src/HOL/Probability/Measure.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 43920 cedb5cb948fd child 44890 22f665a2e91c permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     1 (*  Title:      HOL/Probability/Measure.thy
```
```     2     Author:     Lawrence C Paulson
```
```     3     Author:     Johannes Hölzl, TU München
```
```     4     Author:     Armin Heller, TU München
```
```     5 *)
```
```     6
```
```     7 header {* Properties about measure spaces *}
```
```     8
```
```     9 theory Measure
```
```    10   imports Caratheodory
```
```    11 begin
```
```    12
```
```    13 lemma measure_algebra_more[simp]:
```
```    14   "\<lparr> space = A, sets = B, \<dots> = algebra.more M \<rparr> \<lparr> measure := m \<rparr> =
```
```    15    \<lparr> space = A, sets = B, \<dots> = algebra.more (M \<lparr> measure := m \<rparr>) \<rparr>"
```
```    16   by (cases M) simp
```
```    17
```
```    18 lemma measure_algebra_more_eq[simp]:
```
```    19   "\<And>X. measure \<lparr> space = T, sets = A, \<dots> = algebra.more X \<rparr> = measure X"
```
```    20   unfolding measure_space.splits by simp
```
```    21
```
```    22 lemma measure_sigma[simp]: "measure (sigma A) = measure A"
```
```    23   unfolding sigma_def by simp
```
```    24
```
```    25 lemma algebra_measure_update[simp]:
```
```    26   "algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> algebra M'"
```
```    27   unfolding algebra_iff_Un by simp
```
```    28
```
```    29 lemma sigma_algebra_measure_update[simp]:
```
```    30   "sigma_algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> sigma_algebra M'"
```
```    31   unfolding sigma_algebra_def sigma_algebra_axioms_def by simp
```
```    32
```
```    33 lemma finite_sigma_algebra_measure_update[simp]:
```
```    34   "finite_sigma_algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> finite_sigma_algebra M'"
```
```    35   unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
```
```    36
```
```    37 lemma measurable_cancel_measure[simp]:
```
```    38   "measurable M1 (M2\<lparr>measure := m2\<rparr>) = measurable M1 M2"
```
```    39   "measurable (M2\<lparr>measure := m1\<rparr>) M1 = measurable M2 M1"
```
```    40   unfolding measurable_def by auto
```
```    41
```
```    42 lemma inj_on_image_eq_iff:
```
```    43   assumes "inj_on f S"
```
```    44   assumes "A \<subseteq> S" "B \<subseteq> S"
```
```    45   shows "(f ` A = f ` B) \<longleftrightarrow> (A = B)"
```
```    46 proof -
```
```    47   have "inj_on f (A \<union> B)"
```
```    48     using assms by (auto intro: subset_inj_on)
```
```    49   from inj_on_Un_image_eq_iff[OF this]
```
```    50   show ?thesis .
```
```    51 qed
```
```    52
```
```    53 lemma image_vimage_inter_eq:
```
```    54   assumes "f ` S = T" "X \<subseteq> T"
```
```    55   shows "f ` (f -` X \<inter> S) = X"
```
```    56 proof (intro antisym)
```
```    57   have "f ` (f -` X \<inter> S) \<subseteq> f ` (f -` X)" by auto
```
```    58   also have "\<dots> = X \<inter> range f" by simp
```
```    59   also have "\<dots> = X" using assms by auto
```
```    60   finally show "f ` (f -` X \<inter> S) \<subseteq> X" by auto
```
```    61 next
```
```    62   show "X \<subseteq> f ` (f -` X \<inter> S)"
```
```    63   proof
```
```    64     fix x assume "x \<in> X"
```
```    65     then have "x \<in> T" using `X \<subseteq> T` by auto
```
```    66     then obtain y where "x = f y" "y \<in> S"
```
```    67       using assms by auto
```
```    68     then have "{y} \<subseteq> f -` X \<inter> S" using `x \<in> X` by auto
```
```    69     moreover have "x \<in> f ` {y}" using `x = f y` by auto
```
```    70     ultimately show "x \<in> f ` (f -` X \<inter> S)" by auto
```
```    71   qed
```
```    72 qed
```
```    73
```
```    74 text {*
```
```    75   This formalisation of measure theory is based on the work of Hurd/Coble wand
```
```    76   was later translated by Lawrence Paulson to Isabelle/HOL. Later it was
```
```    77   modified to use the positive infinite reals and to prove the uniqueness of
```
```    78   cut stable measures.
```
```    79 *}
```
```    80
```
```    81 section {* Equations for the measure function @{text \<mu>} *}
```
```    82
```
```    83 lemma (in measure_space) measure_countably_additive:
```
```    84   assumes "range A \<subseteq> sets M" "disjoint_family A"
```
```    85   shows "(\<Sum>i. \<mu> (A i)) = \<mu> (\<Union>i. A i)"
```
```    86 proof -
```
```    87   have "(\<Union> i. A i) \<in> sets M" using assms(1) by (rule countable_UN)
```
```    88   with ca assms show ?thesis by (simp add: countably_additive_def)
```
```    89 qed
```
```    90
```
```    91 lemma (in sigma_algebra) sigma_algebra_cong:
```
```    92   assumes "space N = space M" "sets N = sets M"
```
```    93   shows "sigma_algebra N"
```
```    94   by default (insert sets_into_space, auto simp: assms)
```
```    95
```
```    96 lemma (in measure_space) measure_space_cong:
```
```    97   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M"
```
```    98   shows "measure_space N"
```
```    99 proof -
```
```   100   interpret N: sigma_algebra N by (intro sigma_algebra_cong assms)
```
```   101   show ?thesis
```
```   102   proof
```
```   103     show "positive N (measure N)" using assms by (auto simp: positive_def)
```
```   104     show "countably_additive N (measure N)" unfolding countably_additive_def
```
```   105     proof safe
```
```   106       fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets N" "disjoint_family A"
```
```   107       then have "\<And>i. A i \<in> sets M" "(UNION UNIV A) \<in> sets M" unfolding assms by auto
```
```   108       from measure_countably_additive[of A] A this[THEN assms(1)]
```
```   109       show "(\<Sum>n. measure N (A n)) = measure N (UNION UNIV A)"
```
```   110         unfolding assms by simp
```
```   111     qed
```
```   112   qed
```
```   113 qed
```
```   114
```
```   115 lemma (in measure_space) additive: "additive M \<mu>"
```
```   116   using ca by (auto intro!: countably_additive_additive simp: positive_def)
```
```   117
```
```   118 lemma (in measure_space) measure_additive:
```
```   119      "a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {}
```
```   120       \<Longrightarrow> \<mu> a + \<mu> b = \<mu> (a \<union> b)"
```
```   121   by (metis additiveD additive)
```
```   122
```
```   123 lemma (in measure_space) measure_mono:
```
```   124   assumes "a \<subseteq> b" "a \<in> sets M" "b \<in> sets M"
```
```   125   shows "\<mu> a \<le> \<mu> b"
```
```   126 proof -
```
```   127   have "b = a \<union> (b - a)" using assms by auto
```
```   128   moreover have "{} = a \<inter> (b - a)" by auto
```
```   129   ultimately have "\<mu> b = \<mu> a + \<mu> (b - a)"
```
```   130     using measure_additive[of a "b - a"] Diff[of b a] assms by auto
```
```   131   moreover have "\<mu> a + 0 \<le> \<mu> a + \<mu> (b - a)" using assms by (intro add_mono) auto
```
```   132   ultimately show "\<mu> a \<le> \<mu> b" by auto
```
```   133 qed
```
```   134
```
```   135 lemma (in measure_space) measure_top:
```
```   136   "A \<in> sets M \<Longrightarrow> \<mu> A \<le> \<mu> (space M)"
```
```   137   using sets_into_space[of A] by (auto intro!: measure_mono)
```
```   138
```
```   139 lemma (in measure_space) measure_compl:
```
```   140   assumes s: "s \<in> sets M" and fin: "\<mu> s \<noteq> \<infinity>"
```
```   141   shows "\<mu> (space M - s) = \<mu> (space M) - \<mu> s"
```
```   142 proof -
```
```   143   have s_less_space: "\<mu> s \<le> \<mu> (space M)"
```
```   144     using s by (auto intro!: measure_mono sets_into_space)
```
```   145   from s have "0 \<le> \<mu> s" by auto
```
```   146   have "\<mu> (space M) = \<mu> (s \<union> (space M - s))" using s
```
```   147     by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
```
```   148   also have "... = \<mu> s + \<mu> (space M - s)"
```
```   149     by (rule additiveD [OF additive]) (auto simp add: s)
```
```   150   finally have "\<mu> (space M) = \<mu> s + \<mu> (space M - s)" .
```
```   151   then show ?thesis
```
```   152     using fin `0 \<le> \<mu> s`
```
```   153     unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
```
```   154 qed
```
```   155
```
```   156 lemma (in measure_space) measure_Diff:
```
```   157   assumes finite: "\<mu> B \<noteq> \<infinity>"
```
```   158   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
```
```   159   shows "\<mu> (A - B) = \<mu> A - \<mu> B"
```
```   160 proof -
```
```   161   have "0 \<le> \<mu> B" using assms by auto
```
```   162   have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
```
```   163   then have "\<mu> A = \<mu> ((A - B) \<union> B)" by simp
```
```   164   also have "\<dots> = \<mu> (A - B) + \<mu> B"
```
```   165     using measurable by (subst measure_additive[symmetric]) auto
```
```   166   finally show "\<mu> (A - B) = \<mu> A - \<mu> B"
```
```   167     unfolding ereal_eq_minus_iff
```
```   168     using finite `0 \<le> \<mu> B` by auto
```
```   169 qed
```
```   170
```
```   171 lemma (in measure_space) measure_countable_increasing:
```
```   172   assumes A: "range A \<subseteq> sets M"
```
```   173       and A0: "A 0 = {}"
```
```   174       and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
```
```   175   shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
```
```   176 proof -
```
```   177   { fix n
```
```   178     have "\<mu> (A n) = (\<Sum>i<n. \<mu> (A (Suc i) - A i))"
```
```   179       proof (induct n)
```
```   180         case 0 thus ?case by (auto simp add: A0)
```
```   181       next
```
```   182         case (Suc m)
```
```   183         have "A (Suc m) = A m \<union> (A (Suc m) - A m)"
```
```   184           by (metis ASuc Un_Diff_cancel Un_absorb1)
```
```   185         hence "\<mu> (A (Suc m)) =
```
```   186                \<mu> (A m) + \<mu> (A (Suc m) - A m)"
```
```   187           by (subst measure_additive)
```
```   188              (auto simp add: measure_additive range_subsetD [OF A])
```
```   189         with Suc show ?case
```
```   190           by simp
```
```   191       qed }
```
```   192   note Meq = this
```
```   193   have Aeq: "(\<Union>i. A (Suc i) - A i) = (\<Union>i. A i)"
```
```   194     proof (rule UN_finite2_eq [where k=1], simp)
```
```   195       fix i
```
```   196       show "(\<Union>i\<in>{0..<i}. A (Suc i) - A i) = (\<Union>i\<in>{0..<Suc i}. A i)"
```
```   197         proof (induct i)
```
```   198           case 0 thus ?case by (simp add: A0)
```
```   199         next
```
```   200           case (Suc i)
```
```   201           thus ?case
```
```   202             by (auto simp add: atLeastLessThanSuc intro: subsetD [OF ASuc])
```
```   203         qed
```
```   204     qed
```
```   205   have A1: "\<And>i. A (Suc i) - A i \<in> sets M"
```
```   206     by (metis A Diff range_subsetD)
```
```   207   have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
```
```   208     by (blast intro: range_subsetD [OF A])
```
```   209   have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = (\<Sum>i. \<mu> (A (Suc i) - A i))"
```
```   210     using A by (auto intro!: suminf_ereal_eq_SUPR[symmetric])
```
```   211   also have "\<dots> = \<mu> (\<Union>i. A (Suc i) - A i)"
```
```   212     by (rule measure_countably_additive)
```
```   213        (auto simp add: disjoint_family_Suc ASuc A1 A2)
```
```   214   also have "... =  \<mu> (\<Union>i. A i)"
```
```   215     by (simp add: Aeq)
```
```   216   finally have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = \<mu> (\<Union>i. A i)" .
