author wenzelm
Fri Aug 18 20:47:47 2017 +0200 (2017-08-18)
changeset 66453 cc19f7ca2ed6
parent 64320 ba194424b895
child 67226 ec32cdaab97b
permissions -rw-r--r--
session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
     1 (*  Title:      HOL/Probability/Giry_Monad.thy
     2     Author:     Johannes Hölzl, TU München
     3     Author:     Manuel Eberl, TU München
     5 Defines the subprobability spaces, the subprobability functor and the Giry monad on subprobability
     6 spaces.
     7 *)
     9 theory Giry_Monad
    10   imports Probability_Measure "HOL-Library.Monad_Syntax"
    11 begin
    13 section \<open>Sub-probability spaces\<close>
    15 locale subprob_space = finite_measure +
    16   assumes emeasure_space_le_1: "emeasure M (space M) \<le> 1"
    17   assumes subprob_not_empty: "space M \<noteq> {}"
    19 lemma subprob_spaceI[Pure.intro!]:
    20   assumes *: "emeasure M (space M) \<le> 1"
    21   assumes "space M \<noteq> {}"
    22   shows "subprob_space M"
    23 proof -
    24   interpret finite_measure M
    25   proof
    26     show "emeasure M (space M) \<noteq> \<infinity>" using * by (auto simp: top_unique)
    27   qed
    28   show "subprob_space M" by standard fact+
    29 qed
    31 lemma (in subprob_space) emeasure_subprob_space_less_top: "emeasure M A \<noteq> top"
    32   using emeasure_finite[of A] .
    34 lemma prob_space_imp_subprob_space:
    35   "prob_space M \<Longrightarrow> subprob_space M"
    36   by (rule subprob_spaceI) (simp_all add: prob_space.emeasure_space_1 prob_space.not_empty)
    38 lemma subprob_space_imp_sigma_finite: "subprob_space M \<Longrightarrow> sigma_finite_measure M"
    39   unfolding subprob_space_def finite_measure_def by simp
    41 sublocale prob_space \<subseteq> subprob_space
    42   by (rule subprob_spaceI) (simp_all add: emeasure_space_1 not_empty)
    44 lemma subprob_space_sigma [simp]: "\<Omega> \<noteq> {} \<Longrightarrow> subprob_space (sigma \<Omega> X)"
    45 by(rule subprob_spaceI)(simp_all add: emeasure_sigma space_measure_of_conv)
    47 lemma subprob_space_null_measure: "space M \<noteq> {} \<Longrightarrow> subprob_space (null_measure M)"
    48 by(simp add: null_measure_def)
    50 lemma (in subprob_space) subprob_space_distr:
    51   assumes f: "f \<in> measurable M M'" and "space M' \<noteq> {}" shows "subprob_space (distr M M' f)"
    52 proof (rule subprob_spaceI)
    53   have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
    54   with f show "emeasure (distr M M' f) (space (distr M M' f)) \<le> 1"
    55     by (auto simp: emeasure_distr emeasure_space_le_1)
    56   show "space (distr M M' f) \<noteq> {}" by (simp add: assms)
    57 qed
    59 lemma (in subprob_space) subprob_emeasure_le_1: "emeasure M X \<le> 1"
    60   by (rule order.trans[OF emeasure_space emeasure_space_le_1])
    62 lemma (in subprob_space) subprob_measure_le_1: "measure M X \<le> 1"
    63   using subprob_emeasure_le_1[of X] by (simp add: emeasure_eq_measure)
    65 lemma (in subprob_space) nn_integral_le_const:
    66   assumes "0 \<le> c" "AE x in M. f x \<le> c"
    67   shows "(\<integral>\<^sup>+x. f x \<partial>M) \<le> c"
    68 proof -
    69   have "(\<integral>\<^sup>+ x. f x \<partial>M) \<le> (\<integral>\<^sup>+ x. c \<partial>M)"
    70     by(rule nn_integral_mono_AE) fact
    71   also have "\<dots> \<le> c * emeasure M (space M)"
    72     using \<open>0 \<le> c\<close> by simp
    73   also have "\<dots> \<le> c * 1" using emeasure_space_le_1 \<open>0 \<le> c\<close> by(rule mult_left_mono)
    74   finally show ?thesis by simp
    75 qed
    77 lemma emeasure_density_distr_interval:
    78   fixes h :: "real \<Rightarrow> real" and g :: "real \<Rightarrow> real" and g' :: "real \<Rightarrow> real"
    79   assumes [simp]: "a \<le> b"
    80   assumes Mf[measurable]: "f \<in> borel_measurable borel"
    81   assumes Mg[measurable]: "g \<in> borel_measurable borel"
    82   assumes Mg'[measurable]: "g' \<in> borel_measurable borel"
    83   assumes Mh[measurable]: "h \<in> borel_measurable borel"
    84   assumes prob: "subprob_space (density lborel f)"
    85   assumes nonnegf: "\<And>x. f x \<ge> 0"
    86   assumes derivg: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"
    87   assumes contg': "continuous_on {a..b} g'"
    88   assumes mono: "strict_mono_on g {a..b}" and inv: "\<And>x. h x \<in> {a..b} \<Longrightarrow> g (h x) = x"
    89   assumes range: "{a..b} \<subseteq> range h"
    90   shows "emeasure (distr (density lborel f) lborel h) {a..b} =
    91              emeasure (density lborel (\<lambda>x. f (g x) * g' x)) {a..b}"
    92 proof (cases "a < b")
    93   assume "a < b"
    94   from mono have inj: "inj_on g {a..b}" by (rule strict_mono_on_imp_inj_on)
    95   from mono have mono': "mono_on g {a..b}" by (rule strict_mono_on_imp_mono_on)
    96   from mono' derivg have "\<And>x. x \<in> {a<..<b} \<Longrightarrow> g' x \<ge> 0"
    97     by (rule mono_on_imp_deriv_nonneg) auto
    98   from contg' this have derivg_nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0"
    99     by (rule continuous_ge_on_Ioo) (simp_all add: \<open>a < b\<close>)
   101   from derivg have contg: "continuous_on {a..b} g" by (rule has_real_derivative_imp_continuous_on)
   102   have A: "h -` {a..b} = {g a..g b}"
   103   proof (intro equalityI subsetI)
   104     fix x assume x: "x \<in> h -` {a..b}"
   105     hence "g (h x) \<in> {g a..g b}" by (auto intro: mono_onD[OF mono'])
   106     with inv and x show "x \<in> {g a..g b}" by simp
   107   next
   108     fix y assume y: "y \<in> {g a..g b}"
   109     with IVT'[OF _ _ _ contg, of y] obtain x where "x \<in> {a..b}" "y = g x" by auto
   110     with range and inv show "y \<in> h -` {a..b}" by auto
   111   qed
   113   have prob': "subprob_space (distr (density lborel f) lborel h)"
   114     by (rule subprob_space.subprob_space_distr[OF prob]) (simp_all add: Mh)
   115   have B: "emeasure (distr (density lborel f) lborel h) {a..b} =
   116             \<integral>\<^sup>+x. f x * indicator (h -` {a..b}) x \<partial>lborel"
   117     by (subst emeasure_distr) (simp_all add: emeasure_density Mf Mh measurable_sets_borel[OF Mh])
   118   also note A
   119   also have "emeasure (distr (density lborel f) lborel h) {a..b} \<le> 1"
   120     by (rule subprob_space.subprob_emeasure_le_1) (rule prob')
   121   hence "emeasure (distr (density lborel f) lborel h) {a..b} \<noteq> \<infinity>" by (auto simp: top_unique)
   122   with assms have "(\<integral>\<^sup>+x. f x * indicator {g a..g b} x \<partial>lborel) =
   123                       (\<integral>\<^sup>+x. f (g x) * g' x * indicator {a..b} x \<partial>lborel)"
   124     by (intro nn_integral_substitution_aux)
   125        (auto simp: derivg_nonneg A B emeasure_density mult.commute \<open>a < b\<close>)
   126   also have "... = emeasure (density lborel (\<lambda>x. f (g x) * g' x)) {a..b}"
   127     by (simp add: emeasure_density)
   128   finally show ?thesis .
   129 next
   130   assume "\<not>a < b"
   131   with \<open>a \<le> b\<close> have [simp]: "b = a" by (simp add: not_less del: \<open>a \<le> b\<close>)
   132   from inv and range have "h -` {a} = {g a}" by auto
   133   thus ?thesis by (simp_all add: emeasure_distr emeasure_density measurable_sets_borel[OF Mh])
   134 qed
   136 locale pair_subprob_space =
   137   pair_sigma_finite M1 M2 + M1: subprob_space M1 + M2: subprob_space M2 for M1 M2
   139 sublocale pair_subprob_space \<subseteq> P?: subprob_space "M1 \<Otimes>\<^sub>M M2"
   140 proof
   141   from mult_le_one[OF M1.emeasure_space_le_1 _ M2.emeasure_space_le_1]
   142   show "emeasure (M1 \<Otimes>\<^sub>M M2) (space (M1 \<Otimes>\<^sub>M M2)) \<le> 1"
   143     by (simp add: M2.emeasure_pair_measure_Times space_pair_measure)
   144   from M1.subprob_not_empty and M2.subprob_not_empty show "space (M1 \<Otimes>\<^sub>M M2) \<noteq> {}"
   145     by (simp add: space_pair_measure)
   146 qed
   148 lemma subprob_space_null_measure_iff:
   149     "subprob_space (null_measure M) \<longleftrightarrow> space M \<noteq> {}"
   150   by (auto intro!: subprob_spaceI dest: subprob_space.subprob_not_empty)
   152 lemma subprob_space_restrict_space:
   153   assumes M: "subprob_space M"
   154   and A: "A \<inter> space M \<in> sets M" "A \<inter> space M \<noteq> {}"
   155   shows "subprob_space (restrict_space M A)"
   156 proof(rule subprob_spaceI)
   157   have "emeasure (restrict_space M A) (space (restrict_space M A)) = emeasure M (A \<inter> space M)"
   158     using A by(simp add: emeasure_restrict_space space_restrict_space)
   159   also have "\<dots> \<le> 1" by(rule subprob_space.subprob_emeasure_le_1)(rule M)
   160   finally show "emeasure (restrict_space M A) (space (restrict_space M A)) \<le> 1" .
   161 next
   162   show "space (restrict_space M A) \<noteq> {}"
   163     using A by(simp add: space_restrict_space)
   164 qed
   166 definition subprob_algebra :: "'a measure \<Rightarrow> 'a measure measure" where
   167   "subprob_algebra K =
   168     (SUP A : sets K. vimage_algebra {M. subprob_space M \<and> sets M = sets K} (\<lambda>M. emeasure M A) borel)"
   170 lemma space_subprob_algebra: "space (subprob_algebra A) = {M. subprob_space M \<and> sets M = sets A}"
   171   by (auto simp add: subprob_algebra_def space_Sup_eq_UN)
   173 lemma subprob_algebra_cong: "sets M = sets N \<Longrightarrow> subprob_algebra M = subprob_algebra N"
   174   by (simp add: subprob_algebra_def)
   176 lemma measurable_emeasure_subprob_algebra[measurable]:
   177   "a \<in> sets A \<Longrightarrow> (\<lambda>M. emeasure M a) \<in> borel_measurable (subprob_algebra A)"
   178   by (auto intro!: measurable_Sup1 measurable_vimage_algebra1 simp: subprob_algebra_def)
   180 lemma measurable_measure_subprob_algebra[measurable]:
   181   "a \<in> sets A \<Longrightarrow> (\<lambda>M. measure M a) \<in> borel_measurable (subprob_algebra A)"
   182   unfolding measure_def by measurable
   184 lemma subprob_measurableD:
   185   assumes N: "N \<in> measurable M (subprob_algebra S)" and x: "x \<in> space M"
   186   shows "space (N x) = space S"
   187     and "sets (N x) = sets S"
   188     and "measurable (N x) K = measurable S K"
   189     and "measurable K (N x) = measurable K S"
   190   using measurable_space[OF N x]
   191   by (auto simp: space_subprob_algebra intro!: measurable_cong_sets dest: sets_eq_imp_space_eq)
   193 ML \<open>
   195 fun subprob_cong thm ctxt = (
   196   let
   197     val thm' = Thm.transfer (Proof_Context.theory_of ctxt) thm
   198     val free = thm' |> Thm.concl_of |> HOLogic.dest_Trueprop |> dest_comb |> fst |>
   199       dest_comb |> snd |> strip_abs_body |> head_of |> is_Free
   200   in
   201     if free then ([], Measurable.add_local_cong (thm' RS @{thm subprob_measurableD(2)}) ctxt)
   202             else ([], ctxt)
   203   end
   204   handle THM _ => ([], ctxt) | TERM _ => ([], ctxt))
   206 \<close>
   208 setup \<open>
   209   Context.theory_map (Measurable.add_preprocessor "subprob_cong" subprob_cong)
   210 \<close>
   212 context
   213   fixes K M N assumes K: "K \<in> measurable M (subprob_algebra N)"
   214 begin
   216 lemma subprob_space_kernel: "a \<in> space M \<Longrightarrow> subprob_space (K a)"
   217   using measurable_space[OF K] by (simp add: space_subprob_algebra)
   219 lemma sets_kernel: "a \<in> space M \<Longrightarrow> sets (K a) = sets N"
   220   using measurable_space[OF K] by (simp add: space_subprob_algebra)
   222 lemma measurable_emeasure_kernel[measurable]:
   223     "A \<in> sets N \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
   224   using measurable_compose[OF K measurable_emeasure_subprob_algebra] .
