src/HOL/Probability/Lebesgue_Measure.thy
author hoelzl
Tue Jan 18 21:37:23 2011 +0100 (2011-01-18)
changeset 41654 32fe42892983
parent 41546 2a12c23b7a34
child 41661 baf1964bc468
permissions -rw-r--r--
Gauge measure removed
     1 (*  Author: Robert Himmelmann, TU Muenchen *)
     2 header {* Lebsegue measure *}
     3 theory Lebesgue_Measure
     4   imports Product_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 definition lebesgue :: "'a::ordered_euclidean_space algebra" where
    46   "lebesgue = \<lparr> space = UNIV, sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n} \<rparr>"
    47 
    48 lemma space_lebesgue[simp]: "space lebesgue = UNIV"
    49   unfolding lebesgue_def by simp
    50 
    51 lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n"
    52   unfolding lebesgue_def by simp
    53 
    54 lemma lebesgueI: "(\<And>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n) \<Longrightarrow> A \<in> sets lebesgue"
    55   unfolding lebesgue_def by simp
    56 
    57 lemma absolutely_integrable_on_indicator[simp]:
    58   fixes A :: "'a::ordered_euclidean_space set"
    59   shows "((indicator A :: _ \<Rightarrow> real) absolutely_integrable_on X) \<longleftrightarrow>
    60     (indicator A :: _ \<Rightarrow> real) integrable_on X"
    61   unfolding absolutely_integrable_on_def by simp
    62 
    63 lemma LIMSEQ_indicator_UN:
    64   "(\<lambda>k. indicator (\<Union> i<k. A i) x) ----> (indicator (\<Union>i. A i) x :: real)"
    65 proof cases
    66   assume "\<exists>i. x \<in> A i" then guess i .. note i = this
    67   then have *: "\<And>n. (indicator (\<Union> i<n + Suc i. A i) x :: real) = 1"
    68     "(indicator (\<Union> i. A i) x :: real) = 1" by (auto simp: indicator_def)
    69   show ?thesis
    70     apply (rule LIMSEQ_offset[of _ "Suc i"]) unfolding * by auto
    71 qed (auto simp: indicator_def)
    72 
    73 lemma indicator_add:
    74   "A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x"
    75   unfolding indicator_def by auto
    76 
    77 interpretation lebesgue: sigma_algebra lebesgue
    78 proof (intro sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI lebesgueI)
    79   fix A n assume A: "A \<in> sets lebesgue"
    80   have "indicator (space lebesgue - A) = (\<lambda>x. 1 - indicator A x :: real)"
    81     by (auto simp: fun_eq_iff indicator_def)
    82   then show "(indicator (space lebesgue - A) :: _ \<Rightarrow> real) integrable_on cube n"
    83     using A by (auto intro!: integrable_sub dest: lebesgueD simp: cube_def)
    84 next
    85   fix n show "(indicator {} :: _\<Rightarrow>real) integrable_on cube n"
    86     by (auto simp: cube_def indicator_def_raw)
    87 next
    88   fix A :: "nat \<Rightarrow> 'a set" and n ::nat assume "range A \<subseteq> sets lebesgue"
    89   then have A: "\<And>i. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
    90     by (auto dest: lebesgueD)
    91   show "(indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "?g integrable_on _")
    92   proof (intro dominated_convergence[where g="?g"] ballI)
    93     fix k show "(indicator (\<Union>i<k. A i) :: _ \<Rightarrow> real) integrable_on cube n"
    94     proof (induct k)
    95       case (Suc k)
    96       have *: "(\<Union> i<Suc k. A i) = (\<Union> i<k. A i) \<union> A k"
    97         unfolding lessThan_Suc UN_insert by auto
    98       have *: "(\<lambda>x. max (indicator (\<Union> i<k. A i) x) (indicator (A k) x) :: real) =
    99           indicator (\<Union> i<Suc k. A i)" (is "(\<lambda>x. max (?f x) (?g x)) = _")
   100         by (auto simp: fun_eq_iff * indicator_def)
   101       show ?case
   102         using absolutely_integrable_max[of ?f "cube n" ?g] A Suc by (simp add: *)
   103     qed auto
   104   qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