```
```   217   then show ?thesis by (auto simp add: Meq)
```
```   218 qed
```
```   219
```
```   220 lemma (in measure_space) continuity_from_below:
```
```   221   assumes A: "range A \<subseteq> sets M" and "incseq A"
```
```   222   shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
```
```   223 proof -
```
```   224   have *: "(SUP n. \<mu> (nat_case {} A (Suc n))) = (SUP n. \<mu> (nat_case {} A n))"
```
```   225     using A by (auto intro!: SUPR_eq exI split: nat.split)
```
```   226   have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
```
```   227     by (auto simp add: split: nat.splits)
```
```   228   have meq: "\<And>n. \<mu> (A n) = (\<mu> \<circ> nat_case {} A) (Suc n)"
```
```   229     by simp
```
```   230   have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. nat_case {} A i)"
```
```   231     using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
```
```   232     by (force split: nat.splits intro!: measure_countable_increasing)
```
```   233   also have "\<mu> (\<Union>i. nat_case {} A i) = \<mu> (\<Union>i. A i)"
```
```   234     by (simp add: ueq)
```
```   235   finally have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. A i)" .
```
```   236   thus ?thesis unfolding meq * comp_def .
```
```   237 qed
```
```   238
```
```   239 lemma (in measure_space) measure_incseq:
```
```   240   assumes "range B \<subseteq> sets M" "incseq B"
```
```   241   shows "incseq (\<lambda>i. \<mu> (B i))"
```
```   242   using assms by (auto simp: incseq_def intro!: measure_mono)
```
```   243
```
```   244 lemma (in measure_space) continuity_from_below_Lim:
```
```   245   assumes A: "range A \<subseteq> sets M" "incseq A"
```
```   246   shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Union>i. A i)"
```
```   247   using LIMSEQ_ereal_SUPR[OF measure_incseq, OF A]
```
```   248     continuity_from_below[OF A] by simp
```
```   249
```
```   250 lemma (in measure_space) measure_decseq:
```
```   251   assumes "range B \<subseteq> sets M" "decseq B"
```
```   252   shows "decseq (\<lambda>i. \<mu> (B i))"
```
```   253   using assms by (auto simp: decseq_def intro!: measure_mono)
```
```   254
```
```   255 lemma (in measure_space) continuity_from_above:
```
```   256   assumes A: "range A \<subseteq> sets M" and "decseq A"
```
```   257   and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
```
```   258   shows "(INF n. \<mu> (A n)) = \<mu> (\<Inter>i. A i)"
```
```   259 proof -
```
```   260   have le_MI: "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
```
```   261     using A by (auto intro!: measure_mono)
```
```   262   hence *: "\<mu> (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
```
```   263
```
```   264   have A0: "0 \<le> \<mu> (A 0)" using A by auto
```
```   265
```
```   266   have "\<mu> (A 0) - (INF n. \<mu> (A n)) = \<mu> (A 0) + (SUP n. - \<mu> (A n))"
```
```   267     by (simp add: ereal_SUPR_uminus minus_ereal_def)
```
```   268   also have "\<dots> = (SUP n. \<mu> (A 0) - \<mu> (A n))"
```
```   269     unfolding minus_ereal_def using A0 assms
```
```   270     by (subst SUPR_ereal_add) (auto simp add: measure_decseq)
```
```   271   also have "\<dots> = (SUP n. \<mu> (A 0 - A n))"
```
```   272     using A finite `decseq A`[unfolded decseq_def] by (subst measure_Diff) auto
```
```   273   also have "\<dots> = \<mu> (\<Union>i. A 0 - A i)"
```
```   274   proof (rule continuity_from_below)
```
```   275     show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
```
```   276       using A by auto
```
```   277     show "incseq (\<lambda>n. A 0 - A n)"
```
```   278       using `decseq A` by (auto simp add: incseq_def decseq_def)
```
```   279   qed
```
```   280   also have "\<dots> = \<mu> (A 0) - \<mu> (\<Inter>i. A i)"
```
```   281     using A finite * by (simp, subst measure_Diff) auto
```
```   282   finally show ?thesis
```
```   283     unfolding ereal_minus_eq_minus_iff using finite A0 by auto
```
```   284 qed
```
```   285
```
```   286 lemma (in measure_space) measure_insert:
```
```   287   assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
```
```   288   shows "\<mu> (insert x A) = \<mu> {x} + \<mu> A"
```
```   289 proof -
```
```   290   have "{x} \<inter> A = {}" using `x \<notin> A` by auto
```
```   291   from measure_additive[OF sets this] show ?thesis by simp
```
```   292 qed
```
```   293
```
```   294 lemma (in measure_space) measure_setsum:
```
```   295   assumes "finite S" and "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
```
```   296   assumes disj: "disjoint_family_on A S"
```
```   297   shows "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>i\<in>S. A i)"
```
```   298 using assms proof induct
```
```   299   case (insert i S)
```
```   300   then have "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>a\<in>S. A a)"
```
```   301     by (auto intro: disjoint_family_on_mono)
```
```   302   moreover have "A i \<inter> (\<Union>a\<in>S. A a) = {}"
```
```   303     using `disjoint_family_on A (insert i S)` `i \<notin> S`
```
```   304     by (auto simp: disjoint_family_on_def)
```
```   305   ultimately show ?case using insert
```
```   306     by (auto simp: measure_additive finite_UN)
```
```   307 qed simp
```
```   308
```
```   309 lemma (in measure_space) measure_finite_singleton:
```
```   310   assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```   311   shows "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})"
```
```   312   using measure_setsum[of S "\<lambda>x. {x}", OF assms]
```
```   313   by (auto simp: disjoint_family_on_def)
```
```   314
```
```   315 lemma finite_additivity_sufficient:
```
```   316   assumes "sigma_algebra M"
```
```   317   assumes fin: "finite (space M)" and pos: "positive M (measure M)" and add: "additive M (measure M)"
```
```   318   shows "measure_space M"
```
```   319 proof -
```
```   320   interpret sigma_algebra M by fact
```
```   321   show ?thesis
```
```   322   proof
```
```   323     show [simp]: "positive M (measure M)" using pos by (simp add: positive_def)
```
```   324     show "countably_additive M (measure M)"
```
```   325     proof (auto simp add: countably_additive_def)
```
```   326       fix A :: "nat \<Rightarrow> 'a set"
```
```   327       assume A: "range A \<subseteq> sets M"
```
```   328          and disj: "disjoint_family A"
```
```   329          and UnA: "(\<Union>i. A i) \<in> sets M"
```
```   330       def I \<equiv> "{i. A i \<noteq> {}}"
```
```   331       have "Union (A ` I) \<subseteq> space M" using A
```
```   332         by auto (metis range_subsetD subsetD sets_into_space)
```
```   333       hence "finite (A ` I)"
```
```   334         by (metis finite_UnionD finite_subset fin)
```
```   335       moreover have "inj_on A I" using disj
```
```   336         by (auto simp add: I_def disjoint_family_on_def inj_on_def)
```
```   337       ultimately have finI: "finite I"
```
```   338         by (metis finite_imageD)
```
```   339       hence "\<exists>N. \<forall>m\<ge>N. A m = {}"
```
```   340         proof (cases "I = {}")
```
```   341           case True thus ?thesis by (simp add: I_def)
```
```   342         next
```
```   343           case False
```
```   344           hence "\<forall>i\<in>I. i < Suc(Max I)"
```
```   345             by (simp add: Max_less_iff [symmetric] finI)
```
```   346           hence "\<forall>m \<ge> Suc(Max I). A m = {}"
```
```   347             by (simp add: I_def) (metis less_le_not_le)
```
```   348           thus ?thesis
```
```   349             by blast
```
```   350         qed
```
```   351       then obtain N where N: "\<forall>m\<ge>N. A m = {}" by blast
```
```   352       then have "\<forall>m\<ge>N. measure M (A m) = 0" using pos[unfolded positive_def] by simp
```
```   353       then have "(\<Sum>n. measure M (A n)) = (\<Sum>m<N. measure M (A m))"
```
```   354         by (simp add: suminf_finite)
```
```   355       also have "... = measure M (\<Union>i<N. A i)"
```
```   356         proof (induct N)
```
```   357           case 0 thus ?case using pos[unfolded positive_def] by simp
```
```   358         next
```
```   359           case (Suc n)
```
```   360           have "measure M (A n \<union> (\<Union> x<n. A x)) = measure M (A n) + measure M (\<Union> i<n. A i)"
```
```   361             proof (rule Caratheodory.additiveD [OF add])
```
```   362               show "A n \<inter> (\<Union> x<n. A x) = {}" using disj
```
```   363                 by (auto simp add: disjoint_family_on_def nat_less_le) blast
```
```   364               show "A n \<in> sets M" using A
```
```   365                 by force
```
```   366               show "(\<Union>i<n. A i) \<in> sets M"
```
```   367                 proof (induct n)
```
```   368                   case 0 thus ?case by simp
```
```   369                 next
```
```   370                   case (Suc n) thus ?case using A
```
```   371                     by (simp add: lessThan_Suc Un range_subsetD)
```
```   372                 qed
```
```   373             qed
```
```   374           thus ?case using Suc
```
```   375             by (simp add: lessThan_Suc)
```
```   376         qed
```
```   377       also have "... = measure M (\<Union>i. A i)"
```
```   378         proof -
```
```   379           have "(\<Union> i<N. A i) = (\<Union>i. A i)" using N
```
```   380             by auto (metis Int_absorb N disjoint_iff_not_equal lessThan_iff not_leE)
```
```   381           thus ?thesis by simp
```
```   382         qed
```
```   383       finally show "(\<Sum>n. measure M (A n)) = measure M (\<Union>i. A i)" .