   226 end
   228 lemma measurable_subprob_algebra:
   229   "(\<And>a. a \<in> space M \<Longrightarrow> subprob_space (K a)) \<Longrightarrow>
   230   (\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N) \<Longrightarrow>
   231   (\<And>A. A \<in> sets N \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M) \<Longrightarrow>
   232   K \<in> measurable M (subprob_algebra N)"
   233   by (auto intro!: measurable_Sup2 measurable_vimage_algebra2 simp: subprob_algebra_def)
   235 lemma measurable_submarkov:
   236   "K \<in> measurable M (subprob_algebra M) \<longleftrightarrow>
   237     (\<forall>x\<in>space M. subprob_space (K x) \<and> sets (K x) = sets M) \<and>
   238     (\<forall>A\<in>sets M. (\<lambda>x. emeasure (K x) A) \<in> measurable M borel)"
   239 proof
   240   assume "(\<forall>x\<in>space M. subprob_space (K x) \<and> sets (K x) = sets M) \<and>
   241     (\<forall>A\<in>sets M. (\<lambda>x. emeasure (K x) A) \<in> borel_measurable M)"
   242   then show "K \<in> measurable M (subprob_algebra M)"
   243     by (intro measurable_subprob_algebra) auto
   244 next
   245   assume "K \<in> measurable M (subprob_algebra M)"
   246   then show "(\<forall>x\<in>space M. subprob_space (K x) \<and> sets (K x) = sets M) \<and>
   247     (\<forall>A\<in>sets M. (\<lambda>x. emeasure (K x) A) \<in> borel_measurable M)"
   248     by (auto dest: subprob_space_kernel sets_kernel)
   249 qed
   251 lemma measurable_subprob_algebra_generated:
   252   assumes eq: "sets N = sigma_sets \<Omega> G" and "Int_stable G" "G \<subseteq> Pow \<Omega>"
   253   assumes subsp: "\<And>a. a \<in> space M \<Longrightarrow> subprob_space (K a)"
   254   assumes sets: "\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N"
   255   assumes "\<And>A. A \<in> G \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
   256   assumes \<Omega>: "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M"
   257   shows "K \<in> measurable M (subprob_algebra N)"
   258 proof (rule measurable_subprob_algebra)
   259   fix a assume "a \<in> space M" then show "subprob_space (K a)" "sets (K a) = sets N" by fact+
   260 next
   261   interpret G: sigma_algebra \<Omega> "sigma_sets \<Omega> G"
   262     using \<open>G \<subseteq> Pow \<Omega>\<close> by (rule sigma_algebra_sigma_sets)
   263   fix A assume "A \<in> sets N" with assms(2,3) show "(\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
   264     unfolding \<open>sets N = sigma_sets \<Omega> G\<close>
   265   proof (induction rule: sigma_sets_induct_disjoint)
   266     case (basic A) then show ?case by fact
   267   next
   268     case empty then show ?case by simp
   269   next
   270     case (compl A)
   271     have "(\<lambda>a. emeasure (K a) (\<Omega> - A)) \<in> borel_measurable M \<longleftrightarrow>
   272       (\<lambda>a. emeasure (K a) \<Omega> - emeasure (K a) A) \<in> borel_measurable M"
   273       using G.sets_into_space sets eq compl subprob_space.emeasure_subprob_space_less_top[OF subsp]
   274       by (intro measurable_cong emeasure_Diff) auto
   275     with compl \<Omega> show ?case
   276       by simp
   277   next
   278     case (union F)
   279     moreover have "(\<lambda>a. emeasure (K a) (\<Union>i. F i)) \<in> borel_measurable M \<longleftrightarrow>
   280         (\<lambda>a. \<Sum>i. emeasure (K a) (F i)) \<in> borel_measurable M"
   281       using sets union eq
   282       by (intro measurable_cong suminf_emeasure[symmetric]) auto
   283     ultimately show ?case
   284       by auto
   285   qed
   286 qed
   288 lemma space_subprob_algebra_empty_iff:
   289   "space (subprob_algebra N) = {} \<longleftrightarrow> space N = {}"
   290 proof
   291   have "\<And>x. x \<in> space N \<Longrightarrow> density N (\<lambda>_. 0) \<in> space (subprob_algebra N)"
   292     by (auto simp: space_subprob_algebra emeasure_density intro!: subprob_spaceI)
   293   then show "space (subprob_algebra N) = {} \<Longrightarrow> space N = {}"
   294     by auto
   295 next
   296   assume "space N = {}"
   297   hence "sets N = {{}}" by (simp add: space_empty_iff)
   298   moreover have "\<And>M. subprob_space M \<Longrightarrow> sets M \<noteq> {{}}"
   299     by (simp add: subprob_space.subprob_not_empty space_empty_iff[symmetric])
   300   ultimately show "space (subprob_algebra N) = {}" by (auto simp: space_subprob_algebra)
   301 qed
   303 lemma nn_integral_measurable_subprob_algebra[measurable]:
   304   assumes f: "f \<in> borel_measurable N"
   305   shows "(\<lambda>M. integral\<^sup>N M f) \<in> borel_measurable (subprob_algebra N)" (is "_ \<in> ?B")
   306   using f
   307 proof induct
   308   case (cong f g)
   309   moreover have "(\<lambda>M'. \<integral>\<^sup>+M''. f M'' \<partial>M') \<in> ?B \<longleftrightarrow> (\<lambda>M'. \<integral>\<^sup>+M''. g M'' \<partial>M') \<in> ?B"
   310     by (intro measurable_cong nn_integral_cong cong)
   311        (auto simp: space_subprob_algebra dest!: sets_eq_imp_space_eq)
   312   ultimately show ?case by simp
   313 next
   314   case (set B)
   315   then have "(\<lambda>M'. \<integral>\<^sup>+M''. indicator B M'' \<partial>M') \<in> ?B \<longleftrightarrow> (\<lambda>M'. emeasure M' B) \<in> ?B"
   316     by (intro measurable_cong nn_integral_indicator) (simp add: space_subprob_algebra)
   317   with set show ?case
   318     by (simp add: measurable_emeasure_subprob_algebra)
   319 next
   320   case (mult f c)
   321   then have "(\<lambda>M'. \<integral>\<^sup>+M''. c * f M'' \<partial>M') \<in> ?B \<longleftrightarrow> (\<lambda>M'. c * \<integral>\<^sup>+M''. f M'' \<partial>M') \<in> ?B"
   322     by (intro measurable_cong nn_integral_cmult) (auto simp add: space_subprob_algebra)
   323   with mult show ?case
   324     by simp
   325 next
   326   case (add f g)
   327   then have "(\<lambda>M'. \<integral>\<^sup>+M''. f M'' + g M'' \<partial>M') \<in> ?B \<longleftrightarrow> (\<lambda>M'. (\<integral>\<^sup>+M''. f M'' \<partial>M') + (\<integral>\<^sup>+M''. g M'' \<partial>M')) \<in> ?B"
   328     by (intro measurable_cong nn_integral_add) (auto simp add: space_subprob_algebra)
   329   with add show ?case
   330     by (simp add: ac_simps)
   331 next
   332   case (seq F)
   333   then have "(\<lambda>M'. \<integral>\<^sup>+M''. (SUP i. F i) M'' \<partial>M') \<in> ?B \<longleftrightarrow> (\<lambda>M'. SUP i. (\<integral>\<^sup>+M''. F i M'' \<partial>M')) \<in> ?B"
   334     unfolding SUP_apply
   335     by (intro measurable_cong nn_integral_monotone_convergence_SUP) (auto simp add: space_subprob_algebra)
   336   with seq show ?case
   337     by (simp add: ac_simps)
   338 qed
   340 lemma measurable_distr:
   341   assumes [measurable]: "f \<in> measurable M N"
   342   shows "(\<lambda>M'. distr M' N f) \<in> measurable (subprob_algebra M) (subprob_algebra N)"
   343 proof (cases "space N = {}")
   344   assume not_empty: "space N \<noteq> {}"
   345   show ?thesis
   346   proof (rule measurable_subprob_algebra)
   347     fix A assume A: "A \<in> sets N"
   348     then have "(\<lambda>M'. emeasure (distr M' N f) A) \<in> borel_measurable (subprob_algebra M) \<longleftrightarrow>
   349       (\<lambda>M'. emeasure M' (f -` A \<inter> space M)) \<in> borel_measurable (subprob_algebra M)"
   350       by (intro measurable_cong)
   351          (auto simp: emeasure_distr space_subprob_algebra
   352                intro!: arg_cong2[where f=emeasure] sets_eq_imp_space_eq arg_cong2[where f="op \<inter>"])
   353     also have "\<dots>"
   354       using A by (intro measurable_emeasure_subprob_algebra) simp
   355     finally show "(\<lambda>M'. emeasure (distr M' N f) A) \<in> borel_measurable (subprob_algebra M)" .
   356   qed (auto intro!: subprob_space.subprob_space_distr simp: space_subprob_algebra not_empty cong: measurable_cong_sets)
   357 qed (insert assms, auto simp: measurable_empty_iff space_subprob_algebra_empty_iff)
   359 lemma emeasure_space_subprob_algebra[measurable]:
   360   "(\<lambda>a. emeasure a (space a)) \<in> borel_measurable (subprob_algebra N)"
   361 proof-
   362   have "(\<lambda>a. emeasure a (space N)) \<in> borel_measurable (subprob_algebra N)" (is "?f \<in> ?M")
   363     by (rule measurable_emeasure_subprob_algebra) simp
   364   also have "?f \<in> ?M \<longleftrightarrow> (\<lambda>a. emeasure a (space a)) \<in> ?M"
   365     by (rule measurable_cong) (auto simp: space_subprob_algebra dest: sets_eq_imp_space_eq)
   366   finally show ?thesis .
   367 qed
   369 lemma integrable_measurable_subprob_algebra[measurable]:
   370   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   371   assumes [measurable]: "f \<in> borel_measurable N"
   372   shows "Measurable.pred (subprob_algebra N) (\<lambda>M. integrable M f)"
   373 proof (rule measurable_cong[THEN iffD2])
   374   show "M \<in> space (subprob_algebra N) \<Longrightarrow> integrable M f \<longleftrightarrow> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>" for M
   375     by (auto simp: space_subprob_algebra integrable_iff_bounded)
   376 qed measurable
   378 lemma integral_measurable_subprob_algebra[measurable]:
   379   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   380   assumes f [measurable]: "f \<in> borel_measurable N"
   381   shows "(\<lambda>M. integral\<^sup>L M f) \<in> subprob_algebra N \<rightarrow>\<^sub>M borel"
   382 proof -
   383   from borel_measurable_implies_sequence_metric[OF f, of 0]
   384   obtain F where F: "\<And>i. simple_function N (F i)"
   385     "\<And>x. x \<in> space N \<Longrightarrow> (\<lambda>i. F i x) \<longlonglongrightarrow> f x"
   386     "\<And>i x. x \<in> space N \<Longrightarrow> norm (F i x) \<le> 2 * norm (f x)"
   387     unfolding norm_conv_dist by blast
   389   have [measurable]: "F i \<in> N \<rightarrow>\<^sub>M count_space UNIV" for i
   390     using F(1) by (rule measurable_simple_function)
   392   define F' where [abs_def]:
   393     "F' M i = (if integrable M f then integral\<^sup>L M (F i) else 0)" for M i
   395   have "(\<lambda>M. F' M i) \<in> subprob_algebra N \<rightarrow>\<^sub>M borel" for i
   396   proof (rule measurable_cong[THEN iffD2])
   397     fix M assume "M \<in> space (subprob_algebra N)"
   398     then have [simp]: "sets M = sets N" "space M = space N" "subprob_space M"
   399       by (auto simp: space_subprob_algebra intro!: sets_eq_imp_space_eq)
   400     interpret subprob_space M by fact
   401     have "F' M i = (if integrable M f then Bochner_Integration.simple_bochner_integral M (F i) else 0)"
   402       using F(1)
   403       by (subst simple_bochner_integrable_eq_integral)
   404          (auto simp: simple_bochner_integrable.simps simple_function_def F'_def)
   405     then show "F' M i = (if integrable M f then \<Sum>y\<in>F i ` space N. measure M {x\<in>space N. F i x = y} *\<^sub>R y else 0)"
   406       unfolding simple_bochner_integral_def by simp
   407   qed measurable
   408   moreover
   409   have "F' M \<longlonglongrightarrow> integral\<^sup>L M f" if M: "M \<in> space (subprob_algebra N)" for M
   410   proof cases
   411     from M have [simp]: "sets M = sets N" "space M = space N"
   412       by (auto simp: space_subprob_algebra intro!: sets_eq_imp_space_eq)
   413     assume "integrable M f" then show ?thesis
   414       unfolding F'_def using F(1)[THEN borel_measurable_simple_function] F
   415       by (auto intro!: integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]
   416                cong: measurable_cong_sets)
   417   qed (auto simp: F'_def not_integrable_integral_eq)
   418   ultimately show ?thesis
   419     by (rule borel_measurable_LIMSEQ_metric)
   420 qed
   422 (* TODO: Rename. This name is too general -- Manuel *)
   423 lemma measurable_pair_measure:
   424   assumes f: "f \<in> measurable M (subprob_algebra N)"
   425   assumes g: "g \<in> measurable M (subprob_algebra L)"
   426   shows "(\<lambda>x. f x \<Otimes>\<^sub>M g x) \<in> measurable M (subprob_algebra (N \<Otimes>\<^sub>M L))"
   427 proof (rule measurable_subprob_algebra)
   428   { fix x assume "x \<in> space M"
   429     with measurable_space[OF f] measurable_space[OF g]
   430     have fx: "f x \<in> space (subprob_algebra N)" and gx: "g x \<in> space (subprob_algebra L)"
   431       by auto
   432     interpret F: subprob_space "f x"
   433       using fx by (simp add: space_subprob_algebra)
   434     interpret G: subprob_space "g x"
   435       using gx by (simp add: space_subprob_algebra)
   437     interpret pair_subprob_space "f x" "g x" ..
   438     show "subprob_space (f x \<Otimes>\<^sub>M g x)" by unfold_locales
   439     show sets_eq: "sets (f x \<Otimes>\<^sub>M g x) = sets (N \<Otimes>\<^sub>M L)"
   440       using fx gx by (simp add: space_subprob_algebra)
   442     have 1: "\<And>A B. A \<in> sets N \<Longrightarrow> B \<in> sets L \<Longrightarrow> emeasure (f x \<Otimes>\<^sub>M g x) (A \<times> B) = emeasure (f x) A * emeasure (g x) B"
   443       using fx gx by (intro G.emeasure_pair_measure_Times) (auto simp: space_subprob_algebra)
   444     have "emeasure (f x \<Otimes>\<^sub>M g x) (space (f x \<Otimes>\<^sub>M g x)) =
   445               emeasure (f x) (space (f x)) * emeasure (g x) (space (g x))"
   446       by (subst G.emeasure_pair_measure_Times[symmetric]) (simp_all add: space_pair_measure)
   447     hence 2: "\<And>A. A \<in> sets (N \<Otimes>\<^sub>M L) \<Longrightarrow> emeasure (f x \<Otimes>\<^sub>M g x) (space N \<times> space L - A) =
   448                                              ... - emeasure (f x \<Otimes>\<^sub>M g x) A"
   449       using emeasure_compl[simplified, OF _ P.emeasure_finite]
   450       unfolding sets_eq
   451       unfolding sets_eq_imp_space_eq[OF sets_eq]
   452       by (simp add: space_pair_measure G.emeasure_pair_measure_Times)
   453     note 1 2 sets_eq }
   454   note Times = this(1) and Compl = this(2) and sets_eq = this(3)
   456   fix A assume A: "A \<in> sets (N \<Otimes>\<^sub>M L)"
   457   show "(\<lambda>a. emeasure (f a \<Otimes>\<^sub>M g a) A) \<in> borel_measurable M"
   458     using Int_stable_pair_measure_generator pair_measure_closed A
   459     unfolding sets_pair_measure
   460   proof (induct A rule: sigma_sets_induct_disjoint)
   461     case (basic A) then show ?case
   462       by (auto intro!: borel_measurable_times_ennreal simp: Times cong: measurable_cong)
   463          (auto intro!: measurable_emeasure_kernel f g)
   464   next
   465     case (compl A)
   466     then have A: "A \<in> sets (N \<Otimes>\<^sub>M L)"
   467       by (auto simp: sets_pair_measure)
   468     have "(\<lambda>x. emeasure (f x) (space (f x)) * emeasure (g x) (space (g x)) -
   469                    emeasure (f x \<Otimes>\<^sub>M g x) A) \<in> borel_measurable M" (is "?f \<in> ?M")
   470       using compl(2) f g by measurable
   471     thus ?case by (simp add: Compl A cong: measurable_cong)
   472   next
   473     case (union A)
   474     then have "range A \<subseteq> sets (N \<Otimes>\<^sub>M L)" "disjoint_family A"
   475       by (auto simp: sets_pair_measure)
   476     then have "(\<lambda>a. emeasure (f a \<Otimes>\<^sub>M g a) (\<Union>i. A i)) \<in> borel_measurable M \<longleftrightarrow>
   477       (\<lambda>a. \<Sum>i. emeasure (f a \<Otimes>\<^sub>M g a) (A i)) \<in> borel_measurable M"
   478       by (intro measurable_cong suminf_emeasure[symmetric])
   479          (auto simp: sets_eq)
   480     also have "\<dots>"
   481       using union by auto
   482     finally show ?case .
