src/HOL/Probability/Lebesgue_Measure.thy
 author hoelzl Fri Jan 14 15:59:49 2011 +0100 (2011-01-14) changeset 41546 2a12c23b7a34 parent 41097 a1abfa4e2b44 child 41654 32fe42892983 permissions -rw-r--r--
integral on lebesgue measure is extension of integral on borel measure
```     1 (*  Author: Robert Himmelmann, TU Muenchen *)
```
```     2 header {* Lebsegue measure *}
```
```     3 theory Lebesgue_Measure
```
```     4   imports Product_Measure Gauge_Measure Complete_Measure
```
```     5 begin
```
```     6
```
```     7 subsection {* Standard Cubes *}
```
```     8
```
```     9 definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where
```
```    10   "cube n \<equiv> {\<chi>\<chi> i. - real n .. \<chi>\<chi> i. real n}"
```
```    11
```
```    12 lemma cube_closed[intro]: "closed (cube n)"
```
```    13   unfolding cube_def by auto
```
```    14
```
```    15 lemma cube_subset[intro]: "n \<le> N \<Longrightarrow> cube n \<subseteq> cube N"
```
```    16   by (fastsimp simp: eucl_le[where 'a='a] cube_def)
```
```    17
```
```    18 lemma cube_subset_iff:
```
```    19   "cube n \<subseteq> cube N \<longleftrightarrow> n \<le> N"
```
```    20 proof
```
```    21   assume subset: "cube n \<subseteq> (cube N::'a set)"
```
```    22   then have "((\<chi>\<chi> i. real n)::'a) \<in> cube N"
```
```    23     using DIM_positive[where 'a='a]
```
```    24     by (fastsimp simp: cube_def eucl_le[where 'a='a])
```
```    25   then show "n \<le> N"
```
```    26     by (fastsimp simp: cube_def eucl_le[where 'a='a])
```
```    27 next
```
```    28   assume "n \<le> N" then show "cube n \<subseteq> (cube N::'a set)" by (rule cube_subset)
```
```    29 qed
```
```    30
```
```    31 lemma ball_subset_cube:"ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n"
```
```    32   unfolding cube_def subset_eq mem_interval apply safe unfolding euclidean_lambda_beta'
```
```    33 proof- fix x::'a and i assume as:"x\<in>ball 0 (real n)" "i<DIM('a)"
```
```    34   thus "- real n \<le> x \$\$ i" "real n \<ge> x \$\$ i"
```
```    35     using component_le_norm[of x i] by(auto simp: dist_norm)
```
```    36 qed
```
```    37
```
```    38 lemma mem_big_cube: obtains n where "x \<in> cube n"
```
```    39 proof- from real_arch_lt[of "norm x"] guess n ..
```
```    40   thus ?thesis apply-apply(rule that[where n=n])
```
```    41     apply(rule ball_subset_cube[unfolded subset_eq,rule_format])
```
```    42     by (auto simp add:dist_norm)
```
```    43 qed
```
```    44
```
```    45 lemma Union_inter_cube:"\<Union>{s \<inter> cube n |n. n \<in> UNIV} = s"
```
```    46 proof safe case goal1
```
```    47   from mem_big_cube[of x] guess n . note n=this
```
```    48   show ?case unfolding Union_iff apply(rule_tac x="s \<inter> cube n" in bexI)
```
```    49     using n goal1 by auto
```
```    50 qed
```
```    51
```
```    52 lemma gmeasurable_cube[intro]:"gmeasurable (cube n)"
```
```    53   unfolding cube_def by auto
```
```    54
```
```    55 lemma gmeasure_le_inter_cube[intro]: fixes s::"'a::ordered_euclidean_space set"
```
```    56   assumes "gmeasurable (s \<inter> cube n)" shows "gmeasure (s \<inter> cube n) \<le> gmeasure (cube n::'a set)"
```
```    57   apply(rule has_gmeasure_subset[of "s\<inter>cube n" _ "cube n"])
```
```    58   unfolding has_gmeasure_measure[THEN sym] using assms by auto
```
```    59
```
```    60 lemma has_gmeasure_cube[intro]: "(cube n::('a::ordered_euclidean_space) set)
```
```    61   has_gmeasure ((2 * real n) ^ (DIM('a)))"
```
```    62 proof-
```
```    63   have "content {\<chi>\<chi> i. - real n..(\<chi>\<chi> i. real n)::'a} = (2 * real n) ^ (DIM('a))"
```
```    64     apply(subst content_closed_interval) defer
```
```    65     by (auto simp add:setprod_constant)
```
```    66   thus ?thesis unfolding cube_def
```
```    67     using has_gmeasure_interval(1)[of "(\<chi>\<chi> i. - real n)::'a" "(\<chi>\<chi> i. real n)::'a"]
```
```    68     by auto
```
```    69 qed
```
```    70
```
```    71 lemma gmeasure_cube_eq[simp]:
```
```    72   "gmeasure (cube n::('a::ordered_euclidean_space) set) = (2 * real n) ^ DIM('a)"
```
```    73   by (intro measure_unique) auto
```
```    74
```
```    75 lemma gmeasure_cube_ge_n: "gmeasure (cube n::('a::ordered_euclidean_space) set) \<ge> real n"
```
```    76 proof cases
```
```    77   assume "n = 0" then show ?thesis by simp
```
```    78 next
```
```    79   assume "n \<noteq> 0"
```
```    80   have "real n \<le> (2 * real n)^1" by simp
```
```    81   also have "\<dots> \<le> (2 * real n)^DIM('a)"
```
```    82     using DIM_positive[where 'a='a] `n \<noteq> 0`
```
```    83     by (intro power_increasing) auto
```
```    84   also have "\<dots> = gmeasure (cube n::'a set)" by simp
```
```    85   finally show ?thesis .
```
```    86 qed
```
```    87
```
```    88 lemma gmeasure_setsum:
```
```    89   assumes "finite A" and "\<And>s t. s \<in> A \<Longrightarrow> t \<in> A \<Longrightarrow> s \<noteq> t \<Longrightarrow> f s \<inter> f t = {}"
```
```    90     and "\<And>i. i \<in> A \<Longrightarrow> gmeasurable (f i)"
```
```    91   shows "gmeasure (\<Union>i\<in>A. f i) = (\<Sum>i\<in>A. gmeasure (f i))"
```
```    92 proof -
```
```    93   have "gmeasure (\<Union>i\<in>A. f i) = gmeasure (\<Union>f ` A)" by auto
```
```    94   also have "\<dots> = setsum gmeasure (f ` A)" using assms
```
```    95   proof (intro measure_negligible_unions)
```
```    96     fix X Y assume "X \<in> f`A" "Y \<in> f`A" "X \<noteq> Y"
```
```    97     then have "X \<inter> Y = {}" using assms by auto
```
```    98     then show "negligible (X \<inter> Y)" by auto
```
```    99   qed auto
```
```   100   also have "\<dots> = setsum gmeasure (f ` A - {{}})"
```
```   101     using assms by (intro setsum_mono_zero_cong_right) auto
```
```   102   also have "\<dots> = (\<Sum>i\<in>A - {i. f i = {}}. gmeasure (f i))"
```
```   103   proof (intro setsum_reindex_cong inj_onI)
```
```   104     fix s t assume *: "s \<in> A - {i. f i = {}}" "t \<in> A - {i. f i = {}}" "f s = f t"
```
```   105     show "s = t"
```
```   106     proof (rule ccontr)
```
```   107       assume "s \<noteq> t" with assms(2)[of s t] * show False by auto
```
```   108     qed
```
```   109   qed auto
```
```   110   also have "\<dots> = (\<Sum>i\<in>A. gmeasure (f i))"
```
```   111     using assms by (intro setsum_mono_zero_cong_left) auto
```
```   112   finally show ?thesis .