```
```   384     qed
```
```   385   qed
```
```   386 qed
```
```   387
```
```   388 lemma (in measure_space) measure_setsum_split:
```
```   389   assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
```
```   390   assumes "(\<Union>i\<in>S. B i) = space M"
```
```   391   assumes "disjoint_family_on B S"
```
```   392   shows "\<mu> A = (\<Sum>i\<in>S. \<mu> (A \<inter> (B i)))"
```
```   393 proof -
```
```   394   have *: "\<mu> A = \<mu> (\<Union>i\<in>S. A \<inter> B i)"
```
```   395     using assms by auto
```
```   396   show ?thesis unfolding *
```
```   397   proof (rule measure_setsum[symmetric])
```
```   398     show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
```
```   399       using `disjoint_family_on B S`
```
```   400       unfolding disjoint_family_on_def by auto
```
```   401   qed (insert assms, auto)
```
```   402 qed
```
```   403
```
```   404 lemma (in measure_space) measure_subadditive:
```
```   405   assumes measurable: "A \<in> sets M" "B \<in> sets M"
```
```   406   shows "\<mu> (A \<union> B) \<le> \<mu> A + \<mu> B"
```
```   407 proof -
```
```   408   from measure_additive[of A "B - A"]
```
```   409   have "\<mu> (A \<union> B) = \<mu> A + \<mu> (B - A)"
```
```   410     using assms by (simp add: Diff)
```
```   411   also have "\<dots> \<le> \<mu> A + \<mu> B"
```
```   412     using assms by (auto intro!: add_left_mono measure_mono)
```
```   413   finally show ?thesis .
```
```   414 qed
```
```   415
```
```   416 lemma (in measure_space) measure_subadditive_finite:
```
```   417   assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M"
```
```   418   shows "\<mu> (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu> (A i))"
```
```   419 using assms proof induct
```
```   420   case (insert i I)
```
```   421   then have "\<mu> (\<Union>i\<in>insert i I. A i) = \<mu> (A i \<union> (\<Union>i\<in>I. A i))"
```
```   422     by simp
```
```   423   also have "\<dots> \<le> \<mu> (A i) + \<mu> (\<Union>i\<in>I. A i)"
```
```   424     using insert by (intro measure_subadditive finite_UN) auto
```
```   425   also have "\<dots> \<le> \<mu> (A i) + (\<Sum>i\<in>I. \<mu> (A i))"
```
```   426     using insert by (intro add_mono) auto
```
```   427   also have "\<dots> = (\<Sum>i\<in>insert i I. \<mu> (A i))"
```
```   428     using insert by auto
```
```   429   finally show ?case .
```
```   430 qed simp
```
```   431
```
```   432 lemma (in measure_space) measure_eq_0:
```
```   433   assumes "N \<in> sets M" and "\<mu> N = 0" and "K \<subseteq> N" and "K \<in> sets M"
```
```   434   shows "\<mu> K = 0"
```
```   435   using measure_mono[OF assms(3,4,1)] assms(2) positive_measure[OF assms(4)] by auto
```
```   436
```
```   437 lemma (in measure_space) measure_finitely_subadditive:
```
```   438   assumes "finite I" "A ` I \<subseteq> sets M"
```
```   439   shows "\<mu> (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu> (A i))"
```
```   440 using assms proof induct
```
```   441   case (insert i I)
```
```   442   then have "(\<Union>i\<in>I. A i) \<in> sets M" by (auto intro: finite_UN)
```
```   443   then have "\<mu> (\<Union>i\<in>insert i I. A i) \<le> \<mu> (A i) + \<mu> (\<Union>i\<in>I. A i)"
```
```   444     using insert by (simp add: measure_subadditive)
```
```   445   also have "\<dots> \<le> (\<Sum>i\<in>insert i I. \<mu> (A i))"
```
```   446     using insert by (auto intro!: add_left_mono)
```
```   447   finally show ?case .
```
```   448 qed simp
```
```   449
```
```   450 lemma (in measure_space) measure_countably_subadditive:
```
```   451   assumes "range f \<subseteq> sets M"
```
```   452   shows "\<mu> (\<Union>i. f i) \<le> (\<Sum>i. \<mu> (f i))"
```
```   453 proof -
```
```   454   have "\<mu> (\<Union>i. f i) = \<mu> (\<Union>i. disjointed f i)"
```
```   455     unfolding UN_disjointed_eq ..
```
```   456   also have "\<dots> = (\<Sum>i. \<mu> (disjointed f i))"
```
```   457     using range_disjointed_sets[OF assms] measure_countably_additive
```
```   458     by (simp add:  disjoint_family_disjointed comp_def)
```
```   459   also have "\<dots> \<le> (\<Sum>i. \<mu> (f i))"
```
```   460     using range_disjointed_sets[OF assms] assms
```
```   461     by (auto intro!: suminf_le_pos measure_mono positive_measure disjointed_subset)
```
```   462   finally show ?thesis .
```
```   463 qed
```
```   464
```
```   465 lemma (in measure_space) measure_UN_eq_0:
```
```   466   assumes "\<And>i::nat. \<mu> (N i) = 0" and "range N \<subseteq> sets M"
```
```   467   shows "\<mu> (\<Union> i. N i) = 0"
```
```   468 proof -
```
```   469   have "0 \<le> \<mu> (\<Union> i. N i)" using assms by auto
```
```   470   moreover have "\<mu> (\<Union> i. N i) \<le> 0"
```
```   471     using measure_countably_subadditive[OF assms(2)] assms(1) by simp
```
```   472   ultimately show ?thesis by simp
```
```   473 qed
```
```   474
```
```   475 lemma (in measure_space) measure_inter_full_set:
```
```   476   assumes "S \<in> sets M" "T \<in> sets M" and fin: "\<mu> (T - S) \<noteq> \<infinity>"
```
```   477   assumes T: "\<mu> T = \<mu> (space M)"
```
```   478   shows "\<mu> (S \<inter> T) = \<mu> S"
```
```   479 proof (rule antisym)
```
```   480   show " \<mu> (S \<inter> T) \<le> \<mu> S"
```
```   481     using assms by (auto intro!: measure_mono)
```
```   482
```
```   483   have pos: "0 \<le> \<mu> (T - S)" using assms by auto
```
```   484   show "\<mu> S \<le> \<mu> (S \<inter> T)"
```
```   485   proof (rule ccontr)
```
```   486     assume contr: "\<not> ?thesis"
```
```   487     have "\<mu> (space M) = \<mu> ((T - S) \<union> (S \<inter> T))"
```
```   488       unfolding T[symmetric] by (auto intro!: arg_cong[where f="\<mu>"])
```
```   489     also have "\<dots> \<le> \<mu> (T - S) + \<mu> (S \<inter> T)"
```
```   490       using assms by (auto intro!: measure_subadditive)
```
```   491     also have "\<dots> < \<mu> (T - S) + \<mu> S"
```
```   492       using fin contr pos by (intro ereal_less_add) auto
```
```   493     also have "\<dots> = \<mu> (T \<union> S)"
```
```   494       using assms by (subst measure_additive) auto
```
```   495     also have "\<dots> \<le> \<mu> (space M)"
```
```   496       using assms sets_into_space by (auto intro!: measure_mono)
```
```   497     finally show False ..
```
```   498   qed
```
```   499 qed
```
```   500
```
```   501 lemma measure_unique_Int_stable:
```
```   502   fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
```
```   503   assumes "Int_stable E"
```
```   504   and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E"
```
```   505   and M: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<mu>\<rparr>" (is "measure_space ?M")
```
```   506   and N: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<nu>\<rparr>" (is "measure_space ?N")
```
```   507   and eq: "\<And>X. X \<in> sets E \<Longrightarrow> \<mu> X = \<nu> X"
```
```   508   and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
```
```   509   assumes "X \<in> sets (sigma E)"
```
```   510   shows "\<mu> X = \<nu> X"
```
```   511 proof -
```
```   512   let "?D F" = "{D. D \<in> sets (sigma E) \<and> \<mu> (F \<inter> D) = \<nu> (F \<inter> D)}"
```
```   513   interpret M: measure_space ?M
```
```   514     where "space ?M = space E" and "sets ?M = sets (sigma E)" and "measure ?M = \<mu>" by (simp_all add: M)
```
```   515   interpret N: measure_space ?N
```
```   516     where "space ?N = space E" and "sets ?N = sets (sigma E)" and "measure ?N = \<nu>" by (simp_all add: N)
```
```   517   { fix F assume "F \<in> sets E" and "\<mu> F \<noteq> \<infinity>"
```
```   518     then have [intro]: "F \<in> sets (sigma E)" by auto
```
```   519     have "\<nu> F \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` `F \<in> sets E` eq by simp
```
```   520     interpret D: dynkin_system "\<lparr>space=space E, sets=?D F\<rparr>"
```
```   521     proof (rule dynkin_systemI, simp_all)
```
```   522       fix A assume "A \<in> sets (sigma E) \<and> \<mu> (F \<inter> A) = \<nu> (F \<inter> A)"
```
```   523       then show "A \<subseteq> space E" using M.sets_into_space by auto
```
```   524     next
```
```   525       have "F \<inter> space E = F" using `F \<in> sets E` by auto
```
```   526       then show "\<mu> (F \<inter> space E) = \<nu> (F \<inter> space E)"
```
```   527         using `F \<in> sets E` eq by auto
```
```   528     next
```
```   529       fix A assume *: "A \<in> sets (sigma E) \<and> \<mu> (F \<inter> A) = \<nu> (F \<inter> A)"
```
```   530       then have **: "F \<inter> (space (sigma E) - A) = F - (F \<inter> A)"
```
```   531         and [intro]: "F \<inter> A \<in> sets (sigma E)"
```
```   532         using `F \<in> sets E` M.sets_into_space by auto
```
```   533       have "\<nu> (F \<inter> A) \<le> \<nu> F" by (auto intro!: N.measure_mono)
```
```   534       then have "\<nu> (F \<inter> A) \<noteq> \<infinity>" using `\<nu> F \<noteq> \<infinity>` by auto
```
```   535       have "\<mu> (F \<inter> A) \<le> \<mu> F" by (auto intro!: M.measure_mono)
```
```   536       then have "\<mu> (F \<inter> A) \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` by auto
```
```   537       then have "\<mu> (F \<inter> (space (sigma E) - A)) = \<mu> F - \<mu> (F \<inter> A)" unfolding **
```
```   538         using `F \<inter> A \<in> sets (sigma E)` by (auto intro!: M.measure_Diff)
```
```   539       also have "\<dots> = \<nu> F - \<nu> (F \<inter> A)" using eq `F \<in> sets E` * by simp
```
```   540       also have "\<dots> = \<nu> (F \<inter> (space (sigma E) - A))" unfolding **
```
```   541         using `F \<inter> A \<in> sets (sigma E)` `\<nu> (F \<inter> A) \<noteq> \<infinity>` by (auto intro!: N.measure_Diff[symmetric])
```
```   542       finally show "space E - A \<in> sets (sigma E) \<and> \<mu> (F \<inter> (space E - A)) = \<nu> (F \<inter> (space E - A))"
```
```   543         using * by auto
```
```   544     next
```
```   545       fix A :: "nat \<Rightarrow> 'a set"
```
```   546       assume "disjoint_family A" "range A \<subseteq> {X \<in> sets (sigma E). \<mu> (F \<inter> X) = \<nu> (F \<inter> X)}"
```
```   547       then have A: "range (\<lambda>i. F \<inter> A i) \<subseteq> sets (sigma E)" "F \<inter> (\<Union>x. A x) = (\<Union>x. F \<inter> A x)"
```
```   548         "disjoint_family (\<lambda>i. F \<inter> A i)" "\<And>i. \<mu> (F \<inter> A i) = \<nu> (F \<inter> A i)" "range A \<subseteq> sets (sigma E)"
```
```   549         by (auto simp: disjoint_family_on_def subset_eq)
```
```   550       then show "(\<Union>x. A x) \<in> sets (sigma E) \<and> \<mu> (F \<inter> (\<Union>x. A x)) = \<nu> (F \<inter> (\<Union>x. A x))"
```
```   551         by (auto simp: M.measure_countably_additive[symmetric]
```
```   552                        N.measure_countably_additive[symmetric]
```
```   553             simp del: UN_simps)
```
```   554     qed
```
```   555     have *: "sets (sigma E) = sets \<lparr>space = space E, sets = ?D F\<rparr>"
```
```   556       using `F \<in> sets E` `Int_stable E`
```
```   557       by (intro D.dynkin_lemma)
```
```   558          (auto simp add: sets_sigma Int_stable_def eq intro: sigma_sets.Basic)
```
```   559     have "\<And>D. D \<in> sets (sigma E) \<Longrightarrow> \<mu> (F \<inter> D) = \<nu> (F \<inter> D)"
```
```   560       by (subst (asm) *) auto }
```
```   561   note * = this
```
```   562   let "?A i" = "A i \<inter> X"
```
```   563   have A': "range ?A \<subseteq> sets (sigma E)" "incseq ?A"
```
```   564     using A(1,2) `X \<in> sets (sigma E)` by (auto simp: incseq_def)
```
```   565   { fix i have "\<mu> (?A i) = \<nu> (?A i)"
```
```   566       using *[of "A i" X] `X \<in> sets (sigma E)` A finite by auto }
```
```   567   with M.continuity_from_below[OF A'] N.continuity_from_below[OF A']
```
```   568   show ?thesis using A(3) `X \<in> sets (sigma E)` by auto
```
```   569 qed
```
```   570
```
```   571 section "@{text \<mu>}-null sets"
```
```   572
```
```   573 abbreviation (in measure_space) "null_sets \<equiv> {N\<in>sets M. \<mu> N = 0}"
```
```   574
```
```   575 sublocale measure_space \<subseteq> nullsets!: ring_of_sets "\<lparr> space = space M, sets = null_sets \<rparr>"
```
```   576   where "space \<lparr> space = space M, sets = null_sets \<rparr> = space M"
```
```   577   and "sets \<lparr> space = space M, sets = null_sets \<rparr> = null_sets"
```
```   578 proof -
```
```   579   { fix A B assume sets: "A \<in> sets M" "B \<in> sets M"
```
```   580     moreover then have "\<mu> (A \<union> B) \<le> \<mu> A + \<mu> B" "\<mu> (A - B) \<le> \<mu> A"
```
```   581       by (auto intro!: measure_subadditive measure_mono)
```
```   582     moreover assume "\<mu> B = 0" "\<mu> A = 0"
```
```   583     ultimately have "\<mu> (A - B) = 0" "\<mu> (A \<union> B) = 0"
```
```   584       by (auto intro!: antisym) }
```
```   585   note null = this
```
```   586   show "ring_of_sets \<lparr> space = space M, sets = null_sets \<rparr>"
```
```   587     by default (insert sets_into_space null, auto)
```
```   588 qed simp_all
```
```   589
```
```   590 lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
```
```   591 proof -
```
```   592   have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)"
```
```   593     unfolding SUPR_def image_compose
```
```   594     unfolding surj_from_nat ..