   483   qed simp
   484 qed
   486 lemma restrict_space_measurable:
   487   assumes X: "X \<noteq> {}" "X \<in> sets K"
   488   assumes N: "N \<in> measurable M (subprob_algebra K)"
   489   shows "(\<lambda>x. restrict_space (N x) X) \<in> measurable M (subprob_algebra (restrict_space K X))"
   490 proof (rule measurable_subprob_algebra)
   491   fix a assume a: "a \<in> space M"
   492   from N[THEN measurable_space, OF this]
   493   have "subprob_space (N a)" and [simp]: "sets (N a) = sets K" "space (N a) = space K"
   494     by (auto simp add: space_subprob_algebra dest: sets_eq_imp_space_eq)
   495   then interpret subprob_space "N a"
   496     by simp
   497   show "subprob_space (restrict_space (N a) X)"
   498   proof
   499     show "space (restrict_space (N a) X) \<noteq> {}"
   500       using X by (auto simp add: space_restrict_space)
   501     show "emeasure (restrict_space (N a) X) (space (restrict_space (N a) X)) \<le> 1"
   502       using X by (simp add: emeasure_restrict_space space_restrict_space subprob_emeasure_le_1)
   503   qed
   504   show "sets (restrict_space (N a) X) = sets (restrict_space K X)"
   505     by (intro sets_restrict_space_cong) fact
   506 next
   507   fix A assume A: "A \<in> sets (restrict_space K X)"
   508   show "(\<lambda>a. emeasure (restrict_space (N a) X) A) \<in> borel_measurable M"
   509   proof (subst measurable_cong)
   510     fix a assume "a \<in> space M"
   511     from N[THEN measurable_space, OF this]
   512     have [simp]: "sets (N a) = sets K" "space (N a) = space K"
   513       by (auto simp add: space_subprob_algebra dest: sets_eq_imp_space_eq)
   514     show "emeasure (restrict_space (N a) X) A = emeasure (N a) (A \<inter> X)"
   515       using X A by (subst emeasure_restrict_space) (auto simp add: sets_restrict_space ac_simps)
   516   next
   517     show "(\<lambda>w. emeasure (N w) (A \<inter> X)) \<in> borel_measurable M"
   518       using A X
   519       by (intro measurable_compose[OF N measurable_emeasure_subprob_algebra])
   520          (auto simp: sets_restrict_space)
   521   qed
   522 qed
   524 section \<open>Properties of return\<close>
   526 definition return :: "'a measure \<Rightarrow> 'a \<Rightarrow> 'a measure" where
   527   "return R x = measure_of (space R) (sets R) (\<lambda>A. indicator A x)"
   529 lemma space_return[simp]: "space (return M x) = space M"
   530   by (simp add: return_def)
   532 lemma sets_return[simp]: "sets (return M x) = sets M"
   533   by (simp add: return_def)
   535 lemma measurable_return1[simp]: "measurable (return N x) L = measurable N L"
   536   by (simp cong: measurable_cong_sets)
   538 lemma measurable_return2[simp]: "measurable L (return N x) = measurable L N"
   539   by (simp cong: measurable_cong_sets)
   541 lemma return_sets_cong: "sets M = sets N \<Longrightarrow> return M = return N"
   542   by (auto dest: sets_eq_imp_space_eq simp: fun_eq_iff return_def)
   544 lemma return_cong: "sets A = sets B \<Longrightarrow> return A x = return B x"
   545   by (auto simp add: return_def dest: sets_eq_imp_space_eq)
   547 lemma emeasure_return[simp]:
   548   assumes "A \<in> sets M"
   549   shows "emeasure (return M x) A = indicator A x"
   550 proof (rule emeasure_measure_of[OF return_def])
   551   show "sets M \<subseteq> Pow (space M)" by (rule sets.space_closed)
   552   show "positive (sets (return M x)) (\<lambda>A. indicator A x)" by (simp add: positive_def)
   553   from assms show "A \<in> sets (return M x)" unfolding return_def by simp
   554   show "countably_additive (sets (return M x)) (\<lambda>A. indicator A x)"
   555     by (auto intro!: countably_additiveI suminf_indicator)
   556 qed
   558 lemma prob_space_return: "x \<in> space M \<Longrightarrow> prob_space (return M x)"
   559   by rule simp
   561 lemma subprob_space_return: "x \<in> space M \<Longrightarrow> subprob_space (return M x)"
   562   by (intro prob_space_return prob_space_imp_subprob_space)
   564 lemma subprob_space_return_ne:
   565   assumes "space M \<noteq> {}" shows "subprob_space (return M x)"
   566 proof
   567   show "emeasure (return M x) (space (return M x)) \<le> 1"
   568     by (subst emeasure_return) (auto split: split_indicator)
   569 qed (simp, fact)
   571 lemma measure_return: assumes X: "X \<in> sets M" shows "measure (return M x) X = indicator X x"
   572   unfolding measure_def emeasure_return[OF X, of x] by (simp split: split_indicator)
   574 lemma AE_return:
   575   assumes [simp]: "x \<in> space M" and [measurable]: "Measurable.pred M P"
   576   shows "(AE y in return M x. P y) \<longleftrightarrow> P x"
   577 proof -
   578   have "(AE y in return M x. y \<notin> {x\<in>space M. \<not> P x}) \<longleftrightarrow> P x"
   579     by (subst AE_iff_null_sets[symmetric]) (simp_all add: null_sets_def split: split_indicator)
   580   also have "(AE y in return M x. y \<notin> {x\<in>space M. \<not> P x}) \<longleftrightarrow> (AE y in return M x. P y)"
   581     by (rule AE_cong) auto
   582   finally show ?thesis .
   583 qed
   585 lemma nn_integral_return:
   586   assumes "x \<in> space M" "g \<in> borel_measurable M"
   587   shows "(\<integral>\<^sup>+ a. g a \<partial>return M x) = g x"
   588 proof-
   589   interpret prob_space "return M x" by (rule prob_space_return[OF \<open>x \<in> space M\<close>])
   590   have "(\<integral>\<^sup>+ a. g a \<partial>return M x) = (\<integral>\<^sup>+ a. g x \<partial>return M x)" using assms
   591     by (intro nn_integral_cong_AE) (auto simp: AE_return)
   592   also have "... = g x"
   593     using nn_integral_const[of "return M x"] emeasure_space_1 by simp
   594   finally show ?thesis .
   595 qed
   597 lemma integral_return:
   598   fixes g :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
   599   assumes "x \<in> space M" "g \<in> borel_measurable M"
   600   shows "(\<integral>a. g a \<partial>return M x) = g x"
   601 proof-
   602   interpret prob_space "return M x" by (rule prob_space_return[OF \<open>x \<in> space M\<close>])
   603   have "(\<integral>a. g a \<partial>return M x) = (\<integral>a. g x \<partial>return M x)" using assms
   604     by (intro integral_cong_AE) (auto simp: AE_return)
   605   then show ?thesis
   606     using prob_space by simp
   607 qed
   609 lemma return_measurable[measurable]: "return N \<in> measurable N (subprob_algebra N)"
   610   by (rule measurable_subprob_algebra) (auto simp: subprob_space_return)
   612 lemma distr_return:
   613   assumes "f \<in> measurable M N" and "x \<in> space M"
   614   shows "distr (return M x) N f = return N (f x)"
   615   using assms by (intro measure_eqI) (simp_all add: indicator_def emeasure_distr)
   617 lemma return_restrict_space:
   618   "\<Omega> \<in> sets M \<Longrightarrow> return (restrict_space M \<Omega>) x = restrict_space (return M x) \<Omega>"
   619   by (auto intro!: measure_eqI simp: sets_restrict_space emeasure_restrict_space)
   621 lemma measurable_distr2:
   622   assumes f[measurable]: "case_prod f \<in> measurable (L \<Otimes>\<^sub>M M) N"
   623   assumes g[measurable]: "g \<in> measurable L (subprob_algebra M)"
   624   shows "(\<lambda>x. distr (g x) N (f x)) \<in> measurable L (subprob_algebra N)"
   625 proof -
   626   have "(\<lambda>x. distr (g x) N (f x)) \<in> measurable L (subprob_algebra N)
   627     \<longleftrightarrow> (\<lambda>x. distr (return L x \<Otimes>\<^sub>M g x) N (case_prod f)) \<in> measurable L (subprob_algebra N)"
   628   proof (rule measurable_cong)
   629     fix x assume x: "x \<in> space L"
   630     have gx: "g x \<in> space (subprob_algebra M)"
   631       using measurable_space[OF g x] .
   632     then have [simp]: "sets (g x) = sets M"
   633       by (simp add: space_subprob_algebra)
   634     then have [simp]: "space (g x) = space M"
   635       by (rule sets_eq_imp_space_eq)
   636     let ?R = "return L x"
   637     from measurable_compose_Pair1[OF x f] have f_M': "f x \<in> measurable M N"
   638       by simp
   639     interpret subprob_space "g x"
   640       using gx by (simp add: space_subprob_algebra)
   641     have space_pair_M'[simp]: "\<And>X. space (X \<Otimes>\<^sub>M g x) = space (X \<Otimes>\<^sub>M M)"
   642       by (simp add: space_pair_measure)
   643     show "distr (g x) N (f x) = distr (?R \<Otimes>\<^sub>M g x) N (case_prod f)" (is "?l = ?r")
   644     proof (rule measure_eqI)
   645       show "sets ?l = sets ?r"
   646         by simp
   647     next
   648       fix A assume "A \<in> sets ?l"
   649       then have A[measurable]: "A \<in> sets N"
   650         by simp
   651       then have "emeasure ?r A = emeasure (?R \<Otimes>\<^sub>M g x) ((\<lambda>(x, y). f x y) -` A \<inter> space (?R \<Otimes>\<^sub>M g x))"
   652         by (auto simp add: emeasure_distr f_M' cong: measurable_cong_sets)
   653       also have "\<dots> = (\<integral>\<^sup>+M''. emeasure (g x) (f M'' -` A \<inter> space M) \<partial>?R)"
   654         apply (subst emeasure_pair_measure_alt)
   655         apply (rule measurable_sets[OF _ A])
   656         apply (auto simp add: f_M' cong: measurable_cong_sets)
   657         apply (intro nn_integral_cong arg_cong[where f="emeasure (g x)"])
   658         apply (auto simp: space_subprob_algebra space_pair_measure)
   659         done
   660       also have "\<dots> = emeasure (g x) (f x -` A \<inter> space M)"
   661         by (subst nn_integral_return)
   662            (auto simp: x intro!: measurable_emeasure)
   663       also have "\<dots> = emeasure ?l A"
   664         by (simp add: emeasure_distr f_M' cong: measurable_cong_sets)
   665       finally show "emeasure ?l A = emeasure ?r A" ..
   666     qed
   667   qed
   668   also have "\<dots>"
   669     apply (intro measurable_compose[OF measurable_pair_measure measurable_distr])
   670     apply (rule return_measurable)
   671     apply measurable
   672     done
   673   finally show ?thesis .
