src/HOL/Probability/Measure_Space.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60172 423273355b55 child 60580 7e741e22d7fc permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
```     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 section {* Measure spaces and their properties *}
```
```     8
```
```     9 theory Measure_Space
```
```    10 imports
```
```    11   Measurable "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
```
```    12 begin
```
```    13
```
```    14 subsection "Relate extended reals and the indicator function"
```
```    15
```
```    16 lemma suminf_cmult_indicator:
```
```    17   fixes f :: "nat \<Rightarrow> ereal"
```
```    18   assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
```
```    19   shows "(\<Sum>n. f n * indicator (A n) x) = f i"
```
```    20 proof -
```
```    21   have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)"
```
```    22     using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
```
```    23   then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)"
```
```    24     by (auto simp: setsum.If_cases)
```
```    25   moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
```
```    26   proof (rule SUP_eqI)
```
```    27     fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
```
```    28     from this[of "Suc i"] show "f i \<le> y" by auto
```
```    29   qed (insert assms, simp)
```
```    30   ultimately show ?thesis using assms
```
```    31     by (subst suminf_ereal_eq_SUP) (auto simp: indicator_def)
```
```    32 qed
```
```    33
```
```    34 lemma suminf_indicator:
```
```    35   assumes "disjoint_family A"
```
```    36   shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x"
```
```    37 proof cases
```
```    38   assume *: "x \<in> (\<Union>i. A i)"
```
```    39   then obtain i where "x \<in> A i" by auto
```
```    40   from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
```
```    41   show ?thesis using * by simp
```
```    42 qed simp
```
```    43
```
```    44 text {*
```
```    45   The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to
```
```    46   represent sigma algebras (with an arbitrary emeasure).
```
```    47 *}
```
```    48
```
```    49 subsection "Extend binary sets"
```
```    50
```
```    51 lemma LIMSEQ_binaryset:
```
```    52   assumes f: "f {} = 0"
```
```    53   shows  "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
```
```    54 proof -
```
```    55   have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
```
```    56     proof
```
```    57       fix n
```
```    58       show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
```
```    59         by (induct n)  (auto simp add: binaryset_def f)
```
```    60     qed
```
```    61   moreover
```
```    62   have "... ----> f A + f B" by (rule tendsto_const)
```
```    63   ultimately
```
```    64   have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
```
```    65     by metis
```
```    66   hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
```
```    67     by simp
```
```    68   thus ?thesis by (rule LIMSEQ_offset [where k=2])
```
```    69 qed
```
```    70
```
```    71 lemma binaryset_sums:
```
```    72   assumes f: "f {} = 0"
```
```    73   shows  "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
```
```    74     by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
```
```    75
```
```    76 lemma suminf_binaryset_eq:
```
```    77   fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
```
```    78   shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
```
```    79   by (metis binaryset_sums sums_unique)
```
```    80
```
```    81 subsection {* Properties of a premeasure @{term \<mu>} *}
```
```    82
```
```    83 text {*
```
```    84   The definitions for @{const positive} and @{const countably_additive} should be here, by they are
```
```    85   necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}.
```
```    86 *}
```
```    87
```
```    88 definition additive where
```
```    89   "additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"
```
```    90
```
```    91 definition increasing where
```
```    92   "increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)"
```
```    93
```
```    94 lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def)
```
```    95 lemma positiveD2: "positive M f \<Longrightarrow> A \<in> M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def)
```
```    96
```
```    97 lemma positiveD_empty:
```
```    98   "positive M f \<Longrightarrow> f {} = 0"
```
```    99   by (auto simp add: positive_def)
```
```   100
```
```   101 lemma additiveD:
```
```   102   "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y"
```
```   103   by (auto simp add: additive_def)
```
```   104
```
```   105 lemma increasingD:
```
```   106   "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"
```
```   107   by (auto simp add: increasing_def)
```
```   108
```
```   109 lemma countably_additiveI[case_names countably]:
```
```   110   "(\<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))
```
```   111   \<Longrightarrow> countably_additive M f"
```
```   112   by (simp add: countably_additive_def)
```
```   113
```
```   114 lemma (in ring_of_sets) disjointed_additive:
```
```   115   assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A"
```
```   116   shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
```
```   117 proof (induct n)
```
```   118   case (Suc n)
```
```   119   then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
```
```   120     by simp
```
```   121   also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
```
```   122     using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq)
```
```   123   also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
```
```   124     using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq)
```
```   125   finally show ?case .
```
```   126 qed simp
```
```   127
```
```   128 lemma (in ring_of_sets) additive_sum:
```
```   129   fixes A:: "'i \<Rightarrow> 'a set"
```
```   130   assumes f: "positive M f" and ad: "additive M f" and "finite S"
```
```   131       and A: "A`S \<subseteq> M"
```
```   132       and disj: "disjoint_family_on A S"
```
```   133   shows  "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
```
```   134   using `finite S` disj A
```
```   135 proof induct
```
```   136   case empty show ?case using f by (simp add: positive_def)
```
```   137 next
```
```   138   case (insert s S)
```
```   139   then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
```
```   140     by (auto simp add: disjoint_family_on_def neq_iff)
```
```   141   moreover
```
```   142   have "A s \<in> M" using insert by blast
```
```   143   moreover have "(\<Union>i\<in>S. A i) \<in> M"
```
```   144     using insert `finite S` by auto
```
```   145   ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
```
```   146     using ad UNION_in_sets A by (auto simp add: additive_def)
```
```   147   with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
```
```   148     by (auto simp add: additive_def subset_insertI)
```
```   149 qed
```
```   150
```
```   151 lemma (in ring_of_sets) additive_increasing:
```
```   152   assumes posf: "positive M f" and addf: "additive M f"
```
```   153   shows "increasing M f"
```
```   154 proof (auto simp add: increasing_def)
```
```   155   fix x y
```
```   156   assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y"
```
```   157   then have "y - x \<in> M" by auto
```
```   158   then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
```
```   159   then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
```
```   160   also have "... = f (x \<union> (y-x))" using addf
```
```   161     by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
```
```   162   also have "... = f y"
```
```   163     by (metis Un_Diff_cancel Un_absorb1 xy(3))
```
```   164   finally show "f x \<le> f y" by simp
```
```   165 qed
```
```   166
```
```   167 lemma (in ring_of_sets) subadditive:
```
```   168   assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" and S: "finite S"
```
```   169   shows "f (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. f (A i))"
```
```   170 using S
```
```   171 proof (induct S)
```
```   172   case empty thus ?case using f by (auto simp: positive_def)
```
```   173 next
```
```   174   case (insert x F)
```
```   175   hence in_M: "A x \<in> M" "(\<Union> i\<in>F. A i) \<in> M" "(\<Union> i\<in>F. A i) - A x \<in> M" using A by force+
```
```   176   have subs: "(\<Union> i\<in>F. A i) - A x \<subseteq> (\<Union> i\<in>F. A i)" by auto
```
```   177   have "(\<Union> i\<in>(insert x F). A i) = A x \<union> ((\<Union> i\<in>F. A i) - A x)" by auto
```
```   178   hence "f (\<Union> i\<in>(insert x F). A i) = f (A x \<union> ((\<Union> i\<in>F. A i) - A x))"
```
```   179     by simp
```
```   180   also have "\<dots> = f (A x) + f ((\<Union> i\<in>F. A i) - A x)"
```
```   181     using f(2) by (rule additiveD) (insert in_M, auto)
```
```   182   also have "\<dots> \<le> f (A x) + f (\<Union> i\<in>F. A i)"
```
```   183     using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono)
```
```   184   also have "\<dots> \<le> f (A x) + (\<Sum>i\<in>F. f (A i))" using insert by (auto intro: add_left_mono)
```
```   185   finally show "f (\<Union> i\<in>(insert x F). A i) \<le> (\<Sum>i\<in>(insert x F). f (A i))" using insert by simp
```
```   186 qed
```
```   187
```
```   188 lemma (in ring_of_sets) countably_additive_additive:
```
```   189   assumes posf: "positive M f" and ca: "countably_additive M f"
```
```   190   shows "additive M f"
```
```   191 proof (auto simp add: additive_def)
```
```   192   fix x y
```
```   193   assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
```
```   194   hence "disjoint_family (binaryset x y)"
```
```   195     by (auto simp add: disjoint_family_on_def binaryset_def)
```
```   196   hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
```
```   197          (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
```
```   198          f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
```
```   199     using ca
```
```   200     by (simp add: countably_additive_def)
```
```   201   hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
```
```   202          f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
```
```   203     by (simp add: range_binaryset_eq UN_binaryset_eq)
```
```   204   thus "f (x \<union> y) = f x + f y" using posf x y
```
```   205     by (auto simp add: Un suminf_binaryset_eq positive_def)
```
```   206 qed
```
```   207
```
```   208 lemma (in algebra) increasing_additive_bound:
```
```   209   fixes A:: "nat \<Rightarrow> 'a set" and  f :: "'a set \<Rightarrow> ereal"
```
```   210   assumes f: "positive M f" and ad: "additive M f"
```
```   211       and inc: "increasing M f"
```
```   212       and A: "range A \<subseteq> M"
```
```   213       and disj: "disjoint_family A"
```
```   214   shows  "(\<Sum>i. f (A i)) \<le> f \<Omega>"
```
```   215 proof (safe intro!: suminf_bound)
```
```   216   fix N
```
```   217   note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
```
```   218   have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
```
```   219     using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)
```
```   220   also have "... \<le> f \<Omega>" using space_closed A
```
```   221     by (intro increasingD[OF inc] finite_UN) auto
```
```   222   finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp
```
```   223 qed (insert f A, auto simp: positive_def)
```
```   224
```
```   225 lemma (in ring_of_sets) countably_additiveI_finite:
```
```   226   assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>"
```
```   227   shows "countably_additive M \<mu>"
```
```   228 proof (rule countably_additiveI)
```
```   229   fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F"
```
```   230
```
```   231   have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
```
```   232   from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
```
```   233
```
```   234   have inj_f: "inj_on f {i. F i \<noteq> {}}"
```
```   235   proof (rule inj_onI, simp)
```
```   236     fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
```
```   237     then have "f i \<in> F i" "f j \<in> F j" using f by force+
```
```   238     with disj * show "i = j" by (auto simp: disjoint_family_on_def)
```
```   239   qed
```
```   240   have "finite (\<Union>i. F i)"
```
```   241     by (metis F(2) assms(1) infinite_super sets_into_space)
```
```   242
```
```   243   have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
```
```   244     by (auto simp: positiveD_empty[OF `positive M \<mu>`])
```
```   245   moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"
```
```   246   proof (rule finite_imageD)
```
```   247     from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
```
```   248     then show "finite (f`{i. F i \<noteq> {}})"
```
```   249       by (rule finite_subset) fact
```
```   250   qed fact
```
```   251   ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}"
```
```   252     by (rule finite_subset)
```
```   253
```
```   254   have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}"
```
```   255     using disj by (auto simp: disjoint_family_on_def)
```
```   256
```
```   257   from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"
```
```   258     by (rule suminf_finite) auto
```
```   259   also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
```
```   260     using fin_not_empty F_subset by (rule setsum.mono_neutral_left) auto
```
```   261   also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
```
```   262     using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto
```
```   263   also have "\<dots> = \<mu> (\<Union>i. F i)"
```
```   264     by (rule arg_cong[where f=\<mu>]) auto
```
```   265   finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .
```
```   266 qed
```
```   267
```
```   268 lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:
```
```   269   assumes f: "positive M f" "additive M f"
```
```   270   shows "countably_additive M f \<longleftrightarrow>
```
```   271     (\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))"
```
```   272   unfolding countably_additive_def
```
```   273 proof safe
```
```   274   assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"
```
```   275   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
```
```   276   then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets)
```
```   277   with count_sum[THEN spec, of "disjointed A"] A(3)
```
```   278   have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)"
```
```   279     by (auto simp: UN_disjointed_eq disjoint_family_disjointed)
```
```   280   moreover have "(\<lambda>n. (\<Sum>i<n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))"
```
```   281     using f(1)[unfolded positive_def] dA
```
```   282     by (auto intro!: summable_LIMSEQ summable_ereal_pos)
```
```   283   from LIMSEQ_Suc[OF this]
```
```   284   have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))"
```
```   285     unfolding lessThan_Suc_atMost .