   105 qed simp
   106 
   107 definition "lmeasure A = (SUP n. Real (integral (cube n) (indicator A)))"
   108 
   109 interpretation lebesgue: measure_space lebesgue lmeasure
   110 proof
   111   have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff)
   112   show "lmeasure {} = 0" by (simp add: integral_0 * lmeasure_def)
   113 next
   114   show "countably_additive lebesgue lmeasure"
   115   proof (intro countably_additive_def[THEN iffD2] allI impI)
   116     fix A :: "nat \<Rightarrow> 'b set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A"
   117     then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
   118       by (auto dest: lebesgueD)
   119     let "?m n i" = "integral (cube n) (indicator (A i) :: _\<Rightarrow>real)"
   120     let "?M n I" = "integral (cube n) (indicator (\<Union>i\<in>I. A i) :: _\<Rightarrow>real)"
   121     have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: integral_nonneg)
   122     assume "(\<Union>i. A i) \<in> sets lebesgue"
   123     then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n"
   124       by (auto dest: lebesgueD)
   125     show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (\<Union>i. A i)" unfolding lmeasure_def
   126     proof (subst psuminf_SUP_eq)
   127       fix n i show "Real (?m n i) \<le> Real (?m (Suc n) i)"
   128         using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le)
   129     next
   130       show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (?m n i)) = (SUP n. Real (?M n UNIV))"
   131         unfolding psuminf_def
   132       proof (subst setsum_Real, (intro arg_cong[where f="SUPR UNIV"] ext ballI nn SUP_eq_LIMSEQ[THEN iffD2])+)
   133         fix n :: nat show "mono (\<lambda>m. \<Sum>x<m. ?m n x)"
   134         proof (intro mono_iff_le_Suc[THEN iffD2] allI)
   135           fix m show "(\<Sum>x<m. ?m n x) \<le> (\<Sum>x<Suc m. ?m n x)"
   136             using nn[of n m] by auto
   137         qed
   138         show "0 \<le> ?M n UNIV"
   139           using UN_A by (auto intro!: integral_nonneg)
   140         fix m show "0 \<le> (\<Sum>x<m. ?m n x)" by (auto intro!: setsum_nonneg)
   141       next
   142         fix n
   143         have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto
   144         from lebesgueD[OF this]
   145         have "(\<lambda>m. ?M n {..< m}) ----> ?M n UNIV"
   146           (is "(\<lambda>m. integral _ (?A m)) ----> ?I")
   147           by (intro dominated_convergence(2)[where f="?A" and h="\<lambda>x. 1::real"])
   148              (auto intro: LIMSEQ_indicator_UN simp: cube_def)
   149         moreover
   150         { fix m have *: "(\<Sum>x<m. ?m n x) = ?M n {..< m}"
   151           proof (induct m)
   152             case (Suc m)
   153             have "(\<Union>i<m. A i) \<in> sets lebesgue" using rA by auto
   154             then have "(indicator (\<Union>i<m. A i) :: _\<Rightarrow>real) integrable_on (cube n)"
   155               by (auto dest!: lebesgueD)
   156             moreover
   157             have "(\<Union>i<m. A i) \<inter> A m = {}"
   158               using rA(2)[unfolded disjoint_family_on_def, THEN bspec, of m]
   159               by auto
   160             then have "\<And>x. indicator (\<Union>i<Suc m. A i) x =
   161               indicator (\<Union>i<m. A i) x + (indicator (A m) x :: real)"
   162               by (auto simp: indicator_add lessThan_Suc ac_simps)
   163             ultimately show ?case
   164               using Suc A by (simp add: integral_add[symmetric])
   165           qed auto }
   166         ultimately show "(\<lambda>m. \<Sum>x<m. ?m n x) ----> ?M n UNIV"
   167           by simp
   168       qed
   169     qed
   170   qed
   171 qed
   172 
   173 lemma has_integral_interval_cube:
   174   fixes a b :: "'a::ordered_euclidean_space"