   674 qed
   676 lemma nn_integral_measurable_subprob_algebra2:
   677   assumes f[measurable]: "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M N)"
   678   assumes N[measurable]: "L \<in> measurable M (subprob_algebra N)"
   679   shows "(\<lambda>x. integral\<^sup>N (L x) (f x)) \<in> borel_measurable M"
   680 proof -
   681   note nn_integral_measurable_subprob_algebra[measurable]
   682   note measurable_distr2[measurable]
   683   have "(\<lambda>x. integral\<^sup>N (distr (L x) (M \<Otimes>\<^sub>M N) (\<lambda>y. (x, y))) (\<lambda>(x, y). f x y)) \<in> borel_measurable M"
   684     by measurable
   685   then show "(\<lambda>x. integral\<^sup>N (L x) (f x)) \<in> borel_measurable M"
   686     by (rule measurable_cong[THEN iffD1, rotated])
   687        (simp add: nn_integral_distr)
   688 qed
   690 lemma emeasure_measurable_subprob_algebra2:
   691   assumes A[measurable]: "(SIGMA x:space M. A x) \<in> sets (M \<Otimes>\<^sub>M N)"
   692   assumes L[measurable]: "L \<in> measurable M (subprob_algebra N)"
   693   shows "(\<lambda>x. emeasure (L x) (A x)) \<in> borel_measurable M"
   694 proof -
   695   { fix x assume "x \<in> space M"
   696     then have "Pair x -` Sigma (space M) A = A x"
   697       by auto
   698     with sets_Pair1[OF A, of x] have "A x \<in> sets N"
   699       by auto }
   700   note ** = this
   702   have *: "\<And>x. fst x \<in> space M \<Longrightarrow> snd x \<in> A (fst x) \<longleftrightarrow> x \<in> (SIGMA x:space M. A x)"
   703     by (auto simp: fun_eq_iff)
   704   have "(\<lambda>(x, y). indicator (A x) y::ennreal) \<in> borel_measurable (M \<Otimes>\<^sub>M N)"
   705     apply measurable
   706     apply (subst measurable_cong)
   707     apply (rule *)
   708     apply (auto simp: space_pair_measure)
   709     done
   710   then have "(\<lambda>x. integral\<^sup>N (L x) (indicator (A x))) \<in> borel_measurable M"
   711     by (intro nn_integral_measurable_subprob_algebra2[where N=N] L)
   712   then show "(\<lambda>x. emeasure (L x) (A x)) \<in> borel_measurable M"
   713     apply (rule measurable_cong[THEN iffD1, rotated])
   714     apply (rule nn_integral_indicator)
   715     apply (simp add: subprob_measurableD[OF L] **)
   716     done
   717 qed
   719 lemma measure_measurable_subprob_algebra2:
   720   assumes A[measurable]: "(SIGMA x:space M. A x) \<in> sets (M \<Otimes>\<^sub>M N)"
   721   assumes L[measurable]: "L \<in> measurable M (subprob_algebra N)"
   722   shows "(\<lambda>x. measure (L x) (A x)) \<in> borel_measurable M"
   723   unfolding measure_def
   724   by (intro borel_measurable_enn2real emeasure_measurable_subprob_algebra2[OF assms])
   726 definition "select_sets M = (SOME N. sets M = sets (subprob_algebra N))"
   728 lemma select_sets1:
   729   "sets M = sets (subprob_algebra N) \<Longrightarrow> sets M = sets (subprob_algebra (select_sets M))"
   730   unfolding select_sets_def by (rule someI)
   732 lemma sets_select_sets[simp]:
   733   assumes sets: "sets M = sets (subprob_algebra N)"
   734   shows "sets (select_sets M) = sets N"
   735   unfolding select_sets_def
   736 proof (rule someI2)
   737   show "sets M = sets (subprob_algebra N)"
   738     by fact
   739 next
   740   fix L assume "sets M = sets (subprob_algebra L)"
   741   with sets have eq: "space (subprob_algebra N) = space (subprob_algebra L)"
   742     by (intro sets_eq_imp_space_eq) simp
   743   show "sets L = sets N"
   744   proof cases
   745     assume "space (subprob_algebra N) = {}"
   746     with space_subprob_algebra_empty_iff[of N] space_subprob_algebra_empty_iff[of L]
   747     show ?thesis
   748       by (simp add: eq space_empty_iff)
   749   next
   750     assume "space (subprob_algebra N) \<noteq> {}"
   751     with eq show ?thesis
   752       by (fastforce simp add: space_subprob_algebra)
   753   qed
   754 qed
   756 lemma space_select_sets[simp]:
   757   "sets M = sets (subprob_algebra N) \<Longrightarrow> space (select_sets M) = space N"
   758   by (intro sets_eq_imp_space_eq sets_select_sets)
   760 section \<open>Join\<close>
   762 definition join :: "'a measure measure \<Rightarrow> 'a measure" where
   763   "join M = measure_of (space (select_sets M)) (sets (select_sets M)) (\<lambda>B. \<integral>\<^sup>+ M'. emeasure M' B \<partial>M)"
   765 lemma
   766   shows space_join[simp]: "space (join M) = space (select_sets M)"
   767     and sets_join[simp]: "sets (join M) = sets (select_sets M)"
   768   by (simp_all add: join_def)
   770 lemma emeasure_join:
   771   assumes M[simp, measurable_cong]: "sets M = sets (subprob_algebra N)" and A: "A \<in> sets N"
   772   shows "emeasure (join M) A = (\<integral>\<^sup>+ M'. emeasure M' A \<partial>M)"
   773 proof (rule emeasure_measure_of[OF join_def])
   774   show "countably_additive (sets (join M)) (\<lambda>B. \<integral>\<^sup>+ M'. emeasure M' B \<partial>M)"
   775   proof (rule countably_additiveI)
   776     fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets (join M)" "disjoint_family A"
   777     have "(\<Sum>i. \<integral>\<^sup>+ M'. emeasure M' (A i) \<partial>M) = (\<integral>\<^sup>+M'. (\<Sum>i. emeasure M' (A i)) \<partial>M)"
   778       using A by (subst nn_integral_suminf) (auto simp: measurable_emeasure_subprob_algebra)
   779     also have "\<dots> = (\<integral>\<^sup>+M'. emeasure M' (\<Union>i. A i) \<partial>M)"
   780     proof (rule nn_integral_cong)
   781       fix M' assume "M' \<in> space M"
   782       then show "(\<Sum>i. emeasure M' (A i)) = emeasure M' (\<Union>i. A i)"
   783         using A sets_eq_imp_space_eq[OF M] by (simp add: suminf_emeasure space_subprob_algebra)
   784     qed
   785     finally show "(\<Sum>i. \<integral>\<^sup>+M'. emeasure M' (A i) \<partial>M) = (\<integral>\<^sup>+M'. emeasure M' (\<Union>i. A i) \<partial>M)" .
   786   qed
   787 qed (auto simp: A sets.space_closed positive_def)
   789 lemma measurable_join:
   790   "join \<in> measurable (subprob_algebra (subprob_algebra N)) (subprob_algebra N)"
   791 proof (cases "space N \<noteq> {}", rule measurable_subprob_algebra)
   792   fix A assume "A \<in> sets N"
   793   let ?B = "borel_measurable (subprob_algebra (subprob_algebra N))"
   794   have "(\<lambda>M'. emeasure (join M') A) \<in> ?B \<longleftrightarrow> (\<lambda>M'. (\<integral>\<^sup>+ M''. emeasure M'' A \<partial>M')) \<in> ?B"
   795   proof (rule measurable_cong)
   796     fix M' assume "M' \<in> space (subprob_algebra (subprob_algebra N))"
   797     then show "emeasure (join M') A = (\<integral>\<^sup>+ M''. emeasure M'' A \<partial>M')"
   798       by (intro emeasure_join) (auto simp: space_subprob_algebra \<open>A\<in>sets N\<close>)
   799   qed
   800   also have "(\<lambda>M'. \<integral>\<^sup>+M''. emeasure M'' A \<partial>M') \<in> ?B"
   801     using measurable_emeasure_subprob_algebra[OF \<open>A\<in>sets N\<close>]
   802     by (rule nn_integral_measurable_subprob_algebra)
   803   finally show "(\<lambda>M'. emeasure (join M') A) \<in> borel_measurable (subprob_algebra (subprob_algebra N))" .
   804 next
   805   assume [simp]: "space N \<noteq> {}"
   806   fix M assume M: "M \<in> space (subprob_algebra (subprob_algebra N))"
   807   then have "(\<integral>\<^sup>+M'. emeasure M' (space N) \<partial>M) \<le> (\<integral>\<^sup>+M'. 1 \<partial>M)"
   808     apply (intro nn_integral_mono)
   809     apply (auto simp: space_subprob_algebra
   810                  dest!: sets_eq_imp_space_eq subprob_space.emeasure_space_le_1)
   811     done
   812   with M show "subprob_space (join M)"
   813     by (intro subprob_spaceI)
   814        (auto simp: emeasure_join space_subprob_algebra M dest: subprob_space.emeasure_space_le_1)
   815 next
   816   assume "\<not>(space N \<noteq> {})"
   817   thus ?thesis by (simp add: measurable_empty_iff space_subprob_algebra_empty_iff)
   818 qed (auto simp: space_subprob_algebra)
   820 lemma nn_integral_join:
   821   assumes f: "f \<in> borel_measurable N"
   822     and M[measurable_cong]: "sets M = sets (subprob_algebra N)"
   823   shows "(\<integral>\<^sup>+x. f x \<partial>join M) = (\<integral>\<^sup>+M'. \<integral>\<^sup>+x. f x \<partial>M' \<partial>M)"
   824   using f
   825 proof induct
   826   case (cong f g)
   827   moreover have "integral\<^sup>N (join M) f = integral\<^sup>N (join M) g"
   828     by (intro nn_integral_cong cong) (simp add: M)
   829   moreover from M have "(\<integral>\<^sup>+ M'. integral\<^sup>N M' f \<partial>M) = (\<integral>\<^sup>+ M'. integral\<^sup>N M' g \<partial>M)"
   830     by (intro nn_integral_cong cong)
   831        (auto simp add: space_subprob_algebra dest!: sets_eq_imp_space_eq)
   832   ultimately show ?case
   833     by simp
   834 next
   835   case (set A)
   836   with M have "(\<integral>\<^sup>+ M'. integral\<^sup>N M' (indicator A) \<partial>M) = (\<integral>\<^sup>+ M'. emeasure M' A \<partial>M)"
   837     by (intro nn_integral_cong nn_integral_indicator)
   838        (auto simp: space_subprob_algebra dest!: sets_eq_imp_space_eq)
   839   with set show ?case
   840     using M by (simp add: emeasure_join)
   841 next
   842   case (mult f c)
   843   have "(\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. c * f x \<partial>M' \<partial>M) = (\<integral>\<^sup>+ M'. c * \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M)"
   844     using mult M M[THEN sets_eq_imp_space_eq]
   845     by (intro nn_integral_cong nn_integral_cmult) (auto simp add: space_subprob_algebra)
   846   also have "\<dots> = c * (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M)"
   847     using nn_integral_measurable_subprob_algebra[OF mult(2)]
   848     by (intro nn_integral_cmult mult) (simp add: M)
   849   also have "\<dots> = c * (integral\<^sup>N (join M) f)"
   850     by (simp add: mult)
   851   also have "\<dots> = (\<integral>\<^sup>+ x. c * f x \<partial>join M)"
   852     using mult(2,3) by (intro nn_integral_cmult[symmetric] mult) (simp add: M cong: measurable_cong_sets)
   853   finally show ?case by simp
   854 next
   855   case (add f g)
   856   have "(\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x + g x \<partial>M' \<partial>M) = (\<integral>\<^sup>+ M'. (\<integral>\<^sup>+ x. f x \<partial>M') + (\<integral>\<^sup>+ x. g x \<partial>M') \<partial>M)"
   857     using add M M[THEN sets_eq_imp_space_eq]
   858     by (intro nn_integral_cong nn_integral_add) (auto simp add: space_subprob_algebra)
   859   also have "\<dots> = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M) + (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. g x \<partial>M' \<partial>M)"
   860     using nn_integral_measurable_subprob_algebra[OF add(1)]
   861     using nn_integral_measurable_subprob_algebra[OF add(4)]
   862     by (intro nn_integral_add add) (simp_all add: M)
   863   also have "\<dots> = (integral\<^sup>N (join M) f) + (integral\<^sup>N (join M) g)"
   864     by (simp add: add)
   865   also have "\<dots> = (\<integral>\<^sup>+ x. f x + g x \<partial>join M)"
   866     using add by (intro nn_integral_add[symmetric] add) (simp_all add: M cong: measurable_cong_sets)
   867   finally show ?case by (simp add: ac_simps)
   868 next
   869   case (seq F)
   870   have "(\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. (SUP i. F i) x \<partial>M' \<partial>M) = (\<integral>\<^sup>+ M'. (SUP i. \<integral>\<^sup>+ x. F i x \<partial>M') \<partial>M)"
   871     using seq M M[THEN sets_eq_imp_space_eq] unfolding SUP_apply
   872     by (intro nn_integral_cong nn_integral_monotone_convergence_SUP)
   873        (auto simp add: space_subprob_algebra)
   874   also have "\<dots> = (SUP i. \<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. F i x \<partial>M' \<partial>M)"
   875     using nn_integral_measurable_subprob_algebra[OF seq(1)] seq
   876     by (intro nn_integral_monotone_convergence_SUP)
   877        (simp_all add: M incseq_nn_integral incseq_def le_fun_def nn_integral_mono )
   878   also have "\<dots> = (SUP i. integral\<^sup>N (join M) (F i))"
   879     by (simp add: seq)
   880   also have "\<dots> = (\<integral>\<^sup>+ x. (SUP i. F i x) \<partial>join M)"
   881     using seq by (intro nn_integral_monotone_convergence_SUP[symmetric] seq)
   882                  (simp_all add: M cong: measurable_cong_sets)
   883   finally show ?case by (simp add: ac_simps)
   884 qed
   886 lemma measurable_join1:
   887   "\<lbrakk> f \<in> measurable N K; sets M = sets (subprob_algebra N) \<rbrakk>
   888   \<Longrightarrow> f \<in> measurable (join M) K"
   889 by(simp add: measurable_def)
   891 lemma
   892   fixes f :: "_ \<Rightarrow> real"
   893   assumes f_measurable [measurable]: "f \<in> borel_measurable N"
   894   and f_bounded: "\<And>x. x \<in> space N \<Longrightarrow> \<bar>f x\<bar> \<le> B"
   895   and M [measurable_cong]: "sets M = sets (subprob_algebra N)"
   896   and fin: "finite_measure M"
   897   and M_bounded: "AE M' in M. emeasure M' (space M') \<le> ennreal B'"
   898   shows integrable_join: "integrable (join M) f" (is ?integrable)
   899   and integral_join: "integral\<^sup>L (join M) f = \<integral> M'. integral\<^sup>L M' f \<partial>M" (is ?integral)
   900 proof(case_tac [!] "space N = {}")
   901   assume *: "space N = {}"
   902   show ?integrable
   903     using M * by(simp add: real_integrable_def measurable_def nn_integral_empty)
   904   have "(\<integral> M'. integral\<^sup>L M' f \<partial>M) = (\<integral> M'. 0 \<partial>M)"
   905   proof(rule Bochner_Integration.integral_cong)
   906     fix M'
   907     assume "M' \<in> space M"
   908     with sets_eq_imp_space_eq[OF M] have "space M' = space N"
   909       by(auto simp add: space_subprob_algebra dest: sets_eq_imp_space_eq)
   910     with * show "(\<integral> x. f x \<partial>M') = 0" by(simp add: Bochner_Integration.integral_empty)
   911   qed simp
   912   then show ?integral
   913     using M * by(simp add: Bochner_Integration.integral_empty)
   914 next
   915   assume *: "space N \<noteq> {}"
   917   from * have B [simp]: "0 \<le> B" by(auto dest: f_bounded)
   919   have [measurable]: "f \<in> borel_measurable (join M)" using f_measurable M
   920     by(rule measurable_join1)
   922   { fix f M'
   923     assume [measurable]: "f \<in> borel_measurable N"
   924       and f_bounded: "\<And>x. x \<in> space N \<Longrightarrow> f x \<le> B"
   925       and "M' \<in> space M" "emeasure M' (space M') \<le> ennreal B'"
   926     have "AE x in M'. ennreal (f x) \<le> ennreal B"
   927     proof(rule AE_I2)
   928       fix x
   929       assume "x \<in> space M'"
   930       with \<open>M' \<in> space M\<close> sets_eq_imp_space_eq[OF M]
   931       have "x \<in> space N" by(auto simp add: space_subprob_algebra dest: sets_eq_imp_space_eq)
   932       from f_bounded[OF this] show "ennreal (f x) \<le> ennreal B" by simp
   933     qed
   934     then have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>M') \<le> (\<integral>\<^sup>+ x. ennreal B \<partial>M')"
   935       by(rule nn_integral_mono_AE)
   936     also have "\<dots> = ennreal B * emeasure M' (space M')" by(simp)
   937     also have "\<dots> \<le> ennreal B * ennreal B'" by(rule mult_left_mono)(fact, simp)
   938     also have "\<dots> \<le> ennreal B * ennreal \<bar>B'\<bar>" by(rule mult_left_mono)(simp_all)