```
```   286   moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
```
```   287     using disjointed_additive[OF f A(1,2)] .
```
```   288   ultimately show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" by simp
```
```   289 next
```
```   290   assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
```
```   291   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M"
```
```   292   have *: "(\<Union>n. (\<Union>i<n. A i)) = (\<Union>i. A i)" by auto
```
```   293   have "(\<lambda>n. f (\<Union>i<n. A i)) ----> f (\<Union>i. A i)"
```
```   294   proof (unfold *[symmetric], intro cont[rule_format])
```
```   295     show "range (\<lambda>i. \<Union> i<i. A i) \<subseteq> M" "(\<Union>i. \<Union> i<i. A i) \<in> M"
```
```   296       using A * by auto
```
```   297   qed (force intro!: incseq_SucI)
```
```   298   moreover have "\<And>n. f (\<Union>i<n. A i) = (\<Sum>i<n. f (A i))"
```
```   299     using A
```
```   300     by (intro additive_sum[OF f, of _ A, symmetric])
```
```   301        (auto intro: disjoint_family_on_mono[where B=UNIV])
```
```   302   ultimately
```
```   303   have "(\<lambda>i. f (A i)) sums f (\<Union>i. A i)"
```
```   304     unfolding sums_def by simp
```
```   305   from sums_unique[OF this]
```
```   306   show "(\<Sum>i. f (A i)) = f (\<Union>i. A i)" by simp
```
```   307 qed
```
```   308
```
```   309 lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:
```
```   310   assumes f: "positive M f" "additive M f"
```
```   311   shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))
```
```   312      \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)"
```
```   313 proof safe
```
```   314   assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))"
```
```   315   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>"
```
```   316   with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0"
```
```   317     using `positive M f`[unfolded positive_def] by auto
```
```   318 next
```
```   319   assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
```
```   320   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>"
```
```   321
```
```   322   have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b"
```
```   323     using additive_increasing[OF f] unfolding increasing_def by simp
```
```   324
```
```   325   have decseq_fA: "decseq (\<lambda>i. f (A i))"
```
```   326     using A by (auto simp: decseq_def intro!: f_mono)
```
```   327   have decseq: "decseq (\<lambda>i. A i - (\<Inter>i. A i))"
```
```   328     using A by (auto simp: decseq_def)
```
```   329   then have decseq_f: "decseq (\<lambda>i. f (A i - (\<Inter>i. A i)))"
```
```   330     using A unfolding decseq_def by (auto intro!: f_mono Diff)
```
```   331   have "f (\<Inter>x. A x) \<le> f (A 0)"
```
```   332     using A by (auto intro!: f_mono)
```
```   333   then have f_Int_fin: "f (\<Inter>x. A x) \<noteq> \<infinity>"
```
```   334     using A by auto
```
```   335   { fix i
```
```   336     have "f (A i - (\<Inter>i. A i)) \<le> f (A i)" using A by (auto intro!: f_mono)
```
```   337     then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>"
```
```   338       using A by auto }
```
```   339   note f_fin = this
```
```   340   have "(\<lambda>i. f (A i - (\<Inter>i. A i))) ----> 0"
```
```   341   proof (intro cont[rule_format, OF _ decseq _ f_fin])
```
```   342     show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}"
```
```   343       using A by auto
```
```   344   qed
```
```   345   from INF_Lim_ereal[OF decseq_f this]
```
```   346   have "(INF n. f (A n - (\<Inter>i. A i))) = 0" .
```
```   347   moreover have "(INF n. f (\<Inter>i. A i)) = f (\<Inter>i. A i)"
```
```   348     by auto
```
```   349   ultimately have "(INF n. f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i)) = 0 + f (\<Inter>i. A i)"
```
```   350     using A(4) f_fin f_Int_fin
```
```   351     by (subst INF_ereal_add) (auto simp: decseq_f)
```
```   352   moreover {
```
```   353     fix n
```
```   354     have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))"
```
```   355       using A by (subst f(2)[THEN additiveD]) auto
```
```   356     also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n"
```
```   357       by auto
```
```   358     finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . }
```
```   359   ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)"
```
```   360     by simp
```
```   361   with LIMSEQ_INF[OF decseq_fA]
```
```   362   show "(\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)" by simp
```
```   363 qed
```
```   364
```
```   365 lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:
```
```   366   assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>"
```
```   367   assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0"
```
```   368   assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
```
```   369   shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
```
```   370 proof -
```
```   371   have "\<forall>A\<in>M. \<exists>x. f A = ereal x"
```
```   372   proof
```
```   373     fix A assume "A \<in> M" with f show "\<exists>x. f A = ereal x"
```
```   374       unfolding positive_def by (cases "f A") auto
```
```   375   qed
```
```   376   from bchoice[OF this] guess f' .. note f' = this[rule_format]
```
```   377   from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) ----> 0"
```
```   378     by (intro cont[rule_format]) (auto simp: decseq_def incseq_def)
```
```   379   moreover
```
```   380   { fix i
```
```   381     have "f ((\<Union>i. A i) - A i) + f (A i) = f ((\<Union>i. A i) - A i \<union> A i)"
```
```   382       using A by (intro f(2)[THEN additiveD, symmetric]) auto
```
```   383     also have "(\<Union>i. A i) - A i \<union> A i = (\<Union>i. A i)"
```
```   384       by auto
```
```   385     finally have "f' (\<Union>i. A i) - f' (A i) = f' ((\<Union>i. A i) - A i)"
```
```   386       using A by (subst (asm) (1 2 3) f') auto
```
```   387     then have "f ((\<Union>i. A i) - A i) = ereal (f' (\<Union>i. A i) - f' (A i))"
```
```   388       using A f' by auto }
```
```   389   ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) ----> 0"
```
```   390     by (simp add: zero_ereal_def)
```
```   391   then have "(\<lambda>i. f' (A i)) ----> f' (\<Union>i. A i)"
```
```   392     by (rule Lim_transform2[OF tendsto_const])
```
```   393   then show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)"
```
```   394     using A by (subst (1 2) f') auto
```
```   395 qed
```
```   396
```
```   397 lemma (in ring_of_sets) empty_continuous_imp_countably_additive:
```
```   398   assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>"
```
```   399   assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
```
```   400   shows "countably_additive M f"
```
```   401   using countably_additive_iff_continuous_from_below[OF f]
```
```   402   using empty_continuous_imp_continuous_from_below[OF f fin] cont
```
```   403   by blast
```
```   404
```
```   405 subsection {* Properties of @{const emeasure} *}
```
```   406
```
```   407 lemma emeasure_positive: "positive (sets M) (emeasure M)"
```
```   408   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
```
```   409
```
```   410 lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
```
```   411   using emeasure_positive[of M] by (simp add: positive_def)
```
```   412
```
```   413 lemma emeasure_nonneg[intro!]: "0 \<le> emeasure M A"
```
```   414   using emeasure_notin_sets[of A M] emeasure_positive[of M]
```
```   415   by (cases "A \<in> sets M") (auto simp: positive_def)
```
```   416
```
```   417 lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>"
```
```   418   using emeasure_nonneg[of M A] by auto
```
```   419
```
```   420 lemma emeasure_le_0_iff: "emeasure M A \<le> 0 \<longleftrightarrow> emeasure M A = 0"
```
```   421   using emeasure_nonneg[of M A] by auto
```
```   422
```
```   423 lemma emeasure_less_0_iff: "emeasure M A < 0 \<longleftrightarrow> False"
```
```   424   using emeasure_nonneg[of M A] by auto
```
```   425
```
```   426 lemma emeasure_single_in_space: "emeasure M {x} \<noteq> 0 \<Longrightarrow> x \<in> space M"
```
```   427   using emeasure_notin_sets[of "{x}" M] by (auto dest: sets.sets_into_space)
```
```   428
```
```   429 lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
```
```   430   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
```
```   431
```
```   432 lemma suminf_emeasure:
```
```   433   "range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
```
```   434   using sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M]
```
```   435   by (simp add: countably_additive_def)
```
```   436
```
```   437 lemma sums_emeasure:
```
```   438   "disjoint_family F \<Longrightarrow> (\<And>i. F i \<in> sets M) \<Longrightarrow> (\<lambda>i. emeasure M (F i)) sums emeasure M (\<Union>i. F i)"
```
```   439   unfolding sums_iff by (intro conjI summable_ereal_pos emeasure_nonneg suminf_emeasure) auto
```
```   440
```
```   441 lemma emeasure_additive: "additive (sets M) (emeasure M)"
```
```   442   by (metis sets.countably_additive_additive emeasure_positive emeasure_countably_additive)
```
```   443
```
```   444 lemma plus_emeasure:
```
```   445   "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)"
```
```   446   using additiveD[OF emeasure_additive] ..
```
```   447
```
```   448 lemma setsum_emeasure:
```
```   449   "F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
```
```   450     (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
```
```   451   by (metis sets.additive_sum emeasure_positive emeasure_additive)
```
```   452
```
```   453 lemma emeasure_mono:
```
```   454   "a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b"
```
```   455   by (metis sets.additive_increasing emeasure_additive emeasure_nonneg emeasure_notin_sets
```
```   456             emeasure_positive increasingD)
```
```   457
```
```   458 lemma emeasure_space:
```
```   459   "emeasure M A \<le> emeasure M (space M)"
```
```   460   by (metis emeasure_mono emeasure_nonneg emeasure_notin_sets sets.sets_into_space sets.top)
```
```   461
```
```   462 lemma emeasure_compl:
```
```   463   assumes s: "s \<in> sets M" and fin: "emeasure M s \<noteq> \<infinity>"
```
```   464   shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
```
```   465 proof -
```
```   466   from s have "0 \<le> emeasure M s" by auto
```
```   467   have "emeasure M (space M) = emeasure M (s \<union> (space M - s))" using s
```
```   468     by (metis Un_Diff_cancel Un_absorb1 s sets.sets_into_space)
```
```   469   also have "... = emeasure M s + emeasure M (space M - s)"
```
```   470     by (rule plus_emeasure[symmetric]) (auto simp add: s)
```
```   471   finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" .
