src/HOL/Probability/Measure_Space.thy
 author hoelzl Wed Apr 25 19:26:00 2012 +0200 (2012-04-25) changeset 47761 dfe747e72fa8 parent 47694 05663f75964c child 47762 d31085f07f60 permissions -rw-r--r--
moved lemmas to appropriate places
```     1 (*  Title:      HOL/Probability/Measure_Space.thy
```
```     2     Author:     Lawrence C Paulson
```
```     3     Author:     Johannes Hölzl, TU München
```
```     4     Author:     Armin Heller, TU München
```
```     5 *)
```
```     6
```
```     7 header {* Measure spaces and their properties *}
```
```     8
```
```     9 theory Measure_Space
```
```    10 imports
```
```    11   Sigma_Algebra
```
```    12   "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits"
```
```    13 begin
```
```    14
```
```    15 lemma suminf_cmult_indicator:
```
```    16   fixes f :: "nat \<Rightarrow> ereal"
```
```    17   assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
```
```    18   shows "(\<Sum>n. f n * indicator (A n) x) = f i"
```
```    19 proof -
```
```    20   have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)"
```
```    21     using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
```
```    22   then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)"
```
```    23     by (auto simp: setsum_cases)
```
```    24   moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
```
```    25   proof (rule ereal_SUPI)
```
```    26     fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
```
```    27     from this[of "Suc i"] show "f i \<le> y" by auto
```
```    28   qed (insert assms, simp)
```
```    29   ultimately show ?thesis using assms
```
```    30     by (subst suminf_ereal_eq_SUPR) (auto simp: indicator_def)
```
```    31 qed
```
```    32
```
```    33 lemma suminf_indicator:
```
```    34   assumes "disjoint_family A"
```
```    35   shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x"
```
```    36 proof cases
```
```    37   assume *: "x \<in> (\<Union>i. A i)"
```
```    38   then obtain i where "x \<in> A i" by auto
```
```    39   from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
```
```    40   show ?thesis using * by simp
```
```    41 qed simp
```
```    42
```
```    43 text {*
```
```    44   The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to
```
```    45   represent sigma algebras (with an arbitrary emeasure).
```
```    46 *}
```
```    47
```
```    48 section "Extend binary sets"
```
```    49
```
```    50 lemma LIMSEQ_binaryset:
```
```    51   assumes f: "f {} = 0"
```
```    52   shows  "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
```
```    53 proof -
```
```    54   have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
```
```    55     proof
```
```    56       fix n
```
```    57       show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
```
```    58         by (induct n)  (auto simp add: binaryset_def f)
```
```    59     qed
```
```    60   moreover
```
```    61   have "... ----> f A + f B" by (rule tendsto_const)
```
```    62   ultimately
```
```    63   have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
```
```    64     by metis
```
```    65   hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
```
```    66     by simp
```
```    67   thus ?thesis by (rule LIMSEQ_offset [where k=2])
```
```    68 qed
```
```    69
```
```    70 lemma binaryset_sums:
```
```    71   assumes f: "f {} = 0"
```
```    72   shows  "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
```
```    73     by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
```
```    74
```
```    75 lemma suminf_binaryset_eq:
```
```    76   fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
```
```    77   shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
```
```    78   by (metis binaryset_sums sums_unique)
```
```    79
```
```    80 section {* Properties of a premeasure @{term \<mu>} *}
```
```    81
```
```    82 text {*
```
```    83   The definitions for @{const positive} and @{const countably_additive} should be here, by they are
```
```    84   necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}.
```
```    85 *}
```
```    86
```
```    87 definition additive where
```
```    88   "additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"
```
```    89
```
```    90 definition increasing where
```
```    91   "increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)"
```
```    92
```
```    93 lemma positiveD_empty:
```
```    94   "positive M f \<Longrightarrow> f {} = 0"
```
```    95   by (auto simp add: positive_def)
```
```    96
```
```    97 lemma additiveD:
```
```    98   "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y"
```
```    99   by (auto simp add: additive_def)
```
```   100
```
```   101 lemma increasingD:
```
```   102   "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"
```
```   103   by (auto simp add: increasing_def)
```
```   104
```
```   105 lemma countably_additiveI:
```
```   106   "(\<And>A. range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> M \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i))
```
```   107   \<Longrightarrow> countably_additive M f"
```
```   108   by (simp add: countably_additive_def)
```
```   109
```
```   110 lemma (in ring_of_sets) disjointed_additive:
```
```   111   assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A"
```
```   112   shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
```
```   113 proof (induct n)
```
```   114   case (Suc n)
```
```   115   then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
```
```   116     by simp
```
```   117   also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
```
```   118     using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq)
```
```   119   also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
```
```   120     using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq)
```
```   121   finally show ?case .
```
```   122 qed simp
```
```   123
```
```   124 lemma (in ring_of_sets) additive_sum:
```
```   125   fixes A:: "'i \<Rightarrow> 'a set"
```
```   126   assumes f: "positive M f" and ad: "additive M f" and "finite S"
```
```   127       and A: "A`S \<subseteq> M"
```
```   128       and disj: "disjoint_family_on A S"
```
```   129   shows  "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
```
```   130 using `finite S` disj A proof induct
```
```   131   case empty show ?case using f by (simp add: positive_def)
```
```   132 next
```
```   133   case (insert s S)
```
```   134   then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
```
```   135     by (auto simp add: disjoint_family_on_def neq_iff)
```
```   136   moreover
```
```   137   have "A s \<in> M" using insert by blast
```
```   138   moreover have "(\<Union>i\<in>S. A i) \<in> M"
```
```   139     using insert `finite S` by auto
```
```   140   moreover
```
```   141   ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
```
```   142     using ad UNION_in_sets A by (auto simp add: additive_def)
```
```   143   with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
```
```   144     by (auto simp add: additive_def subset_insertI)
```
```   145 qed
```
```   146
```
```   147 lemma (in ring_of_sets) additive_increasing:
```
```   148   assumes posf: "positive M f" and addf: "additive M f"
```
```   149   shows "increasing M f"
```
```   150 proof (auto simp add: increasing_def)
```
```   151   fix x y
```
```   152   assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y"
```
```   153   then have "y - x \<in> M" by auto
```
```   154   then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
```
```   155   then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
```
```   156   also have "... = f (x \<union> (y-x))" using addf
```
```   157     by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
```
```   158   also have "... = f y"
```
```   159     by (metis Un_Diff_cancel Un_absorb1 xy(3))
```
```   160   finally show "f x \<le> f y" by simp
```
```   161 qed
```
```   162
```
```   163 lemma (in ring_of_sets) countably_additive_additive:
```
```   164   assumes posf: "positive M f" and ca: "countably_additive M f"
```
```   165   shows "additive M f"
```
```   166 proof (auto simp add: additive_def)
```
```   167   fix x y
```
```   168   assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
```
```   169   hence "disjoint_family (binaryset x y)"
```
```   170     by (auto simp add: disjoint_family_on_def binaryset_def)
```
```   171   hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
```
```   172          (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
```
```   173          f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
```
```   174     using ca
```
```   175     by (simp add: countably_additive_def)
```
```   176   hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
```
```   177          f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
```
```   178     by (simp add: range_binaryset_eq UN_binaryset_eq)
```
```   179   thus "f (x \<union> y) = f x + f y" using posf x y
```
```   180     by (auto simp add: Un suminf_binaryset_eq positive_def)
```
```   181 qed
```
```   182
```
```   183 lemma (in algebra) increasing_additive_bound:
```
```   184   fixes A:: "nat \<Rightarrow> 'a set" and  f :: "'a set \<Rightarrow> ereal"
```
```   185   assumes f: "positive M f" and ad: "additive M f"
```
```   186       and inc: "increasing M f"
```
```   187       and A: "range A \<subseteq> M"
```
```   188       and disj: "disjoint_family A"
```
```   189   shows  "(\<Sum>i. f (A i)) \<le> f \<Omega>"
```
```   190 proof (safe intro!: suminf_bound)
```
```   191   fix N
```
```   192   note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
```
```   193   have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
```
```   194     using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)
```
```   195   also have "... \<le> f \<Omega>" using space_closed A
```
```   196     by (intro increasingD[OF inc] finite_UN) auto
```
```   197   finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp
```
```   198 qed (insert f A, auto simp: positive_def)
```
```   199
```
```   200 lemma (in ring_of_sets) countably_additiveI_finite:
```
```   201   assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>"
```
```   202   shows "countably_additive M \<mu>"
```
```   203 proof (rule countably_additiveI)
```
```   204   fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F"
```
```   205
```
```   206   have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
```
```   207   from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
```
```   208
```
```   209   have inj_f: "inj_on f {i. F i \<noteq> {}}"
```
```   210   proof (rule inj_onI, simp)
```
```   211     fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
```
```   212     then have "f i \<in> F i" "f j \<in> F j" using f by force+
```
```   213     with disj * show "i = j" by (auto simp: disjoint_family_on_def)
```
```   214   qed
```
```   215   have "finite (\<Union>i. F i)"
```
```   216     by (metis F(2) assms(1) infinite_super sets_into_space)
```
```   217
```
```   218   have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
```
```   219     by (auto simp: positiveD_empty[OF `positive M \<mu>`])
```
```   220   moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"
```
```   221   proof (rule finite_imageD)
```
```   222     from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
```
```   223     then show "finite (f`{i. F i \<noteq> {}})"
```
```   224       by (rule finite_subset) fact
```
```   225   qed fact
```
```   226   ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}"
```
```   227     by (rule finite_subset)
```
```   228
```
```   229   have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}"
```
```   230     using disj by (auto simp: disjoint_family_on_def)
```
```   231
```
```   232   from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"
```
```   233     by (rule suminf_finite) auto
```
```   234   also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
```
```   235     using fin_not_empty F_subset by (rule setsum_mono_zero_left) auto
```
```   236   also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
```
```   237     using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto
```
```   238   also have "\<dots> = \<mu> (\<Union>i. F i)"
```
```   239     by (rule arg_cong[where f=\<mu>]) auto
```
```   240   finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .
```
```   241 qed
```
```   242
```
```   243 section {* Properties of @{const emeasure} *}
```
```   244
```
```   245 lemma emeasure_positive: "positive (sets M) (emeasure M)"
```
```   246   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
```
```   247
```
```   248 lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
```
```   249   using emeasure_positive[of M] by (simp add: positive_def)
```
```   250
```
```   251 lemma emeasure_nonneg[intro!]: "0 \<le> emeasure M A"
```
```   252   using emeasure_notin_sets[of A M] emeasure_positive[of M]
```
```   253   by (cases "A \<in> sets M") (auto simp: positive_def)
```
```   254
```
```   255 lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>"
```
```   256   using emeasure_nonneg[of M A] by auto
```
```   257
```
```   258 lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
```
```   259   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
```
```   260
```
```   261 lemma suminf_emeasure:
```
```   262   "range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
```
```   263   using countable_UN[of A UNIV M] emeasure_countably_additive[of M]
```
```   264   by (simp add: countably_additive_def)
```
```   265
```
```   266 lemma emeasure_additive: "additive (sets M) (emeasure M)"
```
```   267   by (metis countably_additive_additive emeasure_positive emeasure_countably_additive)
```
```   268
```
```   269 lemma plus_emeasure:
```
```   270   "a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)"
```
```   271   using additiveD[OF emeasure_additive] ..
```
```   272
```
```   273 lemma setsum_emeasure:
```
```   274   "F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
```
```   275     (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
```
```   276   by (metis additive_sum emeasure_positive emeasure_additive)
```
```   277
```
```   278 lemma emeasure_mono:
```
```   279   "a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b"
```
```   280   by (metis additive_increasing emeasure_additive emeasure_nonneg emeasure_notin_sets
```
```   281             emeasure_positive increasingD)
```
```   282
```
```   283 lemma emeasure_space:
```
```   284   "emeasure M A \<le> emeasure M (space M)"
```
```   285   by (metis emeasure_mono emeasure_nonneg emeasure_notin_sets sets_into_space top)
```
```   286
```
```   287 lemma emeasure_compl:
```
```   288   assumes s: "s \<in> sets M" and fin: "emeasure M s \<noteq> \<infinity>"
```
```   289   shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
```
```   290 proof -
```
```   291   from s have "0 \<le> emeasure M s" by auto
```
```   292   have "emeasure M (space M) = emeasure M (s \<union> (space M - s))" using s
```
```   293     by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
```
```   294   also have "... = emeasure M s + emeasure M (space M - s)"
```
```   295     by (rule plus_emeasure[symmetric]) (auto simp add: s)
```
```   296   finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" .
```
```   297   then show ?thesis
```
```   298     using fin `0 \<le> emeasure M s`
```
```   299     unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
```
```   300 qed
```
```   301
```
```   302 lemma emeasure_Diff:
```
```   303   assumes finite: "emeasure M B \<noteq> \<infinity>"
```
```   304   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
```
```   305   shows "emeasure M (A - B) = emeasure M A - emeasure M B"
```
```   306 proof -
```
```   307   have "0 \<le> emeasure M B" using assms by auto
```
```   308   have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
```
```   309   then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp
```
```   310   also have "\<dots> = emeasure M (A - B) + emeasure M B"
```
```   311     using measurable by (subst plus_emeasure[symmetric]) auto
```
```   312   finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
```
```   313     unfolding ereal_eq_minus_iff
```
```   314     using finite `0 \<le> emeasure M B` by auto
```
```   315 qed
```
```   316
```
```   317 lemma emeasure_countable_increasing:
```
```   318   assumes A: "range A \<subseteq> sets M"
```
```   319       and A0: "A 0 = {}"
```
```   320       and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
```
```   321   shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
```
```   322 proof -
```
```   323   { fix n
```
```   324     have "emeasure M (A n) = (\<Sum>i<n. emeasure M (A (Suc i) - A i))"
```
```   325       proof (induct n)
```
```   326         case 0 thus ?case by (auto simp add: A0)
```
```   327       next
```
```   328         case (Suc m)
```
```   329         have "A (Suc m) = A m \<union> (A (Suc m) - A m)"
```
```   330           by (metis ASuc Un_Diff_cancel Un_absorb1)
```
```   331         hence "emeasure M (A (Suc m)) =
```
```   332                emeasure M (A m) + emeasure M (A (Suc m) - A m)"
```
```   333           by (subst plus_emeasure)
```
```   334              (auto simp add: emeasure_additive range_subsetD [OF A])
```
```   335         with Suc show ?case
```
```   336           by simp
```
```   337       qed }
```
```   338   note Meq = this
```
```   339   have Aeq: "(\<Union>i. A (Suc i) - A i) = (\<Union>i. A i)"
```
```   340     proof (rule UN_finite2_eq [where k=1], simp)
```
```   341       fix i
```
```   342       show "(\<Union>i\<in>{0..<i}. A (Suc i) - A i) = (\<Union>i\<in>{0..<Suc i}. A i)"
```
```   343         proof (induct i)
```
```   344           case 0 thus ?case by (simp add: A0)
```
```   345         next
```
```   346           case (Suc i)
```
```   347           thus ?case
```
```   348             by (auto simp add: atLeastLessThanSuc intro: subsetD [OF ASuc])
```
```   349         qed
```
```   350     qed
```
```   351   have A1: "\<And>i. A (Suc i) - A i \<in> sets M"
```
```   352     by (metis A Diff range_subsetD)
```
```   353   have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
```
```   354     by (blast intro: range_subsetD [OF A])
```
```   355   have "(SUP n. \<Sum>i<n. emeasure M (A (Suc i) - A i)) = (\<Sum>i. emeasure M (A (Suc i) - A i))"
```
```   356     using A by (auto intro!: suminf_ereal_eq_SUPR[symmetric])
```
```   357   also have "\<dots> = emeasure M (\<Union>i. A (Suc i) - A i)"
```
```   358     by (rule suminf_emeasure)
```
```   359        (auto simp add: disjoint_family_Suc ASuc A1 A2)
```
```   360   also have "... =  emeasure M (\<Union>i. A i)"
```
```   361     by (simp add: Aeq)
```
```   362   finally have "(SUP n. \<Sum>i<n. emeasure M (A (Suc i) - A i)) = emeasure M (\<Union>i. A i)" .
```
```   363   then show ?thesis by (auto simp add: Meq)
```
```   364 qed
```
```   365
```
```   366 lemma SUP_emeasure_incseq:
```
```   367   assumes A: "range A \<subseteq> sets M" and "incseq A"
```
```   368   shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
```
```   369 proof -
```
```   370   have *: "(SUP n. emeasure M (nat_case {} A (Suc n))) = (SUP n. emeasure M (nat_case {} A n))"
```
```   371     using A by (auto intro!: SUPR_eq exI split: nat.split)
```
```   372   have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
```
```   373     by (auto simp add: split: nat.splits)
```
```   374   have meq: "\<And>n. emeasure M (A n) = (emeasure M \<circ> nat_case {} A) (Suc n)"
```
```   375     by simp
```
```   376   have "(SUP n. emeasure M (nat_case {} A n)) = emeasure M (\<Union>i. nat_case {} A i)"
```
```   377     using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
```
```   378     by (force split: nat.splits intro!: emeasure_countable_increasing)
```
```   379   also have "emeasure M (\<Union>i. nat_case {} A i) = emeasure M (\<Union>i. A i)"
```
```   380     by (simp add: ueq)
```
```   381   finally have "(SUP n. emeasure M (nat_case {} A n)) = emeasure M (\<Union>i. A i)" .