```
```   113 qed
```
```   114
```
```   115 lemma gmeasurable_finite_UNION[intro]:
```
```   116   assumes "\<And>i. i \<in> S \<Longrightarrow> gmeasurable (A i)" "finite S"
```
```   117   shows "gmeasurable (\<Union>i\<in>S. A i)"
```
```   118   unfolding UNION_eq_Union_image using assms
```
```   119   by (intro gmeasurable_finite_unions) auto
```
```   120
```
```   121 lemma gmeasurable_countable_UNION[intro]:
```
```   122   fixes A :: "nat \<Rightarrow> ('a::ordered_euclidean_space) set"
```
```   123   assumes measurable: "\<And>i. gmeasurable (A i)"
```
```   124     and finite: "\<And>n. gmeasure (UNION {.. n} A) \<le> B"
```
```   125   shows "gmeasurable (\<Union>i. A i)"
```
```   126 proof -
```
```   127   have *: "\<And>n. \<Union>{A k |k. k \<le> n} = (\<Union>i\<le>n. A i)"
```
```   128     "(\<Union>{A n |n. n \<in> UNIV}) = (\<Union>i. A i)" by auto
```
```   129   show ?thesis
```
```   130     by (rule gmeasurable_countable_unions_strong[of A B, unfolded *, OF assms])
```
```   131 qed
```
```   132
```
```   133 subsection {* Measurability *}
```
```   134
```
```   135 definition lebesgue :: "'a::ordered_euclidean_space algebra" where
```
```   136   "lebesgue = \<lparr> space = UNIV, sets = {A. \<forall>n. gmeasurable (A \<inter> cube n)} \<rparr>"
```
```   137
```
```   138 lemma space_lebesgue[simp]:"space lebesgue = UNIV"
```
```   139   unfolding lebesgue_def by auto
```
```   140
```
```   141 lemma lebesgueD[dest]: assumes "S \<in> sets lebesgue"
```
```   142   shows "\<And>n. gmeasurable (S \<inter> cube n)"
```
```   143   using assms unfolding lebesgue_def by auto
```
```   144
```
```   145 lemma lebesgueI[intro]: assumes "gmeasurable S"
```
```   146   shows "S \<in> sets lebesgue" unfolding lebesgue_def cube_def
```
```   147   using assms gmeasurable_interval by auto
```
```   148
```
```   149 lemma lebesgueI2: "(\<And>n. gmeasurable (S \<inter> cube n)) \<Longrightarrow> S \<in> sets lebesgue"
```
```   150   using assms unfolding lebesgue_def by auto
```
```   151
```
```   152 interpretation lebesgue: sigma_algebra lebesgue
```
```   153 proof
```
```   154   show "sets lebesgue \<subseteq> Pow (space lebesgue)"
```
```   155     unfolding lebesgue_def by auto
```
```   156   show "{} \<in> sets lebesgue"
```
```   157     using gmeasurable_empty by auto
```
```   158   { fix A B :: "'a set" assume "A \<in> sets lebesgue" "B \<in> sets lebesgue"
```
```   159     then show "A \<union> B \<in> sets lebesgue"
```
```   160       by (auto intro: gmeasurable_union simp: lebesgue_def Int_Un_distrib2) }
```
```   161   { fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets lebesgue"
```
```   162     show "(\<Union>i. A i) \<in> sets lebesgue"
```
```   163     proof (rule lebesgueI2)
```
```   164       fix n show "gmeasurable ((\<Union>i. A i) \<inter> cube n)" unfolding UN_extend_simps
```
```   165         using A
```
```   166         by (intro gmeasurable_countable_UNION[where B="gmeasure (cube n::'a set)"])
```
```   167            (auto intro!: measure_subset gmeasure_setsum simp: UN_extend_simps simp del: gmeasure_cube_eq UN_simps)
```
```   168     qed }
```
```   169   { fix A assume A: "A \<in> sets lebesgue" show "space lebesgue - A \<in> sets lebesgue"
```
```   170     proof (rule lebesgueI2)
```
```   171       fix n
```
```   172       have *: "(space lebesgue - A) \<inter> cube n = cube n - (A \<inter> cube n)"
```
```   173         unfolding lebesgue_def by auto
```
```   174       show "gmeasurable ((space lebesgue - A) \<inter> cube n)" unfolding *
```
```   175         using A by (auto intro!: gmeasurable_diff)
```
```   176     qed }
```
```   177 qed
```
```   178
```
```   179 lemma lebesgueI_borel[intro, simp]: fixes s::"'a::ordered_euclidean_space set"
```
```   180   assumes "s \<in> sets borel" shows "s \<in> sets lebesgue"
```
```   181 proof- let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})"
```
```   182   have *:"?S \<subseteq> sets lebesgue" by auto
```
```   183   have "s \<in> sigma_sets UNIV ?S" using assms
```
```   184     unfolding borel_eq_atLeastAtMost by (simp add: sigma_def)
```
```   185   thus ?thesis
```
```   186     using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *]
```
```   187     by (auto simp: sigma_def)
```
```   188 qed
```
```   189
```
```   190 lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
```
```   191   assumes "negligible s" shows "s \<in> sets lebesgue"
```
```   192 proof (rule lebesgueI2)
```
```   193   fix n
```
```   194   have *:"\<And>x. (if x \<in> cube n then indicator s x else 0) = (if x \<in> s \<inter> cube n then 1 else 0)"
```
```   195     unfolding indicator_def_raw by auto
```
```   196   note assms[unfolded negligible_def,rule_format,of "(\<chi>\<chi> i. - real n)::'a" "\<chi>\<chi> i. real n"]
```
```   197   thus "gmeasurable (s \<inter> cube n)" apply-apply(rule gmeasurableI[of _ 0]) unfolding has_gmeasure_def
```
```   198     apply(subst(asm) has_integral_restrict_univ[THEN sym]) unfolding cube_def[symmetric]
```
```   199     apply(subst has_integral_restrict_univ[THEN sym]) unfolding * .