```
```   472   then show ?thesis
```
```   473     using fin `0 \<le> emeasure M s`
```
```   474     unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
```
```   475 qed
```
```   476
```
```   477 lemma emeasure_Diff:
```
```   478   assumes finite: "emeasure M B \<noteq> \<infinity>"
```
```   479   and [measurable]: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
```
```   480   shows "emeasure M (A - B) = emeasure M A - emeasure M B"
```
```   481 proof -
```
```   482   have "0 \<le> emeasure M B" using assms by auto
```
```   483   have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
```
```   484   then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp
```
```   485   also have "\<dots> = emeasure M (A - B) + emeasure M B"
```
```   486     by (subst plus_emeasure[symmetric]) auto
```
```   487   finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
```
```   488     unfolding ereal_eq_minus_iff
```
```   489     using finite `0 \<le> emeasure M B` by auto
```
```   490 qed
```
```   491
```
```   492 lemma Lim_emeasure_incseq:
```
```   493   "range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)"
```
```   494   using emeasure_countably_additive
```
```   495   by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive
```
```   496     emeasure_additive)
```
```   497
```
```   498 lemma incseq_emeasure:
```
```   499   assumes "range B \<subseteq> sets M" "incseq B"
```
```   500   shows "incseq (\<lambda>i. emeasure M (B i))"
```
```   501   using assms by (auto simp: incseq_def intro!: emeasure_mono)
```
```   502
```
```   503 lemma SUP_emeasure_incseq:
```
```   504   assumes A: "range A \<subseteq> sets M" "incseq A"
```
```   505   shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
```
```   506   using LIMSEQ_SUP[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A]
```
```   507   by (simp add: LIMSEQ_unique)
```
```   508
```
```   509 lemma decseq_emeasure:
```
```   510   assumes "range B \<subseteq> sets M" "decseq B"
```
```   511   shows "decseq (\<lambda>i. emeasure M (B i))"
```
```   512   using assms by (auto simp: decseq_def intro!: emeasure_mono)
```
```   513
```
```   514 lemma INF_emeasure_decseq:
```
```   515   assumes A: "range A \<subseteq> sets M" and "decseq A"
```
```   516   and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```   517   shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
```
```   518 proof -
```
```   519   have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
```
```   520     using A by (auto intro!: emeasure_mono)
```
```   521   hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
```
```   522
```
```   523   have A0: "0 \<le> emeasure M (A 0)" using A by auto
```
```   524
```
```   525   have "emeasure M (A 0) - (INF n. emeasure M (A n)) = emeasure M (A 0) + (SUP n. - emeasure M (A n))"
```
```   526     by (simp add: ereal_SUP_uminus minus_ereal_def)
```
```   527   also have "\<dots> = (SUP n. emeasure M (A 0) - emeasure M (A n))"
```
```   528     unfolding minus_ereal_def using A0 assms
```
```   529     by (subst SUP_ereal_add) (auto simp add: decseq_emeasure)
```
```   530   also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"
```
```   531     using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto
```
```   532   also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"
```
```   533   proof (rule SUP_emeasure_incseq)
```
```   534     show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
```
```   535       using A by auto
```
```   536     show "incseq (\<lambda>n. A 0 - A n)"
```
```   537       using `decseq A` by (auto simp add: incseq_def decseq_def)
```
```   538   qed
```
```   539   also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
```
```   540     using A finite * by (simp, subst emeasure_Diff) auto
```
```   541   finally show ?thesis
```
```   542     unfolding ereal_minus_eq_minus_iff using finite A0 by auto
```
```   543 qed
```
```   544
```
```   545 lemma Lim_emeasure_decseq:
```
```   546   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```   547   shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)"
```
```   548   using LIMSEQ_INF[OF decseq_emeasure, OF A]
```
```   549   using INF_emeasure_decseq[OF A fin] by simp
```
```   550
```
```   551 lemma emeasure_lfp[consumes 1, case_names cont measurable]:
```
```   552   assumes "P M"
```
```   553   assumes cont: "sup_continuous F"
```
```   554   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
```
```   555   shows "emeasure M {x\<in>space M. lfp F x} = (SUP i. emeasure M {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
```
```   556 proof -
```
```   557   have "emeasure M {x\<in>space M. lfp F x} = emeasure M (\<Union>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
```
```   558     using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure])
```
```   559   moreover { fix i from `P M` have "{x\<in>space M. (F ^^ i) (\<lambda>x. False) x} \<in> sets M"
```
```   560     by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }
```
```   561   moreover have "incseq (\<lambda>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
```
```   562   proof (rule incseq_SucI)
```
```   563     fix i
```
```   564     have "(F ^^ i) (\<lambda>x. False) \<le> (F ^^ (Suc i)) (\<lambda>x. False)"
```
```   565     proof (induct i)
```
```   566       case 0 show ?case by (simp add: le_fun_def)
```
```   567     next
```
```   568       case Suc thus ?case using monoD[OF sup_continuous_mono[OF cont] Suc] by auto
```
```   569     qed
```
```   570     then show "{x \<in> space M. (F ^^ i) (\<lambda>x. False) x} \<subseteq> {x \<in> space M. (F ^^ Suc i) (\<lambda>x. False) x}"
```
```   571       by auto
```
```   572   qed
```
```   573   ultimately show ?thesis
```
```   574     by (subst SUP_emeasure_incseq) auto
```
```   575 qed
```
```   576
```
```   577 lemma emeasure_subadditive:
```
```   578   assumes [measurable]: "A \<in> sets M" "B \<in> sets M"
```
```   579   shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
```
```   580 proof -
```
```   581   from plus_emeasure[of A M "B - A"]
```
```   582   have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)" by simp
```
```   583   also have "\<dots> \<le> emeasure M A + emeasure M B"
```
```   584     using assms by (auto intro!: add_left_mono emeasure_mono)
```
```   585   finally show ?thesis .
```
```   586 qed
```
```   587
```
```   588 lemma emeasure_subadditive_finite:
```
```   589   assumes "finite I" "A ` I \<subseteq> sets M"
```
```   590   shows "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
```
```   591 using assms proof induct
```
```   592   case (insert i I)
```
```   593   then have "emeasure M (\<Union>i\<in>insert i I. A i) = emeasure M (A i \<union> (\<Union>i\<in>I. A i))"
```
```   594     by simp
```
```   595   also have "\<dots> \<le> emeasure M (A i) + emeasure M (\<Union>i\<in>I. A i)"
```
```   596     using insert by (intro emeasure_subadditive) auto
```
```   597   also have "\<dots> \<le> emeasure M (A i) + (\<Sum>i\<in>I. emeasure M (A i))"
```
```   598     using insert by (intro add_mono) auto
```
```   599   also have "\<dots> = (\<Sum>i\<in>insert i I. emeasure M (A i))"
```
```   600     using insert by auto
```
```   601   finally show ?case .
```
```   602 qed simp
```
```   603
```
```   604 lemma emeasure_subadditive_countably:
```
```   605   assumes "range f \<subseteq> sets M"
```
```   606   shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))"
```
```   607 proof -
```
```   608   have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)"
```
```   609     unfolding UN_disjointed_eq ..
```
```   610   also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))"
```
```   611     using sets.range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
```
```   612     by (simp add:  disjoint_family_disjointed comp_def)
```
```   613   also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))"
```
```   614     using sets.range_disjointed_sets[OF assms] assms
```
```   615     by (auto intro!: suminf_le_pos emeasure_mono disjointed_subset)
```
```   616   finally show ?thesis .
```
```   617 qed
```
```   618
```
```   619 lemma emeasure_insert:
```
```   620   assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
```
```   621   shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
```
```   622 proof -
```
```   623   have "{x} \<inter> A = {}" using `x \<notin> A` by auto
```
```   624   from plus_emeasure[OF sets this] show ?thesis by simp
```
```   625 qed
```
```   626
```
```   627 lemma emeasure_insert_ne:
```
```   628   "A \<noteq> {} \<Longrightarrow> {x} \<in> sets M \<Longrightarrow> A \<in> sets M \<Longrightarrow> x \<notin> A \<Longrightarrow> emeasure M (insert x A) = emeasure M {x} + emeasure M A"
```
```   629   by (rule emeasure_insert)
```
```   630
```
```   631 lemma emeasure_eq_setsum_singleton:
```
```   632   assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```   633   shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
```
```   634   using setsum_emeasure[of "\<lambda>x. {x}" S M] assms
```
```   635   by (auto simp: disjoint_family_on_def subset_eq)
```
```   636
```
```   637 lemma setsum_emeasure_cover:
```
```   638   assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
```
```   639   assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"
```
```   640   assumes disj: "disjoint_family_on B S"
```
```   641   shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"
```
```   642 proof -
```
```   643   have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
```
```   644   proof (rule setsum_emeasure)
```
```   645     show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
```
```   646       using `disjoint_family_on B S`
```
```   647       unfolding disjoint_family_on_def by auto
```
```   648   qed (insert assms, auto)
```
```   649   also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
```
```   650     using A by auto
```
```   651   finally show ?thesis by simp
```
```   652 qed
```
```   653
```
```   654 lemma emeasure_eq_0:
```
```   655   "N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0"
```
```   656   by (metis emeasure_mono emeasure_nonneg order_eq_iff)
```
```   657
```
```   658 lemma emeasure_UN_eq_0:
```
```   659   assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M"
```
```   660   shows "emeasure M (\<Union> i. N i) = 0"
```
```   661 proof -
```
```   662   have "0 \<le> emeasure M (\<Union> i. N i)" using assms by auto
```
```   663   moreover have "emeasure M (\<Union> i. N i) \<le> 0"
```
```   664     using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
```
```   665   ultimately show ?thesis by simp
```
```   666 qed
```
```   667
```
```   668 lemma measure_eqI_finite:
```
```   669   assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
```
```   670   assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"
```
```   671   shows "M = N"
```
```   672 proof (rule measure_eqI)
```
```   673   fix X assume "X \<in> sets M"
```
```   674   then have X: "X \<subseteq> A" by auto
```
```   675   then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
```
```   676     using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
```
```   677   also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
```
```   678     using X eq by (auto intro!: setsum.cong)
```
```   679   also have "\<dots> = emeasure N X"
```
```   680     using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
```
```   681   finally show "emeasure M X = emeasure N X" .
```
```   682 qed simp
```
```   683
```
```   684 lemma measure_eqI_generator_eq:
```
```   685   fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set"
```
```   686   assumes "Int_stable E" "E \<subseteq> Pow \<Omega>"
```
```   687   and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
```
```   688   and M: "sets M = sigma_sets \<Omega> E"
```
```   689   and N: "sets N = sigma_sets \<Omega> E"
```
```   690   and A: "range A \<subseteq> E" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```   691   shows "M = N"
```
```   692 proof -
```
```   693   let ?\<mu>  = "emeasure M" and ?\<nu> = "emeasure N"
```
```   694   interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
```
```   695   have "space M = \<Omega>"
```
```   696     using sets.top[of M] sets.space_closed[of M] S.top S.space_closed `sets M = sigma_sets \<Omega> E`
```
```   697     by blast
```
```   698
```
```   699   { fix F D assume "F \<in> E" and "?\<mu> F \<noteq> \<infinity>"
```
```   700     then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
```
```   701     have "?\<nu> F \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` `F \<in> E` eq by simp
```
```   702     assume "D \<in> sets M"
```
```   703     with `Int_stable E` `E \<subseteq> Pow \<Omega>` have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
```
```   704       unfolding M
```
```   705     proof (induct rule: sigma_sets_induct_disjoint)
```
```   706       case (basic A)
```
```   707       then have "F \<inter> A \<in> E" using `Int_stable E` `F \<in> E` by (auto simp: Int_stable_def)
```
```   708       then show ?case using eq by auto
```
```   709     next
```
```   710       case empty then show ?case by simp
```
```   711     next
```
```   712       case (compl A)
```
```   713       then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"
```
```   714         and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"
```
```   715         using `F \<in> E` S.sets_into_space by (auto simp: M)
```
```   716       have "?\<nu> (F \<inter> A) \<le> ?\<nu> F" by (auto intro!: emeasure_mono simp: M N)
```
```   717       then have "?\<nu> (F \<inter> A) \<noteq> \<infinity>" using `?\<nu> F \<noteq> \<infinity>` by auto
```
```   718       have "?\<mu> (F \<inter> A) \<le> ?\<mu> F" by (auto intro!: emeasure_mono simp: M N)
```
```   719       then have "?\<mu> (F \<inter> A) \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` by auto
```
```   720       then have "?\<mu> (F \<inter> (\<Omega> - A)) = ?\<mu> F - ?\<mu> (F \<inter> A)" unfolding **
```
```   721         using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N)
```
```   722       also have "\<dots> = ?\<nu> F - ?\<nu> (F \<inter> A)" using eq `F \<in> E` compl by simp
```
```   723       also have "\<dots> = ?\<nu> (F \<inter> (\<Omega> - A))" unfolding **
```
```   724         using `F \<inter> A \<in> sigma_sets \<Omega> E` `?\<nu> (F \<inter> A) \<noteq> \<infinity>`
```
```   725         by (auto intro!: emeasure_Diff[symmetric] simp: M N)
```
```   726       finally show ?case
```
```   727         using `space M = \<Omega>` by auto
```
```   728     next
```
```   729       case (union A)
```
```   730       then have "?\<mu> (\<Union>x. F \<inter> A x) = ?\<nu> (\<Union>x. F \<inter> A x)"
```
```   731         by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N)
```
```   732       with A show ?case
```
```   733         by auto
```
```   734     qed }
```
```   735   note * = this
```
```   736   show "M = N"
```
```   737   proof (rule measure_eqI)
```
```   738     show "sets M = sets N"
```
```   739       using M N by simp
```
```   740     have [simp, intro]: "\<And>i. A i \<in> sets M"
```
```   741       using A(1) by (auto simp: subset_eq M)
```
```   742     fix F assume "F \<in> sets M"
```
```   743     let ?D = "disjointed (\<lambda>i. F \<inter> A i)"
```
```   744     from `space M = \<Omega>` have F_eq: "F = (\<Union>i. ?D i)"
```
```   745       using `F \<in> sets M`[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)
```
```   746     have [simp, intro]: "\<And>i. ?D i \<in> sets M"
```
```   747       using sets.range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] `F \<in> sets M`
```
```   748       by (auto simp: subset_eq)
```
```   749     have "disjoint_family ?D"
```
```   750       by (auto simp: disjoint_family_disjointed)
```
```   751     moreover
```
```   752     have "(\<Sum>i. emeasure M (?D i)) = (\<Sum>i. emeasure N (?D i))"
```
```   753     proof (intro arg_cong[where f=suminf] ext)
```
```   754       fix i
```
```   755       have "A i \<inter> ?D i = ?D i"
```
```   756         by (auto simp: disjointed_def)
```
```   757       then show "emeasure M (?D i) = emeasure N (?D i)"
```
```   758         using *[of "A i" "?D i", OF _ A(3)] A(1) by auto
```
```   759     qed
```
```   760     ultimately show "emeasure M F = emeasure N F"
```
```   761       by (simp add: image_subset_iff `sets M = sets N`[symmetric] F_eq[symmetric] suminf_emeasure)
```
```   762   qed
```
```   763 qed
```
```   764
```
```   765 lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
```
```   766 proof (intro measure_eqI emeasure_measure_of_sigma)
```
```   767   show "sigma_algebra (space M) (sets M)" ..
