src/HOL/Probability/Complete_Measure.thy
author haftmann
Sun Jun 23 21:16:07 2013 +0200 (2013-06-23)
changeset 52435 6646bb548c6b
parent 50244 de72bbe42190
child 56949 d1a937cbf858
permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
     1 (*  Title:      HOL/Probability/Complete_Measure.thy
     2     Author:     Robert Himmelmann, Johannes Hoelzl, TU Muenchen
     3 *)
     4 
     5 theory Complete_Measure
     6 imports Lebesgue_Integration
     7 begin
     8 
     9 definition
    10   "split_completion M A p = (if A \<in> sets M then p = (A, {}) else
    11    \<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and> fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets M)"
    12 
    13 definition
    14   "main_part M A = fst (Eps (split_completion M A))"
    15 
    16 definition
    17   "null_part M A = snd (Eps (split_completion M A))"
    18 
    19 definition completion :: "'a measure \<Rightarrow> 'a measure" where
    20   "completion M = measure_of (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }
    21     (emeasure M \<circ> main_part M)"
    22 
    23 lemma completion_into_space:
    24   "{ S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' } \<subseteq> Pow (space M)"
    25   using sets.sets_into_space by auto
    26 
    27 lemma space_completion[simp]: "space (completion M) = space M"
    28   unfolding completion_def using space_measure_of[OF completion_into_space] by simp
    29 
    30 lemma completionI:
    31   assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
    32   shows "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
    33   using assms by auto
    34 
    35 lemma completionE:
    36   assumes "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
    37   obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
    38   using assms by auto
    39 
    40 lemma sigma_algebra_completion:
    41   "sigma_algebra (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
    42     (is "sigma_algebra _ ?A")
    43   unfolding sigma_algebra_iff2
    44 proof (intro conjI ballI allI impI)
    45   show "?A \<subseteq> Pow (space M)"
    46     using sets.sets_into_space by auto
    47 next
    48   show "{} \<in> ?A" by auto
    49 next
    50   let ?C = "space M"
    51   fix A assume "A \<in> ?A" from completionE[OF this] guess S N N' .
    52   then show "space M - A \<in> ?A"
    53     by (intro completionI[of _ "(?C - S) \<inter> (?C - N')" "(?C - S) \<inter> N' \<inter> (?C - N)"]) auto
    54 next
    55   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> ?A"
    56   then have "\<forall>n. \<exists>S N N'. A n = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'"
    57     by (auto simp: image_subset_iff)
    58   from choice[OF this] guess S ..
    59   from choice[OF this] guess N ..
    60   from choice[OF this] guess N' ..
    61   then show "UNION UNIV A \<in> ?A"
    62     using null_sets_UN[of N']
    63     by (intro completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"]) auto
    64 qed
    65 
    66 lemma sets_completion:
    67   "sets (completion M) = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
    68   using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_completion] by (simp add: completion_def)
    69 
    70 lemma sets_completionE:
    71   assumes "A \<in> sets (completion M)"
    72   obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
    73   using assms unfolding sets_completion by auto
    74 
    75 lemma sets_completionI:
    76   assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
    77   shows "A \<in> sets (completion M)"
    78   using assms unfolding sets_completion by auto
    79 
    80 lemma sets_completionI_sets[intro, simp]:
    81   "A \<in> sets M \<Longrightarrow> A \<in> sets (completion M)"
    82   unfolding sets_completion by force
    83 
    84 lemma null_sets_completion:
    85   assumes "N' \<in> null_sets M" "N \<subseteq> N'" shows "N \<in> sets (completion M)"
    86   using assms by (intro sets_completionI[of N "{}" N N']) auto
    87 
    88 lemma split_completion:
    89   assumes "A \<in> sets (completion M)"
    90   shows "split_completion M A (main_part M A, null_part M A)"
    91 proof cases
    92   assume "A \<in> sets M" then show ?thesis