```
```   382   thus ?thesis unfolding meq * comp_def .
```
```   383 qed
```
```   384
```
```   385 lemma incseq_emeasure:
```
```   386   assumes "range B \<subseteq> sets M" "incseq B"
```
```   387   shows "incseq (\<lambda>i. emeasure M (B i))"
```
```   388   using assms by (auto simp: incseq_def intro!: emeasure_mono)
```
```   389
```
```   390 lemma Lim_emeasure_incseq:
```
```   391   assumes A: "range A \<subseteq> sets M" "incseq A"
```
```   392   shows "(\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)"
```
```   393   using LIMSEQ_ereal_SUPR[OF incseq_emeasure, OF A]
```
```   394     SUP_emeasure_incseq[OF A] by simp
```
```   395
```
```   396 lemma decseq_emeasure:
```
```   397   assumes "range B \<subseteq> sets M" "decseq B"
```
```   398   shows "decseq (\<lambda>i. emeasure M (B i))"
```
```   399   using assms by (auto simp: decseq_def intro!: emeasure_mono)
```
```   400
```
```   401 lemma INF_emeasure_decseq:
```
```   402   assumes A: "range A \<subseteq> sets M" and "decseq A"
```
```   403   and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```   404   shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
```
```   405 proof -
```
```   406   have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
```
```   407     using A by (auto intro!: emeasure_mono)
```
```   408   hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
```
```   409
```
```   410   have A0: "0 \<le> emeasure M (A 0)" using A by auto
```
```   411
```
```   412   have "emeasure M (A 0) - (INF n. emeasure M (A n)) = emeasure M (A 0) + (SUP n. - emeasure M (A n))"
```
```   413     by (simp add: ereal_SUPR_uminus minus_ereal_def)
```
```   414   also have "\<dots> = (SUP n. emeasure M (A 0) - emeasure M (A n))"
```
```   415     unfolding minus_ereal_def using A0 assms
```
```   416     by (subst SUPR_ereal_add) (auto simp add: decseq_emeasure)
```
```   417   also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"
```
```   418     using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto
```
```   419   also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"
```
```   420   proof (rule SUP_emeasure_incseq)
```
```   421     show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
```
```   422       using A by auto
```
```   423     show "incseq (\<lambda>n. A 0 - A n)"
```
```   424       using `decseq A` by (auto simp add: incseq_def decseq_def)
```
```   425   qed
```
```   426   also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
```
```   427     using A finite * by (simp, subst emeasure_Diff) auto
```
```   428   finally show ?thesis
```
```   429     unfolding ereal_minus_eq_minus_iff using finite A0 by auto
```
```   430 qed
```
```   431
```
```   432 lemma Lim_emeasure_decseq:
```
```   433   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```   434   shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)"
```
```   435   using LIMSEQ_ereal_INFI[OF decseq_emeasure, OF A]
```
```   436   using INF_emeasure_decseq[OF A fin] by simp
```
```   437
```
```   438 lemma emeasure_subadditive:
```
```   439   assumes measurable: "A \<in> sets M" "B \<in> sets M"
```
```   440   shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
```
```   441 proof -
```
```   442   from plus_emeasure[of A M "B - A"]
```
```   443   have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
```
```   444     using assms by (simp add: Diff)
```
```   445   also have "\<dots> \<le> emeasure M A + emeasure M B"
```
```   446     using assms by (auto intro!: add_left_mono emeasure_mono)
```
```   447   finally show ?thesis .
```
```   448 qed
```
```   449
```
```   450 lemma emeasure_subadditive_finite:
```
```   451   assumes "finite I" "A ` I \<subseteq> sets M"
```
```   452   shows "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
```
```   453 using assms proof induct
```
```   454   case (insert i I)
```
```   455   then have "emeasure M (\<Union>i\<in>insert i I. A i) = emeasure M (A i \<union> (\<Union>i\<in>I. A i))"
```
```   456     by simp
```
```   457   also have "\<dots> \<le> emeasure M (A i) + emeasure M (\<Union>i\<in>I. A i)"
```
```   458     using insert by (intro emeasure_subadditive finite_UN) auto
```
```   459   also have "\<dots> \<le> emeasure M (A i) + (\<Sum>i\<in>I. emeasure M (A i))"
```
```   460     using insert by (intro add_mono) auto
```
```   461   also have "\<dots> = (\<Sum>i\<in>insert i I. emeasure M (A i))"
```
```   462     using insert by auto
```
```   463   finally show ?case .
```
```   464 qed simp
```
```   465
```
```   466 lemma emeasure_subadditive_countably:
```
```   467   assumes "range f \<subseteq> sets M"
```
```   468   shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))"
```
```   469 proof -
```
```   470   have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)"
```
```   471     unfolding UN_disjointed_eq ..
```
```   472   also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))"
```
```   473     using range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
```
```   474     by (simp add:  disjoint_family_disjointed comp_def)
```
```   475   also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))"
```
```   476     using range_disjointed_sets[OF assms] assms
```
```   477     by (auto intro!: suminf_le_pos emeasure_mono disjointed_subset)
```
```   478   finally show ?thesis .
```
```   479 qed
```
```   480
```
```   481 lemma emeasure_insert:
```
```   482   assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
```
```   483   shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
```
```   484 proof -
```
```   485   have "{x} \<inter> A = {}" using `x \<notin> A` by auto
```
```   486   from plus_emeasure[OF sets this] show ?thesis by simp
```
```   487 qed
```
```   488
```
```   489 lemma emeasure_eq_setsum_singleton:
```
```   490   assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```   491   shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
```
```   492   using setsum_emeasure[of "\<lambda>x. {x}" S M] assms
```
```   493   by (auto simp: disjoint_family_on_def subset_eq)
```
```   494
```
```   495 lemma setsum_emeasure_cover:
```
```   496   assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
```
```   497   assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"
```
```   498   assumes disj: "disjoint_family_on B S"
```
```   499   shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"
```
```   500 proof -
```
```   501   have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
```
```   502   proof (rule setsum_emeasure)
```
```   503     show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
```
```   504       using `disjoint_family_on B S`
```
```   505       unfolding disjoint_family_on_def by auto
```
```   506   qed (insert assms, auto)
```
```   507   also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
```
```   508     using A by auto
```
```   509   finally show ?thesis by simp
```
```   510 qed
```
```   511
```
```   512 lemma emeasure_eq_0:
```
```   513   "N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0"
```
```   514   by (metis emeasure_mono emeasure_nonneg order_eq_iff)
```
```   515
```
```   516 lemma emeasure_UN_eq_0:
```
```   517   assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M"
```
```   518   shows "emeasure M (\<Union> i. N i) = 0"
```
```   519 proof -
```
```   520   have "0 \<le> emeasure M (\<Union> i. N i)" using assms by auto
```
```   521   moreover have "emeasure M (\<Union> i. N i) \<le> 0"
```
```   522     using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
```
```   523   ultimately show ?thesis by simp
```
```   524 qed
```
```   525
```
```   526 lemma measure_eqI_finite:
```
```   527   assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
```
```   528   assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"
```
```   529   shows "M = N"
```
```   530 proof (rule measure_eqI)
```
```   531   fix X assume "X \<in> sets M"
```
```   532   then have X: "X \<subseteq> A" by auto
```
```   533   then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
```
```   534     using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
```
```   535   also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
```
```   536     using X eq by (auto intro!: setsum_cong)
```
```   537   also have "\<dots> = emeasure N X"
```
```   538     using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
```
```   539   finally show "emeasure M X = emeasure N X" .