```
```   768   show "positive (sets M) (emeasure M)"
```
```   769     by (simp add: positive_def emeasure_nonneg)
```
```   770   show "countably_additive (sets M) (emeasure M)"
```
```   771     by (simp add: emeasure_countably_additive)
```
```   772 qed simp_all
```
```   773
```
```   774 subsection {* @{text \<mu>}-null sets *}
```
```   775
```
```   776 definition null_sets :: "'a measure \<Rightarrow> 'a set set" where
```
```   777   "null_sets M = {N\<in>sets M. emeasure M N = 0}"
```
```   778
```
```   779 lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0"
```
```   780   by (simp add: null_sets_def)
```
```   781
```
```   782 lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M"
```
```   783   unfolding null_sets_def by simp
```
```   784
```
```   785 lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M"
```
```   786   unfolding null_sets_def by simp
```
```   787
```
```   788 interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
```
```   789 proof (rule ring_of_setsI)
```
```   790   show "null_sets M \<subseteq> Pow (space M)"
```
```   791     using sets.sets_into_space by auto
```
```   792   show "{} \<in> null_sets M"
```
```   793     by auto
```
```   794   fix A B assume null_sets: "A \<in> null_sets M" "B \<in> null_sets M"
```
```   795   then have sets: "A \<in> sets M" "B \<in> sets M"
```
```   796     by auto
```
```   797   then have *: "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
```
```   798     "emeasure M (A - B) \<le> emeasure M A"
```
```   799     by (auto intro!: emeasure_subadditive emeasure_mono)
```
```   800   then have "emeasure M B = 0" "emeasure M A = 0"
```
```   801     using null_sets by auto
```
```   802   with sets * show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"
```
```   803     by (auto intro!: antisym)
```
```   804 qed
```
```   805
```
```   806 lemma UN_from_nat_into:
```
```   807   assumes I: "countable I" "I \<noteq> {}"
```
```   808   shows "(\<Union>i\<in>I. N i) = (\<Union>i. N (from_nat_into I i))"
```
```   809 proof -
```
```   810   have "(\<Union>i\<in>I. N i) = \<Union>(N ` range (from_nat_into I))"
```
```   811     using I by simp
```
```   812   also have "\<dots> = (\<Union>i. (N \<circ> from_nat_into I) i)"
```
```   813     by (simp only: SUP_def image_comp)
```
```   814   finally show ?thesis by simp
```
```   815 qed
```
```   816
```
```   817 lemma null_sets_UN':
```
```   818   assumes "countable I"
```
```   819   assumes "\<And>i. i \<in> I \<Longrightarrow> N i \<in> null_sets M"
```
```   820   shows "(\<Union>i\<in>I. N i) \<in> null_sets M"
```
```   821 proof cases
```
```   822   assume "I = {}" then show ?thesis by simp
```
```   823 next
```
```   824   assume "I \<noteq> {}"
```
```   825   show ?thesis
```
```   826   proof (intro conjI CollectI null_setsI)
```
```   827     show "(\<Union>i\<in>I. N i) \<in> sets M"
```
```   828       using assms by (intro sets.countable_UN') auto
```
```   829     have "emeasure M (\<Union>i\<in>I. N i) \<le> (\<Sum>n. emeasure M (N (from_nat_into I n)))"
```
```   830       unfolding UN_from_nat_into[OF `countable I` `I \<noteq> {}`]
```
```   831       using assms `I \<noteq> {}` by (intro emeasure_subadditive_countably) (auto intro: from_nat_into)
```
```   832     also have "(\<lambda>n. emeasure M (N (from_nat_into I n))) = (\<lambda>_. 0)"
```
```   833       using assms `I \<noteq> {}` by (auto intro: from_nat_into)
```
```   834     finally show "emeasure M (\<Union>i\<in>I. N i) = 0"
```
```   835       by (intro antisym emeasure_nonneg) simp
```
```   836   qed
```
```   837 qed
```
```   838
```
```   839 lemma null_sets_UN[intro]:
```
```   840   "(\<And>i::'i::countable. N i \<in> null_sets M) \<Longrightarrow> (\<Union>i. N i) \<in> null_sets M"
```
```   841   by (rule null_sets_UN') auto
```
```   842
```
```   843 lemma null_set_Int1:
```
```   844   assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
```
```   845 proof (intro CollectI conjI null_setsI)
```
```   846   show "emeasure M (A \<inter> B) = 0" using assms
```
```   847     by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto
```
```   848 qed (insert assms, auto)
```
```   849
```
```   850 lemma null_set_Int2:
```
```   851   assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
```
```   852   using assms by (subst Int_commute) (rule null_set_Int1)
```
```   853
```
```   854 lemma emeasure_Diff_null_set:
```
```   855   assumes "B \<in> null_sets M" "A \<in> sets M"
```
```   856   shows "emeasure M (A - B) = emeasure M A"
```
```   857 proof -
```
```   858   have *: "A - B = (A - (A \<inter> B))" by auto
```
```   859   have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1)
```
```   860   then show ?thesis
```
```   861     unfolding * using assms
```
```   862     by (subst emeasure_Diff) auto
```
```   863 qed
```
```   864
```
```   865 lemma null_set_Diff:
```
```   866   assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"
```
```   867 proof (intro CollectI conjI null_setsI)
```
```   868   show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
```
```   869 qed (insert assms, auto)
```
```   870
```
```   871 lemma emeasure_Un_null_set:
```
```   872   assumes "A \<in> sets M" "B \<in> null_sets M"
```
```   873   shows "emeasure M (A \<union> B) = emeasure M A"
```
```   874 proof -
```
```   875   have *: "A \<union> B = A \<union> (B - A)" by auto
```
```   876   have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)
```
```   877   then show ?thesis
```
```   878     unfolding * using assms
```
```   879     by (subst plus_emeasure[symmetric]) auto
```
```   880 qed
```
```   881
```
```   882 subsection {* The almost everywhere filter (i.e.\ quantifier) *}
```
```   883
```
```   884 definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where
```
```   885   "ae_filter M = (INF N:null_sets M. principal (space M - N))"
```
```   886
```
```   887 abbreviation almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   888   "almost_everywhere M P \<equiv> eventually P (ae_filter M)"
```
```   889
```
```   890 syntax
```
```   891   "_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
```
```   892
```
```   893 translations
```
```   894   "AE x in M. P" == "CONST almost_everywhere M (\<lambda>x. P)"
```
```   895
```
```   896 lemma eventually_ae_filter: "eventually P (ae_filter M) \<longleftrightarrow> (\<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
```
```   897   unfolding ae_filter_def by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq)
```
```   898
```
```   899 lemma AE_I':
```
```   900   "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"
```
```   901   unfolding eventually_ae_filter by auto
```
```   902
```
```   903 lemma AE_iff_null:
```
```   904   assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
```
```   905   shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"
```
```   906 proof
```
```   907   assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"
```
```   908     unfolding eventually_ae_filter by auto
```
```   909   have "0 \<le> emeasure M ?P" by auto
```
```   910   moreover have "emeasure M ?P \<le> emeasure M N"
```
```   911     using assms N(1,2) by (auto intro: emeasure_mono)
```
```   912   ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto
```
```   913   then show "?P \<in> null_sets M" using assms by auto
```
```   914 next
```
```   915   assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
```
```   916 qed
```
```   917
```
```   918 lemma AE_iff_null_sets:
```
```   919   "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"
```
```   920   using Int_absorb1[OF sets.sets_into_space, of N M]
```
```   921   by (subst AE_iff_null) (auto simp: Int_def[symmetric])
```
```   922
```
```   923 lemma AE_not_in:
```
```   924   "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
```
```   925   by (metis AE_iff_null_sets null_setsD2)
```
```   926
```
```   927 lemma AE_iff_measurable:
```
```   928   "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"
```
```   929   using AE_iff_null[of _ P] by auto
```
```   930
```
```   931 lemma AE_E[consumes 1]:
```
```   932   assumes "AE x in M. P x"
```
```   933   obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
```
```   934   using assms unfolding eventually_ae_filter by auto
```
```   935
```
```   936 lemma AE_E2:
```
```   937   assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"
```
```   938   shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")
```
```   939 proof -
```
```   940   have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto
```
```   941   with AE_iff_null[of M P] assms show ?thesis by auto
```
```   942 qed
```
```   943
```
```   944 lemma AE_I:
```
```   945   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
```
```   946   shows "AE x in M. P x"
```
```   947   using assms unfolding eventually_ae_filter by auto
```
```   948
```
```   949 lemma AE_mp[elim!]:
```
```   950   assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"
```
```   951   shows "AE x in M. Q x"
```
```   952 proof -
```
```   953   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
```
```   954     and A: "A \<in> sets M" "emeasure M A = 0"
```
```   955     by (auto elim!: AE_E)
```
```   956
```
```   957   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
```
```   958     and B: "B \<in> sets M" "emeasure M B = 0"
```
```   959     by (auto elim!: AE_E)
```
```   960
```
```   961   show ?thesis
```
```   962   proof (intro AE_I)
```
```   963     have "0 \<le> emeasure M (A \<union> B)" using A B by auto
```
```   964     moreover have "emeasure M (A \<union> B) \<le> 0"
```
```   965       using emeasure_subadditive[of A M B] A B by auto
```
```   966     ultimately show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0" using A B by auto
```
```   967     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
```
```   968       using P imp by auto
```
```   969   qed
```
```   970 qed
```
```   971
```
```   972 (* depricated replace by laws about eventually *)
```
```   973 lemma
```
```   974   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"
```
```   975     and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"
```
```   976     and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"
```
```   977     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"
```
```   978     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)"
```
```   979   by auto
```
```   980
```
```   981 lemma AE_impI:
```
```   982   "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"
```
```   983   by (cases P) auto
```
```   984
```
```   985 lemma AE_measure:
```
```   986   assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")
```
```   987   shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"
```
```   988 proof -
```
```   989   from AE_E[OF AE] guess N . note N = this
```
```   990   with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"
```
```   991     by (intro emeasure_mono) auto
```
```   992   also have "\<dots> \<le> emeasure M ?P + emeasure M N"
```
```   993     using sets N by (intro emeasure_subadditive) auto
```
```   994   also have "\<dots> = emeasure M ?P" using N by simp
```
```   995   finally show "emeasure M ?P = emeasure M (space M)"
```
```   996     using emeasure_space[of M "?P"] by auto
```
```   997 qed
```
```   998
```
```   999 lemma AE_space: "AE x in M. x \<in> space M"
```
```  1000   by (rule AE_I[where N="{}"]) auto
```
```  1001
```
```  1002 lemma AE_I2[simp, intro]:
```
```  1003   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"
```
```  1004   using AE_space by force
```
```  1005
```
```  1006 lemma AE_Ball_mp:
```
```  1007   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
```
```  1008   by auto
```
```  1009
```
```  1010 lemma AE_cong[cong]:
```
```  1011   "(\<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)"
```
```  1012   by auto
```
```  1013
```
```  1014 lemma AE_all_countable:
```
```  1015   "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
```
```  1016 proof
```
```  1017   assume "\<forall>i. AE x in M. P i x"
```
```  1018   from this[unfolded eventually_ae_filter Bex_def, THEN choice]
```
```  1019   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
```
```  1020   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
```
```  1021   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
```
```  1022   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
```
```  1023   moreover from N have "(\<Union>i. N i) \<in> null_sets M"
```
```  1024     by (intro null_sets_UN) auto
```
```  1025   ultimately show "AE x in M. \<forall>i. P i x"
```
```  1026     unfolding eventually_ae_filter by auto
```
```  1027 qed auto
```
```  1028
```
```  1029 lemma AE_ball_countable:
```
```  1030   assumes [intro]: "countable X"
```
```  1031   shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"
```
```  1032 proof
```
```  1033   assume "\<forall>y\<in>X. AE x in M. P x y"
```
```  1034   from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]
```
```  1035   obtain N where N: "\<And>y. y \<in> X \<Longrightarrow> N y \<in> null_sets M" "\<And>y. y \<in> X \<Longrightarrow> {x\<in>space M. \<not> P x y} \<subseteq> N y"
```
```  1036     by auto
```
```  1037   have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. {x\<in>space M. \<not> P x y})"
```
```  1038     by auto
```
```  1039   also have "\<dots> \<subseteq> (\<Union>y\<in>X. N y)"
```
```  1040     using N by auto
```
```  1041   finally have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. N y)" .