   939     finally have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>M') \<le> ennreal (B * \<bar>B'\<bar>)" by (simp add: ennreal_mult) }
   940   note bounded1 = this
   942   have bounded:
   943     "\<And>f. \<lbrakk> f \<in> borel_measurable N; \<And>x. x \<in> space N \<Longrightarrow> f x \<le> B \<rbrakk>
   944     \<Longrightarrow> (\<integral>\<^sup>+ x. ennreal (f x) \<partial>join M) \<noteq> top"
   945   proof -
   946     fix f
   947     assume [measurable]: "f \<in> borel_measurable N"
   948       and f_bounded: "\<And>x. x \<in> space N \<Longrightarrow> f x \<le> B"
   949     have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>join M) = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. ennreal (f x) \<partial>M' \<partial>M)"
   950       by(rule nn_integral_join[OF _ M]) simp
   951     also have "\<dots> \<le> \<integral>\<^sup>+ M'. B * \<bar>B'\<bar> \<partial>M"
   952       using bounded1[OF \<open>f \<in> borel_measurable N\<close> f_bounded]
   953       by(rule nn_integral_mono_AE[OF AE_mp[OF M_bounded AE_I2], rule_format])
   954     also have "\<dots> = B * \<bar>B'\<bar> * emeasure M (space M)" by simp
   955     also have "\<dots> < \<infinity>"
   956       using finite_measure.finite_emeasure_space[OF fin]
   957       by(simp add: ennreal_mult_less_top less_top)
   958     finally show "?thesis f" by simp
   959   qed
   960   have f_pos: "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>join M) \<noteq> \<infinity>"
   961     and f_neg: "(\<integral>\<^sup>+ x. ennreal (- f x) \<partial>join M) \<noteq> \<infinity>"
   962     using f_bounded by(auto del: notI intro!: bounded simp add: abs_le_iff)
   964   show ?integrable using f_pos f_neg by(simp add: real_integrable_def)
   966   note [measurable] = nn_integral_measurable_subprob_algebra
   968   have int_f: "(\<integral>\<^sup>+ x. f x \<partial>join M) = \<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M"
   969     by(simp add: nn_integral_join[OF _ M])
   970   have int_mf: "(\<integral>\<^sup>+ x. - f x \<partial>join M) = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M)"
   971     by(simp add: nn_integral_join[OF _ M])
   973   have pos_finite: "AE M' in M. (\<integral>\<^sup>+ x. f x \<partial>M') \<noteq> \<infinity>"
   974     using AE_space M_bounded
   975   proof eventually_elim
   976     fix M' assume "M' \<in> space M" "emeasure M' (space M') \<le> ennreal B'"
   977     then have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>M') \<le> ennreal (B * \<bar>B'\<bar>)"
   978       using f_measurable by(auto intro!: bounded1 dest: f_bounded)
   979     then show "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>M') \<noteq> \<infinity>"
   980       by (auto simp: top_unique)
   981   qed
   982   hence [simp]: "(\<integral>\<^sup>+ M'. ennreal (enn2real (\<integral>\<^sup>+ x. f x \<partial>M')) \<partial>M) = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M)"
   983     by (rule nn_integral_cong_AE[OF AE_mp]) (simp add: less_top)
   984   from f_pos have [simp]: "integrable M (\<lambda>M'. enn2real (\<integral>\<^sup>+ x. f x \<partial>M'))"
   985     by(simp add: int_f real_integrable_def nn_integral_0_iff_AE[THEN iffD2] ennreal_neg enn2real_nonneg)
   987   have neg_finite: "AE M' in M. (\<integral>\<^sup>+ x. - f x \<partial>M') \<noteq> \<infinity>"
   988     using AE_space M_bounded
   989   proof eventually_elim
   990     fix M' assume "M' \<in> space M" "emeasure M' (space M') \<le> ennreal B'"
   991     then have "(\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M') \<le> ennreal (B * \<bar>B'\<bar>)"
   992       using f_measurable by(auto intro!: bounded1 dest: f_bounded)
   993     then show "(\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M') \<noteq> \<infinity>"
   994       by (auto simp: top_unique)
   995   qed
   996   hence [simp]: "(\<integral>\<^sup>+ M'. ennreal (enn2real (\<integral>\<^sup>+ x. - f x \<partial>M')) \<partial>M) = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M)"
   997     by (rule nn_integral_cong_AE[OF AE_mp]) (simp add: less_top)
   998   from f_neg have [simp]: "integrable M (\<lambda>M'. enn2real (\<integral>\<^sup>+ x. - f x \<partial>M'))"
   999     by(simp add: int_mf real_integrable_def nn_integral_0_iff_AE[THEN iffD2] ennreal_neg enn2real_nonneg)
  1001   have "(\<integral> x. f x \<partial>join M) = enn2real (\<integral>\<^sup>+ N. \<integral>\<^sup>+x. f x \<partial>N \<partial>M) - enn2real (\<integral>\<^sup>+ N. \<integral>\<^sup>+x. - f x \<partial>N \<partial>M)"
  1002     unfolding real_lebesgue_integral_def[OF \<open>?integrable\<close>] by (simp add: nn_integral_join[OF _ M])
  1003   also have "\<dots> = (\<integral>N. enn2real (\<integral>\<^sup>+x. f x \<partial>N) \<partial>M) - (\<integral>N. enn2real (\<integral>\<^sup>+x. - f x \<partial>N) \<partial>M)"
  1004     using pos_finite neg_finite by (subst (1 2) integral_eq_nn_integral) (auto simp: enn2real_nonneg)
  1005   also have "\<dots> = (\<integral>N. enn2real (\<integral>\<^sup>+x. f x \<partial>N) - enn2real (\<integral>\<^sup>+x. - f x \<partial>N) \<partial>M)"
  1006     by simp
  1007   also have "\<dots> = \<integral>M'. \<integral> x. f x \<partial>M' \<partial>M"
  1008   proof (rule integral_cong_AE)
  1009     show "AE x in M.
  1010         enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>x) - enn2real (\<integral>\<^sup>+ x. ennreal (- f x) \<partial>x) = integral\<^sup>L x f"
  1011       using AE_space M_bounded
  1012     proof eventually_elim
  1013       fix M' assume "M' \<in> space M" "emeasure M' (space M') \<le> B'"
  1014       then interpret subprob_space M'
  1015         by (auto simp: M[THEN sets_eq_imp_space_eq] space_subprob_algebra)
  1017       from \<open>M' \<in> space M\<close> sets_eq_imp_space_eq[OF M]
  1018       have [measurable_cong]: "sets M' = sets N" by(simp add: space_subprob_algebra)
  1019       hence [simp]: "space M' = space N" by(rule sets_eq_imp_space_eq)
  1020       have "integrable M' f"
  1021         by(rule integrable_const_bound[where B=B])(auto simp add: f_bounded)
  1022       then show "enn2real (\<integral>\<^sup>+ x. f x \<partial>M') - enn2real (\<integral>\<^sup>+ x. - f x \<partial>M') = \<integral> x. f x \<partial>M'"
  1023         by(simp add: real_lebesgue_integral_def)
  1024     qed
  1025   qed simp_all
  1026   finally show ?integral by simp
  1027 qed
  1029 lemma join_assoc:
  1030   assumes M[measurable_cong]: "sets M = sets (subprob_algebra (subprob_algebra N))"
  1031   shows "join (distr M (subprob_algebra N) join) = join (join M)"
  1032 proof (rule measure_eqI)
  1033   fix A assume "A \<in> sets (join (distr M (subprob_algebra N) join))"
  1034   then have A: "A \<in> sets N" by simp
  1035   show "emeasure (join (distr M (subprob_algebra N) join)) A = emeasure (join (join M)) A"
  1036     using measurable_join[of N]
  1037     by (auto simp: M A nn_integral_distr emeasure_join measurable_emeasure_subprob_algebra
  1038                    sets_eq_imp_space_eq[OF M] space_subprob_algebra nn_integral_join[OF _ M]
  1039              intro!: nn_integral_cong emeasure_join)
  1040 qed (simp add: M)
  1042 lemma join_return:
  1043   assumes "sets M = sets N" and "subprob_space M"
  1044   shows "join (return (subprob_algebra N) M) = M"
  1045   by (rule measure_eqI)
  1046      (simp_all add: emeasure_join space_subprob_algebra
  1047                     measurable_emeasure_subprob_algebra nn_integral_return assms)
  1049 lemma join_return':
  1050   assumes "sets N = sets M"
  1051   shows "join (distr M (subprob_algebra N) (return N)) = M"
  1052 apply (rule measure_eqI)
  1053 apply (simp add: assms)
  1054 apply (subgoal_tac "return N \<in> measurable M (subprob_algebra N)")
  1055 apply (simp add: emeasure_join nn_integral_distr measurable_emeasure_subprob_algebra assms)
  1056 apply (subst measurable_cong_sets, rule assms[symmetric], rule refl, rule return_measurable)
  1057 done
  1059 lemma join_distr_distr:
  1060   fixes f :: "'a \<Rightarrow> 'b" and M :: "'a measure measure" and N :: "'b measure"
  1061   assumes "sets M = sets (subprob_algebra R)" and "f \<in> measurable R N"
  1062   shows "join (distr M (subprob_algebra N) (\<lambda>M. distr M N f)) = distr (join M) N f" (is "?r = ?l")
  1063 proof (rule measure_eqI)
  1064   fix A assume "A \<in> sets ?r"
  1065   hence A_in_N: "A \<in> sets N" by simp
  1067   from assms have "f \<in> measurable (join M) N"
  1068       by (simp cong: measurable_cong_sets)
  1069   moreover from assms and A_in_N have "f-`A \<inter> space R \<in> sets R"
  1070       by (intro measurable_sets) simp_all
  1071   ultimately have "emeasure (distr (join M) N f) A = \<integral>\<^sup>+M'. emeasure M' (f-`A \<inter> space R) \<partial>M"
  1072       by (simp_all add: A_in_N emeasure_distr emeasure_join assms)
  1074   also have "... = \<integral>\<^sup>+ x. emeasure (distr x N f) A \<partial>M" using A_in_N
  1075   proof (intro nn_integral_cong, subst emeasure_distr)
  1076     fix M' assume "M' \<in> space M"
  1077     from assms have "space M = space (subprob_algebra R)"
  1078         using sets_eq_imp_space_eq by blast
  1079     with \<open>M' \<in> space M\<close> have [simp]: "sets M' = sets R" using space_subprob_algebra by blast
  1080     show "f \<in> measurable M' N" by (simp cong: measurable_cong_sets add: assms)
  1081     have "space M' = space R" by (rule sets_eq_imp_space_eq) simp
  1082     thus "emeasure M' (f -` A \<inter> space R) = emeasure M' (f -` A \<inter> space M')" by simp
  1083   qed
  1085   also have "(\<lambda>M. distr M N f) \<in> measurable M (subprob_algebra N)"
  1086       by (simp cong: measurable_cong_sets add: assms measurable_distr)
  1087   hence "(\<integral>\<^sup>+ x. emeasure (distr x N f) A \<partial>M) =
  1088              emeasure (join (distr M (subprob_algebra N) (\<lambda>M. distr M N f))) A"
  1089       by (simp_all add: emeasure_join assms A_in_N nn_integral_distr measurable_emeasure_subprob_algebra)
  1090   finally show "emeasure ?r A = emeasure ?l A" ..
  1091 qed simp
  1093 definition bind :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b measure) \<Rightarrow> 'b measure" where
  1094   "bind M f = (if space M = {} then count_space {} else
  1095     join (distr M (subprob_algebra (f (SOME x. x \<in> space M))) f))"
  1097 adhoc_overloading Monad_Syntax.bind bind
  1099 lemma bind_empty:
  1100   "space M = {} \<Longrightarrow> bind M f = count_space {}"
  1101   by (simp add: bind_def)
  1103 lemma bind_nonempty:
  1104   "space M \<noteq> {} \<Longrightarrow> bind M f = join (distr M (subprob_algebra (f (SOME x. x \<in> space M))) f)"
  1105   by (simp add: bind_def)
  1107 lemma sets_bind_empty: "sets M = {} \<Longrightarrow> sets (bind M f) = {{}}"
  1108   by (auto simp: bind_def)
  1110 lemma space_bind_empty: "space M = {} \<Longrightarrow> space (bind M f) = {}"
  1111   by (simp add: bind_def)
  1113 lemma sets_bind[simp, measurable_cong]:
  1114   assumes f: "\<And>x. x \<in> space M \<Longrightarrow> sets (f x) = sets N" and M: "space M \<noteq> {}"
  1115   shows "sets (bind M f) = sets N"
  1116   using f [of "SOME x. x \<in> space M"] by (simp add: bind_nonempty M some_in_eq)
  1118 lemma space_bind[simp]:
  1119   assumes "\<And>x. x \<in> space M \<Longrightarrow> sets (f x) = sets N" and "space M \<noteq> {}"
  1120   shows "space (bind M f) = space N"
  1121   using assms by (intro sets_eq_imp_space_eq sets_bind)
  1123 lemma bind_cong_All:
  1124   assumes "\<forall>x \<in> space M. f x = g x"
  1125   shows "bind M f = bind M g"
  1126 proof (cases "space M = {}")
  1127   assume "space M \<noteq> {}"
  1128   hence "(SOME x. x \<in> space M) \<in> space M" by (rule_tac someI_ex) blast
  1129   with assms have "f (SOME x. x \<in> space M) = g (SOME x. x \<in> space M)" by blast
  1130   with \<open>space M \<noteq> {}\<close> and assms show ?thesis by (simp add: bind_nonempty cong: distr_cong)
  1131 qed (simp add: bind_empty)
  1133 lemma bind_cong:
  1134   "M = N \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> bind M f = bind N g"
  1135   using bind_cong_All[of M f g] by auto
  1137 lemma bind_nonempty':
  1138   assumes "f \<in> measurable M (subprob_algebra N)" "x \<in> space M"
  1139   shows "bind M f = join (distr M (subprob_algebra N) f)"
  1140   using assms
  1141   apply (subst bind_nonempty, blast)
  1142   apply (subst subprob_algebra_cong[OF sets_kernel[OF assms(1) someI_ex]], blast)
  1143   apply (simp add: subprob_algebra_cong[OF sets_kernel[OF assms]])
  1144   done
  1146 lemma bind_nonempty'':
  1147   assumes "f \<in> measurable M (subprob_algebra N)" "space M \<noteq> {}"
  1148   shows "bind M f = join (distr M (subprob_algebra N) f)"
  1149   using assms by (auto intro: bind_nonempty')
  1151 lemma emeasure_bind:
  1152     "\<lbrakk>space M \<noteq> {}; f \<in> measurable M (subprob_algebra N);X \<in> sets N\<rbrakk>
  1153       \<Longrightarrow> emeasure (M \<bind> f) X = \<integral>\<^sup>+x. emeasure (f x) X \<partial>M"
  1154   by (simp add: bind_nonempty'' emeasure_join nn_integral_distr measurable_emeasure_subprob_algebra)
  1156 lemma nn_integral_bind:
  1157   assumes f: "f \<in> borel_measurable B"
  1158   assumes N: "N \<in> measurable M (subprob_algebra B)"
  1159   shows "(\<integral>\<^sup>+x. f x \<partial>(M \<bind> N)) = (\<integral>\<^sup>+x. \<integral>\<^sup>+y. f y \<partial>N x \<partial>M)"
  1160 proof cases
  1161   assume M: "space M \<noteq> {}" show ?thesis
  1162     unfolding bind_nonempty''[OF N M] nn_integral_join[OF f sets_distr]
  1163     by (rule nn_integral_distr[OF N])
  1164        (simp add: f nn_integral_measurable_subprob_algebra)
  1165 qed (simp add: bind_empty space_empty[of M] nn_integral_count_space)
  1167 lemma AE_bind:
  1168   assumes N[measurable]: "N \<in> measurable M (subprob_algebra B)"
  1169   assumes P[measurable]: "Measurable.pred B P"
  1170   shows "(AE x in M \<bind> N. P x) \<longleftrightarrow> (AE x in M. AE y in N x. P y)"
  1171 proof cases
  1172   assume M: "space M = {}" show ?thesis
  1173     unfolding bind_empty[OF M] unfolding space_empty[OF M] by (simp add: AE_count_space)
  1174 next
  1175   assume M: "space M \<noteq> {}"
  1176   note sets_kernel[OF N, simp]
  1177   have *: "(\<integral>\<^sup>+x. indicator {x. \<not> P x} x \<partial>(M \<bind> N)) = (\<integral>\<^sup>+x. indicator {x\<in>space B. \<not> P x} x \<partial>(M \<bind> N))"
  1178     by (intro nn_integral_cong) (simp add: space_bind[OF _ M] split: split_indicator)
  1180   have "(AE x in M \<bind> N. P x) \<longleftrightarrow> (\<integral>\<^sup>+ x. integral\<^sup>N (N x) (indicator {x \<in> space B. \<not> P x}) \<partial>M) = 0"
  1181     by (simp add: AE_iff_nn_integral sets_bind[OF _ M] space_bind[OF _ M] * nn_integral_bind[where B=B]
  1182              del: nn_integral_indicator)
  1183   also have "\<dots> = (AE x in M. AE y in N x. P y)"
  1184     apply (subst nn_integral_0_iff_AE)
  1185     apply (rule measurable_compose[OF N nn_integral_measurable_subprob_algebra])
  1186     apply measurable
  1187     apply (intro eventually_subst AE_I2)
  1188     apply (auto simp add: subprob_measurableD(1)[OF N]
  1189                 intro!: AE_iff_measurable[symmetric])
  1190     done
  1191   finally show ?thesis .
