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