    93     by (simp add: split_completion_def[abs_def] main_part_def null_part_def)
    94 next
    95   assume nA: "A \<notin> sets M"
    96   show ?thesis
    97     unfolding main_part_def null_part_def if_not_P[OF nA]
    98   proof (rule someI2_ex)
    99     from assms[THEN sets_completionE] guess S N N' . note A = this
   100     let ?P = "(S, N - S)"
   101     show "\<exists>p. split_completion M A p"
   102       unfolding split_completion_def if_not_P[OF nA] using A
   103     proof (intro exI conjI)
   104       show "A = fst ?P \<union> snd ?P" using A by auto
   105       show "snd ?P \<subseteq> N'" using A by auto
   106    qed auto
   107   qed auto
   108 qed
   109 
   110 lemma
   111   assumes "S \<in> sets (completion M)"
   112   shows main_part_sets[intro, simp]: "main_part M S \<in> sets M"
   113     and main_part_null_part_Un[simp]: "main_part M S \<union> null_part M S = S"
   114     and main_part_null_part_Int[simp]: "main_part M S \<inter> null_part M S = {}"
   115   using split_completion[OF assms]
   116   by (auto simp: split_completion_def split: split_if_asm)
   117 
   118 lemma main_part[simp]: "S \<in> sets M \<Longrightarrow> main_part M S = S"
   119   using split_completion[of S M]
   120   by (auto simp: split_completion_def split: split_if_asm)
   121 
   122 lemma null_part:
   123   assumes "S \<in> sets (completion M)" shows "\<exists>N. N\<in>null_sets M \<and> null_part M S \<subseteq> N"
   124   using split_completion[OF assms] by (auto simp: split_completion_def split: split_if_asm)
   125 
   126 lemma null_part_sets[intro, simp]:
   127   assumes "S \<in> sets M" shows "null_part M S \<in> sets M" "emeasure M (null_part M S) = 0"
   128 proof -
   129   have S: "S \<in> sets (completion M)" using assms by auto
   130   have "S - main_part M S \<in> sets M" using assms by auto
   131   moreover
   132   from main_part_null_part_Un[OF S] main_part_null_part_Int[OF S]
   133   have "S - main_part M S = null_part M S" by auto
   134   ultimately show sets: "null_part M S \<in> sets M" by auto
   135   from null_part[OF S] guess N ..
   136   with emeasure_eq_0[of N _ "null_part M S"] sets
   137   show "emeasure M (null_part M S) = 0" by auto
   138 qed
   139 
   140 lemma emeasure_main_part_UN:
   141   fixes S :: "nat \<Rightarrow> 'a set"
   142   assumes "range S \<subseteq> sets (completion M)"
   143   shows "emeasure M (main_part M (\<Union>i. (S i))) = emeasure M (\<Union>i. main_part M (S i))"
   144 proof -
   145   have S: "\<And>i. S i \<in> sets (completion M)" using assms by auto
   146   then have UN: "(\<Union>i. S i) \<in> sets (completion M)" by auto
   147   have "\<forall>i. \<exists>N. N \<in> null_sets M \<and> null_part M (S i) \<subseteq> N"
   148     using null_part[OF S] by auto
   149   from choice[OF this] guess N .. note N = this
   150   then have UN_N: "(\<Union>i. N i) \<in> null_sets M" by (intro null_sets_UN) auto
   151   have "(\<Union>i. S i) \<in> sets (completion M)" using S by auto
   152   from null_part[OF this] guess N' .. note N' = this
   153   let ?N = "(\<Union>i. N i) \<union> N'"
   154   have null_set: "?N \<in> null_sets M" using N' UN_N by (intro null_sets.Un) auto
   155   have "main_part M (\<Union>i. S i) \<union> ?N = (main_part M (\<Union>i. S i) \<union> null_part M (\<Union>i. S i)) \<union> ?N"
   156     using N' by auto
   157   also have "\<dots> = (\<Union>i. main_part M (S i) \<union> null_part M (S i)) \<union> ?N"
   158     unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto
   159   also have "\<dots> = (\<Union>i. main_part M (S i)) \<union> ?N"
   160     using N by auto
   161   finally have *: "main_part M (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part M (S i)) \<union> ?N" .
   162   have "emeasure M (main_part M (\<Union>i. S i)) = emeasure M (main_part M (\<Union>i. S i) \<union> ?N)"
   163     using null_set UN by (intro emeasure_Un_null_set[symmetric]) auto
   164   also have "\<dots> = emeasure M ((\<Union>i. main_part M (S i)) \<union> ?N)"
   165     unfolding * ..
   166   also have "\<dots> = emeasure M (\<Union>i. main_part M (S i))"
   167     using null_set S by (intro emeasure_Un_null_set) auto
   168   finally show ?thesis .