```
```   200 qed
```
```   201
```
```   202 section {* The Lebesgue Measure *}
```
```   203
```
```   204 definition "lmeasure A = (SUP n. Real (gmeasure (A \<inter> cube n)))"
```
```   205
```
```   206 lemma lmeasure_eq_0: assumes "negligible S" shows "lmeasure S = 0"
```
```   207 proof -
```
```   208   from lebesgueI_negligible[OF assms]
```
```   209   have "\<And>n. gmeasurable (S \<inter> cube n)" by auto
```
```   210   from gmeasurable_measure_eq_0[OF this]
```
```   211   have "\<And>n. gmeasure (S \<inter> cube n) = 0" using assms by auto
```
```   212   then show ?thesis unfolding lmeasure_def by simp
```
```   213 qed
```
```   214
```
```   215 lemma lmeasure_iff_LIMSEQ:
```
```   216   assumes "A \<in> sets lebesgue" "0 \<le> m"
```
```   217   shows "lmeasure A = Real m \<longleftrightarrow> (\<lambda>n. (gmeasure (A \<inter> cube n))) ----> m"
```
```   218   unfolding lmeasure_def using assms cube_subset[where 'a='a]
```
```   219   by (intro SUP_eq_LIMSEQ monoI measure_subset) force+
```
```   220
```
```   221 interpretation lebesgue: measure_space lebesgue lmeasure
```
```   222 proof
```
```   223   show "lmeasure {} = 0"
```
```   224     by (auto intro!: lmeasure_eq_0)
```
```   225   show "countably_additive lebesgue lmeasure"
```
```   226   proof (unfold countably_additive_def, intro allI impI conjI)
```
```   227     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets lebesgue" "disjoint_family A"
```
```   228     then have A: "\<And>i. A i \<in> sets lebesgue" by auto
```
```   229     show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (\<Union>i. A i)" unfolding lmeasure_def
```
```   230     proof (subst psuminf_SUP_eq)
```
```   231       { fix i n
```
```   232         have "gmeasure (A i \<inter> cube n) \<le> gmeasure (A i \<inter> cube (Suc n))"
```
```   233           using A cube_subset[of n "Suc n"] by (auto intro!: measure_subset)
```
```   234         then show "Real (gmeasure (A i \<inter> cube n)) \<le> Real (gmeasure (A i \<inter> cube (Suc n)))"
```
```   235           by auto }
```
```   236       show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = (SUP n. Real (gmeasure ((\<Union>i. A i) \<inter> cube n)))"
```
```   237       proof (intro arg_cong[where f="SUPR UNIV"] ext)
```
```   238         fix n
```
```   239         have sums: "(\<lambda>i. gmeasure (A i \<inter> cube n)) sums gmeasure (\<Union>{A i \<inter> cube n |i. i \<in> UNIV})"
```
```   240         proof (rule has_gmeasure_countable_negligible_unions(2))
```
```   241           fix i show "gmeasurable (A i \<inter> cube n)" using A by auto
```
```   242         next
```
```   243           fix i m :: nat assume "m \<noteq> i"
```
```   244           then have "A m \<inter> cube n \<inter> (A i \<inter> cube n) = {}"
```
```   245             using `disjoint_family A` unfolding disjoint_family_on_def by auto
```
```   246           then show "negligible (A m \<inter> cube n \<inter> (A i \<inter> cube n))" by auto
```
```   247         next
```
```   248           fix i
```
```   249           have "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) = gmeasure (\<Union>k\<le>i . A k \<inter> cube n)"
```
```   250             unfolding atLeast0AtMost using A
```
```   251           proof (intro gmeasure_setsum[symmetric])
```
```   252             fix s t :: nat assume "s \<noteq> t" then have "A t \<inter> A s = {}"
```
```   253               using `disjoint_family A` unfolding disjoint_family_on_def by auto
```
```   254             then show "A s \<inter> cube n \<inter> (A t \<inter> cube n) = {}" by auto
```
```   255           qed auto
```
```   256           also have "\<dots> \<le> gmeasure (cube n :: 'b set)" using A
```
```   257             by (intro measure_subset gmeasurable_finite_UNION) auto
```
```   258           finally show "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) \<le> gmeasure (cube n :: 'b set)" .
```
```   259         qed
```
```   260         show "(\<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = Real (gmeasure ((\<Union>i. A i) \<inter> cube n))"
```
```   261           unfolding psuminf_def
```
```   262           apply (subst setsum_Real)
```
```   263           apply (simp add: measure_pos_le)
```
```   264         proof (rule SUP_eq_LIMSEQ[THEN iffD2])
```
```   265           have "(\<Union>{A i \<inter> cube n |i. i \<in> UNIV}) = (\<Union>i. A i) \<inter> cube n" by auto
```
```   266           with sums show "(\<lambda>i. \<Sum>k<i. gmeasure (A k \<inter> cube n)) ----> gmeasure ((\<Union>i. A i) \<inter> cube n)"
```
```   267             unfolding sums_def atLeast0LessThan by simp
```
```   268         qed (auto intro!: monoI setsum_nonneg setsum_mono2)
```
```   269       qed
```
```   270     qed
```
```   271   qed
```
```   272 qed
```
```   273
```
```   274 lemma lmeasure_finite_has_gmeasure: assumes "s \<in> sets lebesgue" "lmeasure s = Real m" "0 \<le> m"
```
```   275   shows "s has_gmeasure m"
```
```   276 proof-
```
```   277   have *:"(\<lambda>n. (gmeasure (s \<inter> cube n))) ----> m"
```
```   278     using `lmeasure s = Real m` unfolding lmeasure_iff_LIMSEQ[OF `s \<in> sets lebesgue` `0 \<le> m`] .
```
```   279   have s: "\<And>n. gmeasurable (s \<inter> cube n)" using assms by auto
```
```   280   have "(\<lambda>x. if x \<in> s then 1 else (0::real)) integrable_on UNIV \<and>
```
```   281     (\<lambda>k. integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)))
```
```   282     ----> integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)"
```
```   283   proof(rule monotone_convergence_increasing)
```
```   284     have "lmeasure s \<le> Real m" using `lmeasure s = Real m` by simp
```
```   285     then have "\<forall>n. gmeasure (s \<inter> cube n) \<le> m"
```
```   286       unfolding lmeasure_def complete_lattice_class.SUP_le_iff
```
```   287       using `0 \<le> m` by (auto simp: measure_pos_le)
```
```   288     thus *:"bounded {integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)) |k. True}"
```
```   289       unfolding integral_measure_univ[OF s] bounded_def apply-
```
```   290       apply(rule_tac x=0 in exI,rule_tac x=m in exI) unfolding dist_real_def
```
```   291       by (auto simp: measure_pos_le)
```
```   292     show "\<forall>k. (\<lambda>x. if x \<in> s \<inter> cube k then (1::real) else 0) integrable_on UNIV"
```
```   293       unfolding integrable_restrict_univ
```
```   294       using s unfolding gmeasurable_def has_gmeasure_def by auto
```
```   295     have *:"\<And>n. n \<le> Suc n" by auto
```
```   296     show "\<forall>k. \<forall>x\<in>UNIV. (if x \<in> s \<inter> cube k then 1 else 0) \<le> (if x \<in> s \<inter> cube (Suc k) then 1 else (0::real))"
```
```   297       using cube_subset[OF *] by fastsimp
```
```   298     show "\<forall>x\<in>UNIV. (\<lambda>k. if x \<in> s \<inter> cube k then 1 else 0) ----> (if x \<in> s then 1 else (0::real))"
```
```   299       unfolding Lim_sequentially
```
```   300     proof safe case goal1 from real_arch_lt[of "norm x"] guess N .. note N = this
```
```   301       show ?case apply(rule_tac x=N in exI)
```
```   302       proof safe case goal1
```
```   303         have "x \<in> cube n" using cube_subset[OF goal1] N
```
```   304           using ball_subset_cube[of N] by(auto simp: dist_norm)
```
```   305         thus ?case using `e>0` by auto
```
```   306       qed
```
```   307     qed
```
```   308   qed note ** = conjunctD2[OF this]
```
```   309   hence *:"m = integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)" apply-
```
```   310     apply(rule LIMSEQ_unique[OF _ **(2)]) unfolding measure_integral_univ[THEN sym,OF s] using * .
```
```   311   show ?thesis unfolding has_gmeasure * apply(rule integrable_integral) using ** by auto
```
```   312 qed
```
```   313
```
```   314 lemma lmeasure_finite_gmeasurable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>"
```
```   315   shows "gmeasurable s"
```
```   316 proof (cases "lmeasure s")
```
```   317   case (preal m) from lmeasure_finite_has_gmeasure[OF `s \<in> sets lebesgue` this]
```
```   318   show ?thesis unfolding gmeasurable_def by auto
```
```   319 qed (insert assms, auto)
```
```   320
```
```   321 lemma has_gmeasure_lmeasure: assumes "s has_gmeasure m"
```
```   322   shows "lmeasure s = Real m"
```
```   323 proof-
```
```   324   have gmea:"gmeasurable s" using assms by auto
```
```   325   then have s: "s \<in> sets lebesgue" by auto
```
```   326   have m:"m \<ge> 0" using assms by auto
```
```   327   have *:"m = gmeasure (\<Union>{s \<inter> cube n |n. n \<in> UNIV})" unfolding Union_inter_cube
```
```   328     using assms by(rule measure_unique[THEN sym])
```
```   329   show ?thesis
```
```   330     unfolding lmeasure_iff_LIMSEQ[OF s `0 \<le> m`] unfolding *
```
```   331     apply(rule gmeasurable_nested_unions[THEN conjunct2, where B1="gmeasure s"])
```
```   332   proof- fix n::nat show *:"gmeasurable (s \<inter> cube n)"
```
```   333       using gmeasurable_inter[OF gmea gmeasurable_cube] .