```
```   540 qed simp
```
```   541
```
```   542 lemma measure_eqI_generator_eq:
```
```   543   fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set"
```
```   544   assumes "Int_stable E" "E \<subseteq> Pow \<Omega>"
```
```   545   and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
```
```   546   and M: "sets M = sigma_sets \<Omega> E"
```
```   547   and N: "sets N = sigma_sets \<Omega> E"
```
```   548   and A: "range A \<subseteq> E" "incseq A" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```   549   shows "M = N"
```
```   550 proof -
```
```   551   let ?D = "\<lambda>F. {D. D \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)}"
```
```   552   interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
```
```   553   { fix F assume "F \<in> E" and "emeasure M F \<noteq> \<infinity>"
```
```   554     then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
```
```   555     have "emeasure N F \<noteq> \<infinity>" using `emeasure M F \<noteq> \<infinity>` `F \<in> E` eq by simp
```
```   556     interpret D: dynkin_system \<Omega> "?D F"
```
```   557     proof (rule dynkin_systemI, simp_all)
```
```   558       fix A assume "A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> A) = emeasure N (F \<inter> A)"
```
```   559       then show "A \<subseteq> \<Omega>" using S.sets_into_space by auto
```
```   560     next
```
```   561       have "F \<inter> \<Omega> = F" using `F \<in> E` `E \<subseteq> Pow \<Omega>` by auto
```
```   562       then show "emeasure M (F \<inter> \<Omega>) = emeasure N (F \<inter> \<Omega>)"
```
```   563         using `F \<in> E` eq by (auto intro: sigma_sets_top)
```
```   564     next
```
```   565       fix A assume *: "A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> A) = emeasure N (F \<inter> A)"
```
```   566       then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"
```
```   567         and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"
```
```   568         using `F \<in> E` S.sets_into_space by auto
```
```   569       have "emeasure N (F \<inter> A) \<le> emeasure N F" by (auto intro!: emeasure_mono simp: M N)
```
```   570       then have "emeasure N (F \<inter> A) \<noteq> \<infinity>" using `emeasure N F \<noteq> \<infinity>` by auto
```
```   571       have "emeasure M (F \<inter> A) \<le> emeasure M F" by (auto intro!: emeasure_mono simp: M N)
```
```   572       then have "emeasure M (F \<inter> A) \<noteq> \<infinity>" using `emeasure M F \<noteq> \<infinity>` by auto
```
```   573       then have "emeasure M (F \<inter> (\<Omega> - A)) = emeasure M F - emeasure M (F \<inter> A)" unfolding **
```
```   574         using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N)
```
```   575       also have "\<dots> = emeasure N F - emeasure N (F \<inter> A)" using eq `F \<in> E` * by simp
```
```   576       also have "\<dots> = emeasure N (F \<inter> (\<Omega> - A))" unfolding **
```
```   577         using `F \<inter> A \<in> sigma_sets \<Omega> E` `emeasure N (F \<inter> A) \<noteq> \<infinity>`
```
```   578         by (auto intro!: emeasure_Diff[symmetric] simp: M N)
```
```   579       finally show "\<Omega> - A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> (\<Omega> - A)) = emeasure N (F \<inter> (\<Omega> - A))"
```
```   580         using * by auto
```
```   581     next
```
```   582       fix A :: "nat \<Rightarrow> 'a set"
```
```   583       assume "disjoint_family A" "range A \<subseteq> {X \<in> sigma_sets \<Omega> E. emeasure M (F \<inter> X) = emeasure N (F \<inter> X)}"
```
```   584       then have A: "range (\<lambda>i. F \<inter> A i) \<subseteq> sigma_sets \<Omega> E" "F \<inter> (\<Union>x. A x) = (\<Union>x. F \<inter> A x)"
```
```   585         "disjoint_family (\<lambda>i. F \<inter> A i)" "\<And>i. emeasure M (F \<inter> A i) = emeasure N (F \<inter> A i)" "range A \<subseteq> sigma_sets \<Omega> E"
```
```   586         by (auto simp: disjoint_family_on_def subset_eq)
```
```   587       then show "(\<Union>x. A x) \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> (\<Union>x. A x)) = emeasure N (F \<inter> (\<Union>x. A x))"
```
```   588         by (auto simp: M N suminf_emeasure[symmetric] simp del: UN_simps)
```
```   589     qed
```
```   590     have *: "sigma_sets \<Omega> E = ?D F"
```
```   591       using `F \<in> E` `Int_stable E`
```
```   592       by (intro D.dynkin_lemma) (auto simp add: Int_stable_def eq)
```
```   593     have "\<And>D. D \<in> sigma_sets \<Omega> E \<Longrightarrow> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
```
```   594       by (subst (asm) *) auto }
```
```   595   note * = this
```
```   596   show "M = N"
```
```   597   proof (rule measure_eqI)
```
```   598     show "sets M = sets N"
```
```   599       using M N by simp
```
```   600     fix X assume "X \<in> sets M"
```
```   601     then have "X \<in> sigma_sets \<Omega> E"
```
```   602       using M by simp
```
```   603     let ?A = "\<lambda>i. A i \<inter> X"
```
```   604     have "range ?A \<subseteq> sigma_sets \<Omega> E" "incseq ?A"
```
```   605       using A(1,2) `X \<in> sigma_sets \<Omega> E` by (auto simp: incseq_def)
```
```   606     moreover
```
```   607     { fix i have "emeasure M (?A i) = emeasure N (?A i)"
```
```   608         using *[of "A i" X] `X \<in> sigma_sets \<Omega> E` A finite by auto }
```
```   609     ultimately show "emeasure M X = emeasure N X"
```
```   610       using SUP_emeasure_incseq[of ?A M] SUP_emeasure_incseq[of ?A N] A(3) `X \<in> sigma_sets \<Omega> E`
```
```   611       by (auto simp: M N SUP_emeasure_incseq)
```
```   612   qed
```
```   613 qed
```
```   614
```
```   615 lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
```
```   616 proof (intro measure_eqI emeasure_measure_of_sigma)
```
```   617   show "sigma_algebra (space M) (sets M)" ..
```
```   618   show "positive (sets M) (emeasure M)"
```
```   619     by (simp add: positive_def emeasure_nonneg)
```
```   620   show "countably_additive (sets M) (emeasure M)"
```
```   621     by (simp add: emeasure_countably_additive)
```
```   622 qed simp_all
```
```   623
```
```   624 section "@{text \<mu>}-null sets"
```
```   625
```
```   626 definition null_sets :: "'a measure \<Rightarrow> 'a set set" where
```
```   627   "null_sets M = {N\<in>sets M. emeasure M N = 0}"
```
```   628
```
```   629 lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0"
```
```   630   by (simp add: null_sets_def)
```
```   631
```
```   632 lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M"
```
```   633   unfolding null_sets_def by simp
```
```   634
```
```   635 lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M"
```
```   636   unfolding null_sets_def by simp
```
```   637
```
```   638 interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
```
```   639 proof
```
```   640   show "null_sets M \<subseteq> Pow (space M)"
```
```   641     using sets_into_space by auto
```
```   642   show "{} \<in> null_sets M"
```
```   643     by auto
```
```   644   fix A B assume sets: "A \<in> null_sets M" "B \<in> null_sets M"
```
```   645   then have "A \<in> sets M" "B \<in> sets M"
```
```   646     by auto
```
```   647   moreover then have "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
```
```   648     "emeasure M (A - B) \<le> emeasure M A"
```
```   649     by (auto intro!: emeasure_subadditive emeasure_mono)
```
```   650   moreover have "emeasure M B = 0" "emeasure M A = 0"
```
```   651     using sets by auto
```
```   652   ultimately show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"
```
```   653     by (auto intro!: antisym)
```
```   654 qed
```
```   655
```
```   656 lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
```
```   657 proof -
```
```   658   have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)"
```
```   659     unfolding SUP_def image_compose
```
```   660     unfolding surj_from_nat ..