```
```   595   then show ?thesis by simp
```
```   596 qed
```
```   597
```
```   598 lemma (in measure_space) null_sets_UN[intro]:
```
```   599   assumes "\<And>i::'i::countable. N i \<in> null_sets"
```
```   600   shows "(\<Union>i. N i) \<in> null_sets"
```
```   601 proof (intro conjI CollectI)
```
```   602   show "(\<Union>i. N i) \<in> sets M" using assms by auto
```
```   603   then have "0 \<le> \<mu> (\<Union>i. N i)" by simp
```
```   604   moreover have "\<mu> (\<Union>i. N i) \<le> (\<Sum>n. \<mu> (N (Countable.from_nat n)))"
```
```   605     unfolding UN_from_nat[of N]
```
```   606     using assms by (intro measure_countably_subadditive) auto
```
```   607   ultimately show "\<mu> (\<Union>i. N i) = 0" using assms by auto
```
```   608 qed
```
```   609
```
```   610 lemma (in measure_space) null_set_Int1:
```
```   611   assumes "B \<in> null_sets" "A \<in> sets M" shows "A \<inter> B \<in> null_sets"
```
```   612 using assms proof (intro CollectI conjI)
```
```   613   show "\<mu> (A \<inter> B) = 0" using assms by (intro measure_eq_0[of B "A \<inter> B"]) auto
```
```   614 qed auto
```
```   615
```
```   616 lemma (in measure_space) null_set_Int2:
```
```   617   assumes "B \<in> null_sets" "A \<in> sets M" shows "B \<inter> A \<in> null_sets"
```
```   618   using assms by (subst Int_commute) (rule null_set_Int1)
```
```   619
```
```   620 lemma (in measure_space) measure_Diff_null_set:
```
```   621   assumes "B \<in> null_sets" "A \<in> sets M"
```
```   622   shows "\<mu> (A - B) = \<mu> A"
```
```   623 proof -
```
```   624   have *: "A - B = (A - (A \<inter> B))" by auto
```
```   625   have "A \<inter> B \<in> null_sets" using assms by (rule null_set_Int1)
```
```   626   then show ?thesis
```
```   627     unfolding * using assms
```
```   628     by (subst measure_Diff) auto
```
```   629 qed
```
```   630
```
```   631 lemma (in measure_space) null_set_Diff:
```
```   632   assumes "B \<in> null_sets" "A \<in> sets M" shows "B - A \<in> null_sets"
```
```   633 using assms proof (intro CollectI conjI)
```
```   634   show "\<mu> (B - A) = 0" using assms by (intro measure_eq_0[of B "B - A"]) auto
```
```   635 qed auto
```
```   636
```
```   637 lemma (in measure_space) measure_Un_null_set:
```
```   638   assumes "A \<in> sets M" "B \<in> null_sets"
```
```   639   shows "\<mu> (A \<union> B) = \<mu> A"
```
```   640 proof -
```
```   641   have *: "A \<union> B = A \<union> (B - A)" by auto
```
```   642   have "B - A \<in> null_sets" using assms(2,1) by (rule null_set_Diff)
```
```   643   then show ?thesis
```
```   644     unfolding * using assms
```
```   645     by (subst measure_additive[symmetric]) auto
```
```   646 qed
```
```   647
```
```   648 section "Formalise almost everywhere"
```
```   649
```
```   650 definition (in measure_space)
```
```   651   almost_everywhere :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "AE " 10) where
```
```   652   "almost_everywhere P \<longleftrightarrow> (\<exists>N\<in>null_sets. { x \<in> space M. \<not> P x } \<subseteq> N)"
```
```   653
```
```   654 syntax
```
```   655   "_almost_everywhere" :: "'a \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
```
```   656
```
```   657 translations
```
```   658   "AE x in M. P" == "CONST measure_space.almost_everywhere M (%x. P)"
```
```   659
```
```   660 lemma (in measure_space) AE_cong_measure:
```
```   661   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
```
```   662   shows "(AE x in N. P x) \<longleftrightarrow> (AE x. P x)"
```
```   663 proof -
```
```   664   interpret N: measure_space N
```
```   665     by (rule measure_space_cong) fact+
```
```   666   show ?thesis
```
```   667     unfolding N.almost_everywhere_def almost_everywhere_def
```
```   668     by (auto simp: assms)
```
```   669 qed
```
```   670
```
```   671 lemma (in measure_space) AE_I':
```
```   672   "N \<in> null_sets \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x. P x)"
```
```   673   unfolding almost_everywhere_def by auto
```
```   674
```
```   675 lemma (in measure_space) AE_iff_null_set:
```
```   676   assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
```
```   677   shows "(AE x. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets"
```
```   678 proof
```
```   679   assume "AE x. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "\<mu> N = 0"
```
```   680     unfolding almost_everywhere_def by auto
```
```   681   have "0 \<le> \<mu> ?P" using assms by simp
```
```   682   moreover have "\<mu> ?P \<le> \<mu> N"
```
```   683     using assms N(1,2) by (auto intro: measure_mono)
```
```   684   ultimately have "\<mu> ?P = 0" unfolding `\<mu> N = 0` by auto
```
```   685   then show "?P \<in> null_sets" using assms by simp
```
```   686 next
```
```   687   assume "?P \<in> null_sets" with assms show "AE x. P x" by (auto intro: AE_I')
```
```   688 qed
```
```   689
```
```   690 lemma (in measure_space) AE_iff_measurable:
```
```   691   "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x. P x) \<longleftrightarrow> \<mu> N = 0"
```
```   692   using AE_iff_null_set[of P] by simp
```
```   693
```
```   694 lemma (in measure_space) AE_True[intro, simp]: "AE x. True"
```
```   695   unfolding almost_everywhere_def by auto
```
```   696
```
```   697 lemma (in measure_space) AE_E[consumes 1]:
```
```   698   assumes "AE x. P x"
```
```   699   obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "\<mu> N = 0" "N \<in> sets M"
```
```   700   using assms unfolding almost_everywhere_def by auto
```
```   701
```
```   702 lemma (in measure_space) AE_E2:
```
```   703   assumes "AE x. P x" "{x\<in>space M. P x} \<in> sets M"
```
```   704   shows "\<mu> {x\<in>space M. \<not> P x} = 0" (is "\<mu> ?P = 0")
```
```   705 proof -
```
```   706   have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}"
```
```   707     by auto
```
```   708   with AE_iff_null_set[of P] assms show ?thesis by auto
```
```   709 qed
```
```   710
```
```   711 lemma (in measure_space) AE_I:
```
```   712   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "\<mu> N = 0" "N \<in> sets M"
```
```   713   shows "AE x. P x"
```
```   714   using assms unfolding almost_everywhere_def by auto
```
```   715
```
```   716 lemma (in measure_space) AE_mp[elim!]:
```
```   717   assumes AE_P: "AE x. P x" and AE_imp: "AE x. P x \<longrightarrow> Q x"
```
```   718   shows "AE x. Q x"
```
```   719 proof -
```
```   720   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
```
```   721     and A: "A \<in> sets M" "\<mu> A = 0"
```
```   722     by (auto elim!: AE_E)
```
```   723
```
```   724   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
```
```   725     and B: "B \<in> sets M" "\<mu> B = 0"
```
```   726     by (auto elim!: AE_E)
```
```   727
```
```   728   show ?thesis
```
```   729   proof (intro AE_I)
```
```   730     have "0 \<le> \<mu> (A \<union> B)" using A B by auto
```
```   731     moreover have "\<mu> (A \<union> B) \<le> 0"
```
```   732       using measure_subadditive[of A B] A B by auto
```
```   733     ultimately show "A \<union> B \<in> sets M" "\<mu> (A \<union> B) = 0" using A B by auto
```
```   734     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
```
```   735       using P imp by auto
```
```   736   qed
```
```   737 qed
```
```   738
```
```   739 lemma (in measure_space)
```
```   740   shows AE_iffI: "AE x. P x \<Longrightarrow> AE x. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x. Q x"
```
```   741     and AE_disjI1: "AE x. P x \<Longrightarrow> AE x. P x \<or> Q x"
```
