \<equiv> "x - e / 2"
```
```   314     with `0 < e` have "K \<in> ball x e" "K < x" by (auto simp: mem_ball dist_real_def)
```
```   315     ultimately have "K \<in> I" "K < x" "x \<in> I"
```
```   316       using interior_subset[of I] `x \<in> interior I` by auto
```
```   317
```
```   318     have "Inf (?F x) \<le> (f x - f y) / (x - y)"
```
```   319     proof (rule Inf_lower2)
```
```   320       show "(f x - f t) / (x - t) \<in> ?F x"
```
```   321         using `t \<in> I` `x < t` by auto
```
```   322       show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
```
```   323         using `convex_on I f` `x \<in> I` `y \<in> I` `x < t` `t < y` by (rule convex_on_diff)
```
```   324     next
```
```   325       fix y assume "y \<in> ?F x"
```
```   326       with order_trans[OF convex_on_diff[OF `convex_on I f` `K \<in> I` _ `K < x` _]]
```
```   327       show "(f K - f x) / (K - x) \<le> y" by auto
```
```   328     qed
```
```   329     then show ?thesis
```
```   330       using `x < y` by (simp add: field_simps)
```
```   331   next
```
```   332     assume "y < x"
```
```   333     moreover
```
```   334     have "open (interior I)" by auto
```
```   335     from openE[OF this `x \<in> interior I`] guess e . note e = this
```
```   336     moreover def t \<equiv> "x + e / 2"
```
```   337     ultimately have "x < t" "t \<in> ball x e"
```
```   338       by (auto simp: mem_ball dist_real_def field_simps)
```
```   339     with `x \<in> interior I` e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
```
```   340
```
```   341     have "(f x - f y) / (x - y) \<le> Inf (?F x)"
```
```   342     proof (rule Inf_greatest)
```
```   343       have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
```
```   344         using `y < x` by (auto simp: field_simps)
```
```   345       also
```
```   346       fix z  assume "z \<in> ?F x"
```
```   347       with order_trans[OF convex_on_diff[OF `convex_on I f` `y \<in> I` _ `y < x`]]
```
```   348       have "(f y - f x) / (y - x) \<le> z" by auto
```
```   349       finally show "(f x - f y) / (x - y) \<le> z" .
```
```   350     next
```
```   351       have "open (interior I)" by auto
```
```   352       from openE[OF this `x \<in> interior I`] guess e . note e = this
```
```   353       then have "x + e / 2 \<in> ball x e" by (auto simp: mem_ball dist_real_def)
```
```   354       with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I" by auto
```
```   355       then show "?F x \<noteq> {}" by blast
```
```   356     qed
```
```   357     then show ?thesis
```
```   358       using `y < x` by (simp add: field_simps)
```
```   359   qed simp
```
```   360 qed
```
```   361
```
```   362 lemma (in prob_space) jensens_inequality:
```
```   363   fixes a b :: real
```
```   364   assumes X: "integrable M X" "X ` space M \<subseteq> I"
```
```   365   assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
```
```   366   assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
```
```   367   shows "q (expectation X) \<le> expectation (\<lambda>x. q (X x))"
```
```   368 proof -
```
```   369   let "?F x" = "Inf ((\<lambda>t. (q x - q t) / (x - t)) ` ({x<..} \<inter> I))"
```
```   370   from not_empty X(2) have "I \<noteq> {}" by auto
```
```   371
```
```   372   from I have "open I" by auto
```
```   373
```
```   374   note I
```
```   375   moreover
```
```   376   { assume "I \<subseteq> {a <..}"
```
```   377     with X have "a < expectation X"
```
```   378       by (intro expectation_greater) auto }
```
```   379   moreover
```
```   380   { assume "I \<subseteq> {..< b}"
```
```   381     with X have "expectation X < b"
```
```   382       by (intro expectation_less) auto }
```
```   383   ultimately have "expectation X \<in> I"
```
```   384     by (elim disjE)  (auto simp: subset_eq)
```
```   385   moreover
```
```   386   { fix y assume y: "y \<in> I"
```
```   387     with q(2) `open I` have "Sup ((\<lambda>x. q x + ?F x * (y - x)) ` I) = q y"
```
```   388       by (auto intro!: Sup_eq_maximum convex_le_Inf_differential image_eqI[OF _ y] simp: interior_open) }
```
```   389   ultimately have "q (expectation X) = Sup ((\<lambda>x. q x + ?F x * (expectation X - x)) ` I)"
```
```   390     by simp
```
```   391   also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
```
```   392   proof (rule Sup_least)
```
```   393     show "(\<lambda>x. q x + ?F x * (expectation X - x)) ` I \<noteq> {}"
```
```   394       using `I \<noteq> {}` by auto
```
```   395   next
```
```   396     fix k assume "k \<in> (\<lambda>x. q x + ?F x * (expectation X - x)) ` I"
```
```   397     then guess x .. note x = this
```
```   398     have "q x + ?F x * (expectation X  - x) = expectation (\<lambda>w. q x + ?F x * (X w - x))"
```
```   399       using prob_space
```
```   400       by (simp add: integral_add integral_cmult integral_diff lebesgue_integral_const X)
```
```   401     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
```
```   402       using `x \<in> I` `open I` X(2)
```
```   403       by (intro integral_mono integral_add integral_cmult integral_diff
```
```   404                 lebesgue_integral_const X q convex_le_Inf_differential)
```
```   405          (auto simp: interior_open)
```
```   406     finally show "k \<le> expectation (\<lambda>w. q (X w))" using x by auto
```
```   407   qed
```
```   408   finally show "q (expectation X) \<le> expectation (\<lambda>x. q (X x))" .