```
```   661   then show ?thesis by simp
```
```   662 qed
```
```   663
```
```   664 lemma null_sets_UN[intro]:
```
```   665   assumes "\<And>i::'i::countable. N i \<in> null_sets M"
```
```   666   shows "(\<Union>i. N i) \<in> null_sets M"
```
```   667 proof (intro conjI CollectI null_setsI)
```
```   668   show "(\<Union>i. N i) \<in> sets M" using assms by auto
```
```   669   have "0 \<le> emeasure M (\<Union>i. N i)" by (rule emeasure_nonneg)
```
```   670   moreover have "emeasure M (\<Union>i. N i) \<le> (\<Sum>n. emeasure M (N (Countable.from_nat n)))"
```
```   671     unfolding UN_from_nat[of N]
```
```   672     using assms by (intro emeasure_subadditive_countably) auto
```
```   673   ultimately show "emeasure M (\<Union>i. N i) = 0"
```
```   674     using assms by (auto simp: null_setsD1)
```
```   675 qed
```
```   676
```
```   677 lemma null_set_Int1:
```
```   678   assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
```
```   679 proof (intro CollectI conjI null_setsI)
```
```   680   show "emeasure M (A \<inter> B) = 0" using assms
```
```   681     by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto
```
```   682 qed (insert assms, auto)
```
```   683
```
```   684 lemma null_set_Int2:
```
```   685   assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
```
```   686   using assms by (subst Int_commute) (rule null_set_Int1)
```
```   687
```
```   688 lemma emeasure_Diff_null_set:
```
```   689   assumes "B \<in> null_sets M" "A \<in> sets M"
```
```   690   shows "emeasure M (A - B) = emeasure M A"
```
```   691 proof -
```
```   692   have *: "A - B = (A - (A \<inter> B))" by auto
```
```   693   have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1)
```
```   694   then show ?thesis
```
```   695     unfolding * using assms
```
```   696     by (subst emeasure_Diff) auto
```
```   697 qed
```
```   698
```
```   699 lemma null_set_Diff:
```
```   700   assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"
```
```   701 proof (intro CollectI conjI null_setsI)
```
```   702   show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
```
```   703 qed (insert assms, auto)
```
```   704
```
```   705 lemma emeasure_Un_null_set:
```
```   706   assumes "A \<in> sets M" "B \<in> null_sets M"
```
```   707   shows "emeasure M (A \<union> B) = emeasure M A"
```
```   708 proof -
```
```   709   have *: "A \<union> B = A \<union> (B - A)" by auto
```
```   710   have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)
```
```   711   then show ?thesis
```
```   712     unfolding * using assms
```
```   713     by (subst plus_emeasure[symmetric]) auto
```
```   714 qed
```
```   715
```
```   716 section "Formalize almost everywhere"
```
```   717
```
```   718 definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where
```
```   719   "ae_filter M = Abs_filter (\<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
```
```   720
```
```   721 abbreviation
```
```   722   almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   723   "almost_everywhere M P \<equiv> eventually P (ae_filter M)"
```
```   724
```
```   725 syntax
```
```   726   "_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
```
```   727
```
```   728 translations
```
```   729   "AE x in M. P" == "CONST almost_everywhere M (%x. P)"
```
```   730
```
```   731 lemma eventually_ae_filter:
```
```   732   fixes M P
```
```   733   defines [simp]: "F \<equiv> \<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N"
```
```   734   shows "eventually P (ae_filter M) \<longleftrightarrow> F P"
```
```   735   unfolding ae_filter_def F_def[symmetric]
```
```   736 proof (rule eventually_Abs_filter)
```
```   737   show "is_filter F"
```
```   738   proof
```
```   739     fix P Q assume "F P" "F Q"
```
```   740     then obtain N L where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N"
```
```   741       and L: "L \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> L"
```
```   742       by auto
```
```   743     then have "L \<union> N \<in> null_sets M" "{x \<in> space M. \<not> (P x \<and> Q x)} \<subseteq> L \<union> N" by auto
```
```   744     then show "F (\<lambda>x. P x \<and> Q x)" by auto
```
```   745   next
```
```   746     fix P Q assume "F P"
```
```   747     then obtain N where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" by auto
```
```   748     moreover assume "\<forall>x. P x \<longrightarrow> Q x"
```
```   749     ultimately have "N \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> N" by auto
```
```   750     then show "F Q" by auto
```
```   751   qed auto
```
```   752 qed
```
```   753
```
```   754 lemma AE_I':
```
```   755   "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"
```
```   756   unfolding eventually_ae_filter by auto
```
```   757
```
```   758 lemma AE_iff_null:
```
```   759   assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
```
```   760   shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"
```
```   761 proof
```
```   762   assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"
```
```   763     unfolding eventually_ae_filter by auto
```
```   764   have "0 \<le> emeasure M ?P" by auto
```
```   765   moreover have "emeasure M ?P \<le> emeasure M N"
```
```   766     using assms N(1,2) by (auto intro: emeasure_mono)
```
```   767   ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto
```
```   768   then show "?P \<in> null_sets M" using assms by auto
```
```   769 next
```
```   770   assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
```
```   771 qed
```
```   772
```
```   773 lemma AE_iff_null_sets:
```
```   774   "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"
```
```   775   using Int_absorb1[OF sets_into_space, of N M]
```
```   776   by (subst AE_iff_null) (auto simp: Int_def[symmetric])
```
```   777
```
```   778 lemma AE_not_in:
```
```   779   "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
```
```   780   by (metis AE_iff_null_sets null_setsD2)
```
```   781
```
```   782 lemma AE_iff_measurable:
```
```   783   "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> emeasure M N = 0"
```
```   784   using AE_iff_null[of _ P] by auto
```
```   785
```
```   786 lemma AE_E[consumes 1]:
```
```   787   assumes "AE x in M. P x"
```
```   788   obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
```
```   789   using assms unfolding eventually_ae_filter by auto
```
```   790
```
```   791 lemma AE_E2:
```
```   792   assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"
```
```   793   shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")
```
```   794 proof -
```
```   795   have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto
```
```   796   with AE_iff_null[of M P] assms show ?thesis by auto
```
```   797 qed
```
```   798
```
```   799 lemma AE_I:
```
```   800   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
```
```   801   shows "AE x in M. P x"
```
```   802   using assms unfolding eventually_ae_filter by auto
```
```   803
```
```   804 lemma AE_mp[elim!]:
```
```   805   assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"
```
```   806   shows "AE x in M. Q x"
```
```   807 proof -
```
```   808   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
```
```   809     and A: "A \<in> sets M" "emeasure M A = 0"
```
```   810     by (auto elim!: AE_E)
```
```   811
```
```   812   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
```
```   813     and B: "B \<in> sets M" "emeasure M B = 0"
```
```   814     by (auto elim!: AE_E)
```
```   815
```
```   816   show ?thesis
```
```   817   proof (intro AE_I)
```
```   818     have "0 \<le> emeasure M (A \<union> B)" using A B by auto
```
```   819     moreover have "emeasure M (A \<union> B) \<le> 0"
```
```   820       using emeasure_subadditive[of A M B] A B by auto
```
```   821     ultimately show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0" using A B by auto
```
```   822     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
```
```   823       using P imp by auto
```
```   824   qed
```
```   825 qed
```
```   826
```
```   827 (* depricated replace by laws about eventually *)
```
```   828 lemma
```
```   829   shows AE_iffI: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
```
```   830     and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"
```
```   831     and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"
```
```   832     and AE_conjI: "AE x in M. P x \<Longrightarrow> AE x in M. Q x \<Longrightarrow> AE x in M. P x \<and> Q x"
```
```   833     and AE_conj_iff[simp]: "(AE x in M. P x \<and> Q x) \<longleftrightarrow> (AE x in M. P x) \<and> (AE x in M. Q x)"
```
```   834   by auto
```
```   835
```
```   836 lemma AE_impI:
```
```   837   "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"
```
```   838   by (cases P) auto
```
```   839
```
```   840 lemma AE_measure:
```
```   841   assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")
```
```   842   shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"
```
```   843 proof -
```
```   844   from AE_E[OF AE] guess N . note N = this
```
```   845   with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"
```
```   846     by (intro emeasure_mono) auto
```
```   847   also have "\<dots> \<le> emeasure M ?P + emeasure M N"
```
```   848     using sets N by (intro emeasure_subadditive) auto
```
```   849   also have "\<dots> = emeasure M ?P" using N by simp
```
```   850   finally show "emeasure M ?P = emeasure M (space M)"
```
```   851     using emeasure_space[of M "?P"] by auto
```
```   852 qed
```
```   853
```
```   854 lemma AE_space: "AE x in M. x \<in> space M"
```
```   855   by (rule AE_I[where N="{}"]) auto
```
```   856
```
```   857 lemma AE_I2[simp, intro]:
```
```   858   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"
```
```   859   using AE_space by force
```
```   860
```
```   861 lemma AE_Ball_mp:
```
```   862   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
```
```   863   by auto
```
```   864
```
```   865 lemma AE_cong[cong]:
```
```   866   "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in M. Q x)"
```
```   867   by auto
```
```   868
```
```   869 lemma AE_all_countable:
```
```   870   "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
```
```   871 proof
```
```   872   assume "\<forall>i. AE x in M. P i x"
```
```   873   from this[unfolded eventually_ae_filter Bex_def, THEN choice]
```
```   874   obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto
```
```   875   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
```
```   876   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
```
```   877   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
```
```   878   moreover from N have "(\<Union>i. N i) \<in> null_sets M"
```
```   879     by (intro null_sets_UN) auto
```
```   880   ultimately show "AE x in M. \<forall>i. P i x"
```
```   881     unfolding eventually_ae_filter by auto
```
```   882 qed auto
```
```   883
```
```   884 lemma AE_finite_all:
```
```   885   assumes f: "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x in M. P i x)"
```
```   886   using f by induct auto
```
```   887
```
```   888 lemma AE_finite_allI:
```
```   889   assumes "finite S"
```
```   890   shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x"
```
```   891   using AE_finite_all[OF `finite S`] by auto
```
```   892
```
```   893 lemma emeasure_mono_AE:
```
```   894   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
```
```   895     and B: "B \<in> sets M"
```
```   896   shows "emeasure M A \<le> emeasure M B"
```
```   897 proof cases
```
```   898   assume A: "A \<in> sets M"
```
```   899   from imp obtain N where N: "{x\<in>space M. \<not> (x \<in> A \<longrightarrow> x \<in> B)} \<subseteq> N" "N \<in> null_sets M"
```
```   900     by (auto simp: eventually_ae_filter)
```
```   901   have "emeasure M A = emeasure M (A - N)"
```
```   902     using N A by (subst emeasure_Diff_null_set) auto
```
```   903   also have "emeasure M (A - N) \<le> emeasure M (B - N)"
```
```   904     using N A B sets_into_space by (auto intro!: emeasure_mono)
```
```   905   also have "emeasure M (B - N) = emeasure M B"
```
```   906     using N B by (subst emeasure_Diff_null_set) auto
```
```   907   finally show ?thesis .