```   742     and AE_disjI2: "AE x. Q x \<Longrightarrow> AE x. P x \<or> Q x"
```
```   743     and AE_conjI: "AE x. P x \<Longrightarrow> AE x. Q x \<Longrightarrow> AE x. P x \<and> Q x"
```
```   744     and AE_conj_iff[simp]: "(AE x. P x \<and> Q x) \<longleftrightarrow> (AE x. P x) \<and> (AE x. Q x)"
```
```   745   by auto
```
```   746
```
```   747 lemma (in measure_space) AE_measure:
```
```   748   assumes AE: "AE x. P x" and sets: "{x\<in>space M. P x} \<in> sets M"
```
```   749   shows "\<mu> {x\<in>space M. P x} = \<mu> (space M)"
```
```   750 proof -
```
```   751   from AE_E[OF AE] guess N . note N = this
```
```   752   with sets have "\<mu> (space M) \<le> \<mu> ({x\<in>space M. P x} \<union> N)"
```
```   753     by (intro measure_mono) auto
```
```   754   also have "\<dots> \<le> \<mu> {x\<in>space M. P x} + \<mu> N"
```
```   755     using sets N by (intro measure_subadditive) auto
```
```   756   also have "\<dots> = \<mu> {x\<in>space M. P x}" using N by simp
```
```   757   finally show "\<mu> {x\<in>space M. P x} = \<mu> (space M)"
```
```   758     using measure_top[OF sets] by auto
```
```   759 qed
```
```   760
```
```   761 lemma (in measure_space) AE_space: "AE x. x \<in> space M"
```
```   762   by (rule AE_I[where N="{}"]) auto
```
```   763
```
```   764 lemma (in measure_space) AE_I2[simp, intro]:
```
```   765   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x. P x"
```
```   766   using AE_space by auto
```
```   767
```
```   768 lemma (in measure_space) AE_Ball_mp:
```
```   769   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x. P x \<longrightarrow> Q x \<Longrightarrow> AE x. Q x"
```
```   770   by auto
```
```   771
```
```   772 lemma (in measure_space) AE_cong[cong]:
```
```   773   "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x. P x) \<longleftrightarrow> (AE x. Q x)"
```
```   774   by auto
```
```   775
```
```   776 lemma (in measure_space) AE_all_countable:
```
```   777   "(AE x. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x. P i x)"
```
```   778 proof
```
```   779   assume "\<forall>i. AE x. P i x"
```
```   780   from this[unfolded almost_everywhere_def Bex_def, THEN choice]
```
```   781   obtain N where N: "\<And>i. N i \<in> null_sets" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto
```
```   782   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
```
```   783   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
```
```   784   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
```
```   785   moreover from N have "(\<Union>i. N i) \<in> null_sets"
```
```   786     by (intro null_sets_UN) auto
```
```   787   ultimately show "AE x. \<forall>i. P i x"
```
```   788     unfolding almost_everywhere_def by auto
```
```   789 qed auto
```
```   790
```
```   791 lemma (in measure_space) AE_finite_all:
```
```   792   assumes f: "finite S" shows "(AE x. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x. P i x)"
```
```   793   using f by induct auto
```
```   794
```
```   795 lemma (in measure_space) restricted_measure_space:
```
```   796   assumes "S \<in> sets M"
```
```   797   shows "measure_space (restricted_space S)"
```
```   798     (is "measure_space ?r")
```
```   799   unfolding measure_space_def measure_space_axioms_def
```
```   800 proof safe
```
```   801   show "sigma_algebra ?r" using restricted_sigma_algebra[OF assms] .
```
```   802   show "positive ?r (measure ?r)" using `S \<in> sets M` by (auto simp: positive_def)
```
```   803
```
```   804   show "countably_additive ?r (measure ?r)"
```
```   805     unfolding countably_additive_def
```
```   806   proof safe
```
```   807     fix A :: "nat \<Rightarrow> 'a set"
```
```   808     assume *: "range A \<subseteq> sets ?r" and **: "disjoint_family A"
```
```   809     from restriction_in_sets[OF assms *[simplified]] **
```
```   810     show "(\<Sum>n. measure ?r (A n)) = measure ?r (\<Union>i. A i)"
```
```   811       using measure_countably_additive by simp
```
```   812   qed
```
```   813 qed
```
```   814
```
```   815 lemma (in measure_space) AE_restricted:
```
```   816   assumes "A \<in> sets M"
```
```   817   shows "(AE x in restricted_space A. P x) \<longleftrightarrow> (AE x. x \<in> A \<longrightarrow> P x)"
```
```   818 proof -
```
```   819   interpret R: measure_space "restricted_space A"
```
```   820     by (rule restricted_measure_space[OF `A \<in> sets M`])
```
```   821   show ?thesis
```
```   822   proof
```
```   823     assume "AE x in restricted_space A. P x"
```
```   824     from this[THEN R.AE_E] guess N' .
```
```   825     then obtain N where "{x \<in> A. \<not> P x} \<subseteq> A \<inter> N" "\<mu> (A \<inter> N) = 0" "N \<in> sets M"
```
```   826       by auto
```
```   827     moreover then have "{x \<in> space M. \<not> (x \<in> A \<longrightarrow> P x)} \<subseteq> A \<inter> N"
```
```   828       using `A \<in> sets M` sets_into_space by auto
```
```   829     ultimately show "AE x. x \<in> A \<longrightarrow> P x"
```
```   830       using `A \<in> sets M` by (auto intro!: AE_I[where N="A \<inter> N"])
```
```   831   next
```
```   832     assume "AE x. x \<in> A \<longrightarrow> P x"
```
```   833     from this[THEN AE_E] guess N .
```
```   834     then show "AE x in restricted_space A. P x"
```
```   835       using null_set_Int1[OF _ `A \<in> sets M`] `A \<in> sets M`[THEN sets_into_space]
```
```   836       by (auto intro!: R.AE_I[where N="A \<inter> N"] simp: subset_eq)
```
```   837   qed
```
```   838 qed
```
```   839
```
```   840 lemma (in measure_space) measure_space_subalgebra:
```
```   841   assumes "sigma_algebra N" and "sets N \<subseteq> sets M" "space N = space M"
```
```   842   and measure[simp]: "\<And>X. X \<in> sets N \<Longrightarrow> measure N X = measure M X"
```
```   843   shows "measure_space N"
```
```   844 proof -
```
```   845   interpret N: sigma_algebra N by fact
```
```   846   show ?thesis
```
```   847   proof
```
```   848     from `sets N \<subseteq> sets M` have "\<And>A. range A \<subseteq> sets N \<Longrightarrow> range A \<subseteq> sets M" by blast
```
```   849     then show "countably_additive N (measure N)"
```
```   850       by (auto intro!: measure_countably_additive simp: countably_additive_def subset_eq)
```
```   851     show "positive N (measure_space.measure N)"
```
```   852       using assms(2) by (auto simp add: positive_def)
```
```   853   qed
```
```   854 qed
```
```   855
```
```   856 lemma (in measure_space) AE_subalgebra:
```
```   857   assumes ae: "AE x in N. P x"
```
```   858   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
```
```   859   and sa: "sigma_algebra N"
```
```   860   shows "AE x. P x"
```
```   861 proof -
```
```   862   interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
```
```   863   from ae[THEN N.AE_E] guess N .
```
```   864   with N show ?thesis unfolding almost_everywhere_def by auto
```
```   865 qed
```
```   866
```
```   867 section "@{text \<sigma>}-finite Measures"
```
```   868
```
```   869 locale sigma_finite_measure = measure_space +
```
```   870   assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
```
```   871
```
```   872 lemma (in sigma_finite_measure) restricted_sigma_finite_measure:
```
```   873   assumes "S \<in> sets M"
```
```   874   shows "sigma_finite_measure (restricted_space S)"
```
```   875     (is "sigma_finite_measure ?r")
```
```   876   unfolding sigma_finite_measure_def sigma_finite_measure_axioms_def
```
```   877 proof safe
```
```   878   show "measure_space ?r" using restricted_measure_space[OF assms] .