   169 qed
   170 
   171 lemma emeasure_completion[simp]:
   172   assumes S: "S \<in> sets (completion M)" shows "emeasure (completion M) S = emeasure M (main_part M S)"
   173 proof (subst emeasure_measure_of[OF completion_def completion_into_space])
   174   let ?\<mu> = "emeasure M \<circ> main_part M"
   175   show "S \<in> sets (completion M)" "?\<mu> S = emeasure M (main_part M S) " using S by simp_all
   176   show "positive (sets (completion M)) ?\<mu>"
   177     by (simp add: positive_def emeasure_nonneg)
   178   show "countably_additive (sets (completion M)) ?\<mu>"
   179   proof (intro countably_additiveI)
   180     fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets (completion M)" "disjoint_family A"
   181     have "disjoint_family (\<lambda>i. main_part M (A i))"
   182     proof (intro disjoint_family_on_bisimulation[OF A(2)])
   183       fix n m assume "A n \<inter> A m = {}"
   184       then have "(main_part M (A n) \<union> null_part M (A n)) \<inter> (main_part M (A m) \<union> null_part M (A m)) = {}"
   185         using A by (subst (1 2) main_part_null_part_Un) auto
   186       then show "main_part M (A n) \<inter> main_part M (A m) = {}" by auto
   187     qed
   188     then have "(\<Sum>n. emeasure M (main_part M (A n))) = emeasure M (\<Union>i. main_part M (A i))"
   189       using A by (auto intro!: suminf_emeasure)
   190     then show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (UNION UNIV A)"
   191       by (simp add: completion_def emeasure_main_part_UN[OF A(1)])
   192   qed
   193 qed
   194 
   195 lemma emeasure_completion_UN:
   196   "range S \<subseteq> sets (completion M) \<Longrightarrow>
   197     emeasure (completion M) (\<Union>i::nat. (S i)) = emeasure M (\<Union>i. main_part M (S i))"
   198   by (subst emeasure_completion) (auto simp add: emeasure_main_part_UN)
   199 
   200 lemma emeasure_completion_Un:
   201   assumes S: "S \<in> sets (completion M)" and T: "T \<in> sets (completion M)"
   202   shows "emeasure (completion M) (S \<union> T) = emeasure M (main_part M S \<union> main_part M T)"
   203 proof (subst emeasure_completion)
   204   have UN: "(\<Union>i. binary (main_part M S) (main_part M T) i) = (\<Union>i. main_part M (binary S T i))"
   205     unfolding binary_def by (auto split: split_if_asm)
   206   show "emeasure M (main_part M (S \<union> T)) = emeasure M (main_part M S \<union> main_part M T)"
   207     using emeasure_main_part_UN[of "binary S T" M] assms
   208     unfolding range_binary_eq Un_range_binary UN by auto
   209 qed (auto intro: S T)
   210 
   211 lemma sets_completionI_sub:
   212   assumes N: "N' \<in> null_sets M" "N \<subseteq> N'"
   213   shows "N \<in> sets (completion M)"
   214   using assms by (intro sets_completionI[of _ "{}" N N']) auto
   215 
   216 lemma completion_ex_simple_function:
   217   assumes f: "simple_function (completion M) f"
   218   shows "\<exists>f'. simple_function M f' \<and> (AE x in M. f x = f' x)"
   219 proof -
   220   let ?F = "\<lambda>x. f -` {x} \<inter> space M"
   221   have F: "\<And>x. ?F x \<in> sets (completion M)" and fin: "finite (f`space M)"
   222     using simple_functionD[OF f] simple_functionD[OF f] by simp_all
   223   have "\<forall>x. \<exists>N. N \<in> null_sets M \<and> null_part M (?F x) \<subseteq> N"
   224     using F null_part by auto
   225   from choice[OF this] obtain N where
   226     N: "\<And>x. null_part M (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets M" by auto
   227   let ?N = "\<Union>x\<in>f`space M. N x"
   228   let ?f' = "\<lambda>x. if x \<in> ?N then undefined else f x"
   229   have sets: "?N \<in> null_sets M" using N fin by (intro null_sets.finite_UN) auto
   230   show ?thesis unfolding simple_function_def
   231   proof (safe intro!: exI[of _ ?f'])
   232     have "?f' ` space M \<subseteq> f`space M \<union> {undefined}" by auto
   233     from finite_subset[OF this] simple_functionD(1)[OF f]
   234     show "finite (?f' ` space M)" by auto
   235   next