  1192 qed
  1194 lemma measurable_bind':
  1195   assumes M1: "f \<in> measurable M (subprob_algebra N)" and
  1196           M2: "case_prod g \<in> measurable (M \<Otimes>\<^sub>M N) (subprob_algebra R)"
  1197   shows "(\<lambda>x. bind (f x) (g x)) \<in> measurable M (subprob_algebra R)"
  1198 proof (subst measurable_cong)
  1199   fix x assume x_in_M: "x \<in> space M"
  1200   with assms have "space (f x) \<noteq> {}"
  1201       by (blast dest: subprob_space_kernel subprob_space.subprob_not_empty)
  1202   moreover from M2 x_in_M have "g x \<in> measurable (f x) (subprob_algebra R)"
  1203       by (subst measurable_cong_sets[OF sets_kernel[OF M1 x_in_M] refl])
  1204          (auto dest: measurable_Pair2)
  1205   ultimately show "bind (f x) (g x) = join (distr (f x) (subprob_algebra R) (g x))"
  1206       by (simp_all add: bind_nonempty'')
  1207 next
  1208   show "(\<lambda>w. join (distr (f w) (subprob_algebra R) (g w))) \<in> measurable M (subprob_algebra R)"
  1209     apply (rule measurable_compose[OF _ measurable_join])
  1210     apply (rule measurable_distr2[OF M2 M1])
  1211     done
  1212 qed
  1214 lemma measurable_bind[measurable (raw)]:
  1215   assumes M1: "f \<in> measurable M (subprob_algebra N)" and
  1216           M2: "(\<lambda>x. g (fst x) (snd x)) \<in> measurable (M \<Otimes>\<^sub>M N) (subprob_algebra R)"
  1217   shows "(\<lambda>x. bind (f x) (g x)) \<in> measurable M (subprob_algebra R)"
  1218   using assms by (auto intro: measurable_bind' simp: measurable_split_conv)
  1220 lemma measurable_bind2:
  1221   assumes "f \<in> measurable M (subprob_algebra N)" and "g \<in> measurable N (subprob_algebra R)"
  1222   shows "(\<lambda>x. bind (f x) g) \<in> measurable M (subprob_algebra R)"
  1223     using assms by (intro measurable_bind' measurable_const) auto
  1225 lemma subprob_space_bind:
  1226   assumes "subprob_space M" "f \<in> measurable M (subprob_algebra N)"
  1227   shows "subprob_space (M \<bind> f)"
  1228 proof (rule subprob_space_kernel[of "\<lambda>x. x \<bind> f"])
  1229   show "(\<lambda>x. x \<bind> f) \<in> measurable (subprob_algebra M) (subprob_algebra N)"
  1230     by (rule measurable_bind, rule measurable_ident_sets, rule refl,
  1231         rule measurable_compose[OF measurable_snd assms(2)])
  1232   from assms(1) show "M \<in> space (subprob_algebra M)"
  1233     by (simp add: space_subprob_algebra)
  1234 qed
  1236 lemma
  1237   fixes f :: "_ \<Rightarrow> real"
  1238   assumes f_measurable [measurable]: "f \<in> borel_measurable K"
  1239   and f_bounded: "\<And>x. x \<in> space K \<Longrightarrow> \<bar>f x\<bar> \<le> B"
  1240   and N [measurable]: "N \<in> measurable M (subprob_algebra K)"
  1241   and fin: "finite_measure M"
  1242   and M_bounded: "AE x in M. emeasure (N x) (space (N x)) \<le> ennreal B'"
  1243   shows integrable_bind: "integrable (bind M N) f" (is ?integrable)
  1244   and integral_bind: "integral\<^sup>L (bind M N) f = \<integral> x. integral\<^sup>L (N x) f \<partial>M" (is ?integral)
  1245 proof(case_tac [!] "space M = {}")
  1246   assume [simp]: "space M \<noteq> {}"
  1247   interpret finite_measure M by(rule fin)
  1249   have "integrable (join (distr M (subprob_algebra K) N)) f"
  1250     using f_measurable f_bounded
  1251     by(rule integrable_join[where B'=B'])(simp_all add: finite_measure_distr AE_distr_iff M_bounded)
  1252   then show ?integrable by(simp add: bind_nonempty''[where N=K])
  1254   have "integral\<^sup>L (join (distr M (subprob_algebra K) N)) f = \<integral> M'. integral\<^sup>L M' f \<partial>distr M (subprob_algebra K) N"
  1255     using f_measurable f_bounded
  1256     by(rule integral_join[where B'=B'])(simp_all add: finite_measure_distr AE_distr_iff M_bounded)
  1257   also have "\<dots> = \<integral> x. integral\<^sup>L (N x) f \<partial>M"
  1258     by(rule integral_distr)(simp_all add: integral_measurable_subprob_algebra[OF _])
  1259   finally show ?integral by(simp add: bind_nonempty''[where N=K])
  1260 qed(simp_all add: bind_def integrable_count_space lebesgue_integral_count_space_finite Bochner_Integration.integral_empty)
  1262 lemma (in prob_space) prob_space_bind:
  1263   assumes ae: "AE x in M. prob_space (N x)"
  1264     and N[measurable]: "N \<in> measurable M (subprob_algebra S)"
  1265   shows "prob_space (M \<bind> N)"
  1266 proof
  1267   have "emeasure (M \<bind> N) (space (M \<bind> N)) = (\<integral>\<^sup>+x. emeasure (N x) (space (N x)) \<partial>M)"
  1268     by (subst emeasure_bind[where N=S])
  1269        (auto simp: not_empty space_bind[OF sets_kernel] subprob_measurableD[OF N] intro!: nn_integral_cong)
  1270   also have "\<dots> = (\<integral>\<^sup>+x. 1 \<partial>M)"
  1271     using ae by (intro nn_integral_cong_AE, eventually_elim) (rule prob_space.emeasure_space_1)
  1272   finally show "emeasure (M \<bind> N) (space (M \<bind> N)) = 1"
  1273     by (simp add: emeasure_space_1)
  1274 qed
  1276 lemma (in subprob_space) bind_in_space:
  1277   "A \<in> measurable M (subprob_algebra N) \<Longrightarrow> (M \<bind> A) \<in> space (subprob_algebra N)"
  1278   by (auto simp add: space_subprob_algebra subprob_not_empty sets_kernel intro!: subprob_space_bind)
  1279      unfold_locales
  1281 lemma (in subprob_space) measure_bind:
  1282   assumes f: "f \<in> measurable M (subprob_algebra N)" and X: "X \<in> sets N"
  1283   shows "measure (M \<bind> f) X = \<integral>x. measure (f x) X \<partial>M"
  1284 proof -
  1285   interpret Mf: subprob_space "M \<bind> f"
  1286     by (rule subprob_space_bind[OF _ f]) unfold_locales
  1288   { fix x assume "x \<in> space M"
  1289     from f[THEN measurable_space, OF this] interpret subprob_space "f x"
  1290       by (simp add: space_subprob_algebra)
  1291     have "emeasure (f x) X = ennreal (measure (f x) X)" "measure (f x) X \<le> 1"
  1292       by (auto simp: emeasure_eq_measure subprob_measure_le_1) }
  1293   note this[simp]
  1295   have "emeasure (M \<bind> f) X = \<integral>\<^sup>+x. emeasure (f x) X \<partial>M"
  1296     using subprob_not_empty f X by (rule emeasure_bind)
  1297   also have "\<dots> = \<integral>\<^sup>+x. ennreal (measure (f x) X) \<partial>M"
  1298     by (intro nn_integral_cong) simp
  1299   also have "\<dots> = \<integral>x. measure (f x) X \<partial>M"
  1300     by (intro nn_integral_eq_integral integrable_const_bound[where B=1]
  1301               measure_measurable_subprob_algebra2[OF _ f] pair_measureI X)
  1302        (auto simp: measure_nonneg)
  1303   finally show ?thesis
  1304     by (simp add: Mf.emeasure_eq_measure measure_nonneg integral_nonneg)
  1305 qed
  1307 lemma emeasure_bind_const:
  1308     "space M \<noteq> {} \<Longrightarrow> X \<in> sets N \<Longrightarrow> subprob_space N \<Longrightarrow>
  1309          emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X * emeasure M (space M)"
  1310   by (simp add: bind_nonempty emeasure_join nn_integral_distr
  1311                 space_subprob_algebra measurable_emeasure_subprob_algebra)
  1313 lemma emeasure_bind_const':
  1314   assumes "subprob_space M" "subprob_space N"
  1315   shows "emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X * emeasure M (space M)"
  1316 using assms
  1317 proof (case_tac "X \<in> sets N")
  1318   fix X assume "X \<in> sets N"
  1319   thus "emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X * emeasure M (space M)" using assms
  1320       by (subst emeasure_bind_const)
  1321          (simp_all add: subprob_space.subprob_not_empty subprob_space.emeasure_space_le_1)
  1322 next
  1323   fix X assume "X \<notin> sets N"
  1324   with assms show "emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X * emeasure M (space M)"
  1325       by (simp add: sets_bind[of _ _ N] subprob_space.subprob_not_empty
  1326                     space_subprob_algebra emeasure_notin_sets)
  1327 qed
  1329 lemma emeasure_bind_const_prob_space:
  1330   assumes "prob_space M" "subprob_space N"
  1331   shows "emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X"
  1332   using assms by (simp add: emeasure_bind_const' prob_space_imp_subprob_space
  1333                             prob_space.emeasure_space_1)
  1335 lemma bind_return:
  1336   assumes "f \<in> measurable M (subprob_algebra N)" and "x \<in> space M"
  1337   shows "bind (return M x) f = f x"
  1338   using sets_kernel[OF assms] assms
  1339   by (simp_all add: distr_return join_return subprob_space_kernel bind_nonempty'
  1340                cong: subprob_algebra_cong)
  1342 lemma bind_return':
  1343   shows "bind M (return M) = M"
  1344   by (cases "space M = {}")
  1345      (simp_all add: bind_empty space_empty[symmetric] bind_nonempty join_return'
  1346                cong: subprob_algebra_cong)
  1348 lemma distr_bind:
  1349   assumes N: "N \<in> measurable M (subprob_algebra K)" "space M \<noteq> {}"
  1350   assumes f: "f \<in> measurable K R"
  1351   shows "distr (M \<bind> N) R f = (M \<bind> (\<lambda>x. distr (N x) R f))"
  1352   unfolding bind_nonempty''[OF N]
  1353   apply (subst bind_nonempty''[OF measurable_compose[OF N(1) measurable_distr] N(2)])
  1354   apply (rule f)
  1355   apply (simp add: join_distr_distr[OF _ f, symmetric])
  1356   apply (subst distr_distr[OF measurable_distr, OF f N(1)])
  1357   apply (simp add: comp_def)
  1358   done
  1360 lemma bind_distr:
  1361   assumes f[measurable]: "f \<in> measurable M X"
  1362   assumes N[measurable]: "N \<in> measurable X (subprob_algebra K)" and "space M \<noteq> {}"
  1363   shows "(distr M X f \<bind> N) = (M \<bind> (\<lambda>x. N (f x)))"
  1364 proof -
  1365   have "space X \<noteq> {}" "space M \<noteq> {}"
  1366     using \<open>space M \<noteq> {}\<close> f[THEN measurable_space] by auto
  1367   then show ?thesis
  1368     by (simp add: bind_nonempty''[where N=K] distr_distr comp_def)
  1369 qed
  1371 lemma bind_count_space_singleton:
  1372   assumes "subprob_space (f x)"
  1373   shows "count_space {x} \<bind> f = f x"
  1374 proof-
  1375   have A: "\<And>A. A \<subseteq> {x} \<Longrightarrow> A = {} \<or> A = {x}" by auto
  1376   have "count_space {x} = return (count_space {x}) x"
  1377     by (intro measure_eqI) (auto dest: A)
  1378   also have "... \<bind> f = f x"
  1379     by (subst bind_return[of _ _ "f x"]) (auto simp: space_subprob_algebra assms)
  1380   finally show ?thesis .