```
```   409 qed
```
```   410
```
```   411 lemma (in prob_space) distribution_eq_translated_integral:
```
```   412   assumes "random_variable S X" "A \<in> sets S"
```
```   413   shows "distribution X A = integral\<^isup>P (S\<lparr>measure := ereal \<circ> distribution X\<rparr>) (indicator A)"
```
```   414 proof -
```
```   415   interpret S: prob_space "S\<lparr>measure := ereal \<circ> distribution X\<rparr>"
```
```   416     using assms(1) by (rule distribution_prob_space)
```
```   417   show ?thesis
```
```   418     using S.positive_integral_indicator(1)[of A] assms by simp
```
```   419 qed
```
```   420
```
```   421 lemma (in prob_space) finite_expectation1:
```
```   422   assumes f: "finite (X`space M)" and rv: "random_variable borel X"
```
```   423   shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r")
```
```   424 proof (subst integral_on_finite)
```
```   425   show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto
```
```   426   show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r"
```
```   427     "\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>"
```
```   428     using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto
```
```   429 qed
```
```   430
```
```   431 lemma (in prob_space) finite_expectation:
```
```   432   assumes "finite (X`space M)" "random_variable borel X"
```
```   433   shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})"
```
```   434   using assms unfolding distribution_def using finite_expectation1 by auto
```
```   435
```
```   436 lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0:
```
```   437   assumes "{x} \<in> events"
```
```   438   assumes "prob {x} = 1"
```
```   439   assumes "{y} \<in> events"
```
```   440   assumes "y \<noteq> x"
```
```   441   shows "prob {y} = 0"
```
```   442   using prob_one_inter[of "{y}" "{x}"] assms by auto
```
```   443
```
```   444 lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0"
```
```   445   unfolding distribution_def by simp
```
```   446
```
```   447 lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1"
```
```   448 proof -
```
```   449   have "X -` X ` space M \<inter> space M = space M" by auto
```
```   450   thus ?thesis unfolding distribution_def by (simp add: prob_space)
```
```   451 qed
```
```   452
```
```   453 lemma (in prob_space) distribution_one:
```
```   454   assumes "random_variable M' X" and "A \<in> sets M'"
```
```   455   shows "distribution X A \<le> 1"
```
```   456 proof -
```
```   457   have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def
```
```   458     using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono)
```
```   459   thus ?thesis by (simp add: prob_space)
```
```   460 qed
```
```   461
```
```   462 lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0:
```
```   463   assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X"
```
```   464     (is "random_variable ?S X")
```
```   465   assumes "distribution X {x} = 1"
```
```   466   assumes "y \<noteq> x"
```
```   467   shows "distribution X {y} = 0"
```
```   468 proof cases
```
```   469   { fix x have "X -` {x} \<inter> space M \<in> sets M"
```
```   470     proof cases
```
```   471       assume "x \<in> X`space M" with X show ?thesis
```
```   472         by (auto simp: measurable_def image_iff)
```
```   473     next
```
```   474       assume "x \<notin> X`space M" then have "X -` {x} \<inter> space M = {}" by auto
```
```   475       then show ?thesis by auto
```
```   476     qed } note single = this
```
```   477   have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> space M"
```
```   478     "X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> space M) = {}"
```
```   479     using `y \<noteq> x` by auto
```
```   480   with finite_measure_inter_full_set[OF single single, of x y] assms(2)
```
```   481   show ?thesis by (auto simp: distribution_def prob_space)
```
```   482 next
```
```   483   assume "{y} \<notin> sets ?S"
```
```   484   then have "X -` {y} \<inter> space M = {}" by auto
```
```   485   thus "distribution X {y} = 0" unfolding distribution_def by auto
```
```   486 qed
```
```   487
```
```   488 lemma (in prob_space) joint_distribution_Times_le_fst:
```
```   489   assumes X: "random_variable MX X" and Y: "random_variable MY Y"
```
```   490     and A: "A \<in> sets MX" and B: "B \<in> sets MY"
```
```   491   shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
```
```   492   unfolding distribution_def
```
```   493 proof (intro finite_measure_mono)
```
```   494   show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
```
```   495   show "X -` A \<inter> space M \<in> events"
```
```   496     using X A unfolding measurable_def by simp
```
```   497   have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
```
```   498     (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
```
```   499 qed
```
```   500
```
```   501 lemma (in prob_space) joint_distribution_commute:
```
```   502   "joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
```
```   503   unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
```
```   504
```
```   505 lemma (in prob_space) joint_distribution_Times_le_snd:
```
```   506   assumes X: "random_variable MX X" and Y: "random_variable MY Y"
```
```   507     and A: "A \<in> sets MX" and B: "B \<in> sets MY"
```
```   508   shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
```
```   509   using assms
```
```   510   by (subst joint_distribution_commute)
```
```   511      (simp add: swap_product joint_distribution_Times_le_fst)
```
```   512
```
```   513 lemma (in prob_space) random_variable_pairI:
```
```   514   assumes "random_variable MX X"
```
```   515   assumes "random_variable MY Y"
```
```   516   shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
```
```   517 proof
```
```   518   interpret MX: sigma_algebra MX using assms by simp
```
```   519   interpret MY: sigma_algebra MY using assms by simp
```
```   520   interpret P: pair_sigma_algebra MX MY by default
```
```   521   show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
```
```   522   have sa: "sigma_algebra M" by default
```
```   523   show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
```
```   524     unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
```
```   525 qed
```
```   526
```
```   527 lemma (in prob_space) joint_distribution_commute_singleton:
```
```   528   "joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}"