```
```   879 next
```
```   880   obtain A :: "nat \<Rightarrow> 'a set" where
```
```   881       "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
```
```   882     using sigma_finite by auto
```
```   883   show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets ?r \<and> (\<Union>i. A i) = space ?r \<and> (\<forall>i. measure ?r (A i) \<noteq> \<infinity>)"
```
```   884   proof (safe intro!: exI[of _ "\<lambda>i. A i \<inter> S"] del: notI)
```
```   885     fix i
```
```   886     show "A i \<inter> S \<in> sets ?r"
```
```   887       using `range A \<subseteq> sets M` `S \<in> sets M` by auto
```
```   888   next
```
```   889     fix x i assume "x \<in> S" thus "x \<in> space ?r" by simp
```
```   890   next
```
```   891     fix x assume "x \<in> space ?r" thus "x \<in> (\<Union>i. A i \<inter> S)"
```
```   892       using `(\<Union>i. A i) = space M` `S \<in> sets M` by auto
```
```   893   next
```
```   894     fix i
```
```   895     have "\<mu> (A i \<inter> S) \<le> \<mu> (A i)"
```
```   896       using `range A \<subseteq> sets M` `S \<in> sets M` by (auto intro!: measure_mono)
```
```   897     then show "measure ?r (A i \<inter> S) \<noteq> \<infinity>" using `\<mu> (A i) \<noteq> \<infinity>` by auto
```
```   898   qed
```
```   899 qed
```
```   900
```
```   901 lemma (in sigma_finite_measure) sigma_finite_measure_cong:
```
```   902   assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
```
```   903   shows "sigma_finite_measure M'"
```
```   904 proof -
```
```   905   interpret M': measure_space M' by (intro measure_space_cong cong)
```
```   906   from sigma_finite guess A .. note A = this
```
```   907   then have "\<And>i. A i \<in> sets M" by auto
```
```   908   with A have fin: "\<forall>i. measure M' (A i) \<noteq> \<infinity>" using cong by auto
```
```   909   show ?thesis
```
```   910     apply default
```
```   911     using A fin cong by auto
```
```   912 qed
```
```   913
```
```   914 lemma (in sigma_finite_measure) disjoint_sigma_finite:
```
```   915   "\<exists>A::nat\<Rightarrow>'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and>
```
```   916     (\<forall>i. \<mu> (A i) \<noteq> \<infinity>) \<and> disjoint_family A"
```
```   917 proof -
```
```   918   obtain A :: "nat \<Rightarrow> 'a set" where
```
```   919     range: "range A \<subseteq> sets M" and
```
```   920     space: "(\<Union>i. A i) = space M" and
```
```   921     measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
```
```   922     using sigma_finite by auto
```
```   923   note range' = range_disjointed_sets[OF range] range
```
```   924   { fix i
```
```   925     have "\<mu> (disjointed A i) \<le> \<mu> (A i)"
```
```   926       using range' disjointed_subset[of A i] by (auto intro!: measure_mono)
```
```   927     then have "\<mu> (disjointed A i) \<noteq> \<infinity>"
```
```   928       using measure[of i] by auto }
```
```   929   with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
```
```   930   show ?thesis by (auto intro!: exI[of _ "disjointed A"])
```
```   931 qed
```
```   932
```
```   933 lemma (in sigma_finite_measure) sigma_finite_up:
```
```   934   "\<exists>F. range F \<subseteq> sets M \<and> incseq F \<and> (\<Union>i. F i) = space M \<and> (\<forall>i. \<mu> (F i) \<noteq> \<infinity>)"
```
```   935 proof -
```
```   936   obtain F :: "nat \<Rightarrow> 'a set" where
```
```   937     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
```
```   938     using sigma_finite by auto
```
```   939   then show ?thesis
```
```   940   proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
```
```   941     from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
```
```   942     then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
```
```   943       using F by fastsimp
```
```   944   next
```
```   945     fix n
```
```   946     have "\<mu> (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. \<mu> (F i))" using F
```
```   947       by (auto intro!: measure_finitely_subadditive)
```
```   948     also have "\<dots> < \<infinity>"
```
```   949       using F by (auto simp: setsum_Pinfty)
```
```   950     finally show "\<mu> (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
```
```   951   qed (force simp: incseq_def)+
```
```   952 qed
```
```   953
```
```   954 section {* Measure preserving *}
```
```   955
```
```   956 definition "measure_preserving A B =
```
```   957     {f \<in> measurable A B. (\<forall>y \<in> sets B. measure B y = measure A (f -` y \<inter> space A))}"
```
```   958
```
```   959 lemma measure_preservingI[intro?]:
```
```   960   assumes "f \<in> measurable A B"
```
```   961     and "\<And>y. y \<in> sets B \<Longrightarrow> measure A (f -` y \<inter> space A) = measure B y"
```
```   962   shows "f \<in> measure_preserving A B"
```
```   963   unfolding measure_preserving_def using assms by auto
```
```   964
```
```   965 lemma (in measure_space) measure_space_vimage:
```
```   966   fixes M' :: "('c, 'd) measure_space_scheme"
```
```   967   assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
```
```   968   shows "measure_space M'"
```
```   969 proof -
```
```   970   interpret M': sigma_algebra M' by fact
```
```   971   show ?thesis
```
```   972   proof
```
```   973     show "positive M' (measure M')" using T
```
```   974       by (auto simp: measure_preserving_def positive_def measurable_sets)
```
```   975
```
```   976     show "countably_additive M' (measure M')"
```
```   977     proof (intro countably_additiveI)
```
```   978       fix A :: "nat \<Rightarrow> 'c set" assume "range A \<subseteq> sets M'" "disjoint_family A"
```
```   979       then have A: "\<And>i. A i \<in> sets M'" "(\<Union>i. A i) \<in> sets M'" by auto
```
```   980       then have *: "range (\<lambda>i. T -` (A i) \<inter> space M) \<subseteq> sets M"
```
```   981         using T by (auto simp: measurable_def measure_preserving_def)
```
```   982       moreover have "(\<Union>i. T -`  A i \<inter> space M) \<in> sets M"
```
```   983         using * by blast
```
```   984       moreover have **: "disjoint_family (\<lambda>i. T -` A i \<inter> space M)"
```
```   985         using `disjoint_family A` by (auto simp: disjoint_family_on_def)
```
```   986       ultimately show "(\<Sum>i. measure M' (A i)) = measure M' (\<Union>i. A i)"
```
```   987         using measure_countably_additive[OF _ **] A T
```
```   988         by (auto simp: comp_def vimage_UN measure_preserving_def)
```
```   989     qed
```
```   990   qed
```
```   991 qed
```
```   992
```
```   993 lemma (in measure_space) almost_everywhere_vimage:
```
```   994   assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
```
```   995     and AE: "measure_space.almost_everywhere M' P"
```
```   996   shows "AE x. P (T x)"
```
```   997 proof -
```
```   998   interpret M': measure_space M' using T by (rule measure_space_vimage)
```
```   999   from AE[THEN M'.AE_E] guess N .
```
```  1000   then show ?thesis
```
```  1001     unfolding almost_everywhere_def M'.almost_everywhere_def
```
```  1002     using T(2) unfolding measurable_def measure_preserving_def
```
```  1003     by (intro bexI[of _ "T -` N \<inter> space M"]) auto
```
```  1004 qed
```
```  1005
```
```  1006 lemma measure_unique_Int_stable_vimage:
```
```  1007   fixes A :: "nat \<Rightarrow> 'a set"
```
```  1008   assumes E: "Int_stable E"
```
```  1009   and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E" "\<And>i. measure M (A i) \<noteq> \<infinity>"
```
```  1010   and N: "measure_space N" "T \<in> measurable N M"
```
```  1011   and M: "measure_space M" "sets (sigma E) = sets M" "space E = space M"
```
```  1012   and eq: "\<And>X. X \<in> sets E \<Longrightarrow> measure M X = measure N (T -` X \<inter> space N)"
```
```  1013   assumes X: "X \<in> sets (sigma E)"
```
```  1014   shows "measure M X = measure N (T -` X \<inter> space N)"
```
```  1015 proof (rule measure_unique_Int_stable[OF E A(1,2,3) _ _ eq _ X])
```
```  1016   interpret M: measure_space M by fact
```
```  1017   interpret N: measure_space N by fact
```
```  1018   let "?T X" = "T -` X \<inter> space N"
```
```  1019   show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = measure M\<rparr>"
```
```  1020     by (rule M.measure_space_cong) (auto simp: M)
```
```  1021   show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<lambda>X. measure N (?T X)\<rparr>" (is "measure_space ?E")
```
```  1022   proof (rule N.measure_space_vimage)
```
```  1023     show "sigma_algebra ?E"
```
```  1024       by (rule M.sigma_algebra_cong) (auto simp: M)
```
```  1025     show "T \<in> measure_preserving N ?E"
```
```  1026       using `T \<in> measurable N M` by (auto simp: M measurable_def measure_preserving_def)
```
```  1027   qed
```
```  1028   show "\<And>i. M.\<mu> (A i) \<noteq> \<infinity>" by fact
```
```  1029 qed
```
```  1030
```
```  1031 lemma (in measure_space) measure_preserving_Int_stable:
```
```  1032   fixes A :: "nat \<Rightarrow> 'a set"
```
```  1033   assumes E: "Int_stable E" "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E" "\<And>i. measure E (A i) \<noteq> \<infinity>"
```
```  1034   and N: "measure_space (sigma E)"
```
```  1035   and T: "T \<in> measure_preserving M E"
```
```  1036   shows "T \<in> measure_preserving M (sigma E)"
```
```  1037 proof
```
```  1038   interpret E: measure_space "sigma E" by fact
```
```  1039   show "T \<in> measurable M (sigma E)"
```
```  1040     using T E.sets_into_space
```
```  1041     by (intro measurable_sigma) (auto simp: measure_preserving_def measurable_def)
```
```  1042   fix X assume "X \<in> sets (sigma E)"
```
```  1043   show "\<mu> (T -` X \<inter> space M) = E.\<mu> X"
```
```  1044   proof (rule measure_unique_Int_stable_vimage[symmetric])
```
```  1045     show "sets (sigma E) = sets (sigma E)" "space E = space (sigma E)"
```
```  1046       "\<And>i. E.\<mu> (A i) \<noteq> \<infinity>" using E by auto
```
```  1047     show "measure_space M" by default
```
```  1048   next
```
```  1049     fix X assume "X \<in> sets E" then show "E.\<mu> X = \<mu> (T -` X \<inter> space M)"
```
```  1050       using T unfolding measure_preserving_def by auto
```
```  1051   qed fact+
```
```  1052 qed
```
```  1053
```
```  1054 section "Real measure values"
```
```  1055
```
```  1056 lemma (in measure_space) real_measure_Union:
```
```  1057   assumes finite: "\<mu> A \<noteq> \<infinity>" "\<mu> B \<noteq> \<infinity>"
```
```  1058   and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
```
```  1059   shows "real (\<mu> (A \<union> B)) = real (\<mu> A) + real (\<mu> B)"
```
```  1060   unfolding measure_additive[symmetric, OF measurable]
```
```  1061   using measurable(1,2)[THEN positive_measure]
```
```  1062   using finite by (cases rule: ereal2_cases[of "\<mu> A" "\<mu> B"]) auto
```
```  1063
```
```  1064 lemma (in measure_space) real_measure_finite_Union:
```
```  1065   assumes measurable:
```
```  1066     "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M" "disjoint_family_on A S"
```
```  1067   assumes finite: "\<And>i. i \<in> S \<Longrightarrow> \<mu> (A i) \<noteq> \<infinity>"
```
```  1068   shows "real (\<mu> (\<Union>i\<in>S. A i)) = (\<Sum>i\<in>S. real (\<mu> (A i)))"