  1381 qed
  1383 lemma restrict_space_bind:
  1384   assumes N: "N \<in> measurable M (subprob_algebra K)"
  1385   assumes "space M \<noteq> {}"
  1386   assumes X[simp]: "X \<in> sets K" "X \<noteq> {}"
  1387   shows "restrict_space (bind M N) X = bind M (\<lambda>x. restrict_space (N x) X)"
  1388 proof (rule measure_eqI)
  1389   note N_sets = sets_bind[OF sets_kernel[OF N] assms(2), simp]
  1390   note N_space = sets_eq_imp_space_eq[OF N_sets, simp]
  1391   show "sets (restrict_space (bind M N) X) = sets (bind M (\<lambda>x. restrict_space (N x) X))"
  1392     by (simp add: sets_restrict_space assms(2) sets_bind[OF sets_kernel[OF restrict_space_measurable[OF assms(4,3,1)]]])
  1393   fix A assume "A \<in> sets (restrict_space (M \<bind> N) X)"
  1394   with X have "A \<in> sets K" "A \<subseteq> X"
  1395     by (auto simp: sets_restrict_space)
  1396   then show "emeasure (restrict_space (M \<bind> N) X) A = emeasure (M \<bind> (\<lambda>x. restrict_space (N x) X)) A"
  1397     using assms
  1398     apply (subst emeasure_restrict_space)
  1399     apply (simp_all add: emeasure_bind[OF assms(2,1)])
  1400     apply (subst emeasure_bind[OF _ restrict_space_measurable[OF _ _ N]])
  1401     apply (auto simp: sets_restrict_space emeasure_restrict_space space_subprob_algebra
  1402                 intro!: nn_integral_cong dest!: measurable_space)
  1403     done
  1404 qed
  1406 lemma bind_restrict_space:
  1407   assumes A: "A \<inter> space M \<noteq> {}" "A \<inter> space M \<in> sets M"
  1408   and f: "f \<in> measurable (restrict_space M A) (subprob_algebra N)"
  1409   shows "restrict_space M A \<bind> f = M \<bind> (\<lambda>x. if x \<in> A then f x else null_measure (f (SOME x. x \<in> A \<and> x \<in> space M)))"
  1410   (is "?lhs = ?rhs" is "_ = M \<bind> ?f")
  1411 proof -
  1412   let ?P = "\<lambda>x. x \<in> A \<and> x \<in> space M"
  1413   let ?x = "Eps ?P"
  1414   let ?c = "null_measure (f ?x)"
  1415   from A have "?P ?x" by-(rule someI_ex, blast)
  1416   hence "?x \<in> space (restrict_space M A)" by(simp add: space_restrict_space)
  1417   with f have "f ?x \<in> space (subprob_algebra N)" by(rule measurable_space)
  1418   hence sps: "subprob_space (f ?x)"
  1419     and sets: "sets (f ?x) = sets N"
  1420     by(simp_all add: space_subprob_algebra)
  1421   have "space (f ?x) \<noteq> {}" using sps by(rule subprob_space.subprob_not_empty)
  1422   moreover have "sets ?c = sets N" by(simp add: null_measure_def)(simp add: sets)
  1423   ultimately have c: "?c \<in> space (subprob_algebra N)"
  1424     by(simp add: space_subprob_algebra subprob_space_null_measure)
  1425   from f A c have f': "?f \<in> measurable M (subprob_algebra N)"
  1426     by(simp add: measurable_restrict_space_iff)
  1428   from A have [simp]: "space M \<noteq> {}" by blast
  1430   have "?lhs = join (distr (restrict_space M A) (subprob_algebra N) f)"
  1431     using assms by(simp add: space_restrict_space bind_nonempty'')
  1432   also have "\<dots> = join (distr M (subprob_algebra N) ?f)"
  1433     by(rule measure_eqI)(auto simp add: emeasure_join nn_integral_distr nn_integral_restrict_space f f' A intro: nn_integral_cong)
  1434   also have "\<dots> = ?rhs" using f' by(simp add: bind_nonempty'')
  1435   finally show ?thesis .
  1436 qed
  1438 lemma bind_const': "\<lbrakk>prob_space M; subprob_space N\<rbrakk> \<Longrightarrow> M \<bind> (\<lambda>x. N) = N"
  1439   by (intro measure_eqI)
  1440      (simp_all add: space_subprob_algebra prob_space.not_empty emeasure_bind_const_prob_space)
  1442 lemma bind_return_distr:
  1443     "space M \<noteq> {} \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> bind M (return N \<circ> f) = distr M N f"
  1444   apply (simp add: bind_nonempty)
  1445   apply (subst subprob_algebra_cong)
  1446   apply (rule sets_return)
  1447   apply (subst distr_distr[symmetric])
  1448   apply (auto intro!: return_measurable simp: distr_distr[symmetric] join_return')
  1449   done
  1451 lemma bind_return_distr':
  1452   "space M \<noteq> {} \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> bind M (\<lambda>x. return N (f x)) = distr M N f"
  1453   using bind_return_distr[of M f N] by (simp add: comp_def)
  1455 lemma bind_assoc:
  1456   fixes f :: "'a \<Rightarrow> 'b measure" and g :: "'b \<Rightarrow> 'c measure"
  1457   assumes M1: "f \<in> measurable M (subprob_algebra N)" and M2: "g \<in> measurable N (subprob_algebra R)"
  1458   shows "bind (bind M f) g = bind M (\<lambda>x. bind (f x) g)"
  1459 proof (cases "space M = {}")
  1460   assume [simp]: "space M \<noteq> {}"
  1461   from assms have [simp]: "space N \<noteq> {}" "space R \<noteq> {}"
  1462       by (auto simp: measurable_empty_iff space_subprob_algebra_empty_iff)
  1463   from assms have sets_fx[simp]: "\<And>x. x \<in> space M \<Longrightarrow> sets (f x) = sets N"
  1464       by (simp add: sets_kernel)
  1465   have ex_in: "\<And>A. A \<noteq> {} \<Longrightarrow> \<exists>x. x \<in> A" by blast
  1466   note sets_some[simp] = sets_kernel[OF M1 someI_ex[OF ex_in[OF \<open>space M \<noteq> {}\<close>]]]
  1467                          sets_kernel[OF M2 someI_ex[OF ex_in[OF \<open>space N \<noteq> {}\<close>]]]
  1468   note space_some[simp] = sets_eq_imp_space_eq[OF this(1)] sets_eq_imp_space_eq[OF this(2)]
  1470   have "bind M (\<lambda>x. bind (f x) g) =
  1471         join (distr M (subprob_algebra R) (join \<circ> (\<lambda>x. (distr x (subprob_algebra R) g)) \<circ> f))"
  1472     by (simp add: sets_eq_imp_space_eq[OF sets_fx] bind_nonempty o_def
  1473              cong: subprob_algebra_cong distr_cong)
  1474   also have "distr M (subprob_algebra R) (join \<circ> (\<lambda>x. (distr x (subprob_algebra R) g)) \<circ> f) =
  1475              distr (distr (distr M (subprob_algebra N) f)
  1476                           (subprob_algebra (subprob_algebra R))
  1477                           (\<lambda>x. distr x (subprob_algebra R) g))
  1478                    (subprob_algebra R) join"
  1479       apply (subst distr_distr,
  1480              (blast intro: measurable_comp measurable_distr measurable_join M1 M2)+)+
  1481       apply (simp add: o_assoc)
  1482       done
  1483   also have "join ... = bind (bind M f) g"
  1484       by (simp add: join_assoc join_distr_distr M2 bind_nonempty cong: subprob_algebra_cong)
  1485   finally show ?thesis ..
  1486 qed (simp add: bind_empty)
  1488 lemma double_bind_assoc:
  1489   assumes Mg: "g \<in> measurable N (subprob_algebra N')"
  1490   assumes Mf: "f \<in> measurable M (subprob_algebra M')"
  1491   assumes Mh: "case_prod h \<in> measurable (M \<Otimes>\<^sub>M M') N"
  1492   shows "do {x \<leftarrow> M; y \<leftarrow> f x; g (h x y)} = do {x \<leftarrow> M; y \<leftarrow> f x; return N (h x y)} \<bind> g"
  1493 proof-
  1494   have "do {x \<leftarrow> M; y \<leftarrow> f x; return N (h x y)} \<bind> g =
  1495             do {x \<leftarrow> M; do {y \<leftarrow> f x; return N (h x y)} \<bind> g}"
  1496     using Mh by (auto intro!: bind_assoc measurable_bind'[OF Mf] Mf Mg
  1497                       measurable_compose[OF _ return_measurable] simp: measurable_split_conv)
  1498   also from Mh have "\<And>x. x \<in> space M \<Longrightarrow> h x \<in> measurable M' N" by measurable
  1499   hence "do {x \<leftarrow> M; do {y \<leftarrow> f x; return N (h x y)} \<bind> g} =
  1500             do {x \<leftarrow> M; y \<leftarrow> f x; return N (h x y) \<bind> g}"
  1501     apply (intro ballI bind_cong refl bind_assoc)
  1502     apply (subst measurable_cong_sets[OF sets_kernel[OF Mf] refl], simp)
  1503     apply (rule measurable_compose[OF _ return_measurable], auto intro: Mg)
  1504     done
  1505   also have "\<And>x. x \<in> space M \<Longrightarrow> space (f x) = space M'"
  1506     by (intro sets_eq_imp_space_eq sets_kernel[OF Mf])
  1507   with measurable_space[OF Mh]
  1508     have "do {x \<leftarrow> M; y \<leftarrow> f x; return N (h x y) \<bind> g} = do {x \<leftarrow> M; y \<leftarrow> f x; g (h x y)}"
  1509     by (intro ballI bind_cong bind_return[OF Mg]) (auto simp: space_pair_measure)
  1510   finally show ?thesis ..
  1511 qed
  1513 lemma (in prob_space) M_in_subprob[measurable (raw)]: "M \<in> space (subprob_algebra M)"
  1514   by (simp add: space_subprob_algebra) unfold_locales
  1516 lemma (in pair_prob_space) pair_measure_eq_bind:
  1517   "(M1 \<Otimes>\<^sub>M M2) = (M1 \<bind> (\<lambda>x. M2 \<bind> (\<lambda>y. return (M1 \<Otimes>\<^sub>M M2) (x, y))))"
  1518 proof (rule measure_eqI)
  1519   have ps_M2: "prob_space M2" by unfold_locales
  1520   note return_measurable[measurable]
  1521   show "sets (M1 \<Otimes>\<^sub>M M2) = sets (M1 \<bind> (\<lambda>x. M2 \<bind> (\<lambda>y. return (M1 \<Otimes>\<^sub>M M2) (x, y))))"
  1522     by (simp_all add: M1.not_empty M2.not_empty)
  1523   fix A assume [measurable]: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
  1524   show "emeasure (M1 \<Otimes>\<^sub>M M2) A = emeasure (M1 \<bind> (\<lambda>x. M2 \<bind> (\<lambda>y. return (M1 \<Otimes>\<^sub>M M2) (x, y)))) A"
  1525     by (auto simp: M2.emeasure_pair_measure M1.not_empty M2.not_empty emeasure_bind[where N="M1 \<Otimes>\<^sub>M M2"]
  1526              intro!: nn_integral_cong)
  1527 qed
  1529 lemma (in pair_prob_space) bind_rotate:
  1530   assumes C[measurable]: "(\<lambda>(x, y). C x y) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (subprob_algebra N)"
  1531   shows "(M1 \<bind> (\<lambda>x. M2 \<bind> (\<lambda>y. C x y))) = (M2 \<bind> (\<lambda>y. M1 \<bind> (\<lambda>x. C x y)))"
  1532 proof -
  1533   interpret swap: pair_prob_space M2 M1 by unfold_locales
  1534   note measurable_bind[where N="M2", measurable]
  1535   note measurable_bind[where N="M1", measurable]
  1536   have [simp]: "M1 \<in> space (subprob_algebra M1)" "M2 \<in> space (subprob_algebra M2)"
  1537     by (auto simp: space_subprob_algebra) unfold_locales
  1538   have "(M1 \<bind> (\<lambda>x. M2 \<bind> (\<lambda>y. C x y))) =
  1539     (M1 \<bind> (\<lambda>x. M2 \<bind> (\<lambda>y. return (M1 \<Otimes>\<^sub>M M2) (x, y)))) \<bind> (\<lambda>(x, y). C x y)"
  1540     by (auto intro!: bind_cong simp: bind_return[where N=N] space_pair_measure bind_assoc[where N="M1 \<Otimes>\<^sub>M M2" and R=N])
  1541   also have "\<dots> = (distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))) \<bind> (\<lambda>(x, y). C x y)"
  1542     unfolding pair_measure_eq_bind[symmetric] distr_pair_swap[symmetric] ..
  1543   also have "\<dots> = (M2 \<bind> (\<lambda>x. M1 \<bind> (\<lambda>y. return (M2 \<Otimes>\<^sub>M M1) (x, y)))) \<bind> (\<lambda>(y, x). C x y)"
  1544     unfolding swap.pair_measure_eq_bind[symmetric]
  1545     by (auto simp add: space_pair_measure M1.not_empty M2.not_empty bind_distr[OF _ C] intro!: bind_cong)
  1546   also have "\<dots> = (M2 \<bind> (\<lambda>y. M1 \<bind> (\<lambda>x. C x y)))"
  1547     by (auto intro!: bind_cong simp: bind_return[where N=N] space_pair_measure bind_assoc[where N="M2 \<Otimes>\<^sub>M M1" and R=N])
  1548   finally show ?thesis .