```
```   529   unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
```
```   530
```
```   531 lemma (in prob_space) joint_distribution_assoc_singleton:
```
```   532   "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} =
```
```   533    joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}"
```
```   534   unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
```
```   535
```
```   536 locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2
```
```   537
```
```   538 sublocale pair_prob_space \<subseteq> pair_sigma_finite M1 M2 by default
```
```   539
```
```   540 sublocale pair_prob_space \<subseteq> P: prob_space P
```
```   541 by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure)
```
```   542
```
```   543 lemma countably_additiveI[case_names countably]:
```
```   544   assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow>
```
```   545     (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
```
```   546   shows "countably_additive M \<mu>"
```
```   547   using assms unfolding countably_additive_def by auto
```
```   548
```
```   549 lemma (in prob_space) joint_distribution_prob_space:
```
```   550   assumes "random_variable MX X" "random_variable MY Y"
```
```   551   shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)"
```
```   552   using random_variable_pairI[OF assms] by (rule distribution_prob_space)
```
```   553
```
```   554
```
```   555 locale finite_product_prob_space =
```
```   556   fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme"
```
```   557     and I :: "'i set"
```
```   558   assumes prob_space: "\<And>i. prob_space (M i)" and finite_index: "finite I"
```
```   559
```
```   560 sublocale finite_product_prob_space \<subseteq> M: prob_space "M i" for i
```
```   561   by (rule prob_space)
```
```   562
```
```   563 sublocale finite_product_prob_space \<subseteq> finite_product_sigma_finite M I
```
```   564   by default (rule finite_index)
```
```   565
```
```   566 sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
```
```   567   proof qed (simp add: measure_times M.measure_space_1 setprod_1)
```
```   568
```
```   569 lemma (in finite_product_prob_space) prob_times:
```
```   570   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
```
```   571   shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
```
```   572 proof -
```
```   573   have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
```
```   574     using X by (intro finite_measure_eq[symmetric] in_P) auto
```
```   575   also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
```
```   576     using measure_times X by simp
```
```   577   also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
```
```   578     using X by (simp add: M.finite_measure_eq setprod_ereal)
```
```   579   finally show ?thesis by simp
```
```   580 qed
```
```   581
```
```   582 lemma (in prob_space) random_variable_restrict:
```
```   583   assumes I: "finite I"
```
```   584   assumes X: "\<And>i. i \<in> I \<Longrightarrow> random_variable (N i) (X i)"
```
```   585   shows "random_variable (Pi\<^isub>M I N) (\<lambda>x. \<lambda>i\<in>I. X i x)"
```
```   586 proof
```
```   587   { fix i assume "i \<in> I"
```
```   588     with X interpret N: sigma_algebra "N i" by simp
```
```   589     have "sets (N i) \<subseteq> Pow (space (N i))" by (rule N.space_closed) }
```
```   590   note N_closed = this
```
```   591   then show "sigma_algebra (Pi\<^isub>M I N)"
```
```   592     by (simp add: product_algebra_def)
```
```   593        (intro sigma_algebra_sigma product_algebra_generator_sets_into_space)
```
```   594   show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
```
```   595     using X by (intro measurable_restrict[OF I N_closed]) auto
```
```   596 qed
```
```   597
```
```   598 section "Probability spaces on finite sets"
```
```   599
```
```   600 locale finite_prob_space = prob_space + finite_measure_space
```
```   601
```
```   602 abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'"
```
```   603
```
```   604 lemma (in prob_space) finite_random_variableD:
```
```   605   assumes "finite_random_variable M' X" shows "random_variable M' X"
```
```   606 proof -
```
```   607   interpret M': finite_sigma_algebra M' using assms by simp
```
```   608   then show "random_variable M' X" using assms by simp default
```
```   609 qed
```
```   610
```
```   611 lemma (in prob_space) distribution_finite_prob_space:
```
```   612   assumes "finite_random_variable MX X"
```
```   613   shows "finite_prob_space (MX\<lparr>measure := ereal \<circ> distribution X\<rparr>)"
```
```   614 proof -
```
```   615   interpret X: prob_space "MX\<lparr>measure := ereal \<circ> distribution X\<rparr>"
```
```   616     using assms[THEN finite_random_variableD] by (rule distribution_prob_space)
```
```   617   interpret MX: finite_sigma_algebra MX
```
```   618     using assms by auto
```
```   619   show ?thesis by default (simp_all add: MX.finite_space)
```
```   620 qed
```
```   621
```
```   622 lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]:
```
```   623   assumes "simple_function M X"
```
```   624   shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X"
```
```   625     (is "finite_random_variable ?X _")
```
```   626 proof (intro conjI)
```
```   627   have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by simp
```
```   628   interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow)
```
```   629   show "finite_sigma_algebra ?X"
```
```   630     by default auto
```
```   631   show "X \<in> measurable M ?X"
```
```   632   proof (unfold measurable_def, clarsimp)
```
```   633     fix A assume A: "A \<subseteq> X`space M"
```
```   634     then have "finite A" by (rule finite_subset) simp
```
```   635     then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events"
```
```   636       unfolding vimage_UN UN_extend_simps
```
```   637       apply (rule finite_UN)
```
```   638       using A assms unfolding simple_function_def by auto
```
```   639     then show "X -` A \<inter> space M \<in> events" by simp
```
```   640   qed
```
```   641 qed
```
```   642
```
```   643 lemma (in prob_space) simple_function_imp_random_variable[simp, intro]:
```
```   644   assumes "simple_function M X"
```
```   645   shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X"
```
```   646   using simple_function_imp_finite_random_variable[OF assms, of ext]
```