```
```  1069   using finite measurable(2)[THEN positive_measure]
```
```  1070   by (force intro!: setsum_real_of_ereal[symmetric]
```
```  1071             simp: measure_setsum[OF measurable, symmetric])
```
```  1072
```
```  1073 lemma (in measure_space) real_measure_Diff:
```
```  1074   assumes finite: "\<mu> A \<noteq> \<infinity>"
```
```  1075   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
```
```  1076   shows "real (\<mu> (A - B)) = real (\<mu> A) - real (\<mu> B)"
```
```  1077 proof -
```
```  1078   have "\<mu> (A - B) \<le> \<mu> A" "\<mu> B \<le> \<mu> A"
```
```  1079     using measurable by (auto intro!: measure_mono)
```
```  1080   hence "real (\<mu> ((A - B) \<union> B)) = real (\<mu> (A - B)) + real (\<mu> B)"
```
```  1081     using measurable finite by (rule_tac real_measure_Union) auto
```
```  1082   thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
```
```  1083 qed
```
```  1084
```
```  1085 lemma (in measure_space) real_measure_UNION:
```
```  1086   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
```
```  1087   assumes finite: "\<mu> (\<Union>i. A i) \<noteq> \<infinity>"
```
```  1088   shows "(\<lambda>i. real (\<mu> (A i))) sums (real (\<mu> (\<Union>i. A i)))"
```
```  1089 proof -
```
```  1090   have "\<And>i. 0 \<le> \<mu> (A i)" using measurable by auto
```
```  1091   with summable_sums[OF summable_ereal_pos, of "\<lambda>i. \<mu> (A i)"]
```
```  1092      measure_countably_additive[OF measurable]
```
```  1093   have "(\<lambda>i. \<mu> (A i)) sums (\<mu> (\<Union>i. A i))" by simp
```
```  1094   moreover
```
```  1095   { fix i
```
```  1096     have "\<mu> (A i) \<le> \<mu> (\<Union>i. A i)"
```
```  1097       using measurable by (auto intro!: measure_mono)
```
```  1098     moreover have "0 \<le> \<mu> (A i)" using measurable by auto
```
```  1099     ultimately have "\<mu> (A i) = ereal (real (\<mu> (A i)))"
```
```  1100       using finite by (cases "\<mu> (A i)") auto }
```
```  1101   moreover
```
```  1102   have "0 \<le> \<mu> (\<Union>i. A i)" using measurable by auto
```
```  1103   then have "\<mu> (\<Union>i. A i) = ereal (real (\<mu> (\<Union>i. A i)))"
```
```  1104     using finite by (cases "\<mu> (\<Union>i. A i)") auto
```
```  1105   ultimately show ?thesis
```
```  1106     unfolding sums_ereal[symmetric] by simp
```
```  1107 qed
```
```  1108
```
```  1109 lemma (in measure_space) real_measure_subadditive:
```
```  1110   assumes measurable: "A \<in> sets M" "B \<in> sets M"
```
```  1111   and fin: "\<mu> A \<noteq> \<infinity>" "\<mu> B \<noteq> \<infinity>"
```
```  1112   shows "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
```
```  1113 proof -
```
```  1114   have "0 \<le> \<mu> (A \<union> B)" using measurable by auto
```
```  1115   then show "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
```
```  1116     using measure_subadditive[OF measurable] fin
```
```  1117     by (cases rule: ereal3_cases[of "\<mu> (A \<union> B)" "\<mu> A" "\<mu> B"]) auto
```
```  1118 qed
```
```  1119
```
```  1120 lemma (in measure_space) real_measure_setsum_singleton:
```
```  1121   assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```  1122   and fin: "\<And>x. x \<in> S \<Longrightarrow> \<mu> {x} \<noteq> \<infinity>"
```
```  1123   shows "real (\<mu> S) = (\<Sum>x\<in>S. real (\<mu> {x}))"
```
```  1124   using measure_finite_singleton[OF S] fin
```
```  1125   using positive_measure[OF S(2)]
```
```  1126   by (force intro!: setsum_real_of_ereal[symmetric])
```
```  1127
```
```  1128 lemma (in measure_space) real_continuity_from_below:
```
```  1129   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "\<mu> (\<Union>i. A i) \<noteq> \<infinity>"
```
```  1130   shows "(\<lambda>i. real (\<mu> (A i))) ----> real (\<mu> (\<Union>i. A i))"
```
```  1131 proof -
```
```  1132   have "0 \<le> \<mu> (\<Union>i. A i)" using A by auto
```
```  1133   then have "ereal (real (\<mu> (\<Union>i. A i))) = \<mu> (\<Union>i. A i)"
```
```  1134     using fin by (auto intro: ereal_real')
```
```  1135   then show ?thesis
```
```  1136     using continuity_from_below_Lim[OF A]
```
```  1137     by (intro lim_real_of_ereal) simp
```
```  1138 qed
```
```  1139
```
```  1140 lemma (in measure_space) continuity_from_above_Lim:
```
```  1141   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
```
```  1142   shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Inter>i. A i)"
```
```  1143   using LIMSEQ_ereal_INFI[OF measure_decseq, OF A]
```
```  1144   using continuity_from_above[OF A fin] by simp
```
```  1145
```
```  1146 lemma (in measure_space) real_continuity_from_above:
```
```  1147   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
```
```  1148   shows "(\<lambda>n. real (\<mu> (A n))) ----> real (\<mu> (\<Inter>i. A i))"
```
```  1149 proof -
```
```  1150   have "0 \<le> \<mu> (\<Inter>i. A i)" using A by auto
```
```  1151   moreover
```
```  1152   have "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
```
```  1153     using A by (auto intro!: measure_mono)
```
```  1154   ultimately have "ereal (real (\<mu> (\<Inter>i. A i))) = \<mu> (\<Inter>i. A i)"
```
```  1155     using fin by (auto intro: ereal_real')
```
```  1156   then show ?thesis
```
```  1157     using continuity_from_above_Lim[OF A fin]
```
```  1158     by (intro lim_real_of_ereal) simp
```
```  1159 qed
```
```  1160
```
```  1161 lemma (in measure_space) real_measure_countably_subadditive:
```
```  1162   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. \<mu> (A i)) \<noteq> \<infinity>"
```
```  1163   shows "real (\<mu> (\<Union>i. A i)) \<le> (\<Sum>i. real (\<mu> (A i)))"
```
```  1164 proof -
```
```  1165   { fix i
```
```  1166     have "0 \<le> \<mu> (A i)" using A by auto
```
```  1167     moreover have "\<mu> (A i) \<noteq> \<infinity>" using A by (intro suminf_PInfty[OF _ fin]) auto
```
```  1168     ultimately have "\<bar>\<mu> (A i)\<bar> \<noteq> \<infinity>" by auto }
```
```  1169   moreover have "0 \<le> \<mu> (\<Union>i. A i)" using A by auto
```
```  1170   ultimately have "ereal (real (\<mu> (\<Union>i. A i))) \<le> (\<Sum>i. ereal (real (\<mu> (A i))))"
```
```  1171     using measure_countably_subadditive[OF A] by (auto simp: ereal_real)
```
```  1172   also have "\<dots> = ereal (\<Sum>i. real (\<mu> (A i)))"
```
```  1173     using A
```
```  1174     by (auto intro!: sums_unique[symmetric] sums_ereal[THEN iffD2] summable_sums summable_real_of_ereal fin)
```
```  1175   finally show ?thesis by simp
```
```  1176 qed
```
```  1177
```
```  1178 locale finite_measure = measure_space +
```
```  1179   assumes finite_measure_of_space: "\<mu> (space M) \<noteq> \<infinity>"
```
```  1180
```
```  1181 sublocale finite_measure < sigma_finite_measure
```
```  1182 proof
```
```  1183   show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
```
```  1184     using finite_measure_of_space by (auto intro!: exI[of _ "\<lambda>x. space M"])
```
```  1185 qed
```
```  1186
```
```  1187 lemma (in finite_measure) finite_measure[simp, intro]:
```
```  1188   assumes "A \<in> sets M"
```
```  1189   shows "\<mu> A \<noteq> \<infinity>"
```
```  1190 proof -
```
```  1191   from `A \<in> sets M` have "A \<subseteq> space M"
```
```  1192     using sets_into_space by blast
```
```  1193   then have "\<mu> A \<le> \<mu> (space M)"
```
```  1194     using assms top by (rule measure_mono)
```
```  1195   then show ?thesis
```
```  1196     using finite_measure_of_space by auto
```
```  1197 qed
```
```  1198
```
```  1199 definition (in finite_measure)
```
```  1200   "\<mu>' A = (if A \<in> sets M then real (\<mu> A) else 0)"
```
```  1201
```
```  1202 lemma (in finite_measure) finite_measure_eq: "A \<in> sets M \<Longrightarrow> \<mu> A = ereal (\<mu>' A)"
```
```  1203   by (auto simp: \<mu>'_def ereal_real)
```
```  1204
```
```  1205 lemma (in finite_measure) positive_measure'[simp, intro]: "0 \<le> \<mu>' A"
```
```  1206   unfolding \<mu>'_def by (auto simp: real_of_ereal_pos)
```
```  1207
```
```  1208 lemma (in finite_measure) real_measure:
```
```  1209   assumes A: "A \<in> sets M" shows "\<exists>r. 0 \<le> r \<and> \<mu> A = ereal r"
```
```  1210   using finite_measure[OF A] positive_measure[OF A] by (cases "\<mu> A") auto
```
```  1211
```
```  1212 lemma (in finite_measure) bounded_measure: "\<mu>' A \<le> \<mu>' (space M)"
```
```  1213 proof cases
```
```  1214   assume "A \<in> sets M"
```
```  1215   moreover then have "\<mu> A \<le> \<mu> (space M)"
```
```  1216     using sets_into_space by (auto intro!: measure_mono)
```
```  1217   ultimately show ?thesis
```
```  1218     by (auto simp: \<mu>'_def intro!: real_of_ereal_positive_mono)
```
```  1219 qed (simp add: \<mu>'_def real_of_ereal_pos)
```
```  1220
```
```  1221 lemma (in finite_measure) restricted_finite_measure:
```
```  1222   assumes "S \<in> sets M"
```
```  1223   shows "finite_measure (restricted_space S)"
```
```  1224     (is "finite_measure ?r")
```
```  1225   unfolding finite_measure_def finite_measure_axioms_def
```
```  1226 proof (intro conjI)
```
```  1227   show "measure_space ?r" using restricted_measure_space[OF assms] .
```
```  1228 next
```
```  1229   show "measure ?r (space ?r) \<noteq> \<infinity>" using finite_measure[OF `S \<in> sets M`] by auto
```
```  1230 qed
```
```  1231
```
```  1232 lemma (in measure_space) restricted_to_finite_measure:
```
```  1233   assumes "S \<in> sets M" "\<mu> S \<noteq> \<infinity>"
```
```  1234   shows "finite_measure (restricted_space S)"
```
```  1235 proof -
```
```  1236   have "measure_space (restricted_space S)"
```
```  1237     using `S \<in> sets M` by (rule restricted_measure_space)
```
```  1238   then show ?thesis
```
```  1239     unfolding finite_measure_def finite_measure_axioms_def
```
```  1240     using assms by auto
```
```  1241 qed
```
```  1242
```
```  1243 lemma (in finite_measure) finite_measure_Diff:
```
```  1244   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
```
```  1245   shows "\<mu>' (A - B) = \<mu>' A - \<mu>' B"
```
```  1246   using sets[THEN finite_measure_eq]
```
```  1247   using Diff[OF sets, THEN finite_measure_eq]
```
```  1248   using measure_Diff[OF _ assms] by simp
```
```  1249
```
```  1250 lemma (in finite_measure) finite_measure_Union:
```
```  1251   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
```
```  1252   shows "\<mu>' (A \<union> B) = \<mu>' A + \<mu>' B"
```
```  1253   using measure_additive[OF assms]
```
```  1254   using sets[THEN finite_measure_eq]
```
```  1255   using Un[OF sets, THEN finite_measure_eq]
```
```  1256   by simp
```
```  1257
```
```  1258 lemma (in finite_measure) finite_measure_finite_Union:
```
```  1259   assumes S: "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
```
```  1260   and dis: "disjoint_family_on A S"
```
```  1261   shows "\<mu>' (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. \<mu>' (A i))"
```
```  1262   using measure_setsum[OF assms]
```
```  1263   using finite_UN[of S A, OF S, THEN finite_measure_eq]
```
```  1264   using S(2)[THEN finite_measure_eq]
```
```  1265   by simp
```
```  1266
```
```  1267 lemma (in finite_measure) finite_measure_UNION:
```
```  1268   assumes A: "range A \<subseteq> sets M" "disjoint_family A"
```
```  1269   shows "(\<lambda>i. \<mu>' (A i)) sums (\<mu>' (\<Union>i. A i))"
```
```  1270   using real_measure_UNION[OF A]
```
```  1271   using countable_UN[OF A(1), THEN finite_measure_eq]
```
```  1272   using A(1)[THEN subsetD, THEN finite_measure_eq]
```
```  1273   by auto
```
```  1274
```
```  1275 lemma (in finite_measure) finite_measure_mono:
```
```  1276   assumes B: "B \<in> sets M" and "A \<subseteq> B" shows "\<mu>' A \<le> \<mu>' B"
```
```  1277 proof cases
```
```  1278   assume "A \<in> sets M"
```
```  1279   from this[THEN finite_measure_eq] B[THEN finite_measure_eq]
```
```  1280   show ?thesis using measure_mono[OF `A \<subseteq> B` `A \<in> sets M` `B \<in> sets M`] by simp
```
```  1281 next
```
```  1282   assume "A \<notin> sets M" then show ?thesis
```
```  1283     using positive_measure'[of B] unfolding \<mu>'_def by auto
```
```  1284 qed
```
```  1285
```
```  1286 lemma (in finite_measure) finite_measure_subadditive:
```
```  1287   assumes m: "A \<in> sets M" "B \<in> sets M"
```
```  1288   shows "\<mu>' (A \<union> B) \<le> \<mu>' A + \<mu>' B"
```
```  1289   using measure_subadditive[OF m]
```
```  1290   using m[THEN finite_measure_eq] Un[OF m, THEN finite_measure_eq] by simp
```
```  1291
```
```  1292 lemma (in finite_measure) finite_measure_subadditive_finite:
```
```  1293   assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M"
```
```  1294   shows "\<mu>' (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu>' (A i))"
```
```  1295   using measure_subadditive_finite[OF assms] assms
```
```  1296   by (simp add: finite_measure_eq finite_UN)
```
```  1297
```
```  1298 lemma (in finite_measure) finite_measure_countably_subadditive:
```
```  1299   assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. \<mu>' (A i))"
```
```  1300   shows "\<mu>' (\<Union>i. A i) \<le> (\<Sum>i. \<mu>' (A i))"
```
```  1301 proof -
```
```  1302   note A[THEN subsetD, THEN finite_measure_eq, simp]
```
```  1303   note countable_UN[OF A, THEN finite_measure_eq, simp]
```
```  1304   from `summable (\<lambda>i. \<mu>' (A i))`
```
```  1305   have "(\<lambda>i. ereal (\<mu>' (A i))) sums ereal (\<Sum>i. \<mu>' (A i))"
```
```  1306     by (simp add: sums_ereal) (rule summable_sums)
```
```  1307   from sums_unique[OF this, symmetric]
```
```  1308        measure_countably_subadditive[OF A]
```
```  1309   show ?thesis by simp
```
```  1310 qed
```
```  1311
```
```  1312 lemma (in finite_measure) finite_measure_finite_singleton:
```
```  1313   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
```
```  1314   shows "\<mu>' S = (\<Sum>x\<in>S. \<mu>' {x})"
```
```  1315   using real_measure_setsum_singleton[OF assms]
```
```  1316   using *[THEN finite_measure_eq]
```
```  1317   using finite_UN[of S "\<lambda>x. {x}", OF assms, THEN finite_measure_eq]
```
```  1318   by simp
```
```  1319
```
```  1320 lemma (in finite_measure) finite_continuity_from_below:
```
```  1321   assumes A: "range A \<subseteq> sets M" and "incseq A"
```
```  1322   shows "(\<lambda>i. \<mu>' (A i)) ----> \<mu>' (\<Union>i. A i)"
```
```  1323   using real_continuity_from_below[OF A, OF `incseq A` finite_measure] assms
```
```  1324   using A[THEN subsetD, THEN finite_measure_eq]
```
```  1325   using countable_UN[OF A, THEN finite_measure_eq]
```
```  1326   by auto
```
```  1327
```
```  1328 lemma (in finite_measure) finite_continuity_from_above:
```
```  1329   assumes A: "range A \<subseteq> sets M" and "decseq A"
```
```  1330   shows "(\<lambda>n. \<mu>' (A n)) ----> \<mu>' (\<Inter>i. A i)"
```
```  1331   using real_continuity_from_above[OF A, OF `decseq A` finite_measure] assms
```
```  1332   using A[THEN subsetD, THEN finite_measure_eq]
```
```  1333   using countable_INT[OF A, THEN finite_measure_eq]
```
```  1334   by auto
```
```  1335
```
```  1336 lemma (in finite_measure) finite_measure_compl:
```
```  1337   assumes S: "S \<in> sets M"
```
```  1338   shows "\<mu>' (space M - S) = \<mu>' (space M) - \<mu>' S"
```
```  1339   using measure_compl[OF S, OF finite_measure, OF S]
```
```  1340   using S[THEN finite_measure_eq]
```
```  1341   using compl_sets[OF S, THEN finite_measure_eq]
```
```  1342   using top[THEN finite_measure_eq]
```
```  1343   by simp
```
```  1344
```
```  1345 lemma (in finite_measure) finite_measure_inter_full_set:
```
```  1346   assumes S: "S \<in> sets M" "T \<in> sets M"
```
```  1347   assumes T: "\<mu>' T = \<mu>' (space M)"
```
```  1348   shows "\<mu>' (S \<inter> T) = \<mu>' S"
```
```  1349   using measure_inter_full_set[OF S finite_measure]
```
```  1350   using T Diff[OF S(2,1)] Diff[OF S, THEN finite_measure_eq]
```
```  1351   using Int[OF S, THEN finite_measure_eq]
```
```  1352   using S[THEN finite_measure_eq] top[THEN finite_measure_eq]
```
```  1353   by simp
```
```  1354
```
```  1355 lemma (in finite_measure) empty_measure'[simp]: "\<mu>' {} = 0"
```
```  1356   unfolding \<mu>'_def by simp
```
```  1357
```
```  1358 section "Finite spaces"
```
```  1359
```
```  1360 locale finite_measure_space = measure_space + finite_sigma_algebra +
```
```  1361   assumes finite_single_measure[simp]: "\<And>x. x \<in> space M \<Longrightarrow> \<mu> {x} \<noteq> \<infinity>"
```
```  1362
```
```  1363 lemma (in finite_measure_space) sum_over_space: "(\<Sum>x\<in>space M. \<mu> {x}) = \<mu> (space M)"
```
```  1364   using measure_setsum[of "space M" "\<lambda>i. {i}"]
```
```  1365   by (simp add: sets_eq_Pow disjoint_family_on_def finite_space)
```
```  1366
```
```  1367 lemma finite_measure_spaceI:
```
```  1368   assumes "finite (space M)" "sets M = Pow(space M)" and space: "measure M (space M) \<noteq> \<infinity>"
```
```  1369     and add: "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B"
```
```  1370     and "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A"
```
```  1371   shows "finite_measure_space M"
```
```  1372     unfolding finite_measure_space_def finite_measure_space_axioms_def
```
```  1373 proof (intro allI impI conjI)
```
```  1374   show "measure_space M"
```
```  1375   proof (rule finite_additivity_sufficient)
```
```  1376     have *: "\<lparr>space = space M, sets = Pow (space M), \<dots> = algebra.more M\<rparr> = M"
```
```  1377       unfolding assms(2)[symmetric] by (auto intro!: algebra.equality)
```
```  1378     show "sigma_algebra M"
```
```  1379       using sigma_algebra_Pow[of "space M" "algebra.more M"]
```
```  1380       unfolding * .
```
```  1381     show "finite (space M)" by fact
```
```  1382     show "positive M (measure M)" unfolding positive_def using assms by auto
```
```  1383     show "additive M (measure M)" unfolding additive_def using assms by simp
```
```  1384   qed
```
```  1385   then interpret measure_space M .
```
```  1386   show "finite_sigma_algebra M"
```
```  1387   proof
```
```  1388     show "finite (space M)" by fact
```
```  1389     show "sets M = Pow (space M)" using assms by auto
```
```  1390   qed
```
```  1391   { fix x assume *: "x \<in> space M"
```
```  1392     with add[of "{x}" "space M - {x}"] space
```
```  1393     show "\<mu> {x} \<noteq> \<infinity>" by (auto simp: insert_absorb[OF *] Diff_subset) }
```
```  1394 qed
```
```  1395
```
```  1396 sublocale finite_measure_space \<subseteq> finite_measure
```
```  1397 proof
```
```  1398   show "\<mu> (space M) \<noteq> \<infinity>"
```
```  1399     unfolding sum_over_space[symmetric] setsum_Pinfty
```
```  1400     using finite_space finite_single_measure by auto
```
```  1401 qed
```
```  1402
```
```  1403 lemma finite_measure_space_iff:
```
```  1404   "finite_measure_space M \<longleftrightarrow>
```
```  1405     finite (space M) \<and> sets M = Pow(space M) \<and> measure M (space M) \<noteq> \<infinity> \<and>
```
```  1406     measure M {} = 0 \<and> (\<forall>A\<subseteq>space M. 0 \<le> measure M A) \<and>
```
```  1407     (\<forall>A\<subseteq>space M. \<forall>B\<subseteq>space M. A \<inter> B = {} \<longrightarrow> measure M (A \<union> B) = measure M A + measure M B)"
```
```  1408     (is "_ = ?rhs")
```
```  1409 proof (intro iffI)
```
```  1410   assume "finite_measure_space M"
```
```  1411   then interpret finite_measure_space M .
```
```  1412   show ?rhs
```
```  1413     using finite_space sets_eq_Pow measure_additive empty_measure finite_measure
```
```  1414     by auto
```
```  1415 next
```
```  1416   assume ?rhs then show "finite_measure_space M"
```
```  1417     by (auto intro!: finite_measure_spaceI)
```
```  1418 qed
```
```  1419
```
```  1420 lemma (in finite_measure_space) finite_measure_singleton:
```
```  1421   assumes A: "A \<subseteq> space M" shows "\<mu>' A = (\<Sum>x\<in>A. \<mu>' {x})"
```
```  1422   using A finite_subset[OF A finite_space]
```
```  1423   by (intro finite_measure_finite_singleton) auto
```
```  1424
```
```  1425 lemma (in finite_measure_space) finite_measure_subadditive_setsum:
```
```  1426   assumes "finite I"
```
```  1427   shows "\<mu>' (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu>' (A i))"
```
```  1428 proof cases
```
```  1429   assume "(\<Union>i\<in>I. A i) \<subseteq> space M"
```
```  1430   then have "\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M" by auto
```
```  1431   from finite_measure_subadditive_finite[OF `finite I` this]
```
```  1432   show ?thesis by auto
```
```  1433 next
```
```  1434   assume "\<not> (\<Union>i\<in>I. A i) \<subseteq> space M"
```
```  1435   then have "\<mu>' (\<Union>i\<in>I. A i) = 0"
```
```  1436     by (simp add: \<mu>'_def)
```
```  1437   also have "0 \<le> (\<Sum>i\<in>I. \<mu>' (A i))"
```
```  1438     by (auto intro!: setsum_nonneg)
```
```  1439   finally show ?thesis .
```
```  1440 qed
```
```  1441
```
```  1442 lemma suminf_cmult_indicator:
```
```  1443   fixes f :: "nat \<Rightarrow> ereal"
```
```  1444   assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
```
```  1445   shows "(\<Sum>n. f n * indicator (A n) x) = f i"
```
```  1446 proof -
```
```  1447   have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)"
```
```  1448     using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
```
```  1449   then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)"
```
```  1450     by (auto simp: setsum_cases)
```
```  1451   moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
```
```  1452   proof (rule ereal_SUPI)
```
```  1453     fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
```
```  1454     from this[of "Suc i"] show "f i \<le> y" by auto
```
```  1455   qed (insert assms, simp)
```
```  1456   ultimately show ?thesis using assms
```
```  1457     by (subst suminf_ereal_eq_SUPR) (auto simp: indicator_def)
```
```  1458 qed
```
```  1459
```
```  1460 lemma suminf_indicator:
```
```  1461   assumes "disjoint_family A"
```
```  1462   shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x"
```
```  1463 proof cases
```
```  1464   assume *: "x \<in> (\<Union>i. A i)"
```
```  1465   then obtain i where "x \<in> A i" by auto
```
```  1466   from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
```
```  1467   show ?thesis using * by simp
```
```  1468 qed simp
```
```  1469
```
`  1470 end`