```
```  1042   moreover from N have "(\<Union>y\<in>X. N y) \<in> null_sets M"
```
```  1043     by (intro null_sets_UN') auto
```
```  1044   ultimately show "AE x in M. \<forall>y\<in>X. P x y"
```
```  1045     unfolding eventually_ae_filter by auto
```
```  1046 qed auto
```
```  1047
```
```  1048 lemma AE_discrete_difference:
```
```  1049   assumes X: "countable X"
```
```  1050   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"
```
```  1051   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
```
```  1052   shows "AE x in M. x \<notin> X"
```
```  1053 proof -
```
```  1054   have "(\<Union>x\<in>X. {x}) \<in> null_sets M"
```
```  1055     using assms by (intro null_sets_UN') auto
```
```  1056   from AE_not_in[OF this] show "AE x in M. x \<notin> X"
```
```  1057     by auto
```
```  1058 qed
```
```  1059
```
```  1060 lemma AE_finite_all:
```
```  1061   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)"
```
```  1062   using f by induct auto
```
```  1063
```
```  1064 lemma AE_finite_allI:
```
```  1065   assumes "finite S"
```
```  1066   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"
```
```  1067   using AE_finite_all[OF `finite S`] by auto
```
```  1068
```
```  1069 lemma emeasure_mono_AE:
```
```  1070   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
```
```  1071     and B: "B \<in> sets M"
```
```  1072   shows "emeasure M A \<le> emeasure M B"
```
```  1073 proof cases
```
```  1074   assume A: "A \<in> sets M"
```
```  1075   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"
```
```  1076     by (auto simp: eventually_ae_filter)
```
```  1077   have "emeasure M A = emeasure M (A - N)"
```
```  1078     using N A by (subst emeasure_Diff_null_set) auto
```
```  1079   also have "emeasure M (A - N) \<le> emeasure M (B - N)"
```
```  1080     using N A B sets.sets_into_space by (auto intro!: emeasure_mono)
```
```  1081   also have "emeasure M (B - N) = emeasure M B"
```
```  1082     using N B by (subst emeasure_Diff_null_set) auto
```
```  1083   finally show ?thesis .
```
```  1084 qed (simp add: emeasure_nonneg emeasure_notin_sets)
```
```  1085
```
```  1086 lemma emeasure_eq_AE:
```
```  1087   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
```
```  1088   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
```
```  1089   shows "emeasure M A = emeasure M B"
```
```  1090   using assms by (safe intro!: antisym emeasure_mono_AE) auto
```
```  1091
```
```  1092 lemma emeasure_Collect_eq_AE:
```
```  1093   "AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> Measurable.pred M Q \<Longrightarrow> Measurable.pred M P \<Longrightarrow>
```
```  1094    emeasure M {x\<in>space M. P x} = emeasure M {x\<in>space M. Q x}"
```
```  1095    by (intro emeasure_eq_AE) auto
```
```  1096
```
```  1097 lemma emeasure_eq_0_AE: "AE x in M. \<not> P x \<Longrightarrow> emeasure M {x\<in>space M. P x} = 0"
```
```  1098   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"]
```
```  1099   by (cases "{x\<in>space M. P x} \<in> sets M") (simp_all add: emeasure_notin_sets)
```
```  1100
```
```  1101 subsection {* @{text \<sigma>}-finite Measures *}
```
```  1102
```
```  1103 locale sigma_finite_measure =
```
```  1104   fixes M :: "'a measure"
```
```  1105   assumes sigma_finite_countable:
```
```  1106     "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets M \<and> (\<Union>A) = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
```
```  1107
```
```  1108 lemma (in sigma_finite_measure) sigma_finite:
```
```  1109   obtains A :: "nat \<Rightarrow> 'a set"
```
```  1110   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```  1111 proof -
```
```  1112   obtain A :: "'a set set" where
```
```  1113     [simp]: "countable A" and
```
```  1114     A: "A \<subseteq> sets M" "(\<Union>A) = space M" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
```
```  1115     using sigma_finite_countable by metis
```
```  1116   show thesis
```
```  1117   proof cases
```
```  1118     assume "A = {}" with `(\<Union>A) = space M` show thesis
```
```  1119       by (intro that[of "\<lambda>_. {}"]) auto
```
```  1120   next
```
```  1121     assume "A \<noteq> {}"
```
```  1122     show thesis
```
```  1123     proof
```
```  1124       show "range (from_nat_into A) \<subseteq> sets M"
```
```  1125         using `A \<noteq> {}` A by auto
```
```  1126       have "(\<Union>i. from_nat_into A i) = \<Union>A"
```
```  1127         using range_from_nat_into[OF `A \<noteq> {}` `countable A`] by auto
```
```  1128       with A show "(\<Union>i. from_nat_into A i) = space M"
```
```  1129         by auto
```
```  1130     qed (intro A from_nat_into `A \<noteq> {}`)
```
```  1131   qed
```
```  1132 qed
```
```  1133
```
```  1134 lemma (in sigma_finite_measure) sigma_finite_disjoint:
```
```  1135   obtains A :: "nat \<Rightarrow> 'a set"
```
```  1136   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"
```
```  1137 proof atomize_elim
```
```  1138   case goal1
```
```  1139   obtain A :: "nat \<Rightarrow> 'a set" where
```
```  1140     range: "range A \<subseteq> sets M" and
```
```  1141     space: "(\<Union>i. A i) = space M" and
```
```  1142     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```  1143     using sigma_finite by auto
```
```  1144   note range' = sets.range_disjointed_sets[OF range] range
```
```  1145   { fix i
```
```  1146     have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
```
```  1147       using range' disjointed_subset[of A i] by (auto intro!: emeasure_mono)
```
```  1148     then have "emeasure M (disjointed A i) \<noteq> \<infinity>"
```
```  1149       using measure[of i] by auto }
```
```  1150   with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
```
```  1151   show ?case by (auto intro!: exI[of _ "disjointed A"])
```
```  1152 qed
```
```  1153
```
```  1154 lemma (in sigma_finite_measure) sigma_finite_incseq:
```
```  1155   obtains A :: "nat \<Rightarrow> 'a set"
```
```  1156   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
```
```  1157 proof atomize_elim
```
```  1158   case goal1
```
```  1159   obtain F :: "nat \<Rightarrow> 'a set" where
```
```  1160     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"
```
```  1161     using sigma_finite by auto
```
```  1162   then show ?case
```
```  1163   proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
```
```  1164     from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
```
```  1165     then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
```
```  1166       using F by fastforce
```
```  1167   next
```
```  1168     fix n
```
```  1169     have "emeasure M (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))" using F
```
```  1170       by (auto intro!: emeasure_subadditive_finite)
```
```  1171     also have "\<dots> < \<infinity>"
```
```  1172       using F by (auto simp: setsum_Pinfty)
```
```  1173     finally show "emeasure M (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
```
```  1174   qed (force simp: incseq_def)+
```
```  1175 qed
```
```  1176
```
```  1177 subsection {* Measure space induced by distribution of @{const measurable}-functions *}
```
```  1178
```
```  1179 definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
```
```  1180   "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
```
```  1181
```
```  1182 lemma
```
```  1183   shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N"
```
```  1184     and space_distr[simp]: "space (distr M N f) = space N"
```
```  1185   by (auto simp: distr_def)
```
```  1186
```
```  1187 lemma
```
```  1188   shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
```
```  1189     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
```
```  1190   by (auto simp: measurable_def)
```
```  1191
```
```  1192 lemma distr_cong:
```
```  1193   "M = K \<Longrightarrow> sets N = sets L \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> distr M N f = distr K L g"
```
```  1194   using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)
```
```  1195
```
```  1196 lemma emeasure_distr:
```
```  1197   fixes f :: "'a \<Rightarrow> 'b"
```
```  1198   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
```
```  1199   shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A")
```
```  1200   unfolding distr_def
```
```  1201 proof (rule emeasure_measure_of_sigma)
```
```  1202   show "positive (sets N) ?\<mu>"
```
```  1203     by (auto simp: positive_def)
```
```  1204
```
```  1205   show "countably_additive (sets N) ?\<mu>"
```
```  1206   proof (intro countably_additiveI)
```
```  1207     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
```
```  1208     then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
```
```  1209     then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M"
```
```  1210       using f by (auto simp: measurable_def)
```
```  1211     moreover have "(\<Union>i. f -`  A i \<inter> space M) \<in> sets M"
```
```  1212       using * by blast
```
```  1213     moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
```
```  1214       using `disjoint_family A` by (auto simp: disjoint_family_on_def)
```
```  1215     ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
```
```  1216       using suminf_emeasure[OF _ **] A f
```
```  1217       by (auto simp: comp_def vimage_UN)
```
```  1218   qed
```
```  1219   show "sigma_algebra (space N) (sets N)" ..