```
```   334     show "gmeasure (s \<inter> cube n) \<le> gmeasure s" apply(rule measure_subset)
```
```   335       apply(rule * gmea)+ by auto
```
```   336     show "s \<inter> cube n \<subseteq> s \<inter> cube (Suc n)" using cube_subset[of n "Suc n"] by auto
```
```   337   qed
```
```   338 qed
```
```   339
```
```   340 lemma has_gmeasure_iff_lmeasure:
```
```   341   "A has_gmeasure m \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m)"
```
```   342 proof
```
```   343   assume "A has_gmeasure m"
```
```   344   with has_gmeasure_lmeasure[OF this]
```
```   345   have "gmeasurable A" "0 \<le> m" "lmeasure A = Real m" by auto
```
```   346   then show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m" by auto
```
```   347 next
```
```   348   assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m"
```
```   349   then show "A has_gmeasure m" by (intro lmeasure_finite_has_gmeasure) auto
```
```   350 qed
```
```   351
```
```   352 lemma gmeasure_lmeasure: assumes "gmeasurable s" shows "lmeasure s = Real (gmeasure s)"
```
```   353 proof -
```
```   354   note has_gmeasure_measureI[OF assms]
```
```   355   note has_gmeasure_lmeasure[OF this]
```
```   356   thus ?thesis .
```
```   357 qed
```
```   358
```
```   359 lemma lebesgue_simple_function_indicator:
```
```   360   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
```
```   361   assumes f:"lebesgue.simple_function f"
```
```   362   shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
```
```   363   apply(rule,subst lebesgue.simple_function_indicator_representation[OF f]) by auto
```
```   364
```
```   365 lemma lmeasure_gmeasure:
```
```   366   "gmeasurable s \<Longrightarrow> gmeasure s = real (lmeasure s)"
```
```   367   by (subst gmeasure_lmeasure) auto
```
```   368
```
```   369 lemma lmeasure_finite: assumes "gmeasurable s" shows "lmeasure s \<noteq> \<omega>"
```
```   370   using gmeasure_lmeasure[OF assms] by auto
```
```   371
```
```   372 lemma negligible_iff_lebesgue_null_sets:
```
```   373   "negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets"
```
```   374 proof
```
```   375   assume "negligible A"
```
```   376   from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
```
```   377   show "A \<in> lebesgue.null_sets" by auto
```
```   378 next
```
```   379   assume A: "A \<in> lebesgue.null_sets"
```
```   380   then have *:"gmeasurable A" using lmeasure_finite_gmeasurable[of A] by auto
```
```   381   show "negligible A"
```
```   382     unfolding gmeasurable_measure_eq_0[OF *, symmetric]
```
```   383     unfolding lmeasure_gmeasure[OF *] using A by auto
```
```   384 qed
```
```   385
```
```   386 lemma
```
```   387   fixes a b ::"'a::ordered_euclidean_space"
```
```   388   shows lmeasure_atLeastAtMost[simp]: "lmeasure {a..b} = Real (content {a..b})"
```
```   389     and lmeasure_greaterThanLessThan[simp]: "lmeasure {a <..< b} = Real (content {a..b})"
```
```   390   using has_gmeasure_interval[of a b] by (auto intro!: has_gmeasure_lmeasure)
```
```   391
```
```   392 lemma lmeasure_cube:
```
```   393   "lmeasure (cube n::('a::ordered_euclidean_space) set) = (Real ((2 * real n) ^ (DIM('a))))"
```
```   394   by (intro has_gmeasure_lmeasure) auto
```
```   395
```
```   396 lemma lmeasure_UNIV[intro]: "lmeasure UNIV = \<omega>"
```
```   397   unfolding lmeasure_def SUP_\<omega>
```
```   398 proof (intro allI impI)
```
```   399   fix x assume "x < \<omega>"
```
```   400   then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto
```
```   401   then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto
```
```   402   show "\<exists>i\<in>UNIV. x < Real (gmeasure (UNIV \<inter> cube i))"
```
```   403   proof (intro bexI[of _ n])
```
```   404     have "x < Real (of_nat n)" using n r by auto
```
```   405     also have "Real (of_nat n) \<le> Real (gmeasure (UNIV \<inter> cube n))"
```
```   406       using gmeasure_cube_ge_n[of n] by (auto simp: real_eq_of_nat[symmetric])
```
```   407     finally show "x < Real (gmeasure (UNIV \<inter> cube n))" .
```
```   408   qed auto
```
```   409 qed
```
```   410
```
```   411 lemma atLeastAtMost_singleton_euclidean[simp]:
```
```   412   fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
```
```   413   by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a])
```
```   414
```
```   415 lemma content_singleton[simp]: "content {a} = 0"
```
```   416 proof -
```
```   417   have "content {a .. a} = 0"
```
```   418     by (subst content_closed_interval) auto
```
```   419   then show ?thesis by simp
```
```   420 qed
```
```   421
```
```   422 lemma lmeasure_singleton[simp]:
```
```   423   fixes a :: "'a::ordered_euclidean_space" shows "lmeasure {a} = 0"
```
```   424   using has_gmeasure_interval[of a a] unfolding zero_pextreal_def
```
```   425   by (intro has_gmeasure_lmeasure)
```
```   426      (simp add: content_closed_interval DIM_positive)
```
```   427
```
```   428 declare content_real[simp]
```
```   429
```
```   430 lemma
```
```   431   fixes a b :: real
```
```   432   shows lmeasure_real_greaterThanAtMost[simp]:
```
```   433     "lmeasure {a <.. b} = Real (if a \<le> b then b - a else 0)"
```
```   434 proof cases
```
```   435   assume "a < b"
```
```   436   then have "lmeasure {a <.. b} = lmeasure {a <..< b} + lmeasure {b}"
```
```   437     by (subst lebesgue.measure_additive)
```
```   438        (auto intro!: lebesgueI_borel arg_cong[where f=lmeasure])
```
```   439   then show ?thesis by auto
```
```   440 qed auto
```
```   441
```
```   442 lemma
```
```   443   fixes a b :: real
```
```   444   shows lmeasure_real_atLeastLessThan[simp]:
```
```   445     "lmeasure {a ..< b} = Real (if a \<le> b then b - a else 0)" (is ?eqlt)
```
```   446 proof cases
```
```   447   assume "a < b"
```
```   448   then have "lmeasure {a ..< b} = lmeasure {a} + lmeasure {a <..< b}"
```
```   449     by (subst lebesgue.measure_additive)
```
```   450        (auto intro!: lebesgueI_borel arg_cong[where f=lmeasure])
```
```   451   then show ?thesis by auto
```
```   452 qed auto
```
```   453
```
```   454 interpretation borel: measure_space borel lmeasure
```
```   455 proof
```
```   456   show "countably_additive borel lmeasure"
```
```   457     using lebesgue.ca unfolding countably_additive_def
```
```   458     apply safe apply (erule_tac x=A in allE) by auto
```
```   459 qed auto
```
```   460
```
```   461 interpretation borel: sigma_finite_measure borel lmeasure
```
```   462 proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
```
```   463   show "range cube \<subseteq> sets borel" by (auto intro: borel_closed)
```
```   464   { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
```
```   465   thus "(\<Union>i. cube i) = space borel" by auto
```
```   466   show "\<forall>i. lmeasure (cube i) \<noteq> \<omega>" unfolding lmeasure_cube by auto
```
```   467 qed
```
```   468
```
```   469 interpretation lebesgue: sigma_finite_measure lebesgue lmeasure
```
```   470 proof
```
```   471   from borel.sigma_finite guess A ..