```   647   by (auto dest!: finite_random_variableD)
```
```   648
```
```   649 lemma (in prob_space) sum_over_space_real_distribution:
```
```   650   "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1"
```
```   651   unfolding distribution_def prob_space[symmetric]
```
```   652   by (subst finite_measure_finite_Union[symmetric])
```
```   653      (auto simp add: disjoint_family_on_def simple_function_def
```
```   654            intro!: arg_cong[where f=prob])
```
```   655
```
```   656 lemma (in prob_space) finite_random_variable_pairI:
```
```   657   assumes "finite_random_variable MX X"
```
```   658   assumes "finite_random_variable MY Y"
```
```   659   shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
```
```   660 proof
```
```   661   interpret MX: finite_sigma_algebra MX using assms by simp
```
```   662   interpret MY: finite_sigma_algebra MY using assms by simp
```
```   663   interpret P: pair_finite_sigma_algebra MX MY by default
```
```   664   show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
```
```   665   have sa: "sigma_algebra M" by default
```
```   666   show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
```
```   667     unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
```
```   668 qed
```
```   669
```
```   670 lemma (in prob_space) finite_random_variable_imp_sets:
```
```   671   "finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX"
```
```   672   unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
```
```   673
```
```   674 lemma (in prob_space) finite_random_variable_measurable:
```
```   675   assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events"
```
```   676 proof -
```
```   677   interpret X: finite_sigma_algebra MX using X by simp
```
```   678   from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and
```
```   679     "X \<in> space M \<rightarrow> space MX"
```
```   680     by (auto simp: measurable_def)
```
```   681   then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M"
```
```   682     by auto
```
```   683   show "X -` A \<inter> space M \<in> events"
```
```   684     unfolding * by (intro vimage) auto
```
```   685 qed
```
```   686
```
```   687 lemma (in prob_space) joint_distribution_finite_Times_le_fst:
```
```   688   assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
```
```   689   shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
```
```   690   unfolding distribution_def
```
```   691 proof (intro finite_measure_mono)
```
```   692   show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
```
```   693   show "X -` A \<inter> space M \<in> events"
```
```   694     using finite_random_variable_measurable[OF X] .
```
```   695   have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
```
```   696     (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
```
```   697 qed
```
```   698
```
```   699 lemma (in prob_space) joint_distribution_finite_Times_le_snd:
```
```   700   assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
```
```   701   shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
```
```   702   using assms
```
```   703   by (subst joint_distribution_commute)
```
```   704      (simp add: swap_product joint_distribution_finite_Times_le_fst)
```
```   705
```
```   706 lemma (in prob_space) finite_distribution_order:
```
```   707   fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
```
```   708   assumes "finite_random_variable MX X" "finite_random_variable MY Y"
```
```   709   shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
```
```   710     and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
```
```   711     and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
```
```   712     and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
```
```   713     and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
```
```   714     and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
```
```   715   using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"]
```
```   716   using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"]
```
```   717   by (auto intro: antisym)
```
```   718
```
```   719 lemma (in prob_space) setsum_joint_distribution:
```
```   720   assumes X: "finite_random_variable MX X"
```
```   721   assumes Y: "random_variable MY Y" "B \<in> sets MY"
```
```   722   shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B"
```
```   723   unfolding distribution_def
```
```   724 proof (subst finite_measure_finite_Union[symmetric])
```
```   725   interpret MX: finite_sigma_algebra MX using X by auto
```
```   726   show "finite (space MX)" using MX.finite_space .
```
```   727   let "?d i" = "(\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M"
```
```   728   { fix i assume "i \<in> space MX"
```
```   729     moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
```
```   730     ultimately show "?d i \<in> events"
```
```   731       using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y
```
```   732       using MX.sets_eq_Pow by auto }
```
```   733   show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def)
```
```   734   show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)"
```
```   735     using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>'])
```
```   736 qed
```
```   737
```
```   738 lemma (in prob_space) setsum_joint_distribution_singleton:
```
```   739   assumes X: "finite_random_variable MX X"
```
```   740   assumes Y: "finite_random_variable MY Y" "b \<in> space MY"
```
```   741   shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}"
```
```   742   using setsum_joint_distribution[OF X
```
```   743     finite_random_variableD[OF Y(1)]
```
```   744     finite_random_variable_imp_sets[OF Y]] by simp
```
```   745
```
```   746 locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2
```
```   747
```
```   748 sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default
```
```   749 sublocale pair_finite_prob_space \<subseteq> pair_finite_space M1 M2  by default
```
```   750 sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default
```
```   751
```
```   752 locale product_finite_prob_space =
```
```   753   fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme"
```
```   754     and I :: "'i set"
```
```   755   assumes finite_space: "\<And>i. finite_prob_space (M i)" and finite_index: "finite I"
```
```   756
```
```   757 sublocale product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space .