```
```  1220 qed fact
```
```  1221
```
```  1222 lemma emeasure_Collect_distr:
```
```  1223   assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"
```
```  1224   shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"
```
```  1225   by (subst emeasure_distr)
```
```  1226      (auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])
```
```  1227
```
```  1228 lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:
```
```  1229   assumes "P M"
```
```  1230   assumes cont: "sup_continuous F"
```
```  1231   assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"
```
```  1232   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
```
```  1233   shows "emeasure M' {x\<in>space M'. lfp F (f x)} = (SUP i. emeasure M' {x\<in>space M'. (F ^^ i) (\<lambda>x. False) (f x)})"
```
```  1234 proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])
```
```  1235   show "f \<in> measurable M' M"  "f \<in> measurable M' M"
```
```  1236     using f[OF `P M`] by auto
```
```  1237   { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"
```
```  1238     using `P M` by (induction i arbitrary: M) (auto intro!: *) }
```
```  1239   show "Measurable.pred M (lfp F)"
```
```  1240     using `P M` cont * by (rule measurable_lfp_coinduct[of P])
```
```  1241
```
```  1242   have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =
```
```  1243     (SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"
```
```  1244     using `P M`
```
```  1245   proof (coinduction arbitrary: M rule: emeasure_lfp)
```
```  1246     case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"
```
```  1247       by metis
```
```  1248     then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"
```
```  1249       by simp
```
```  1250     with `P N`[THEN *] show ?case
```
```  1251       by auto
```
```  1252   qed fact
```
```  1253   then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =
```
```  1254     (SUP i. emeasure (distr M' M f) {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"
```
```  1255    by simp
```
```  1256 qed
```
```  1257
```
```  1258 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
```
```  1259   by (rule measure_eqI) (auto simp: emeasure_distr)
```
```  1260
```
```  1261 lemma measure_distr:
```
```  1262   "f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f -` S \<inter> space M)"
```
```  1263   by (simp add: emeasure_distr measure_def)
```
```  1264
```
```  1265 lemma distr_cong_AE:
```
```  1266   assumes 1: "M = K" "sets N = sets L" and
```
```  1267     2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"
```
```  1268   shows "distr M N f = distr K L g"
```
```  1269 proof (rule measure_eqI)
```
```  1270   fix A assume "A \<in> sets (distr M N f)"
```
```  1271   with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"
```
```  1272     by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)
```
```  1273 qed (insert 1, simp)
```
```  1274
```
```  1275 lemma AE_distrD:
```
```  1276   assumes f: "f \<in> measurable M M'"
```
```  1277     and AE: "AE x in distr M M' f. P x"
```
```  1278   shows "AE x in M. P (f x)"
```
```  1279 proof -
```
```  1280   from AE[THEN AE_E] guess N .
```
```  1281   with f show ?thesis
```
```  1282     unfolding eventually_ae_filter
```
```  1283     by (intro bexI[of _ "f -` N \<inter> space M"])
```
```  1284        (auto simp: emeasure_distr measurable_def)
```
```  1285 qed
```
```  1286
```
```  1287 lemma AE_distr_iff:
```
```  1288   assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N"
```
```  1289   shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"
```
```  1290 proof (subst (1 2) AE_iff_measurable[OF _ refl])
```
```  1291   have "f -` {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"
```
```  1292     using f[THEN measurable_space] by auto
```
```  1293   then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =
```
```  1294     (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"
```
```  1295     by (simp add: emeasure_distr)
```
```  1296 qed auto
```
```  1297
```
```  1298 lemma null_sets_distr_iff:
```
```  1299   "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"
```
```  1300   by (auto simp add: null_sets_def emeasure_distr)
```
```  1301
```
```  1302 lemma distr_distr:
```
```  1303   "g \<in> measurable N L \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> distr (distr M N f) L g = distr M L (g \<circ> f)"
```
```  1304   by (auto simp add: emeasure_distr measurable_space
```
```  1305            intro!: arg_cong[where f="emeasure M"] measure_eqI)
```
```  1306
```
```  1307 subsection {* Real measure values *}
```
```  1308
```
```  1309 lemma measure_nonneg: "0 \<le> measure M A"
```
```  1310   using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos)
```
```  1311
```
```  1312 lemma measure_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"
```
```  1313   using measure_nonneg[of M X] by auto
```
```  1314
```
```  1315 lemma measure_empty[simp]: "measure M {} = 0"
```
```  1316   unfolding measure_def by simp
```
```  1317
```
```  1318 lemma emeasure_eq_ereal_measure:
```
```  1319   "emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M A = ereal (measure M A)"
```
```  1320   using emeasure_nonneg[of M A]
```
```  1321   by (cases "emeasure M A") (auto simp: measure_def)
```
```  1322
```
```  1323 lemma measure_Union:
```
```  1324   assumes finite: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
```
```  1325   and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
```
```  1326   shows "measure M (A \<union> B) = measure M A + measure M B"
```
```  1327   unfolding measure_def
```
```  1328   using plus_emeasure[OF measurable, symmetric] finite
```
```  1329   by (simp add: emeasure_eq_ereal_measure)
```
```  1330
```
```  1331 lemma measure_finite_Union:
```
```  1332   assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
```
```  1333   assumes finite: "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
```
```  1334   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
```
```  1335   unfolding measure_def
```
```  1336   using setsum_emeasure[OF measurable, symmetric] finite
```
```  1337   by (simp add: emeasure_eq_ereal_measure)
```
```  1338
```
```  1339 lemma measure_Diff:
```
```  1340   assumes finite: "emeasure M A \<noteq> \<infinity>"
```
```  1341   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
```
```  1342   shows "measure M (A - B) = measure M A - measure M B"
```
```  1343 proof -
```
```  1344   have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"
```
```  1345     using measurable by (auto intro!: emeasure_mono)
```
```  1346   hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
```
```  1347     using measurable finite by (rule_tac measure_Union) auto
```
```  1348   thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
```
```  1349 qed
```
```  1350
```
```  1351 lemma measure_UNION:
```
```  1352   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
```
```  1353   assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
```
```  1354   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
```
```  1355 proof -
```
```  1356   from summable_sums[OF summable_ereal_pos, of "\<lambda>i. emeasure M (A i)"]
```
```  1357        suminf_emeasure[OF measurable] emeasure_nonneg[of M]
```
```  1358   have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))" by simp
```
```  1359   moreover
```
```  1360   { fix i
```
```  1361     have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
```
```  1362       using measurable by (auto intro!: emeasure_mono)
```
```  1363     then have "emeasure M (A i) = ereal ((measure M (A i)))"
```
```  1364       using finite by (intro emeasure_eq_ereal_measure) auto }
```
```  1365   ultimately show ?thesis using finite
```
```  1366     unfolding sums_ereal[symmetric] by (simp add: emeasure_eq_ereal_measure)
```
```  1367 qed
```
```  1368
```
```  1369 lemma measure_subadditive:
```
```  1370   assumes measurable: "A \<in> sets M" "B \<in> sets M"
```
```  1371   and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
```
```  1372   shows "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
```
```  1373 proof -
```
```  1374   have "emeasure M (A \<union> B) \<noteq> \<infinity>"
```
```  1375     using emeasure_subadditive[OF measurable] fin by auto
```
```  1376   then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
```
```  1377     using emeasure_subadditive[OF measurable] fin
```
```  1378     by (auto simp: emeasure_eq_ereal_measure)
```
```  1379 qed
```
```  1380
```
```  1381 lemma measure_subadditive_finite:
```
```  1382   assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
```
```  1383   shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
```
```  1384 proof -
```
```  1385   { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
```
```  1386       using emeasure_subadditive_finite[OF A] .
```
```  1387     also have "\<dots> < \<infinity>"
```
```  1388       using fin by (simp add: setsum_Pinfty)
```
```  1389     finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> \<infinity>" by simp }
```
```  1390   then show ?thesis
```
```  1391     using emeasure_subadditive_finite[OF A] fin
```
```  1392     unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
```
```  1393 qed
```
```  1394
```
```  1395 lemma measure_subadditive_countably:
```
```  1396   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
```
```  1397   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
```
```  1398 proof -
```
```  1399   from emeasure_nonneg fin have "\<And>i. emeasure M (A i) \<noteq> \<infinity>" by (rule suminf_PInfty)
```
```  1400   moreover
```
```  1401   { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"
```
```  1402       using emeasure_subadditive_countably[OF A] .
```
```  1403     also have "\<dots> < \<infinity>"
```
```  1404       using fin by simp
```
```  1405     finally have "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" by simp }
```
```  1406   ultimately  show ?thesis
```
```  1407     using emeasure_subadditive_countably[OF A] fin
```
```  1408     unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
```
```  1409 qed
```
```  1410
```
```  1411 lemma measure_eq_setsum_singleton:
```
```  1412   assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```  1413   and fin: "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>"
```
```  1414   shows "(measure M S) = (\<Sum>x\<in>S. (measure M {x}))"
```
```  1415   unfolding measure_def
```
```  1416   using emeasure_eq_setsum_singleton[OF S] fin
```
```  1417   by simp (simp add: emeasure_eq_ereal_measure)
```
```  1418
```
```  1419 lemma Lim_measure_incseq:
```
```  1420   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
```
```  1421   shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))"
```
```  1422 proof -
```
```  1423   have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)"
```
```  1424     using fin by (auto simp: emeasure_eq_ereal_measure)
```
```  1425   then show ?thesis
```
```  1426     using Lim_emeasure_incseq[OF A]
```
```  1427     unfolding measure_def
```
```  1428     by (intro lim_real_of_ereal) simp
```
```  1429 qed
```
```  1430
```
```  1431 lemma Lim_measure_decseq:
```
```  1432   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
```
```  1433   shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
```
```  1434 proof -
```
```  1435   have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
```
```  1436     using A by (auto intro!: emeasure_mono)
```
```  1437   also have "\<dots> < \<infinity>"
```
```  1438     using fin[of 0] by auto
```
```  1439   finally have "ereal ((measure M (\<Inter>i. A i))) = emeasure M (\<Inter>i. A i)"
```
```  1440     by (auto simp: emeasure_eq_ereal_measure)
```
```  1441   then show ?thesis
```
```  1442     unfolding measure_def
```
```  1443     using Lim_emeasure_decseq[OF A fin]
```
```  1444     by (intro lim_real_of_ereal) simp
```
```  1445 qed
```
```  1446
```
```  1447 subsection {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *}
```
```  1448
```
```  1449 locale finite_measure = sigma_finite_measure M for M +
```
```  1450   assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>"
```
```  1451
```
```  1452 lemma finite_measureI[Pure.intro!]:
```
```  1453   "emeasure M (space M) \<noteq> \<infinity> \<Longrightarrow> finite_measure M"
```
```  1454   proof qed (auto intro!: exI[of _ "{space M}"])
```
```  1455
```
```  1456 lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> \<infinity>"
```
```  1457   using finite_emeasure_space emeasure_space[of M A] by auto
```
```  1458
```
```  1459 lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ereal (measure M A)"
```
```  1460   unfolding measure_def by (simp add: emeasure_eq_ereal_measure)
```
```  1461
```
```  1462 lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ereal r"
```
```  1463   using emeasure_finite[of A] emeasure_nonneg[of M A] by (cases "emeasure M A") auto
```
```  1464
```
```  1465 lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"
```
```  1466   using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)
```
```  1467
```
```  1468 lemma (in finite_measure) finite_measure_Diff:
```
```  1469   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
```
```  1470   shows "measure M (A - B) = measure M A - measure M B"
```
```  1471   using measure_Diff[OF _ assms] by simp
```
```  1472
```
```  1473 lemma (in finite_measure) finite_measure_Union:
```
```  1474   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
```
```  1475   shows "measure M (A \<union> B) = measure M A + measure M B"
```
```  1476   using measure_Union[OF _ _ assms] by simp
```
```  1477
```
```  1478 lemma (in finite_measure) finite_measure_finite_Union:
```
```  1479   assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
```
```  1480   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
```
```  1481   using measure_finite_Union[OF assms] by simp
```
```  1482
```
```  1483 lemma (in finite_measure) finite_measure_UNION:
```
```  1484   assumes A: "range A \<subseteq> sets M" "disjoint_family A"
```
```  1485   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
```
```  1486   using measure_UNION[OF A] by simp
```
```  1487
```
```  1488 lemma (in finite_measure) finite_measure_mono:
```
```  1489   assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
```
```  1490   using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
```
```  1491
```
```  1492 lemma (in finite_measure) finite_measure_subadditive:
```
```  1493   assumes m: "A \<in> sets M" "B \<in> sets M"
```
```  1494   shows "measure M (A \<union> B) \<le> measure M A + measure M B"
```
```  1495   using measure_subadditive[OF m] by simp
```
```  1496
```
```  1497 lemma (in finite_measure) finite_measure_subadditive_finite:
```
```  1498   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))"
```
```  1499   using measure_subadditive_finite[OF assms] by simp
```
```  1500
```
```  1501 lemma (in finite_measure) finite_measure_subadditive_countably:
```
```  1502   assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))"
```
```  1503   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
```
```  1504 proof -
```
```  1505   from `summable (\<lambda>i. measure M (A i))`
```
```  1506   have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))"
```
```  1507     by (simp add: sums_ereal) (rule summable_sums)
```
```  1508   from sums_unique[OF this, symmetric]
```
```  1509        measure_subadditive_countably[OF A]
```
```  1510   show ?thesis by (simp add: emeasure_eq_measure)
```
```  1511 qed
```
```  1512
```
```  1513 lemma (in finite_measure) finite_measure_eq_setsum_singleton:
```
```  1514   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```  1515   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
```
```  1516   using measure_eq_setsum_singleton[OF assms] by simp
```
```  1517
```
```  1518 lemma (in finite_measure) finite_Lim_measure_incseq:
```
```  1519   assumes A: "range A \<subseteq> sets M" "incseq A"
```
```  1520   shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)"
```
```  1521   using Lim_measure_incseq[OF A] by simp
```
```  1522
```
```  1523 lemma (in finite_measure) finite_Lim_measure_decseq:
```
```  1524   assumes A: "range A \<subseteq> sets M" "decseq A"
```
```  1525   shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
```
```  1526   using Lim_measure_decseq[OF A] by simp
```
```  1527
```
```  1528 lemma (in finite_measure) finite_measure_compl:
```
```  1529   assumes S: "S \<in> sets M"
```
```  1530   shows "measure M (space M - S) = measure M (space M) - measure M S"
```
```  1531   using measure_Diff[OF _ sets.top S sets.sets_into_space] S by simp
```
```  1532
```
```  1533 lemma (in finite_measure) finite_measure_mono_AE:
```
```  1534   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"
```
```  1535   shows "measure M A \<le> measure M B"
```
```  1536   using assms emeasure_mono_AE[OF imp B]
```
```  1537   by (simp add: emeasure_eq_measure)
```
```  1538
```
```  1539 lemma (in finite_measure) finite_measure_eq_AE:
```
```  1540   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
```
```  1541   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
```
```  1542   shows "measure M A = measure M B"
```
```  1543   using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)
```
```  1544
```
```  1545 lemma (in finite_measure) measure_increasing: "increasing M (measure M)"
```
```  1546   by (auto intro!: finite_measure_mono simp: increasing_def)
```
```  1547
```
```  1548 lemma (in finite_measure) measure_zero_union:
```
```  1549   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"
```
```  1550   shows "measure M (s \<union> t) = measure M s"
```
```  1551 using assms
```
```  1552 proof -
```
```  1553   have "measure M (s \<union> t) \<le> measure M s"
```
```  1554     using finite_measure_subadditive[of s t] assms by auto
```
```  1555   moreover have "measure M (s \<union> t) \<ge> measure M s"
```
```  1556     using assms by (blast intro: finite_measure_mono)
```
```  1557   ultimately show ?thesis by simp
```
```  1558 qed
```
```  1559
```
```  1560 lemma (in finite_measure) measure_eq_compl:
```
```  1561   assumes "s \<in> sets M" "t \<in> sets M"
```
```  1562   assumes "measure M (space M - s) = measure M (space M - t)"
```
```  1563   shows "measure M s = measure M t"
```
```  1564   using assms finite_measure_compl by auto
```
```  1565
```
```  1566 lemma (in finite_measure) measure_eq_bigunion_image:
```
```  1567   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"
```
```  1568   assumes "disjoint_family f" "disjoint_family g"
```
```  1569   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"
```
```  1570   shows "measure M (\<Union> i. f i) = measure M (\<Union> i. g i)"
```
```  1571 using assms
```
```  1572 proof -
```
```  1573   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union> i. f i))"
```
```  1574     by (rule finite_measure_UNION[OF assms(1,3)])
```
```  1575   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union> i. g i))"
```
```  1576     by (rule finite_measure_UNION[OF assms(2,4)])
```
```  1577   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
```
```  1578 qed
```
```  1579
```
```  1580 lemma (in finite_measure) measure_countably_zero:
```
```  1581   assumes "range c \<subseteq> sets M"
```
```  1582   assumes "\<And> i. measure M (c i) = 0"
```
```  1583   shows "measure M (\<Union> i :: nat. c i) = 0"
```
```  1584 proof (rule antisym)
```
```  1585   show "measure M (\<Union> i :: nat. c i) \<le> 0"
```
```  1586     using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))
```
```  1587 qed (simp add: measure_nonneg)
```
```  1588
```
```  1589 lemma (in finite_measure) measure_space_inter:
```
```  1590   assumes events:"s \<in> sets M" "t \<in> sets M"
```
```  1591   assumes "measure M t = measure M (space M)"
```
```  1592   shows "measure M (s \<inter> t) = measure M s"
```
```  1593 proof -
```
```  1594   have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"
```
```  1595     using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)
```
```  1596   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
```
```  1597     by blast
```
```  1598   finally show "measure M (s \<inter> t) = measure M s"
```
```  1599     using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])
```
```  1600 qed
```
```  1601
```
```  1602 lemma (in finite_measure) measure_equiprobable_finite_unions:
```
```  1603   assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"
```
```  1604   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"
```
```  1605   shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"
```
```  1606 proof cases
```
```  1607   assume "s \<noteq> {}"
```
```  1608   then have "\<exists> x. x \<in> s" by blast
```
```  1609   from someI_ex[OF this] assms
```
```  1610   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast
```
```  1611   have "measure M s = (\<Sum> x \<in> s. measure M {x})"
```
```  1612     using finite_measure_eq_setsum_singleton[OF s] by simp
```
```  1613   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto
```
```  1614   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"
```
```  1615     using setsum_constant assms by (simp add: real_eq_of_nat)
```
```  1616   finally show ?thesis by simp
```
```  1617 qed simp
```
```  1618
```
```  1619 lemma (in finite_measure) measure_real_sum_image_fn:
```
```  1620   assumes "e \<in> sets M"
```
```  1621   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"
```
```  1622   assumes "finite s"
```
```  1623   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
```
```  1624   assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)"
```
```  1625   shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
```
```  1626 proof -
```
```  1627   have e: "e = (\<Union> i \<in> s. e \<inter> f i)"
```
```  1628     using `e \<in> sets M` sets.sets_into_space upper by blast
```
```  1629   hence "measure M e = measure M (\<Union> i \<in> s. e \<inter> f i)" by simp
```
```  1630   also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
```
```  1631   proof (rule finite_measure_finite_Union)
```
```  1632     show "finite s" by fact
```
```  1633     show "(\<lambda>i. e \<inter> f i)`s \<subseteq> sets M" using assms(2) by auto
```
```  1634     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
```
```  1635       using disjoint by (auto simp: disjoint_family_on_def)
```
```  1636   qed
```
```  1637   finally show ?thesis .
```
```  1638 qed
```
```  1639
```
```  1640 lemma (in finite_measure) measure_exclude:
```
```  1641   assumes "A \<in> sets M" "B \<in> sets M"
```
```  1642   assumes "measure M A = measure M (space M)" "A \<inter> B = {}"
```
```  1643   shows "measure M B = 0"
```
```  1644   using measure_space_inter[of B A] assms by (auto simp: ac_simps)
```
```  1645 lemma (in finite_measure) finite_measure_distr:
```
```  1646   assumes f: "f \<in> measurable M M'"
```
```  1647   shows "finite_measure (distr M M' f)"
```
```  1648 proof (rule finite_measureI)
```
```  1649   have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
```
```  1650   with f show "emeasure (distr M M' f) (space (distr M M' f)) \<noteq> \<infinity>" by (auto simp: emeasure_distr)
```
```  1651 qed
```
```  1652
```
```  1653 subsection {* Counting space *}
```
```  1654
```
```  1655 lemma strict_monoI_Suc:
```
```  1656   assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"
```
```  1657   unfolding strict_mono_def
```
```  1658 proof safe
```
```  1659   fix n m :: nat assume "n < m" then show "f n < f m"
```
```  1660     by (induct m) (auto simp: less_Suc_eq intro: less_trans ord)
```
```  1661 qed
```
```  1662
```
```  1663 lemma emeasure_count_space:
```
```  1664   assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then ereal (card X) else \<infinity>)"
```
```  1665     (is "_ = ?M X")
```
```  1666   unfolding count_space_def
```
```  1667 proof (rule emeasure_measure_of_sigma)
```
```  1668   show "X \<in> Pow A" using `X \<subseteq> A` by auto
```
```  1669   show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
```
```  1670   show positive: "positive (Pow A) ?M"
```
```  1671     by (auto simp: positive_def)
```
```  1672   have additive: "additive (Pow A) ?M"
```
```  1673     by (auto simp: card_Un_disjoint additive_def)
```
```  1674
```
```  1675   interpret ring_of_sets A "Pow A"
```
```  1676     by (rule ring_of_setsI) auto
```
```  1677   show "countably_additive (Pow A) ?M"
```
```  1678     unfolding countably_additive_iff_continuous_from_below[OF positive additive]
```
```  1679   proof safe
```
```  1680     fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"
```
```  1681     show "(\<lambda>i. ?M (F i)) ----> ?M (\<Union>i. F i)"
```
```  1682     proof cases
```
```  1683       assume "\<exists>i. \<forall>j\<ge>i. F i = F j"
```
```  1684       then guess i .. note i = this
```
```  1685       { fix j from i `incseq F` have "F j \<subseteq> F i"
```
```  1686           by (cases "i \<le> j") (auto simp: incseq_def) }
```
```  1687       then have eq: "(\<Union>i. F i) = F i"
```
```  1688         by auto
```
```  1689       with i show ?thesis
```
```  1690         by (auto intro!: Lim_eventually eventually_sequentiallyI[where c=i])
```
```  1691     next
```
```  1692       assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"
```
```  1693       then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis
```
```  1694       then have "\<And>i. F i \<subseteq> F (f i)" using `incseq F` by (auto simp: incseq_def)
```
```  1695       with f have *: "\<And>i. F i \<subset> F (f i)" by auto
```
```  1696
```
```  1697       have "incseq (\<lambda>i. ?M (F i))"
```
```  1698         using `incseq F` unfolding incseq_def by (auto simp: card_mono dest: finite_subset)
```
```  1699       then have "(\<lambda>i. ?M (F i)) ----> (SUP n. ?M (F n))"
```
```  1700         by (rule LIMSEQ_SUP)
```
```  1701
```
```  1702       moreover have "(SUP n. ?M (F n)) = \<infinity>"
```
```  1703       proof (rule SUP_PInfty)
```
```  1704         fix n :: nat show "\<exists>k::nat\<in>UNIV. ereal n \<le> ?M (F k)"
```
```  1705         proof (induct n)
```
```  1706           case (Suc n)
```
```  1707           then guess k .. note k = this
```
```  1708           moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"
```
```  1709             using `F k \<subset> F (f k)` by (simp add: psubset_card_mono)
```
```  1710           moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"
```
```  1711             using `k \<le> f k` `incseq F` by (auto simp: incseq_def dest: finite_subset)
```
```  1712           ultimately show ?case
```
```  1713             by (auto intro!: exI[of _ "f k"])
```
```  1714         qed auto
```
```  1715       qed
```
```  1716
```
```  1717       moreover
```
```  1718       have "inj (\<lambda>n. F ((f ^^ n) 0))"
```
```  1719         by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)
```
```  1720       then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))"
```
```  1721         by (rule range_inj_infinite)
```
```  1722       have "infinite (Pow (\<Union>i. F i))"
```
```  1723         by (rule infinite_super[OF _ 1]) auto
```
```  1724       then have "infinite (\<Union>i. F i)"
```
```  1725         by auto
```
```  1726
```
```  1727       ultimately show ?thesis by auto
```
```  1728     qed
```
```  1729   qed
```
```  1730 qed
```
```  1731
```
```  1732 lemma distr_bij_count_space:
```
```  1733   assumes f: "bij_betw f A B"
```
```  1734   shows "distr (count_space A) (count_space B) f = count_space B"
```
```  1735 proof (rule measure_eqI)
```
```  1736   have f': "f \<in> measurable (count_space A) (count_space B)"
```
```  1737     using f unfolding Pi_def bij_betw_def by auto
```
```  1738   fix X assume "X \<in> sets (distr (count_space A) (count_space B) f)"
```
```  1739   then have X: "X \<in> sets (count_space B)" by auto
```
```  1740   moreover then have "f -` X \<inter> A = the_inv_into A f ` X"
```
```  1741     using f by (auto simp: bij_betw_def subset_image_iff image_iff the_inv_into_f_f intro: the_inv_into_f_f[symmetric])
```
```  1742   moreover have "inj_on (the_inv_into A f) B"
```
```  1743     using X f by (auto simp: bij_betw_def inj_on_the_inv_into)
```
```  1744   with X have "inj_on (the_inv_into A f) X"
```
```  1745     by (auto intro: subset_inj_on)
```
```  1746   ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"
```
```  1747     using f unfolding emeasure_distr[OF f' X]
```
```  1748     by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)
```
```  1749 qed simp
```
```  1750
```
```  1751 lemma emeasure_count_space_finite[simp]:
```
```  1752   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = ereal (card X)"
```
```  1753   using emeasure_count_space[of X A] by simp
```
```  1754
```
```  1755 lemma emeasure_count_space_infinite[simp]:
```
```  1756   "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"
```
```  1757   using emeasure_count_space[of X A] by simp
```
```  1758
```
```  1759 lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then card X else 0)"
```
```  1760   unfolding measure_def
```
```  1761   by (cases "finite X") (simp_all add: emeasure_notin_sets)
```
```  1762
```
```  1763 lemma emeasure_count_space_eq_0:
```
```  1764   "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"
```
```  1765 proof cases
```
```  1766   assume X: "X \<subseteq> A"
```
```  1767   then show ?thesis
```
```  1768   proof (intro iffI impI)
```
```  1769     assume "emeasure (count_space A) X = 0"
```
```  1770     with X show "X = {}"
```
```  1771       by (subst (asm) emeasure_count_space) (auto split: split_if_asm)
```
```  1772   qed simp
```
```  1773 qed (simp add: emeasure_notin_sets)
```
```  1774
```
```  1775 lemma space_empty: "space M = {} \<Longrightarrow> M = count_space {}"
```
```  1776   by (rule measure_eqI) (simp_all add: space_empty_iff)
```
```  1777
```
```  1778 lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
```
```  1779   unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
```
```  1780
```
```  1781 lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
```
```  1782   unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)
```
```  1783
```
```  1784 lemma sigma_finite_measure_count_space_countable:
```
```  1785   assumes A: "countable A"
```
```  1786   shows "sigma_finite_measure (count_space A)"
```
```  1787   proof qed (auto intro!: exI[of _ "(\<lambda>a. {a}) ` A"] simp: A)
```
```  1788
```
```  1789 lemma sigma_finite_measure_count_space:
```
```  1790   fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"
```
```  1791   by (rule sigma_finite_measure_count_space_countable) auto
```
```  1792
```
```  1793 lemma finite_measure_count_space:
```
```  1794   assumes [simp]: "finite A"
```
```  1795   shows "finite_measure (count_space A)"
```
```  1796   by rule simp
```
```  1797
```
```  1798 lemma sigma_finite_measure_count_space_finite:
```
```  1799   assumes A: "finite A" shows "sigma_finite_measure (count_space A)"
```
```  1800 proof -
```
```  1801   interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)
```
```  1802   show "sigma_finite_measure (count_space A)" ..