```
```   472   moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
```
```   473   ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lmeasure (A i) \<noteq> \<omega>)"
```
```   474     by auto
```
```   475 qed
```
```   476
```
```   477 lemma simple_function_has_integral:
```
```   478   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
```
```   479   assumes f:"lebesgue.simple_function f"
```
```   480   and f':"\<forall>x. f x \<noteq> \<omega>"
```
```   481   and om:"\<forall>x\<in>range f. lmeasure (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
```
```   482   shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
```
```   483   unfolding lebesgue.simple_integral_def
```
```   484   apply(subst lebesgue_simple_function_indicator[OF f])
```
```   485 proof- case goal1
```
```   486   have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
```
```   487     "\<forall>x\<in>range f. x * lmeasure (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
```
```   488     using f' om unfolding indicator_def by auto
```
```   489   show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym]
```
```   490     unfolding real_of_pextreal_setsum'[OF *(2),THEN sym]
```
```   491     unfolding real_of_pextreal_setsum space_lebesgue
```
```   492     apply(rule has_integral_setsum)
```
```   493   proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD)
```
```   494     fix y::'a show "((\<lambda>x. real (f y * indicator (f -` {f y}) x)) has_integral
```
```   495       real (f y * lmeasure (f -` {f y} \<inter> UNIV))) UNIV"
```
```   496     proof(cases "f y = 0") case False
```
```   497       have mea:"gmeasurable (f -` {f y})" apply(rule lmeasure_finite_gmeasurable)
```
```   498         using assms unfolding lebesgue.simple_function_def using False by auto
```
```   499       have *:"\<And>x. real (indicator (f -` {f y}) x::pextreal) = (if x \<in> f -` {f y} then 1 else 0)" by auto
```
```   500       show ?thesis unfolding real_of_pextreal_mult[THEN sym]
```
```   501         apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def])
```
```   502         unfolding Int_UNIV_right lmeasure_gmeasure[OF mea,THEN sym]
```
```   503         unfolding measure_integral_univ[OF mea] * apply(rule integrable_integral)
```
```   504         unfolding gmeasurable_integrable[THEN sym] using mea .
```
```   505     qed auto
```
```   506   qed qed
```
```   507
```
```   508 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
```
```   509   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
```
```   510   using assms by auto
```
```   511
```
```   512 lemma simple_function_has_integral':
```
```   513   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
```
```   514   assumes f:"lebesgue.simple_function f"
```
```   515   and i: "lebesgue.simple_integral f \<noteq> \<omega>"
```
```   516   shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
```
```   517 proof- let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x"
```
```   518   { fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this
```
```   519   have **:"{x. f x \<noteq> ?f x} = f -` {\<omega>}" by auto
```
```   520   have **:"lmeasure {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
```
```   521     using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**)
```
```   522   show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **])
```
```   523     apply(rule lebesgue.simple_function_compose1[OF f])
```
```   524     unfolding * defer apply(rule simple_function_has_integral)
```
```   525   proof-
```
```   526     show "lebesgue.simple_function ?f"
```
```   527       using lebesgue.simple_function_compose1[OF f] .
```
```   528     show "\<forall>x. ?f x \<noteq> \<omega>" by auto
```
```   529     show "\<forall>x\<in>range ?f. lmeasure (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
```
```   530     proof (safe, simp, safe, rule ccontr)
```
```   531       fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0"
```
```   532       hence "(\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y} = f -` {f y}"
```
```   533         by (auto split: split_if_asm)
```
```   534       moreover assume "lmeasure ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
```
```   535       ultimately have "lmeasure (f -` {f y}) = \<omega>" by simp
```
```   536       moreover
```
```   537       have "f y * lmeasure (f -` {f y}) \<noteq> \<omega>" using i f
```
```   538         unfolding lebesgue.simple_integral_def setsum_\<omega> lebesgue.simple_function_def
```
```   539         by auto
```
```   540       ultimately have "f y = 0" by (auto split: split_if_asm)
```
```   541       then show False using `f y \<noteq> 0` by simp
```
```   542     qed
```
```   543   qed
```
```   544 qed
```
```   545
```
```   546 lemma (in measure_space) positive_integral_monotone_convergence:
```
```   547   fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pextreal"
```
```   548   assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)"
```
```   549   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
```
```   550   shows "u \<in> borel_measurable M"
```
```   551   and "(\<lambda>i. positive_integral (f i)) ----> positive_integral u" (is ?ilim)
```
```   552 proof -
```
```   553   from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
```
```   554   show ?ilim using mono lim i by auto
```
```   555   have "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal
```
```   556     unfolding fun_eq_iff mono_def by auto
```
```   557   moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
```
```   558     using i by auto
```
```   559   ultimately show "u \<in> borel_measurable M" by simp
```
```   560 qed
```
```   561
```
```   562 lemma positive_integral_has_integral:
```
```   563   fixes f::"'a::ordered_euclidean_space => pextreal"
```
```   564   assumes f:"f \<in> borel_measurable lebesgue"
```
```   565   and int_om:"lebesgue.positive_integral f \<noteq> \<omega>"
```
```   566   and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
```
```   567   shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.positive_integral f))) UNIV"
```
```   568 proof- let ?i = "lebesgue.positive_integral f"
```
```   569   from lebesgue.borel_measurable_implies_simple_function_sequence[OF f]
```
```   570   guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2)
```
```   571   let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)"
```
```   572   have u_simple:"\<And>k. lebesgue.simple_integral (u k) = lebesgue.positive_integral (u k)"
```
```   573     apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) ..
```
```   574   have int_u_le:"\<And>k. lebesgue.simple_integral (u k) \<le> lebesgue.positive_integral f"
```
```   575     unfolding u_simple apply(rule lebesgue.positive_integral_mono)
```
```   576     using isoton_Sup[OF u(3)] unfolding le_fun_def by auto
```
```   577   have u_int_om:"\<And>i. lebesgue.simple_integral (u i) \<noteq> \<omega>"
```
```   578   proof- case goal1 thus ?case using int_u_le[of i] int_om by auto qed
```
```   579
```
```   580   note u_int = simple_function_has_integral'[OF u(1) this]
```
```   581   have "(\<lambda>x. real (f x)) integrable_on UNIV \<and>
```
```   582     (\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) ----> Integration.integral UNIV (\<lambda>x. real (f x))"
```
```   583     apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int)
```
```   584   proof safe case goal1 show ?case apply(rule real_of_pextreal_mono) using u(2,3) by auto
```
```   585   next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym])
```
```   586       prefer 3 apply(subst Real_real') defer apply(subst Real_real')
```
```   587       using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto
```
```   588   next case goal3
```
```   589     show ?case apply(rule bounded_realI[where B="real (lebesgue.positive_integral f)"])
```
```   590       apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int)
```
```   591       unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le])
```
```   592       using u int_om by auto
```
```   593   qed note int = conjunctD2[OF this]
```
```   594
```
```   595   have "(\<lambda>i. lebesgue.simple_integral (u i)) ----> ?i" unfolding u_simple
```
```   596     apply(rule lebesgue.positive_integral_monotone_convergence(2))
```
```   597     apply(rule lebesgue.borel_measurable_simple_function[OF u(1)])
```
```   598     using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto
```
```   599   hence "(\<lambda>i. real (lebesgue.simple_integral (u i))) ----> real ?i" apply-
```
```   600     apply(subst lim_Real[THEN sym]) prefer 3
```
```   601     apply(subst Real_real') defer apply(subst Real_real')
```
```   602     using u f_om int_om u_int_om by auto
```
```   603   note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]]
```
```   604   show ?thesis unfolding * by(rule integrable_integral[OF int(1)])
```
```   605 qed
```
```   606
```
```   607 lemma lebesgue_integral_has_integral:
```
```   608   fixes f::"'a::ordered_euclidean_space => real"
```
```   609   assumes f:"lebesgue.integrable f"
```
```   610   shows "(f has_integral (lebesgue.integral f)) UNIV"
```
```   611 proof- let ?n = "\<lambda>x. - min (f x) 0" and ?p = "\<lambda>x. max (f x) 0"
```
```   612   have *:"f = (\<lambda>x. ?p x - ?n x)" apply rule by auto
```
```   613   note f = lebesgue.integrableD[OF f]
```
```   614   show ?thesis unfolding lebesgue.integral_def apply(subst *)
```
```   615   proof(rule has_integral_sub) case goal1
```
```   616     have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto
```
```   617     note lebesgue.borel_measurable_Real[OF f(1)]
```
```   618     from positive_integral_has_integral[OF this f(2) *]
```
```   619     show ?case unfolding real_Real_max .