```
```   908 qed (simp add: emeasure_nonneg emeasure_notin_sets)
```
```   909
```
```   910 lemma emeasure_eq_AE:
```
```   911   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
```
```   912   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
```
```   913   shows "emeasure M A = emeasure M B"
```
```   914   using assms by (safe intro!: antisym emeasure_mono_AE) auto
```
```   915
```
```   916 section {* @{text \<sigma>}-finite Measures *}
```
```   917
```
```   918 locale sigma_finite_measure =
```
```   919   fixes M :: "'a measure"
```
```   920   assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set.
```
```   921     range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
```
```   922
```
```   923 lemma (in sigma_finite_measure) sigma_finite_disjoint:
```
```   924   obtains A :: "nat \<Rightarrow> 'a set"
```
```   925   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"
```
```   926 proof atomize_elim
```
```   927   case goal1
```
```   928   obtain A :: "nat \<Rightarrow> 'a set" where
```
```   929     range: "range A \<subseteq> sets M" and
```
```   930     space: "(\<Union>i. A i) = space M" and
```
```   931     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```   932     using sigma_finite by auto
```
```   933   note range' = range_disjointed_sets[OF range] range
```
```   934   { fix i
```
```   935     have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
```
```   936       using range' disjointed_subset[of A i] by (auto intro!: emeasure_mono)
```
```   937     then have "emeasure M (disjointed A i) \<noteq> \<infinity>"
```
```   938       using measure[of i] by auto }
```
```   939   with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
```
```   940   show ?case by (auto intro!: exI[of _ "disjointed A"])
```
```   941 qed
```
```   942
```
```   943 lemma (in sigma_finite_measure) sigma_finite_incseq:
```
```   944   obtains A :: "nat \<Rightarrow> 'a set"
```
```   945   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
```
```   946 proof atomize_elim
```
```   947   case goal1
```
```   948   obtain F :: "nat \<Rightarrow> 'a set" where
```
```   949     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"
```
```   950     using sigma_finite by auto
```
```   951   then show ?case
```
```   952   proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
```
```   953     from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
```
```   954     then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
```
```   955       using F by fastforce
```
```   956   next
```
```   957     fix n
```
```   958     have "emeasure M (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))" using F
```
```   959       by (auto intro!: emeasure_subadditive_finite)
```
```   960     also have "\<dots> < \<infinity>"
```
```   961       using F by (auto simp: setsum_Pinfty)
```
```   962     finally show "emeasure M (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
```
```   963   qed (force simp: incseq_def)+
```
```   964 qed
```
```   965
```
```   966 section {* Measure space induced by distribution of @{const measurable}-functions *}
```
```   967
```
```   968 definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
```
```   969   "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
```
```   970
```
```   971 lemma
```
```   972   shows sets_distr[simp]: "sets (distr M N f) = sets N"
```
```   973     and space_distr[simp]: "space (distr M N f) = space N"
```
```   974   by (auto simp: distr_def)
```
```   975
```
```   976 lemma
```
```   977   shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
```
```   978     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
```
```   979   by (auto simp: measurable_def)
```
```   980
```
```   981 lemma emeasure_distr:
```
```   982   fixes f :: "'a \<Rightarrow> 'b"
```
```   983   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
```
```   984   shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A")
```
```   985   unfolding distr_def
```
```   986 proof (rule emeasure_measure_of_sigma)
```
```   987   show "positive (sets N) ?\<mu>"
```
```   988     by (auto simp: positive_def)
```
```   989
```
```   990   show "countably_additive (sets N) ?\<mu>"
```
```   991   proof (intro countably_additiveI)
```
```   992     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
```
```   993     then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
```
```   994     then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M"
```
```   995       using f by (auto simp: measurable_def)
```
```   996     moreover have "(\<Union>i. f -`  A i \<inter> space M) \<in> sets M"
```
```   997       using * by blast
```
```   998     moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
```
```   999       using `disjoint_family A` by (auto simp: disjoint_family_on_def)
```
```  1000     ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
```
```  1001       using suminf_emeasure[OF _ **] A f
```
```  1002       by (auto simp: comp_def vimage_UN)
```
```  1003   qed
```
```  1004   show "sigma_algebra (space N) (sets N)" ..
```
```  1005 qed fact
```
```  1006
```
```  1007 lemma AE_distrD:
```
```  1008   assumes f: "f \<in> measurable M M'"
```
```  1009     and AE: "AE x in distr M M' f. P x"
```
```  1010   shows "AE x in M. P (f x)"
```
```  1011 proof -
```
```  1012   from AE[THEN AE_E] guess N .
```
```  1013   with f show ?thesis
```
```  1014     unfolding eventually_ae_filter
```
```  1015     by (intro bexI[of _ "f -` N \<inter> space M"])
```
```  1016        (auto simp: emeasure_distr measurable_def)
```
```  1017 qed
```
```  1018
```
```  1019 lemma null_sets_distr_iff:
```
```  1020   "f \<in> measurable M N \<Longrightarrow> A \<in> null_sets (distr M N f) \<longleftrightarrow> f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N"
```
```  1021   by (auto simp add: null_sets_def emeasure_distr measurable_sets)
```
```  1022
```
```  1023 lemma distr_distr:
```
```  1024   assumes f: "g \<in> measurable N L" and g: "f \<in> measurable M N"
```
```  1025   shows "distr (distr M N f) L g = distr M L (g \<circ> f)" (is "?L = ?R")
```
```  1026   using measurable_comp[OF g f] f g
```
```  1027   by (auto simp add: emeasure_distr measurable_sets measurable_space
```
```  1028            intro!: arg_cong[where f="emeasure M"] measure_eqI)
```
```  1029
```
```  1030 section {* Real measure values *}
```
```  1031
```
```  1032 lemma measure_nonneg: "0 \<le> measure M A"
```
```  1033   using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos)
```
```  1034
```
```  1035 lemma measure_empty[simp]: "measure M {} = 0"
```
```  1036   unfolding measure_def by simp
```
```  1037
```
```  1038 lemma emeasure_eq_ereal_measure:
```
```  1039   "emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M A = ereal (measure M A)"
```
```  1040   using emeasure_nonneg[of M A]
```
```  1041   by (cases "emeasure M A") (auto simp: measure_def)
```
```  1042
```
```  1043 lemma measure_Union:
```
```  1044   assumes finite: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
```
```  1045   and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
```
```  1046   shows "measure M (A \<union> B) = measure M A + measure M B"
```
```  1047   unfolding measure_def
```
```  1048   using plus_emeasure[OF measurable, symmetric] finite
```
```  1049   by (simp add: emeasure_eq_ereal_measure)
```
```  1050
```
```  1051 lemma measure_finite_Union:
```
```  1052   assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
```
```  1053   assumes finite: "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
```
```  1054   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
```
```  1055   unfolding measure_def
```
```  1056   using setsum_emeasure[OF measurable, symmetric] finite
```
```  1057   by (simp add: emeasure_eq_ereal_measure)
```
```  1058
```
```  1059 lemma measure_Diff:
```
```  1060   assumes finite: "emeasure M A \<noteq> \<infinity>"
```
```  1061   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
```
```  1062   shows "measure M (A - B) = measure M A - measure M B"
```
```  1063 proof -
```
```  1064   have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"
```
```  1065     using measurable by (auto intro!: emeasure_mono)
```
```  1066   hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
```
```  1067     using measurable finite by (rule_tac measure_Union) auto
```
```  1068   thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
```
```  1069 qed
```
```  1070
```
```  1071 lemma measure_UNION:
```
```  1072   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
```
```  1073   assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
```
```  1074   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
```
```  1075 proof -
```
```  1076   from summable_sums[OF summable_ereal_pos, of "\<lambda>i. emeasure M (A i)"]
```
```  1077        suminf_emeasure[OF measurable] emeasure_nonneg[of M]
```
```  1078   have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))" by simp
```
```  1079   moreover
```
```  1080   { fix i
```
```  1081     have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
```
```  1082       using measurable by (auto intro!: emeasure_mono)
```
```  1083     then have "emeasure M (A i) = ereal ((measure M (A i)))"
```
```  1084       using finite by (intro emeasure_eq_ereal_measure) auto }
```
```  1085   ultimately show ?thesis using finite
```
```  1086     unfolding sums_ereal[symmetric] by (simp add: emeasure_eq_ereal_measure)
```
```  1087 qed
```
```  1088
```
```  1089 lemma measure_subadditive:
```
```  1090   assumes measurable: "A \<in> sets M" "B \<in> sets M"
```
```  1091   and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
```
```  1092   shows "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
```
```  1093 proof -
```
```  1094   have "emeasure M (A \<union> B) \<noteq> \<infinity>"
```
```  1095     using emeasure_subadditive[OF measurable] fin by auto
```
```  1096   then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
```
```  1097     using emeasure_subadditive[OF measurable] fin
```
```  1098     by (auto simp: emeasure_eq_ereal_measure)
```
```  1099 qed
```
```  1100
```
```  1101 lemma measure_subadditive_finite:
```
```  1102   assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
```
```  1103   shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
```
```  1104 proof -
```
```  1105   { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
```
```  1106       using emeasure_subadditive_finite[OF A] .