```
```  1803 qed
```
```  1804
```
```  1805 subsection {* Measure restricted to space *}
```
```  1806
```
```  1807 lemma emeasure_restrict_space:
```
```  1808   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
```
```  1809   shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
```
```  1810 proof cases
```
```  1811   assume "A \<in> sets M"
```
```  1812   show ?thesis
```
```  1813   proof (rule emeasure_measure_of[OF restrict_space_def])
```
```  1814     show "op \<inter> \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"
```
```  1815       using `A \<subseteq> \<Omega>` `A \<in> sets M` sets.space_closed by (auto simp: sets_restrict_space)
```
```  1816     show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"
```
```  1817       by (auto simp: positive_def emeasure_nonneg)
```
```  1818     show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"
```
```  1819     proof (rule countably_additiveI)
```
```  1820       fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"
```
```  1821       with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"
```
```  1822         by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff
```
```  1823                       dest: sets.sets_into_space)+
```
```  1824       then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
```
```  1825         by (subst suminf_emeasure) (auto simp: disjoint_family_subset)
```
```  1826     qed
```
```  1827   qed
```
```  1828 next
```
```  1829   assume "A \<notin> sets M"
```
```  1830   moreover with assms have "A \<notin> sets (restrict_space M \<Omega>)"
```
```  1831     by (simp add: sets_restrict_space_iff)
```
```  1832   ultimately show ?thesis
```
```  1833     by (simp add: emeasure_notin_sets)
```
```  1834 qed
```
```  1835
```
```  1836 lemma measure_restrict_space:
```
```  1837   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
```
```  1838   shows "measure (restrict_space M \<Omega>) A = measure M A"
```
```  1839   using emeasure_restrict_space[OF assms] by (simp add: measure_def)
```
```  1840
```
```  1841 lemma AE_restrict_space_iff:
```
```  1842   assumes "\<Omega> \<inter> space M \<in> sets M"
```
```  1843   shows "(AE x in restrict_space M \<Omega>. P x) \<longleftrightarrow> (AE x in M. x \<in> \<Omega> \<longrightarrow> P x)"
```
```  1844 proof -
```
```  1845   have ex_cong: "\<And>P Q f. (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> (\<And>x. Q x \<Longrightarrow> P (f x)) \<Longrightarrow> (\<exists>x. P x) \<longleftrightarrow> (\<exists>x. Q x)"
```
```  1846     by auto
```
```  1847   { fix X assume X: "X \<in> sets M" "emeasure M X = 0"
```
```  1848     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"
```
```  1849       by (intro emeasure_mono) auto
```
```  1850     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"
```
```  1851       using X by (auto intro!: antisym) }
```
```  1852   with assms show ?thesis
```
```  1853     unfolding eventually_ae_filter
```
```  1854     by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff
```
```  1855                        emeasure_restrict_space cong: conj_cong
```
```  1856              intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])
```
```  1857 qed
```
```  1858
```
```  1859 lemma restrict_restrict_space:
```
```  1860   assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"
```
```  1861   shows "restrict_space (restrict_space M A) B = restrict_space M (A \<inter> B)" (is "?l = ?r")
```
```  1862 proof (rule measure_eqI[symmetric])
```
```  1863   show "sets ?r = sets ?l"
```
```  1864     unfolding sets_restrict_space image_comp by (intro image_cong) auto
```
```  1865 next
```
```  1866   fix X assume "X \<in> sets (restrict_space M (A \<inter> B))"
```
```  1867   then obtain Y where "Y \<in> sets M" "X = Y \<inter> A \<inter> B"
```
```  1868     by (auto simp: sets_restrict_space)
```
```  1869   with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"
```
```  1870     by (subst (1 2) emeasure_restrict_space)
```
```  1871        (auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)
```
```  1872 qed
```
```  1873
```
```  1874 lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)"
```
```  1875 proof (rule measure_eqI)
```
```  1876   show "sets (restrict_space (count_space B) A) = sets (count_space (A \<inter> B))"
```
```  1877     by (subst sets_restrict_space) auto
```
```  1878   moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"
```
```  1879   ultimately have "X \<subseteq> A \<inter> B" by auto
```
```  1880   then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"
```
```  1881     by (cases "finite X") (auto simp add: emeasure_restrict_space)
```
```  1882 qed
```
```  1883
```
```  1884 lemma sigma_finite_measure_restrict_space:
```
```  1885   assumes "sigma_finite_measure M"
```
```  1886   and A: "A \<in> sets M"
```
```  1887   shows "sigma_finite_measure (restrict_space M A)"
```
```  1888 proof -
```
```  1889   interpret sigma_finite_measure M by fact
```
```  1890   from sigma_finite_countable obtain C
```
```  1891     where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"
```
```  1892     by blast
```
```  1893   let ?C = "op \<inter> A ` C"
```
```  1894   from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"
```
```  1895     by(auto simp add: sets_restrict_space space_restrict_space)
```
```  1896   moreover {
```
```  1897     fix a
```
```  1898     assume "a \<in> ?C"
```
```  1899     then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..
```
```  1900     then have "emeasure (restrict_space M A) a \<le> emeasure M a'"
```
```  1901       using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)
```
```  1902     also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by simp
```
```  1903     finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }
```
```  1904   ultimately show ?thesis
```
```  1905     by unfold_locales (rule exI conjI|assumption|blast)+
```
```  1906 qed
```
```  1907
```
```  1908 lemma finite_measure_restrict_space:
```
```  1909   assumes "finite_measure M"
```
```  1910   and A: "A \<in> sets M"
```
```  1911   shows "finite_measure (restrict_space M A)"
```
```  1912 proof -
```
```  1913   interpret finite_measure M by fact
```
```  1914   show ?thesis
```
```  1915     by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)
```
```  1916 qed
```
```  1917
```
```  1918 lemma restrict_distr:
```
```  1919   assumes [measurable]: "f \<in> measurable M N"
```
```  1920   assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"
```
```  1921   shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"
```
```  1922   (is "?l = ?r")
```
```  1923 proof (rule measure_eqI)
```
```  1924   fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"
```
```  1925   with restrict show "emeasure ?l A = emeasure ?r A"
```
```  1926     by (subst emeasure_distr)
```
```  1927        (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr
```
```  1928              intro!: measurable_restrict_space2)
```
```  1929 qed (simp add: sets_restrict_space)
```
```  1930
```
```  1931 lemma measure_eqI_restrict_generator:
```
```  1932   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
```
```  1933   assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"
```
```  1934   assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"
```
```  1935   assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"
```
```  1936   assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"
```
```  1937   assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
```
```  1938   shows "M = N"
```
```  1939 proof (rule measure_eqI)
```
```  1940   fix X assume X: "X \<in> sets M"
```
```  1941   then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"
```
```  1942     using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)
```
```  1943   also have "restrict_space M \<Omega> = restrict_space N \<Omega>"
```
```  1944   proof (rule measure_eqI_generator_eq)
```
```  1945     fix X assume "X \<in> E"
```
```  1946     then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"
```
```  1947       using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])
```
```  1948   next
```
```  1949     show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"
```
```  1950       unfolding Sup_image_eq[symmetric, where f="from_nat_into A"] using A by auto
```
```  1951   next
```
```  1952     fix i
```
```  1953     have "emeasure (restrict_space M \<Omega>) (from_nat_into A i) = emeasure M (from_nat_into A i)"
```
```  1954       using A \<Omega> by (subst emeasure_restrict_space)
```
```  1955                    (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)
```
```  1956     with A show "emeasure (restrict_space M \<Omega>) (from_nat_into A i) \<noteq> \<infinity>"
```
```  1957       by (auto intro: from_nat_into)
```
```  1958   qed fact+
```
```  1959   also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"
```
```  1960     using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)
```
```  1961   finally show "emeasure M X = emeasure N X" .
```
```  1962 qed fact
```
```  1963
```
```  1964 subsection {* Null measure *}
```
```  1965
```
```  1966 definition "null_measure M = sigma (space M) (sets M)"
```
```  1967
```
```  1968 lemma space_null_measure[simp]: "space (null_measure M) = space M"
```
```  1969   by (simp add: null_measure_def)
```
```  1970
```
```  1971 lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"
```
```  1972   by (simp add: null_measure_def)
```
```  1973
```
```  1974 lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"
```
```  1975   by (cases "X \<in> sets M", rule emeasure_measure_of)
```
```  1976      (auto simp: positive_def countably_additive_def emeasure_notin_sets null_measure_def
```
```  1977            dest: sets.sets_into_space)
```
```  1978
```
```  1979 lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"
```
```  1980   by (simp add: measure_def)
```
```  1981
```
```  1982 end
```
```  1983
```