```
```   758 sublocale product_finite_prob_space \<subseteq> finite_product_sigma_finite M I by default (rule finite_index)
```
```   759 sublocale product_finite_prob_space \<subseteq> prob_space "Pi\<^isub>M I M"
```
```   760 proof
```
```   761   show "\<mu> (space P) = 1"
```
```   762     using measure_times[OF M.top] M.measure_space_1
```
```   763     by (simp add: setprod_1 space_product_algebra)
```
```   764 qed
```
```   765
```
```   766 lemma funset_eq_UN_fun_upd_I:
```
```   767   assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
```
```   768   and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
```
```   769   and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
```
```   770   shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
```
```   771 proof safe
```
```   772   fix f assume f: "f \<in> F (insert a A)"
```
```   773   show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
```
```   774   proof (rule UN_I[of "f(a := d)"])
```
```   775     show "f(a := d) \<in> F A" using *[OF f] .
```
```   776     show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
```
```   777     proof (rule image_eqI[of _ _ "f a"])
```
```   778       show "f a \<in> G (f(a := d))" using **[OF f] .
```
```   779     qed simp
```
```   780   qed
```
```   781 next
```
```   782   fix f x assume "f \<in> F A" "x \<in> G f"
```
```   783   from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
```
```   784 qed
```
```   785
```
```   786 lemma extensional_funcset_insert_eq[simp]:
```
```   787   assumes "a \<notin> A"
```
```   788   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)"
```
```   789   apply (rule funset_eq_UN_fun_upd_I)
```
```   790   using assms
```
```   791   by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
```
```   792
```
```   793 lemma finite_extensional_funcset[simp, intro]:
```
```   794   assumes "finite A" "finite B"
```
```   795   shows "finite (extensional A \<inter> (A \<rightarrow> B))"
```
```   796   using assms by induct (auto simp: extensional_funcset_insert_eq)
```
```   797
```
```   798 lemma finite_PiE[simp, intro]:
```
```   799   assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
```
```   800   shows "finite (Pi\<^isub>E A B)"
```
```   801 proof -
```
```   802   have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
```
```   803   show ?thesis
```
```   804     using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
```
```   805 qed
```
```   806
```
```   807 lemma (in product_finite_prob_space) singleton_eq_product:
```
```   808   assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})"
```
```   809 proof (safe intro!: ext[of _ x])
```
```   810   fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I"
```
```   811   with x show "y i = x i"
```
```   812     by (cases "i \<in> I") (auto simp: extensional_def)
```
```   813 qed (insert x, auto)
```
```   814
```
```   815 sublocale product_finite_prob_space \<subseteq> finite_prob_space "Pi\<^isub>M I M"
```
```   816 proof
```
```   817   show "finite (space P)"
```
```   818     using finite_index M.finite_space by auto
```
```   819
```
```   820   { fix x assume "x \<in> space P"
```
```   821     with this[THEN singleton_eq_product]
```
```   822     have "{x} \<in> sets P"
```
```   823       by (auto intro!: in_P) }
```
```   824   note x_in_P = this
```
```   825
```
```   826   have "Pow (space P) \<subseteq> sets P"
```
```   827   proof
```
```   828     fix X assume "X \<in> Pow (space P)"
```
```   829     moreover then have "finite X"
```
```   830       using `finite (space P)` by (blast intro: finite_subset)
```
```   831     ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P"
```
```   832       by (intro finite_UN x_in_P) auto
```
```   833     then show "X \<in> sets P" by simp
```
```   834   qed
```
```   835   with space_closed show [simp]: "sets P = Pow (space P)" ..
```
```   836
```
```   837   { fix x assume "x \<in> space P"
```
```   838     from this top have "\<mu> {x} \<le> \<mu> (space P)" by (intro measure_mono) auto
```
```   839     then show "\<mu> {x} \<noteq> \<infinity>"
```
```   840       using measure_space_1 by auto }
```
```   841 qed
```
```   842
```
```   843 lemma (in product_finite_prob_space) measure_finite_times:
```
```   844   "(\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)) \<Longrightarrow> \<mu> (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.\<mu> i (X i))"
```
```   845   by (rule measure_times) simp
```
```   846
```
```   847 lemma (in product_finite_prob_space) measure_singleton_times:
```
```   848   assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})"
```
```   849   unfolding singleton_eq_product[OF x] using x
```
```   850   by (intro measure_finite_times) auto
```
```   851
```
```   852 lemma (in product_finite_prob_space) prob_finite_times:
```
```   853   assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)"
```
```   854   shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
```
```   855 proof -
```
```   856   have "ereal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
```
```   857     using X by (intro finite_measure_eq[symmetric] in_P) auto
```
```   858   also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
```
```   859     using measure_finite_times X by simp
```
```   860   also have "\<dots> = ereal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
```
```   861     using X by (simp add: M.finite_measure_eq setprod_ereal)
```
```   862   finally show ?thesis by simp
```
```   863 qed
```
```   864
```
```   865 lemma (in product_finite_prob_space) prob_singleton_times:
```
```   866   assumes x: "x \<in> space P"
```
```   867   shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})"
```
```   868   unfolding singleton_eq_product[OF x] using x
```
```   869   by (intro prob_finite_times) auto
```
```   870
```
```   871 lemma (in product_finite_prob_space) prob_finite_product:
```
```   872   "A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})"
```