  1549 qed
  1551 lemma bind_return'': "sets M = sets N \<Longrightarrow> M \<bind> return N = M"
  1552    by (cases "space M = {}")
  1553       (simp_all add: bind_empty space_empty[symmetric] bind_nonempty join_return'
  1554                 cong: subprob_algebra_cong)
  1556 lemma (in prob_space) distr_const[simp]:
  1557   "c \<in> space N \<Longrightarrow> distr M N (\<lambda>x. c) = return N c"
  1558   by (rule measure_eqI) (auto simp: emeasure_distr emeasure_space_1)
  1560 lemma return_count_space_eq_density:
  1561     "return (count_space M) x = density (count_space M) (indicator {x})"
  1562   by (rule measure_eqI)
  1563      (auto simp: indicator_inter_arith[symmetric] emeasure_density split: split_indicator)
  1565 lemma null_measure_in_space_subprob_algebra [simp]:
  1566   "null_measure M \<in> space (subprob_algebra M) \<longleftrightarrow> space M \<noteq> {}"
  1567 by(simp add: space_subprob_algebra subprob_space_null_measure_iff)
  1569 subsection \<open>Giry monad on probability spaces\<close>
  1571 definition prob_algebra :: "'a measure \<Rightarrow> 'a measure measure" where
  1572   "prob_algebra K = restrict_space (subprob_algebra K) {M. prob_space M}"
  1574 lemma space_prob_algebra: "space (prob_algebra M) = {N. sets N = sets M \<and> prob_space N}"
  1575   unfolding prob_algebra_def by (auto simp: space_subprob_algebra space_restrict_space prob_space_imp_subprob_space)
  1577 lemma measurable_measure_prob_algebra[measurable]:
  1578   "a \<in> sets A \<Longrightarrow> (\<lambda>M. Sigma_Algebra.measure M a) \<in> prob_algebra A \<rightarrow>\<^sub>M borel"
  1579   unfolding prob_algebra_def by (intro measurable_restrict_space1 measurable_measure_subprob_algebra)
  1581 lemma measurable_prob_algebraD:
  1582   "f \<in> N \<rightarrow>\<^sub>M prob_algebra M \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M subprob_algebra M"
  1583   unfolding prob_algebra_def measurable_restrict_space2_iff by auto
  1585 lemma measure_measurable_prob_algebra2:
  1586   "Sigma (space M) A \<in> sets (M \<Otimes>\<^sub>M N) \<Longrightarrow> L \<in> M \<rightarrow>\<^sub>M prob_algebra N \<Longrightarrow>
  1587     (\<lambda>x. Sigma_Algebra.measure (L x) (A x)) \<in> borel_measurable M"
  1588   using measure_measurable_subprob_algebra2[of M A N L] by (auto intro: measurable_prob_algebraD)
  1590 lemma measurable_prob_algebraI:
  1591   "(\<And>x. x \<in> space N \<Longrightarrow> prob_space (f x)) \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M subprob_algebra M \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M prob_algebra M"
  1592   unfolding prob_algebra_def by (intro measurable_restrict_space2) auto
  1594 lemma measurable_distr_prob_space:
  1595   assumes f: "f \<in> M \<rightarrow>\<^sub>M N"
  1596   shows "(\<lambda>M'. distr M' N f) \<in> prob_algebra M \<rightarrow>\<^sub>M prob_algebra N"
  1597   unfolding prob_algebra_def measurable_restrict_space2_iff
  1598 proof (intro conjI measurable_restrict_space1 measurable_distr f)
  1599   show "(\<lambda>M'. distr M' N f) \<in> space (restrict_space (subprob_algebra M) (Collect prob_space)) \<rightarrow> Collect prob_space"
  1600     using f by (auto simp: space_restrict_space space_subprob_algebra intro!: prob_space.prob_space_distr)
  1601 qed
  1603 lemma measurable_return_prob_space[measurable]: "return N \<in> N \<rightarrow>\<^sub>M prob_algebra N"
  1604   by (rule measurable_prob_algebraI) (auto simp: prob_space_return)
  1606 lemma measurable_distr_prob_space2[measurable (raw)]:
  1607   assumes f: "g \<in> L \<rightarrow>\<^sub>M prob_algebra M" "(\<lambda>(x, y). f x y) \<in> L \<Otimes>\<^sub>M M \<rightarrow>\<^sub>M N"
  1608   shows "(\<lambda>x. distr (g x) N (f x)) \<in> L \<rightarrow>\<^sub>M prob_algebra N"
  1609   unfolding prob_algebra_def measurable_restrict_space2_iff
  1610 proof (intro conjI measurable_restrict_space1 measurable_distr2[where M=M] f measurable_prob_algebraD)
  1611   show "(\<lambda>x. distr (g x) N (f x)) \<in> space L \<rightarrow> Collect prob_space"
  1612     using f subprob_measurableD[OF measurable_prob_algebraD[OF f(1)]]
  1613     by (auto simp: measurable_restrict_space2_iff prob_algebra_def
  1614              intro!: prob_space.prob_space_distr)
  1615 qed
  1617 lemma measurable_bind_prob_space:
  1618   assumes f: "f \<in> M \<rightarrow>\<^sub>M prob_algebra N" and g: "g \<in> N \<rightarrow>\<^sub>M prob_algebra R"
  1619   shows "(\<lambda>x. bind (f x) g) \<in> M \<rightarrow>\<^sub>M prob_algebra R"
  1620   unfolding prob_algebra_def measurable_restrict_space2_iff
  1621 proof (intro conjI measurable_restrict_space1 measurable_bind2[where N=N] f g measurable_prob_algebraD)
  1622   show "(\<lambda>x. f x \<bind> g) \<in> space M \<rightarrow> Collect prob_space"
  1623     using g f subprob_measurableD[OF measurable_prob_algebraD[OF f]]
  1624     by (auto simp: measurable_restrict_space2_iff prob_algebra_def
  1625                 intro!: prob_space.prob_space_bind[where S=R] AE_I2)
  1626 qed
  1628 lemma measurable_bind_prob_space2[measurable (raw)]:
  1629   assumes f: "f \<in> M \<rightarrow>\<^sub>M prob_algebra N" and g: "(\<lambda>(x, y). g x y) \<in> (M \<Otimes>\<^sub>M N) \<rightarrow>\<^sub>M prob_algebra R"
  1630   shows "(\<lambda>x. bind (f x) (g x)) \<in> M \<rightarrow>\<^sub>M prob_algebra R"
  1631   unfolding prob_algebra_def measurable_restrict_space2_iff
  1632 proof (intro conjI measurable_restrict_space1 measurable_bind[where N=N] f g measurable_prob_algebraD)
  1633   show "(\<lambda>x. f x \<bind> g x) \<in> space M \<rightarrow> Collect prob_space"
  1634     using g f subprob_measurableD[OF measurable_prob_algebraD[OF f]]
  1635       using measurable_space[OF g]
  1636     by (auto simp: measurable_restrict_space2_iff prob_algebra_def space_pair_measure Pi_iff
  1637                 intro!: prob_space.prob_space_bind[where S=R] AE_I2)
  1638 qed (insert g, simp)
  1641 lemma measurable_prob_algebra_generated:
  1642   assumes eq: "sets N = sigma_sets \<Omega> G" and "Int_stable G" "G \<subseteq> Pow \<Omega>"
  1643   assumes subsp: "\<And>a. a \<in> space M \<Longrightarrow> prob_space (K a)"
  1644   assumes sets: "\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N"
  1645   assumes "\<And>A. A \<in> G \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
  1646   shows "K \<in> measurable M (prob_algebra N)"
  1647   unfolding measurable_restrict_space2_iff prob_algebra_def
  1648 proof
  1649   show "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
  1650   proof (rule measurable_subprob_algebra_generated[OF assms(1,2,3) _ assms(5,6)])
  1651     fix a assume "a \<in> space M" then show "subprob_space (K a)"
  1652       using subsp[of a] by (intro prob_space_imp_subprob_space)
  1653   next
  1654     have "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M \<longleftrightarrow> (\<lambda>a. 1::ennreal) \<in> borel_measurable M"
  1655       using sets_eq_imp_space_eq[of "sigma \<Omega> G" N] \<open>G \<subseteq> Pow \<Omega>\<close> eq sets_eq_imp_space_eq[OF sets]
  1656         prob_space.emeasure_space_1[OF subsp]
  1657       by (intro measurable_cong) auto
  1658     then show "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M" by simp
  1659   qed
  1660 qed (insert subsp, auto)
  1662 lemma in_space_prob_algebra:
  1663   "x \<in> space (prob_algebra M) \<Longrightarrow> emeasure x (space M) = 1"
  1664   unfolding prob_algebra_def space_restrict_space space_subprob_algebra
  1665   by (auto dest!: prob_space.emeasure_space_1 sets_eq_imp_space_eq)
  1667 lemma prob_space_pair:
  1668   assumes "prob_space M" "prob_space N" shows "prob_space (M \<Otimes>\<^sub>M N)"
  1669 proof -
  1670   interpret M: prob_space M by fact
  1671   interpret N: prob_space N by fact
  1672   interpret P: pair_prob_space M N proof qed
  1673   show ?thesis
  1674     by unfold_locales
  1675 qed
  1677 lemma measurable_pair_prob[measurable]:
  1678   "f \<in> M \<rightarrow>\<^sub>M prob_algebra N \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M prob_algebra L \<Longrightarrow> (\<lambda>x. f x \<Otimes>\<^sub>M g x) \<in> M \<rightarrow>\<^sub>M prob_algebra (N \<Otimes>\<^sub>M L)"
  1679   unfolding prob_algebra_def measurable_restrict_space2_iff
  1680   by (auto intro!: measurable_pair_measure prob_space_pair)
  1682 lemma emeasure_bind_prob_algebra:
  1683   assumes A: "A \<in> space (prob_algebra N)"
  1684   assumes B: "B \<in> N \<rightarrow>\<^sub>M prob_algebra L"
  1685   assumes X: "X \<in> sets L"
  1686   shows "emeasure (bind A B) X = (\<integral>\<^sup>+x. emeasure (B x) X \<partial>A)"
  1687   using A B
  1688   by (intro emeasure_bind[OF _ _ X])
  1689      (auto simp: space_prob_algebra measurable_prob_algebraD cong: measurable_cong_sets intro!: prob_space.not_empty)
  1691 lemma prob_space_bind':
  1692   assumes A: "A \<in> space (prob_algebra M)" and B: "B \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "prob_space (A \<bind> B)"
  1693   using measurable_bind_prob_space[OF measurable_const, OF A B, THEN measurable_space, of undefined "count_space UNIV"]
  1694   by (simp add: space_prob_algebra)
  1696 lemma sets_bind':
  1697   assumes A: "A \<in> space (prob_algebra M)" and B: "B \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "sets (A \<bind> B) = sets N"
  1698   using measurable_bind_prob_space[OF measurable_const, OF A B, THEN measurable_space, of undefined "count_space UNIV"]
  1699   by (simp add: space_prob_algebra)
  1701 lemma bind_cong_AE':
  1702   assumes M: "M \<in> space (prob_algebra L)"
  1703     and f: "f \<in> L \<rightarrow>\<^sub>M prob_algebra N" and g: "g \<in> L \<rightarrow>\<^sub>M prob_algebra N"
  1704     and ae: "AE x in M. f x = g x"
  1705   shows "bind M f = bind M g"
  1706 proof (rule measure_eqI)
  1707   show "sets (M \<bind> f) = sets (M \<bind> g)"
  1708     unfolding sets_bind'[OF M f] sets_bind'[OF M g] ..
  1709   show "A \<in> sets (M \<bind> f) \<Longrightarrow> emeasure (M \<bind> f) A = emeasure (M \<bind> g) A" for A
  1710     unfolding sets_bind'[OF M f]
  1711     using emeasure_bind_prob_algebra[OF M f, of A] emeasure_bind_prob_algebra[OF M g, of A] ae
  1712     by (auto intro: nn_integral_cong_AE)
  1713 qed
  1715 lemma density_discrete:
  1716   "countable A \<Longrightarrow> sets N = Set.Pow A \<Longrightarrow> (\<And>x. f x \<ge> 0) \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x = emeasure N {x}) \<Longrightarrow>
  1717     density (count_space A) f = N"
  1718   by (rule measure_eqI_countable[of _ A]) (auto simp: emeasure_density)
  1720 lemma distr_density_discrete:
  1721   fixes f'
  1722   assumes "countable A"
  1723   assumes "f' \<in> borel_measurable M"
  1724   assumes "g \<in> measurable M (count_space A)"
  1725   defines "f \<equiv> \<lambda>x. \<integral>\<^sup>+t. (if g t = x then 1 else 0) * f' t \<partial>M"
  1726   assumes "\<And>x.  x \<in> space M \<Longrightarrow> g x \<in> A"
  1727   shows "density (count_space A) (\<lambda>x. f x) = distr (density M f') (count_space A) g"
  1728 proof (rule density_discrete)
  1729   fix x assume x: "x \<in> A"
  1730   have "f x = \<integral>\<^sup>+t. indicator (g -` {x} \<inter> space M) t * f' t \<partial>M" (is "_ = ?I") unfolding f_def
  1731     by (intro nn_integral_cong) (simp split: split_indicator)
  1732   also from x have in_sets: "g -` {x} \<inter> space M \<in> sets M"
  1733     by (intro measurable_sets[OF assms(3)]) simp
  1734   have "?I = emeasure (density M f') (g -` {x} \<inter> space M)" unfolding f_def
  1735     by (subst emeasure_density[OF assms(2) in_sets], subst mult.commute) (rule refl)
  1736   also from assms(3) x have "... = emeasure (distr (density M f') (count_space A) g) {x}"
  1737     by (subst emeasure_distr) simp_all
  1738   finally show "f x = emeasure (distr (density M f') (count_space A) g) {x}" .
  1739 qed (insert assms, auto)
  1741 lemma bind_cong_AE:
  1742   assumes "M = N"
  1743   assumes f: "f \<in> measurable N (subprob_algebra B)"
  1744   assumes g: "g \<in> measurable N (subprob_algebra B)"
  1745   assumes ae: "AE x in N. f x = g x"
  1746   shows "bind M f = bind N g"
  1747 proof cases
  1748   assume "space N = {}" then show ?thesis
  1749     using `M = N` by (simp add: bind_empty)
  1750 next
  1751   assume "space N \<noteq> {}"
  1752   show ?thesis unfolding `M = N`
  1753   proof (rule measure_eqI)
  1754     have *: "sets (N \<bind> f) = sets B"
  1755       using sets_bind[OF sets_kernel[OF f] `space N \<noteq> {}`] by simp
  1756     then show "sets (N \<bind> f) = sets (N \<bind> g)"
  1757       using sets_bind[OF sets_kernel[OF g] `space N \<noteq> {}`] by auto
  1758     fix A assume "A \<in> sets (N \<bind> f)"
  1759     then have "A \<in> sets B"
  1760       unfolding * .
  1761     with ae f g `space N \<noteq> {}` show "emeasure (N \<bind> f) A = emeasure (N \<bind> g) A"
  1762       by (subst (1 2) emeasure_bind[where N=B]) (auto intro!: nn_integral_cong_AE)
  1763   qed
  1764 qed
  1766 lemma bind_cong_strong: "M = N \<Longrightarrow> (\<And>x. x\<in>space M =simp=> f x = g x) \<Longrightarrow> bind M f = bind N g"
  1767   by (auto simp: simp_implies_def intro!: bind_cong)
  1769 lemma sets_bind_measurable:
  1770   assumes f: "f \<in> measurable M (subprob_algebra B)"
  1771   assumes M: "space M \<noteq> {}"
  1772   shows "sets (M \<bind> f) = sets B"
  1773   using M by (intro sets_bind[OF sets_kernel[OF f]]) auto
  1775 lemma space_bind_measurable:
  1776   assumes f: "f \<in> measurable M (subprob_algebra B)"
  1777   assumes M: "space M \<noteq> {}"
  1778   shows "space (M \<bind> f) = space B"
  1779   using M by (intro space_bind[OF sets_kernel[OF f]]) auto
  1781 lemma bind_distr_return:
  1782   "f \<in> M \<rightarrow>\<^sub>M N \<Longrightarrow> g \<in> N \<rightarrow>\<^sub>M L \<Longrightarrow> space M \<noteq> {} \<Longrightarrow>
  1783     distr M N f \<bind> (\<lambda>x. return L (g x)) = distr M L (\<lambda>x. g (f x))"
  1784   by (subst bind_distr[OF _ measurable_compose[OF _ return_measurable]])
  1785      (auto intro!: bind_return_distr')
  1787 end