```
```   620   next case goal2
```
```   621     have *:"\<forall>x. Real (- f x) \<noteq> \<omega>" by auto
```
```   622     note lebesgue.borel_measurable_uminus[OF f(1)]
```
```   623     note lebesgue.borel_measurable_Real[OF this]
```
```   624     from positive_integral_has_integral[OF this f(3) *]
```
```   625     show ?case unfolding real_Real_max minus_min_eq_max by auto
```
```   626   qed
```
```   627 qed
```
```   628
```
```   629 lemma lebesgue_positive_integral_eq_borel:
```
```   630   "f \<in> borel_measurable borel \<Longrightarrow> lebesgue.positive_integral f = borel.positive_integral f "
```
```   631   by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
```
```   632
```
```   633 lemma lebesgue_integral_eq_borel:
```
```   634   assumes "f \<in> borel_measurable borel"
```
```   635   shows "lebesgue.integrable f = borel.integrable f" (is ?P)
```
```   636     and "lebesgue.integral f = borel.integral f" (is ?I)
```
```   637 proof -
```
```   638   have *: "sigma_algebra borel" by default
```
```   639   have "sets borel \<subseteq> sets lebesgue" by auto
```
```   640   from lebesgue.integral_subalgebra[OF assms this _ *]
```
```   641   show ?P ?I by auto
```
```   642 qed
```
```   643
```
```   644 lemma borel_integral_has_integral:
```
```   645   fixes f::"'a::ordered_euclidean_space => real"
```
```   646   assumes f:"borel.integrable f"
```
```   647   shows "(f has_integral (borel.integral f)) UNIV"
```
```   648 proof -
```
```   649   have borel: "f \<in> borel_measurable borel"
```
```   650     using f unfolding borel.integrable_def by auto
```
```   651   from f show ?thesis
```
```   652     using lebesgue_integral_has_integral[of f]
```
```   653     unfolding lebesgue_integral_eq_borel[OF borel] by simp
```
```   654 qed
```
```   655
```
```   656 lemma continuous_on_imp_borel_measurable:
```
```   657   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
```
```   658   assumes "continuous_on UNIV f"
```
```   659   shows "f \<in> borel_measurable borel"
```
```   660   apply(rule borel.borel_measurableI)
```
```   661   using continuous_open_preimage[OF assms] unfolding vimage_def by auto
```
```   662
```
```   663 lemma (in measure_space) integral_monotone_convergence_pos':
```
```   664   assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
```
```   665   and pos: "\<And>x i. 0 \<le> f i x"
```
```   666   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
```
```   667   and ilim: "(\<lambda>i. integral (f i)) ----> x"
```
```   668   shows "integrable u \<and> integral u = x"
```
```   669   using integral_monotone_convergence_pos[OF assms] by auto
```
```   670
```
```   671 definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where
```
```   672   "e2p x = (\<lambda>i\<in>{..<DIM('a)}. x\$\$i)"
```
```   673
```
```   674 definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where
```
```   675   "p2e x = (\<chi>\<chi> i. x i)"
```
```   676
```
```   677 lemma e2p_p2e[simp]:
```
```   678   "x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x"
```
```   679   by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def)
```
```   680
```
```   681 lemma p2e_e2p[simp]:
```
```   682   "p2e (e2p x) = (x::'a::ordered_euclidean_space)"
```
```   683   by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def)
```
```   684
```
```   685 lemma bij_inv_p2e_e2p:
```
```   686   shows "bij_inv ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) (UNIV :: 'a::ordered_euclidean_space set)
```
```   687      p2e e2p" (is "bij_inv ?P ?U _ _")
```
```   688 proof (rule bij_invI)
```
```   689   show "p2e \<in> ?P \<rightarrow> ?U" "e2p \<in> ?U \<rightarrow> ?P" by (auto simp: e2p_def)
```
```   690 qed auto
```
```   691
```
```   692 interpretation borel_product: product_sigma_finite "\<lambda>x. borel::real algebra" "\<lambda>x. lmeasure"
```
```   693   by default
```
```   694
```
```   695 lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
```
```   696   unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
```
```   697
```
```   698 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
```
```   699   unfolding Pi_def by auto
```
```   700
```
```   701 lemma measurable_e2p_on_generator:
```
```   702   "e2p \<in> measurable \<lparr> space = UNIV::'a set, sets = range lessThan \<rparr>
```
```   703   (product_algebra
```
```   704     (\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>)
```
```   705     {..<DIM('a::ordered_euclidean_space)})"
```
```   706   (is "e2p \<in> measurable ?E ?P")
```
```   707 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   708   show "e2p \<in> space ?E \<rightarrow> space ?P" by (auto simp: e2p_def)
```
```   709   fix A assume "A \<in> sets ?P"
```
```   710   then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)"
```
```   711     and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)"
```
```   712     by (auto elim!: product_algebraE)
```
```   713   then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto
```
```   714   from this[THEN bchoice] guess xs ..