   236     fix x assume "x \<in> space M"
   237     have "?f' -` {?f' x} \<inter> space M =
   238       (if x \<in> ?N then ?F undefined \<union> ?N
   239        else if f x = undefined then ?F (f x) \<union> ?N
   240        else ?F (f x) - ?N)"
   241       using N(2) sets.sets_into_space by (auto split: split_if_asm simp: null_sets_def)
   242     moreover { fix y have "?F y \<union> ?N \<in> sets M"
   243       proof cases
   244         assume y: "y \<in> f`space M"
   245         have "?F y \<union> ?N = (main_part M (?F y) \<union> null_part M (?F y)) \<union> ?N"
   246           using main_part_null_part_Un[OF F] by auto
   247         also have "\<dots> = main_part M (?F y) \<union> ?N"
   248           using y N by auto
   249         finally show ?thesis
   250           using F sets by auto
   251       next
   252         assume "y \<notin> f`space M" then have "?F y = {}" by auto
   253         then show ?thesis using sets by auto
   254       qed }
   255     moreover {
   256       have "?F (f x) - ?N = main_part M (?F (f x)) \<union> null_part M (?F (f x)) - ?N"
   257         using main_part_null_part_Un[OF F] by auto
   258       also have "\<dots> = main_part M (?F (f x)) - ?N"
   259         using N `x \<in> space M` by auto
   260       finally have "?F (f x) - ?N \<in> sets M"
   261         using F sets by auto }
   262     ultimately show "?f' -` {?f' x} \<inter> space M \<in> sets M" by auto
   263   next
   264     show "AE x in M. f x = ?f' x"
   265       by (rule AE_I', rule sets) auto
   266   qed
   267 qed
   268 
   269 lemma completion_ex_borel_measurable_pos:
   270   fixes g :: "'a \<Rightarrow> ereal"
   271   assumes g: "g \<in> borel_measurable (completion M)" and "\<And>x. 0 \<le> g x"
   272   shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
   273 proof -
   274   from g[THEN borel_measurable_implies_simple_function_sequence'] guess f . note f = this
   275   from this(1)[THEN completion_ex_simple_function]
   276   have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x in M. f i x = f' x)" ..
   277   from this[THEN choice] obtain f' where
   278     sf: "\<And>i. simple_function M (f' i)" and
   279     AE: "\<forall>i. AE x in M. f i x = f' i x" by auto
   280   show ?thesis
   281   proof (intro bexI)
   282     from AE[unfolded AE_all_countable[symmetric]]
   283     show "AE x in M. g x = (SUP i. f' i x)" (is "AE x in M. g x = ?f x")
   284     proof (elim AE_mp, safe intro!: AE_I2)
   285       fix x assume eq: "\<forall>i. f i x = f' i x"
   286       moreover have "g x = (SUP i. f i x)"
   287         unfolding f using `0 \<le> g x` by (auto split: split_max)
   288       ultimately show "g x = ?f x" by auto
   289     qed
   290     show "?f \<in> borel_measurable M"
   291       using sf by (auto intro: borel_measurable_simple_function)
   292   qed
   293 qed
   294 
   295 lemma completion_ex_borel_measurable:
   296   fixes g :: "'a \<Rightarrow> ereal"
   297   assumes g: "g \<in> borel_measurable (completion M)"
   298   shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
   299 proof -
   300   have "(\<lambda>x. max 0 (g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (g x)" using g by auto
   301   from completion_ex_borel_measurable_pos[OF this] guess g_pos ..
   302   moreover
   303   have "(\<lambda>x. max 0 (- g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (- g x)" using g by auto
   304   from completion_ex_borel_measurable_pos[OF this] guess g_neg ..
   305   ultimately
   306   show ?thesis
   307   proof (safe intro!: bexI[of _ "\<lambda>x. g_pos x - g_neg x"])
   308     show "AE x in M. max 0 (- g x) = g_neg x \<longrightarrow> max 0 (g x) = g_pos x \<longrightarrow> g x = g_pos x - g_neg x"
   309     proof (intro AE_I2 impI)
   310       fix x assume g: "max 0 (- g x) = g_neg x" "max 0 (g x) = g_pos x"
   311       show "g x = g_pos x - g_neg x" unfolding g[symmetric]
   312         by (cases "g x") (auto split: split_max)
   313     qed
   314   qed auto
   315 qed
   316 
   317 end