```   873   by (auto simp add: finite_measure_singleton prob_singleton_times
```
```   874            simp del: space_product_algebra
```
```   875            intro!: setsum_cong prob_singleton_times)
```
```   876
```
```   877 lemma (in prob_space) joint_distribution_finite_prob_space:
```
```   878   assumes X: "finite_random_variable MX X"
```
```   879   assumes Y: "finite_random_variable MY Y"
```
```   880   shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := ereal \<circ> joint_distribution X Y\<rparr>)"
```
```   881   by (intro distribution_finite_prob_space finite_random_variable_pairI X Y)
```
```   882
```
```   883 lemma finite_prob_space_eq:
```
```   884   "finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1"
```
```   885   unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def
```
```   886   by auto
```
```   887
```
```   888 lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1"
```
```   889   using measure_space_1 sum_over_space by simp
```
```   890
```
```   891 lemma (in finite_prob_space) joint_distribution_restriction_fst:
```
```   892   "joint_distribution X Y A \<le> distribution X (fst ` A)"
```
```   893   unfolding distribution_def
```
```   894 proof (safe intro!: finite_measure_mono)
```
```   895   fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
```
```   896   show "x \<in> X -` fst ` A"
```
```   897     by (auto intro!: image_eqI[OF _ *])
```
```   898 qed (simp_all add: sets_eq_Pow)
```
```   899
```
```   900 lemma (in finite_prob_space) joint_distribution_restriction_snd:
```
```   901   "joint_distribution X Y A \<le> distribution Y (snd ` A)"
```
```   902   unfolding distribution_def
```
```   903 proof (safe intro!: finite_measure_mono)
```
```   904   fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
```
```   905   show "x \<in> Y -` snd ` A"
```
```   906     by (auto intro!: image_eqI[OF _ *])
```
```   907 qed (simp_all add: sets_eq_Pow)
```
```   908
```
```   909 lemma (in finite_prob_space) distribution_order:
```
```   910   shows "0 \<le> distribution X x'"
```
```   911   and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')"
```
```   912   and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
```
```   913   and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
```
```   914   and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
```
```   915   and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
```
```   916   and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
```
```   917   and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
```
```   918   using
```
```   919     joint_distribution_restriction_fst[of X Y "{(x, y)}"]
```
```   920     joint_distribution_restriction_snd[of X Y "{(x, y)}"]
```
```   921   by (auto intro: antisym)
```
```   922
```
```   923 lemma (in finite_prob_space) distribution_mono:
```
```   924   assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
```
```   925   shows "distribution X x \<le> distribution Y y"
```
```   926   unfolding distribution_def
```
```   927   using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono)
```
```   928
```
```   929 lemma (in finite_prob_space) distribution_mono_gt_0:
```
```   930   assumes gt_0: "0 < distribution X x"
```
```   931   assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
```
```   932   shows "0 < distribution Y y"
```
```   933   by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
```
```   934
```
```   935 lemma (in finite_prob_space) sum_over_space_distrib:
```
```   936   "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
```
```   937   unfolding distribution_def prob_space[symmetric] using finite_space
```
```   938   by (subst finite_measure_finite_Union[symmetric])
```
```   939      (auto simp add: disjoint_family_on_def sets_eq_Pow
```
```   940            intro!: arg_cong[where f=\<mu>'])
```
```   941
```
```   942 lemma (in finite_prob_space) finite_sum_over_space_eq_1:
```
```   943   "(\<Sum>x\<in>space M. prob {x}) = 1"
```
```   944   using prob_space finite_space
```
```   945   by (subst (asm) finite_measure_finite_singleton) auto
```
```   946
```
```   947 lemma (in prob_space) distribution_remove_const:
```
```   948   shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
```
```   949   and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
```
```   950   and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
```
```   951   and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
```
```   952   and "distribution (\<lambda>x. ()) {()} = 1"
```
```   953   by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric])
```
```   954
```
```   955 lemma (in finite_prob_space) setsum_distribution_gen:
```
```   956   assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
```
```   957   and "inj_on f (X`space M)"
```
```   958   shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
```
```   959   unfolding distribution_def assms
```
```   960   using finite_space assms
```
```   961   by (subst finite_measure_finite_Union[symmetric])
```
```   962      (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
```
```   963       intro!: arg_cong[where f=prob])
```
```   964
```
```   965 lemma (in finite_prob_space) setsum_distribution:
```
```   966   "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
```
```   967   "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
```
```   968   "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
```
```   969   "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
```
```   970   "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
```
```   971   by (auto intro!: inj_onI setsum_distribution_gen)
```
```   972
```
```   973 lemma (in finite_prob_space) uniform_prob:
```
```   974   assumes "x \<in> space M"
```
```   975   assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
```
```   976   shows "prob {x} = 1 / card (space M)"
```
```   977 proof -
```
```   978   have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
```