```
```  1107     also have "\<dots> < \<infinity>"
```
```  1108       using fin by (simp add: setsum_Pinfty)
```
```  1109     finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> \<infinity>" by simp }
```
```  1110   then show ?thesis
```
```  1111     using emeasure_subadditive_finite[OF A] fin
```
```  1112     unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
```
```  1113 qed
```
```  1114
```
```  1115 lemma measure_subadditive_countably:
```
```  1116   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
```
```  1117   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
```
```  1118 proof -
```
```  1119   from emeasure_nonneg fin have "\<And>i. emeasure M (A i) \<noteq> \<infinity>" by (rule suminf_PInfty)
```
```  1120   moreover
```
```  1121   { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"
```
```  1122       using emeasure_subadditive_countably[OF A] .
```
```  1123     also have "\<dots> < \<infinity>"
```
```  1124       using fin by simp
```
```  1125     finally have "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" by simp }
```
```  1126   ultimately  show ?thesis
```
```  1127     using emeasure_subadditive_countably[OF A] fin
```
```  1128     unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
```
```  1129 qed
```
```  1130
```
```  1131 lemma measure_eq_setsum_singleton:
```
```  1132   assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```  1133   and fin: "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>"
```
```  1134   shows "(measure M S) = (\<Sum>x\<in>S. (measure M {x}))"
```
```  1135   unfolding measure_def
```
```  1136   using emeasure_eq_setsum_singleton[OF S] fin
```
```  1137   by simp (simp add: emeasure_eq_ereal_measure)
```
```  1138
```
```  1139 lemma Lim_measure_incseq:
```
```  1140   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
```
```  1141   shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))"
```
```  1142 proof -
```
```  1143   have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)"
```
```  1144     using fin by (auto simp: emeasure_eq_ereal_measure)
```
```  1145   then show ?thesis
```
```  1146     using Lim_emeasure_incseq[OF A]
```
```  1147     unfolding measure_def
```
```  1148     by (intro lim_real_of_ereal) simp
```
```  1149 qed
```
```  1150
```
```  1151 lemma Lim_measure_decseq:
```
```  1152   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```  1153   shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
```
```  1154 proof -
```
```  1155   have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
```
```  1156     using A by (auto intro!: emeasure_mono)
```
```  1157   also have "\<dots> < \<infinity>"
```
```  1158     using fin[of 0] by auto
```
```  1159   finally have "ereal ((measure M (\<Inter>i. A i))) = emeasure M (\<Inter>i. A i)"
```
```  1160     by (auto simp: emeasure_eq_ereal_measure)
```
```  1161   then show ?thesis
```
```  1162     unfolding measure_def
```
```  1163     using Lim_emeasure_decseq[OF A fin]
```
```  1164     by (intro lim_real_of_ereal) simp
```
```  1165 qed
```
```  1166
```
```  1167 section {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *}
```
```  1168
```
```  1169 locale finite_measure = sigma_finite_measure M for M +
```
```  1170   assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>"
```
```  1171
```
```  1172 lemma finite_measureI[Pure.intro!]:
```
```  1173   assumes *: "emeasure M (space M) \<noteq> \<infinity>"
```
```  1174   shows "finite_measure M"
```
```  1175 proof
```
```  1176   show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
```
```  1177     using * by (auto intro!: exI[of _ "\<lambda>_. space M"])
```
```  1178 qed fact
```
```  1179
```
```  1180 lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> \<infinity>"
```
```  1181   using finite_emeasure_space emeasure_space[of M A] by auto
```
```  1182
```
```  1183 lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ereal (measure M A)"
```
```  1184   unfolding measure_def by (simp add: emeasure_eq_ereal_measure)
```
```  1185
```
```  1186 lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ereal r"
```
```  1187   using emeasure_finite[of A] emeasure_nonneg[of M A] by (cases "emeasure M A") auto
```
```  1188
```
```  1189 lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"
```
```  1190   using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)
```
```  1191
```
```  1192 lemma (in finite_measure) finite_measure_Diff:
```
```  1193   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
```
```  1194   shows "measure M (A - B) = measure M A - measure M B"
```
```  1195   using measure_Diff[OF _ assms] by simp
```
```  1196
```
```  1197 lemma (in finite_measure) finite_measure_Union:
```
```  1198   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
```
```  1199   shows "measure M (A \<union> B) = measure M A + measure M B"
```
```  1200   using measure_Union[OF _ _ assms] by simp
```
```  1201
```
```  1202 lemma (in finite_measure) finite_measure_finite_Union:
```
```  1203   assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
```
```  1204   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
```
```  1205   using measure_finite_Union[OF assms] by simp
```
```  1206
```
```  1207 lemma (in finite_measure) finite_measure_UNION:
```
```  1208   assumes A: "range A \<subseteq> sets M" "disjoint_family A"
```
```  1209   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
```
```  1210   using measure_UNION[OF A] by simp
```
```  1211
```
```  1212 lemma (in finite_measure) finite_measure_mono:
```
```  1213   assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
```
```  1214   using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
```
```  1215
```
```  1216 lemma (in finite_measure) finite_measure_subadditive:
```
```  1217   assumes m: "A \<in> sets M" "B \<in> sets M"
```
```  1218   shows "measure M (A \<union> B) \<le> measure M A + measure M B"
```
```  1219   using measure_subadditive[OF m] by simp
```
```  1220
```
```  1221 lemma (in finite_measure) finite_measure_subadditive_finite:
```
```  1222   assumes "finite I" "A`I \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
```
```  1223   using measure_subadditive_finite[OF assms] by simp
```
```  1224
```
```  1225 lemma (in finite_measure) finite_measure_subadditive_countably:
```
```  1226   assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))"
```
```  1227   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
```
```  1228 proof -
```
```  1229   from `summable (\<lambda>i. measure M (A i))`
```
```  1230   have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))"
```
```  1231     by (simp add: sums_ereal) (rule summable_sums)
```
```  1232   from sums_unique[OF this, symmetric]
```
```  1233        measure_subadditive_countably[OF A]
```
```  1234   show ?thesis by (simp add: emeasure_eq_measure)
```
```  1235 qed
```
```  1236
```
```  1237 lemma (in finite_measure) finite_measure_eq_setsum_singleton:
```
```  1238   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```  1239   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
```
```  1240   using measure_eq_setsum_singleton[OF assms] by simp
```
```  1241
```
```  1242 lemma (in finite_measure) finite_Lim_measure_incseq:
```
```  1243   assumes A: "range A \<subseteq> sets M" "incseq A"
```
```  1244   shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)"
```
```  1245   using Lim_measure_incseq[OF A] by simp
```
```  1246
```
```  1247 lemma (in finite_measure) finite_Lim_measure_decseq:
```
```  1248   assumes A: "range A \<subseteq> sets M" "decseq A"
```
```  1249   shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
```
```  1250   using Lim_measure_decseq[OF A] by simp
```
```  1251
```
```  1252 lemma (in finite_measure) finite_measure_compl:
```
```  1253   assumes S: "S \<in> sets M"
```
```  1254   shows "measure M (space M - S) = measure M (space M) - measure M S"
```
```  1255   using measure_Diff[OF _ top S sets_into_space] S by simp
```
```  1256
```
```  1257 lemma (in finite_measure) finite_measure_mono_AE:
```
```  1258   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"
```
```  1259   shows "measure M A \<le> measure M B"
```
```  1260   using assms emeasure_mono_AE[OF imp B]
```
```  1261   by (simp add: emeasure_eq_measure)
```
```  1262
```
```  1263 lemma (in finite_measure) finite_measure_eq_AE:
```
```  1264   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
```
```  1265   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
```
```  1266   shows "measure M A = measure M B"
```
```  1267   using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)
```
```  1268
```
```  1269 section {* Counting space *}
```
```  1270
```
```  1271 definition count_space :: "'a set \<Rightarrow> 'a measure" where
```
```  1272   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
```
```  1273
```
```  1274 lemma
```
```  1275   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
```
```  1276     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
```
```  1277   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