```
```   715   then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})"
```
```   716     using A by auto
```
```   717   have "e2p -` A = {..< (\<chi>\<chi> i. xs i) :: 'a}"
```
```   718     using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq
```
```   719       euclidean_eq[where 'a='a] eucl_less[where 'a='a])
```
```   720   then show "e2p -` A \<inter> space ?E \<in> sets ?E" by simp
```
```   721 qed
```
```   722
```
```   723 lemma measurable_p2e_on_generator:
```
```   724   "p2e \<in> measurable
```
```   725     (product_algebra
```
```   726       (\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>)
```
```   727       {..<DIM('a::ordered_euclidean_space)})
```
```   728     \<lparr> space = UNIV::'a set, sets = range lessThan \<rparr>"
```
```   729   (is "p2e \<in> measurable ?P ?E")
```
```   730 proof (unfold measurable_def, intro CollectI conjI ballI)
```
```   731   show "p2e \<in> space ?P \<rightarrow> space ?E" by simp
```
```   732   fix A assume "A \<in> sets ?E"
```
```   733   then obtain x where "A = {..<x}" by auto
```
```   734   then have "p2e -` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x \$\$ i})"
```
```   735     using DIM_positive
```
```   736     by (auto simp: Pi_iff set_eq_iff p2e_def
```
```   737                    euclidean_eq[where 'a='a] eucl_less[where 'a='a])
```
```   738   then show "p2e -` A \<inter> space ?P \<in> sets ?P" by auto
```
```   739 qed
```
```   740
```
```   741 lemma borel_vimage_algebra_eq:
```
```   742   defines "F \<equiv> product_algebra (\<lambda>x. \<lparr> space = (UNIV::real set), sets = range lessThan \<rparr>) {..<DIM('a::ordered_euclidean_space)}"
```
```   743   shows "sigma_algebra.vimage_algebra (borel::'a::ordered_euclidean_space algebra) (space (sigma F)) p2e = sigma F"
```
```   744   unfolding borel_eq_lessThan
```
```   745 proof (intro vimage_algebra_sigma)
```
```   746   let ?E = "\<lparr>space = (UNIV::'a set), sets = range lessThan\<rparr>"
```
```   747   show "bij_inv (space (sigma F)) (space (sigma ?E)) p2e e2p"
```
```   748     using bij_inv_p2e_e2p unfolding F_def by simp
```
```   749   show "sets F \<subseteq> Pow (space F)" "sets ?E \<subseteq> Pow (space ?E)" unfolding F_def
```
```   750     by (intro product_algebra_sets_into_space) auto
```
```   751   show "p2e \<in> measurable F ?E"
```
```   752     "e2p \<in> measurable ?E F"
```
```   753     unfolding F_def using measurable_p2e_on_generator measurable_e2p_on_generator by auto
```
```   754 qed
```
```   755
```
```   756 lemma product_borel_eq_vimage:
```
```   757   "sigma (product_algebra (\<lambda>x. borel) {..<DIM('a::ordered_euclidean_space)}) =
```
```   758   sigma_algebra.vimage_algebra borel (extensional {..<DIM('a)})
```
```   759   (p2e:: _ \<Rightarrow> 'a::ordered_euclidean_space)"
```
```   760   unfolding borel_vimage_algebra_eq[simplified]
```
```   761   unfolding borel_eq_lessThan
```
```   762   apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>i. \<lambda>n. lessThan (real n)"])
```
```   763   unfolding lessThan_iff
```
```   764 proof- fix i assume i:"i<DIM('a)"
```
```   765   show "(\<lambda>n. {..<real n}) \<up> space \<lparr>space = UNIV, sets = range lessThan\<rparr>"
```
```   766     by(auto intro!:real_arch_lt isotoneI)
```
```   767 qed auto
```
```   768
```
```   769 lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
```
```   770   apply(rule image_Int[THEN sym])
```
```   771   using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)]
```
```   772   unfolding bij_betw_def by auto
```
```   773
```
```   774 lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space"
```
```   775   shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>"
```
```   776   unfolding Int_stable_def algebra.select_convs
```
```   777 proof safe fix a b x y::'a
```
```   778   have *:"e2p ` {a..b} \<inter> e2p ` {x..y} =
```
```   779     (\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a \$\$ i) (x \$\$ i), \<chi>\<chi> i. min (b \$\$ i) (y \$\$ i)::'a)"
```
```   780     unfolding e2p_Int inter_interval by auto
```
```   781   show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding *
```
```   782     apply(rule range_eqI) ..
```
```   783 qed
```
```   784
```
```   785 lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space"
```
```   786   shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>"
```
```   787   unfolding Int_stable_def algebra.select_convs
```
```   788   apply safe unfolding inter_interval by auto
```
```   789
```
```   790 lemma inj_on_disjoint_family_on: assumes "disjoint_family_on A S" "inj f"
```
```   791   shows "disjoint_family_on (\<lambda>x. f ` A x) S"
```
```   792   unfolding disjoint_family_on_def
```
```   793 proof(rule,rule,rule)
```
```   794   fix x1 x2 assume x:"x1 \<in> S" "x2 \<in> S" "x1 \<noteq> x2"
```
```   795   show "f ` A x1 \<inter> f ` A x2 = {}"
```
```   796   proof(rule ccontr) case goal1
```
```   797     then obtain z where z:"z \<in> f ` A x1 \<inter> f ` A x2" by auto
```
```   798     then obtain z1 z2 where z12:"z1 \<in> A x1" "z2 \<in> A x2" "f z1 = z" "f z2 = z" by auto
```
```   799     hence "z1 = z2" using assms(2) unfolding inj_on_def by blast
```
```   800     hence "x1 = x2" using z12(1-2) using assms[unfolded disjoint_family_on_def] using x by auto
```
```   801     thus False using x(3) by auto
```
```   802   qed
```
```   803 qed
```
```   804
```
```   805 declare restrict_extensional[intro]
```
```   806
```
```   807 lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}"
```
```   808   unfolding e2p_def by auto
```
```   809
```
```   810 lemma e2p_image_vimage: fixes A::"'a::ordered_euclidean_space set"
```
```   811   shows "e2p ` A = p2e -` A \<inter> extensional {..<DIM('a)}"
```
```   812 proof(rule set_eqI,rule)
```
```   813   fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this
```
```   814   show "x \<in> p2e -` A \<inter> extensional {..<DIM('a)}"
```
```   815     apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto
```
```   816 next fix x assume "x \<in> p2e -` A \<inter> extensional {..<DIM('a)}"
```
```   817   thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto
```
```   818 qed
```
```   819
```
```   820 lemma lmeasure_measure_eq_borel_prod:
```
```   821   fixes A :: "('a::ordered_euclidean_space) set"
```
```   822   assumes "A \<in> sets borel"
```
```   823   shows "lmeasure A = borel_product.product_measure {..<DIM('a)} (e2p ` A :: (nat \<Rightarrow> real) set)"
```
```   824 proof (rule measure_unique_Int_stable[where X=A and A=cube])
```
```   825   interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
```
```   826   show "Int_stable \<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
```
```   827     (is "Int_stable ?E" ) using Int_stable_cuboids' .
```
```   828   show "borel = sigma ?E" using borel_eq_atLeastAtMost .
```
```   829   show "\<And>i. lmeasure (cube i) \<noteq> \<omega>" unfolding lmeasure_cube by auto
```
```   830   show "\<And>X. X \<in> sets ?E \<Longrightarrow>
```
```   831     lmeasure X = borel_product.product_measure {..<DIM('a)} (e2p ` X :: (nat \<Rightarrow> real) set)"
```
```   832   proof- case goal1 then obtain a b where X:"X = {a..b}" by auto
```
```   833     { presume *:"X \<noteq> {} \<Longrightarrow> ?case"
```
```   834       show ?case apply(cases,rule *,assumption) by auto }
```
```   835     def XX \<equiv> "\<lambda>i. {a \$\$ i .. b \$\$ i}" assume  "X \<noteq> {}"  note X' = this[unfolded X interval_ne_empty]
```
```   836     have *:"Pi\<^isub>E {..<DIM('a)} XX = e2p ` X" apply(rule set_eqI)
```
```   837     proof fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} XX"
```
```   838       thus "x \<in> e2p ` X" unfolding image_iff apply(rule_tac x="\<chi>\<chi> i. x i" in bexI)
```
```   839         unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by rule auto
```
```   840     next fix x assume "x \<in> e2p ` X" then guess y unfolding image_iff .. note y = this
```
```   841       show "x \<in> Pi\<^isub>E {..<DIM('a)} XX" unfolding y using y(1)
```
```   842         unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by auto
```
```   843     qed
```
```   844     have "lmeasure X = (\<Prod>x<DIM('a). Real (b \$\$ x - a \$\$ x))"  using X' apply- unfolding X
```
```   845       unfolding lmeasure_atLeastAtMost content_closed_interval apply(subst Real_setprod) by auto
```
```   846     also have "... = (\<Prod>i<DIM('a). lmeasure (XX i))" apply(rule setprod_cong2)
```
```   847       unfolding XX_def lmeasure_atLeastAtMost apply(subst content_real) using X' by auto
```
```   848     also have "... = borel_product.product_measure {..<DIM('a)} (e2p ` X)" unfolding *[THEN sym]
```
```   849       apply(rule fprod.measure_times[THEN sym]) unfolding XX_def by auto
```
```   850     finally show ?case .