```   979     using assms(2)[OF _ `x \<in> space M`] by blast
```
```   980   have "1 = prob (space M)"
```
```   981     using prob_space by auto
```
```   982   also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
```
```   983     using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
```
```   984       sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
```
```   985       finite_space unfolding disjoint_family_on_def  prob_space[symmetric]
```
```   986     by (auto simp add:setsum_restrict_set)
```
```   987   also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
```
```   988     using prob_x by auto
```
```   989   also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
```
```   990   finally have one: "1 = real (card (space M)) * prob {x}"
```
```   991     using real_eq_of_nat by auto
```
```   992   hence two: "real (card (space M)) \<noteq> 0" by fastsimp
```
```   993   from one have three: "prob {x} \<noteq> 0" by fastsimp
```
```   994   thus ?thesis using one two three divide_cancel_right
```
```   995     by (auto simp:field_simps)
```
```   996 qed
```
```   997
```
```   998 lemma (in prob_space) prob_space_subalgebra:
```
```   999   assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M"
```
```  1000     and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
```
```  1001   shows "prob_space N"
```
```  1002 proof -
```
```  1003   interpret N: measure_space N
```
```  1004     by (rule measure_space_subalgebra[OF assms])
```
```  1005   show ?thesis
```
```  1006   proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1)
```
```  1007 qed
```
```  1008
```
```  1009 lemma (in prob_space) prob_space_of_restricted_space:
```
```  1010   assumes "\<mu> A \<noteq> 0" "A \<in> sets M"
```
```  1011   shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)"
```
```  1012     (is "prob_space ?P")
```
```  1013 proof -
```
```  1014   interpret A: measure_space "restricted_space A"
```
```  1015     using `A \<in> sets M` by (rule restricted_measure_space)
```
```  1016   interpret A': sigma_algebra ?P
```
```  1017     by (rule A.sigma_algebra_cong) auto
```
```  1018   show "prob_space ?P"
```
```  1019   proof
```
```  1020     show "measure ?P (space ?P) = 1"
```
```  1021       using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto
```
```  1022     show "positive ?P (measure ?P)"
```
```  1023     proof (simp add: positive_def, safe)
```
```  1024       show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_ereal_def)
```
```  1025       fix B assume "B \<in> events"
```
```  1026       with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M`
```
```  1027       show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int)
```
```  1028     qed
```
```  1029     show "countably_additive ?P (measure ?P)"
```
```  1030     proof (simp add: countably_additive_def, safe)
```
```  1031       fix B and F :: "nat \<Rightarrow> 'a set"
```
```  1032       assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F"
```
```  1033       { fix i
```
```  1034         from F have "F i \<in> op \<inter> A ` events" by auto
```
```  1035         with `A \<in> events` have "F i \<in> events" by auto }
```
```  1036       moreover then have "range F \<subseteq> events" by auto
```
```  1037       moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S"
```
```  1038         by (simp add: mult_commute divide_ereal_def)
```
```  1039       moreover have "0 \<le> inverse (\<mu> A)"
```
```  1040         using real_measure[OF `A \<in> events`] by auto
```
```  1041       ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A"
```
```  1042         using measure_countably_additive[of F] F
```
```  1043         by (auto simp: suminf_cmult_ereal)
```
```  1044     qed
```
```  1045   qed
```
```  1046 qed
```
```  1047
```
```  1048 lemma finite_prob_spaceI:
```
```  1049   assumes "finite (space M)" "sets M = Pow(space M)"
```
```  1050     and "measure M (space M) = 1" "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A"
```
```  1051     and "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B"
```
```  1052   shows "finite_prob_space M"
```
```  1053   unfolding finite_prob_space_eq
```
```  1054 proof
```
```  1055   show "finite_measure_space M" using assms
```
```  1056     by (auto intro!: finite_measure_spaceI)
```
```  1057   show "measure M (space M) = 1" by fact
```
```  1058 qed
```
```  1059
```
```  1060 lemma (in finite_prob_space) finite_measure_space:
```
```  1061   fixes X :: "'a \<Rightarrow> 'x"
```
```  1062   shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X\<rparr>"
```
```  1063     (is "finite_measure_space ?S")
```
```  1064 proof (rule finite_measure_spaceI, simp_all)
```
```  1065   show "finite (X ` space M)" using finite_space by simp
```
```  1066 next
```
```  1067   fix A B :: "'x set" assume "A \<inter> B = {}"
```
```  1068   then show "distribution X (A \<union> B) = distribution X A + distribution X B"
```
```  1069     unfolding distribution_def
```
```  1070     by (subst finite_measure_Union[symmetric])
```
```  1071        (auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow)
```
```  1072 qed
```
```  1073
```
```  1074 lemma (in finite_prob_space) finite_prob_space_of_images:
```
```  1075   "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = ereal \<circ> distribution X \<rparr>"
```
```  1076   by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_ereal_def)
```
```  1077
```
```  1078 lemma (in finite_prob_space) finite_product_measure_space:
```
```  1079   fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
```
```  1080   assumes "finite s1" "finite s2"
```
```  1081   shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = ereal \<circ> joint_distribution X Y\<rparr>"
```
```  1082     (is "finite_measure_space ?M")
```
```  1083 proof (rule finite_measure_spaceI, simp_all)
```
```  1084   show "finite (s1 \<times> s2)"
```
```  1085     using assms by auto
```
```  1086 next
```
```  1087   fix A B :: "('x*'y) set" assume "A \<inter> B = {}"
```