```
```  1278   by (auto simp: count_space_def)
```
```  1279
```
```  1280 lemma measurable_count_space_eq1[simp]:
```
```  1281   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
```
```  1282  unfolding measurable_def by simp
```
```  1283
```
```  1284 lemma measurable_count_space_eq2[simp]:
```
```  1285   assumes "finite A"
```
```  1286   shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
```
```  1287 proof -
```
```  1288   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
```
```  1289     with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
```
```  1290       by (auto dest: finite_subset)
```
```  1291     moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
```
```  1292     ultimately have "f -` X \<inter> space M \<in> sets M"
```
```  1293       using `X \<subseteq> A` by (auto intro!: finite_UN simp del: UN_simps) }
```
```  1294   then show ?thesis
```
```  1295     unfolding measurable_def by auto
```
```  1296 qed
```
```  1297
```
```  1298 lemma emeasure_count_space:
```
```  1299   assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then ereal (card X) else \<infinity>)"
```
```  1300     (is "_ = ?M X")
```
```  1301   unfolding count_space_def
```
```  1302 proof (rule emeasure_measure_of_sigma)
```
```  1303   show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
```
```  1304
```
```  1305   show "positive (Pow A) ?M"
```
```  1306     by (auto simp: positive_def)
```
```  1307
```
```  1308   show "countably_additive (Pow A) ?M"
```
```  1309   proof (unfold countably_additive_def, safe)
```
```  1310       fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F"
```
```  1311       show "(\<Sum>i. ?M (F i)) = ?M (\<Union>i. F i)"
```
```  1312       proof cases
```
```  1313         assume "\<forall>i. finite (F i)"
```
```  1314         then have finite_F: "\<And>i. finite (F i)" by auto
```
```  1315         have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
```
```  1316         from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
```
```  1317
```
```  1318         have inj_f: "inj_on f {i. F i \<noteq> {}}"
```
```  1319         proof (rule inj_onI, simp)
```
```  1320           fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
```
```  1321           then have "f i \<in> F i" "f j \<in> F j" using f by force+
```
```  1322           with disj * show "i = j" by (auto simp: disjoint_family_on_def)
```
```  1323         qed
```
```  1324         have fin_eq: "finite (\<Union>i. F i) \<longleftrightarrow> finite {i. F i \<noteq> {}}"
```
```  1325         proof
```
```  1326           assume "finite (\<Union>i. F i)"
```
```  1327           show "finite {i. F i \<noteq> {}}"
```
```  1328           proof (rule finite_imageD)
```
```  1329             from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
```
```  1330             then show "finite (f`{i. F i \<noteq> {}})"
```
```  1331               by (rule finite_subset) fact
```
```  1332           qed fact
```
```  1333         next
```
```  1334           assume "finite {i. F i \<noteq> {}}"
```
```  1335           with finite_F have "finite (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
```
```  1336             by auto
```
```  1337           also have "(\<Union>i\<in>{i. F i \<noteq> {}}. F i) = (\<Union>i. F i)"
```
```  1338             by auto
```
```  1339           finally show "finite (\<Union>i. F i)" .
```
```  1340         qed
```
```  1341
```
```  1342         show ?thesis
```
```  1343         proof cases
```
```  1344           assume *: "finite (\<Union>i. F i)"
```
```  1345           with finite_F have "finite {i. ?M (F i) \<noteq> 0} "
```
```  1346             by (simp add: fin_eq)
```
```  1347           then have "(\<Sum>i. ?M (F i)) = (\<Sum>i | ?M (F i) \<noteq> 0. ?M (F i))"
```
```  1348             by (rule suminf_finite) auto
```
```  1349           also have "\<dots> = ereal (\<Sum>i | F i \<noteq> {}. card (F i))"
```
```  1350             using finite_F by simp
```
```  1351           also have "\<dots> = ereal (card (\<Union>i \<in> {i. F i \<noteq> {}}. F i))"
```
```  1352             using * finite_F disj
```
```  1353             by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def fin_eq)
```
```  1354           also have "\<dots> = ?M (\<Union>i. F i)"
```
```  1355             using * by (auto intro!: arg_cong[where f=card])
```
```  1356           finally show ?thesis .
```
```  1357         next
```
```  1358           assume inf: "infinite (\<Union>i. F i)"
```
```  1359           { fix i
```
```  1360             have "\<exists>N. i \<le> (\<Sum>i<N. card (F i))"
```
```  1361             proof (induct i)
```
```  1362               case (Suc j)
```
```  1363               from Suc obtain N where N: "j \<le> (\<Sum>i<N. card (F i))" by auto
```
```  1364               have "infinite ({i. F i \<noteq> {}} - {..< N})"
```
```  1365                 using inf by (auto simp: fin_eq)
```
```  1366               then have "{i. F i \<noteq> {}} - {..< N} \<noteq> {}"
```
```  1367                 by (metis finite.emptyI)
```
```  1368               then obtain i where i: "F i \<noteq> {}" "N \<le> i"
```
```  1369                 by (auto simp: not_less[symmetric])
```
```  1370
```
```  1371               note N
```
```  1372               also have "(\<Sum>i<N. card (F i)) \<le> (\<Sum>i<i. card (F i))"
```
```  1373                 by (rule setsum_mono2) (auto simp: i)
```
```  1374               also have "\<dots> < (\<Sum>i<i. card (F i)) + card (F i)"
```
```  1375                 using finite_F `F i \<noteq> {}` by (simp add: card_gt_0_iff)
```
```  1376               finally have "j < (\<Sum>i<Suc i. card (F i))"
```
```  1377                 by simp
```
```  1378               then show ?case unfolding Suc_le_eq by blast
```
```  1379             qed simp }
```
```  1380           with finite_F inf show ?thesis
```
```  1381             by (auto simp del: real_of_nat_setsum intro!: SUP_PInfty
```
```  1382                      simp add: suminf_ereal_eq_SUPR real_of_nat_setsum[symmetric])
```
```  1383         qed
```
```  1384       next
```
```  1385         assume "\<not> (\<forall>i. finite (F i))"
```
```  1386         then obtain j where j: "infinite (F j)" by auto
```
```  1387         then have "infinite (\<Union>i. F i)"
```
```  1388           using finite_subset[of "F j" "\<Union>i. F i"] by auto
```
```  1389         moreover have "\<And>i. 0 \<le> ?M (F i)" by auto
```
```  1390         ultimately show ?thesis
```
```  1391           using suminf_PInfty[of "\<lambda>i. ?M (F i)" j] j by auto
```
```  1392       qed
```
```  1393   qed
```
```  1394   show "X \<in> Pow A" using `X \<subseteq> A` by simp
```
```  1395 qed
```
```  1396
```
```  1397 lemma emeasure_count_space_finite[simp]:
```
```  1398   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = ereal (card X)"
```
```  1399   using emeasure_count_space[of X A] by simp
```
```  1400
```
```  1401 lemma emeasure_count_space_infinite[simp]:
```
```  1402   "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"
```
```  1403   using emeasure_count_space[of X A] by simp
```
```  1404
```
```  1405 lemma emeasure_count_space_eq_0:
```
```  1406   "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"
```
```  1407 proof cases
```
```  1408   assume X: "X \<subseteq> A"
```
```  1409   then show ?thesis
```
```  1410   proof (intro iffI impI)
```
```  1411     assume "emeasure (count_space A) X = 0"
```
```  1412     with X show "X = {}"
```
```  1413       by (subst (asm) emeasure_count_space) (auto split: split_if_asm)
```
```  1414   qed simp
```
```  1415 qed (simp add: emeasure_notin_sets)
```
```  1416
```
```  1417 lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
```
```  1418   unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
```
```  1419
```
```  1420 lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
```
```  1421   unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)
```
```  1422
```
```  1423 lemma sigma_finite_measure_count_space:
```
```  1424   fixes A :: "'a::countable set"
```
```  1425   shows "sigma_finite_measure (count_space A)"
```
```  1426 proof
```
```  1427   show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (count_space A) \<and> (\<Union>i. F i) = space (count_space A) \<and>
```
```  1428      (\<forall>i. emeasure (count_space A) (F i) \<noteq> \<infinity>)"
```
```  1429      using surj_from_nat by (intro exI[of _ "\<lambda>i. {from_nat i} \<inter> A"]) (auto simp del: surj_from_nat)
```
```  1430 qed
```
```  1431
```
```  1432 lemma finite_measure_count_space:
```
```  1433   assumes [simp]: "finite A"
```
```  1434   shows "finite_measure (count_space A)"
```
```  1435   by rule simp
```
```  1436
```
```  1437 lemma sigma_finite_measure_count_space_finite:
```
```  1438   assumes A: "finite A" shows "sigma_finite_measure (count_space A)"
```
```  1439 proof -
```
```  1440   interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)
```
```  1441   show "sigma_finite_measure (count_space A)" ..
```
```  1442 qed
```
```  1443
```
```  1444 end
```
```  1445
```