```
```   851   qed
```
```   852
```
```   853   show "range cube \<subseteq> sets \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
```
```   854     unfolding cube_def_raw by auto
```
```   855   have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp
```
```   856   thus "cube \<up> space \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
```
```   857     apply-apply(rule isotoneI) apply(rule cube_subset_Suc) by auto
```
```   858   show "A \<in> sets borel " by fact
```
```   859   show "measure_space borel lmeasure" by default
```
```   860   show "measure_space borel
```
```   861      (\<lambda>a::'a set. finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` a))"
```
```   862     apply default unfolding countably_additive_def
```
```   863   proof safe fix A::"nat \<Rightarrow> 'a set" assume A:"range A \<subseteq> sets borel" "disjoint_family A"
```
```   864       "(\<Union>i. A i) \<in> sets borel"
```
```   865     note fprod.ca[unfolded countably_additive_def,rule_format]
```
```   866     note ca = this[of "\<lambda> n. e2p ` (A n)"]
```
```   867     show "(\<Sum>\<^isub>\<infinity>n. finite_product_sigma_finite.measure
```
```   868         (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` A n)) =
```
```   869            finite_product_sigma_finite.measure (\<lambda>x. borel)
```
```   870             (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` (\<Union>i. A i))" unfolding image_UN
```
```   871     proof(rule ca) show "range (\<lambda>n. e2p ` A n) \<subseteq> sets
```
```   872        (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
```
```   873         unfolding product_borel_eq_vimage
```
```   874       proof case goal1
```
```   875         then guess y unfolding image_iff .. note y=this(2)
```
```   876         show ?case unfolding borel.in_vimage_algebra y apply-
```
```   877           apply(rule_tac x="A y" in bexI,rule e2p_image_vimage)
```
```   878           using A(1) by auto
```
```   879       qed
```
```   880
```
```   881       show "disjoint_family (\<lambda>n. e2p ` A n)" apply(rule inj_on_disjoint_family_on)
```
```   882         using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)] using A(2) unfolding bij_betw_def by auto
```
```   883       show "(\<Union>n. e2p ` A n) \<in> sets (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
```
```   884         unfolding product_borel_eq_vimage borel.in_vimage_algebra
```
```   885       proof(rule bexI[OF _ A(3)],rule set_eqI,rule)
```
```   886         fix x assume x:"x \<in> (\<Union>n. e2p ` A n)" hence "p2e x \<in> (\<Union>i. A i)" by auto
```
```   887         moreover have "x \<in> extensional {..<DIM('a)}"
```
```   888           using x unfolding extensional_def e2p_def_raw by auto
```
```   889         ultimately show "x \<in> p2e -` (\<Union>i. A i) \<inter> extensional {..<DIM('a)}" by auto
```
```   890       next fix x assume x:"x \<in> p2e -` (\<Union>i. A i) \<inter> extensional {..<DIM('a)}"
```
```   891         hence "p2e x \<in> (\<Union>i. A i)" by auto
```
```   892         hence "\<exists>n. x \<in> e2p ` A n" apply safe apply(rule_tac x=i in exI)
```
```   893           unfolding image_iff apply(rule_tac x="p2e x" in bexI)
```
```   894           apply(subst e2p_p2e) using x by auto
```
```   895         thus "x \<in> (\<Union>n. e2p ` A n)" by auto
```
```   896       qed
```
```   897     qed
```
```   898   qed auto
```
```   899 qed
```
```   900
```
```   901 lemma e2p_p2e'[simp]: fixes x::"'a::ordered_euclidean_space"
```
```   902   assumes "A \<subseteq> extensional {..<DIM('a)}"
```
```   903   shows "e2p ` (p2e ` A ::'a set) = A"
```
```   904   apply(rule set_eqI) unfolding image_iff Bex_def apply safe defer
```
```   905   apply(rule_tac x="p2e x" in exI,safe) using assms by auto
```
```   906
```
```   907 lemma range_p2e:"range (p2e::_\<Rightarrow>'a::ordered_euclidean_space) = UNIV"
```
```   908   apply safe defer unfolding image_iff apply(rule_tac x="\<lambda>i. x \$\$ i" in bexI)
```
```   909   unfolding p2e_def by auto
```
```   910
```
```   911 lemma p2e_inv_extensional:"(A::'a::ordered_euclidean_space set)
```
```   912   = p2e ` (p2e -` A \<inter> extensional {..<DIM('a)})"
```
```   913   unfolding p2e_def_raw apply safe unfolding image_iff
```
```   914 proof- fix x assume "x\<in>A"
```
```   915   let ?y = "\<lambda>i. if i<DIM('a) then x\$\$i else undefined"
```
```   916   have *:"Chi ?y = x" apply(subst euclidean_eq) by auto
```
```   917   show "\<exists>xa\<in>Chi -` A \<inter> extensional {..<DIM('a)}. x = Chi xa" apply(rule_tac x="?y" in bexI)
```
```   918     apply(subst euclidean_eq) unfolding extensional_def using `x\<in>A` by(auto simp: *)
```
```   919 qed
```
```   920
```
```   921 lemma borel_fubini_positiv_integral:
```
```   922   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal"
```
```   923   assumes f: "f \<in> borel_measurable borel"
```
```   924   shows "borel.positive_integral f =
```
```   925           borel_product.product_positive_integral {..<DIM('a)} (f \<circ> p2e)"
```
```   926 proof- def U \<equiv> "extensional {..<DIM('a)} :: (nat \<Rightarrow> real) set"
```
```   927   interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
```
```   928   have *:"sigma_algebra.vimage_algebra borel U (p2e:: _ \<Rightarrow> 'a)
```
```   929     = sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)})"
```
```   930     unfolding U_def product_borel_eq_vimage[symmetric] ..
```
```   931   show ?thesis
```
```   932     unfolding borel.positive_integral_vimage[unfolded space_borel, OF bij_inv_p2e_e2p[THEN bij_inv_bij_betw(1)]]
```
```   933     apply(subst fprod.positive_integral_cong_measure[THEN sym, of "\<lambda>A. lmeasure (p2e ` A)"])
```
```   934     unfolding U_def[symmetric] *[THEN sym] o_def
```
```   935   proof- fix A assume A:"A \<in> sets (sigma_algebra.vimage_algebra borel U (p2e ::_ \<Rightarrow> 'a))"
```
```   936     hence *:"A \<subseteq> extensional {..<DIM('a)}" unfolding U_def by auto
```
```   937     from A guess B unfolding borel.in_vimage_algebra U_def ..
```
```   938     then have "(p2e ` A::'a set) \<in> sets borel"
```
```   939       by (simp add: p2e_inv_extensional[of B, symmetric])
```
```   940     from lmeasure_measure_eq_borel_prod[OF this] show "lmeasure (p2e ` A::'a set) =
```
```   941       finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} A"
```
```   942       unfolding e2p_p2e'[OF *] .
```
```   943   qed auto
```
```   944 qed
```
```   945
```
```   946 lemma borel_fubini:
```
```   947   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
```
```   948   assumes f: "f \<in> borel_measurable borel"
```
```   949   shows "borel.integral f = borel_product.product_integral {..<DIM('a)} (f \<circ> p2e)"
```
```   950 proof- interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
```
```   951   have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto
```
```   952   have 2:"(\<lambda>x. Real (- f x)) \<in> borel_measurable borel" using f by auto
```
```   953   show ?thesis unfolding fprod.integral_def borel.integral_def
```
```   954     unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2]
```
```   955     unfolding o_def ..
```
```   956 qed
```
```   957
```
```   958 end
```