```  1088   then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B"
```
```  1089     unfolding distribution_def
```
```  1090     by (subst finite_measure_Union[symmetric])
```
```  1091        (auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow)
```
```  1092 qed
```
```  1093
```
```  1094 lemma (in finite_prob_space) finite_product_measure_space_of_images:
```
```  1095   shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` space M,
```
```  1096                                 sets = Pow (X ` space M \<times> Y ` space M),
```
```  1097                                 measure = ereal \<circ> joint_distribution X Y \<rparr>"
```
```  1098   using finite_space by (auto intro!: finite_product_measure_space)
```
```  1099
```
```  1100 lemma (in finite_prob_space) finite_product_prob_space_of_images:
```
```  1101   "finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M),
```
```  1102                        measure = ereal \<circ> joint_distribution X Y \<rparr>"
```
```  1103   (is "finite_prob_space ?S")
```
```  1104 proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_ereal_def)
```
```  1105   have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto
```
```  1106   thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1"
```
```  1107     by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
```
```  1108 qed
```
```  1109
```
```  1110 subsection "Borel Measure on {0 ..< 1}"
```
```  1111
```
```  1112 definition pborel :: "real measure_space" where
```
```  1113   "pborel = lborel.restricted_space {0 ..< 1}"
```
```  1114
```
```  1115 lemma space_pborel[simp]:
```
```  1116   "space pborel = {0 ..< 1}"
```
```  1117   unfolding pborel_def by auto
```
```  1118
```
```  1119 lemma sets_pborel:
```
```  1120   "A \<in> sets pborel \<longleftrightarrow> A \<in> sets borel \<and> A \<subseteq> { 0 ..< 1}"
```
```  1121   unfolding pborel_def by auto
```
```  1122
```
```  1123 lemma in_pborel[intro, simp]:
```
```  1124   "A \<subseteq> {0 ..< 1} \<Longrightarrow> A \<in> sets borel \<Longrightarrow> A \<in> sets pborel"
```
```  1125   unfolding pborel_def by auto
```
```  1126
```
```  1127 interpretation pborel: measure_space pborel
```
```  1128   using lborel.restricted_measure_space[of "{0 ..< 1}"]
```
```  1129   by (simp add: pborel_def)
```
```  1130
```
```  1131 interpretation pborel: prob_space pborel
```
```  1132   by default (simp add: one_ereal_def pborel_def)
```
```  1133
```
```  1134 lemma pborel_prob: "pborel.prob A = (if A \<in> sets borel \<and> A \<subseteq> {0 ..< 1} then real (lborel.\<mu> A) else 0)"
```
```  1135   unfolding pborel.\<mu>'_def by (auto simp: pborel_def)
```
```  1136
```
```  1137 lemma pborel_singelton[simp]: "pborel.prob {a} = 0"
```
```  1138   by (auto simp: pborel_prob)
```
```  1139
```
```  1140 lemma
```
```  1141   shows pborel_atLeastAtMost[simp]: "pborel.\<mu>' {a .. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)"
```
```  1142     and pborel_atLeastLessThan[simp]: "pborel.\<mu>' {a ..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
```
```  1143     and pborel_greaterThanAtMost[simp]: "pborel.\<mu>' {a <.. b} = (if 0 \<le> a \<and> a \<le> b \<and> b < 1 then b - a else 0)"
```
```  1144     and pborel_greaterThanLessThan[simp]: "pborel.\<mu>' {a <..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
```
```  1145   unfolding pborel_prob
```
```  1146   by (auto simp: atLeastAtMost_subseteq_atLeastLessThan_iff
```
```  1147     greaterThanAtMost_subseteq_atLeastLessThan_iff greaterThanLessThan_subseteq_atLeastLessThan_iff)
```
```  1148
```
```  1149 lemma pborel_lebesgue_measure:
```
```  1150   "A \<in> sets pborel \<Longrightarrow> pborel.prob A = real (measure lebesgue A)"
```
```  1151   by (simp add: sets_pborel pborel_prob)
```
```  1152
```
```  1153 lemma pborel_alt:
```
```  1154   "pborel = sigma \<lparr>
```
```  1155     space = {0..<1},
```
```  1156     sets = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1}),
```
```  1157     measure = measure lborel \<rparr>" (is "_ = ?R")
```
```  1158 proof -
```
```  1159   have *: "{0..<1::real} \<in> sets borel" by auto
```
```  1160   have **: "op \<inter> {0..<1::real} ` range (\<lambda>(x, y). {x..<y}) = range (\<lambda>(x,y). {x..<y} \<inter> {0..<1})"
```
```  1161     unfolding image_image by (intro arg_cong[where f=range]) auto
```
```  1162   have "pborel = algebra.restricted_space (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a ..< b :: real}),
```
```  1163     measure = measure pborel\<rparr>) {0 ..< 1}"
```
```  1164     by (simp add: sigma_def lebesgue_def pborel_def borel_eq_atLeastLessThan lborel_def)
```
```  1165   also have "\<dots> = ?R"
```
```  1166     by (subst restricted_sigma)
```
```  1167        (simp_all add: sets_sigma sigma_sets.Basic ** pborel_def image_eqI[of _ _ "(0,1)"])
```
```  1168   finally show ?thesis .
```
```  1169 qed
```
```  1170
```
```  1171 subsection "Bernoulli space"
```
```  1172
```
```  1173 definition "bernoulli_space p = \<lparr> space = UNIV, sets = UNIV,
```
```  1174   measure = ereal \<circ> setsum (\<lambda>b. if b then min 1 (max 0 p) else 1 - min 1 (max 0 p)) \<rparr>"
```
```  1175
```
```  1176 interpretation bernoulli: finite_prob_space "bernoulli_space p" for p
```
```  1177   by (rule finite_prob_spaceI)
```
```  1178      (auto simp: bernoulli_space_def UNIV_bool one_ereal_def setsum_Un_disjoint intro!: setsum_nonneg)
```
```  1179
```
```  1180 lemma bernoulli_measure:
```
```  1181   "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p B = (\<Sum>b\<in>B. if b then p else 1 - p)"
```
```  1182   unfolding bernoulli.\<mu>'_def unfolding bernoulli_space_def by (auto intro!: setsum_cong)
```
```  1183
```
```  1184 lemma bernoulli_measure_True: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {True} = p"
```
```  1185   and bernoulli_measure_False: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {False} = 1 - p"
```
```  1186   unfolding bernoulli_measure by simp_all
```
```  1187
```
```  1188 end
```