   175   shows "(indicator {a .. b} has_integral
   176     content ({\<chi>\<chi> i. max (- real n) (a $$ i) .. \<chi>\<chi> i. min (real n) (b $$ i)} :: 'a set)) (cube n)"
   177     (is "(?I has_integral content ?R) (cube n)")
   178 proof -
   179   let "{?N .. ?P}" = ?R
   180   have [simp]: "(\<lambda>x. if x \<in> cube n then ?I x else 0) = indicator ?R"
   181     by (auto simp: indicator_def cube_def fun_eq_iff eucl_le[where 'a='a])
   182   have "(?I has_integral content ?R) (cube n) \<longleftrightarrow> (indicator ?R has_integral content ?R) UNIV"
   183     unfolding has_integral_restrict_univ[where s="cube n", symmetric] by simp
   184   also have "\<dots> \<longleftrightarrow> ((\<lambda>x. 1) has_integral content ?R) ?R"
   185     unfolding indicator_def_raw has_integral_restrict_univ ..
   186   finally show ?thesis
   187     using has_integral_const[of "1::real" "?N" "?P"] by simp
   188 qed
   189 
   190 lemma lebesgueI_borel[intro, simp]:
   191   fixes s::"'a::ordered_euclidean_space set"
   192   assumes "s \<in> sets borel" shows "s \<in> sets lebesgue"
   193 proof -
   194   let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})"
   195   have *:"?S \<subseteq> sets lebesgue"
   196   proof (safe intro!: lebesgueI)
   197     fix n :: nat and a b :: 'a
   198     let ?N = "\<chi>\<chi> i. max (- real n) (a $$ i)"
   199     let ?P = "\<chi>\<chi> i. min (real n) (b $$ i)"
   200     show "(indicator {a..b} :: 'a\<Rightarrow>real) integrable_on cube n"
   201       unfolding integrable_on_def
   202       using has_integral_interval_cube[of a b] by auto
   203   qed
   204   have "s \<in> sigma_sets UNIV ?S" using assms
   205     unfolding borel_eq_atLeastAtMost by (simp add: sigma_def)
   206   thus ?thesis
   207     using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *]
   208     by (auto simp: sigma_def)
   209 qed
   210 
   211 lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
   212   assumes "negligible s" shows "s \<in> sets lebesgue"
   213   using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI)
   214 
   215 lemma lmeasure_eq_0:
   216   fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "lmeasure S = 0"
   217 proof -
   218   have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0"
   219     unfolding integral_def using assms
   220     by (intro some1_equality ex_ex1I has_integral_unique)
   221        (auto simp: cube_def negligible_def intro: )
   222   then show ?thesis unfolding lmeasure_def by auto
   223 qed
   224 
   225 lemma lmeasure_iff_LIMSEQ:
   226   assumes "A \<in> sets lebesgue" "0 \<le> m"
   227   shows "lmeasure A = Real m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
   228   unfolding lmeasure_def
   229 proof (intro SUP_eq_LIMSEQ)
   230   show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))"
   231     using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD)
   232   fix n show "0 \<le> integral (cube n) (indicator A::_=>real)"
   233     using assms by (auto dest!: lebesgueD intro!: integral_nonneg)
   234 qed fact
   235 
   236 lemma has_integral_indicator_UNIV:
   237   fixes s A :: "'a::ordered_euclidean_space set" and x :: real
   238   shows "((indicator (s \<inter> A) :: 'a\<Rightarrow>real) has_integral x) UNIV = ((indicator s :: _\<Rightarrow>real) has_integral x) A"
   239 proof -
   240   have "(\<lambda>x. if x \<in> A then indicator s x else 0) = (indicator (s \<inter> A) :: _\<Rightarrow>real)"
   241     by (auto simp: fun_eq_iff indicator_def)
   242   then show ?thesis
   243     unfolding has_integral_restrict_univ[where s=A, symmetric] by simp
   244 qed
   245 
   246 lemma
   247   fixes s a :: "'a::ordered_euclidean_space set"
   248   shows integral_indicator_UNIV:
   249     "integral UNIV (indicator (s \<inter> A) :: 'a\<Rightarrow>real) = integral A (indicator s :: _\<Rightarrow>real)"
   250   and integrable_indicator_UNIV:
   251     "(indicator (s \<inter> A) :: 'a\<Rightarrow>real) integrable_on UNIV \<longleftrightarrow> (indicator s :: 'a\<Rightarrow>real) integrable_on A"
   252   unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto
   253 
   254 lemma lmeasure_finite_has_integral:
   255   fixes s :: "'a::ordered_euclidean_space set"
   256   assumes "s \<in> sets lebesgue" "lmeasure s = Real m" "0 \<le> m"
   257   shows "(indicator s has_integral m) UNIV"
   258 proof -
   259   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
   260   have **: "(?I s) integrable_on UNIV \<and> (\<lambda>k. integral UNIV (?I (s \<inter> cube k))) ----> integral UNIV (?I s)"
   261   proof (intro monotone_convergence_increasing allI ballI)
   262     have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) ----> m"
   263       using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1, 3)] .
   264     { fix n have "integral (cube n) (?I s) \<le> m"
   265         using cube_subset assms
   266         by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI)
   267            (auto dest!: lebesgueD) }
   268     moreover
   269     { fix n have "0 \<le> integral (cube n) (?I s)"
   270       using assms by (auto dest!: lebesgueD intro!: integral_nonneg) }
   271     ultimately
   272     show "bounded {integral UNIV (?I (s \<inter> cube k)) |k. True}"
   273       unfolding bounded_def
   274       apply (rule_tac exI[of _ 0])
   275       apply (rule_tac exI[of _ m])
   276       by (auto simp: dist_real_def integral_indicator_UNIV)
   277     fix k show "?I (s \<inter> cube k) integrable_on UNIV"
   278       unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD)
   279     fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x"
   280       using cube_subset[of k "Suc k"] by (auto simp: indicator_def)
   281   next
   282     fix x :: 'a
   283     from mem_big_cube obtain k where k: "x \<in> cube k" .
   284     { fix n have "?I (s \<inter> cube (n + k)) x = ?I s x"
   285       using k cube_subset[of k "n + k"] by (auto simp: indicator_def) }
   286     note * = this
   287     show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x"
   288       by (rule LIMSEQ_offset[where k=k]) (auto simp: *)
   289   qed
   290   note ** = conjunctD2[OF this]
   291   have m: "m = integral UNIV (?I s)"
   292     apply (intro LIMSEQ_unique[OF _ **(2)])
   293     using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1,3)] integral_indicator_UNIV .
   294   show ?thesis
   295     unfolding m by (intro integrable_integral **)
   296 qed
   297 
   298 lemma lmeasure_finite_integrable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>"
   299   shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
   300 proof (cases "lmeasure s")
   301   case (preal m) from lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
   302   show ?thesis unfolding integrable_on_def by auto
   303 qed (insert assms, auto)
   304 
   305 lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
   306   shows "s \<in> sets lebesgue"
   307 proof (intro lebesgueI)
   308   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
   309   fix n show "(?I s) integrable_on cube n" unfolding cube_def
   310   proof (intro integrable_on_subinterval)
   311     show "(?I s) integrable_on UNIV"
   312       unfolding integrable_on_def using assms by auto
   313   qed auto
   314 qed
   315 
   316 lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
   317   shows "lmeasure s = Real m"
   318 proof (intro lmeasure_iff_LIMSEQ[THEN iffD2])
   319   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
   320   show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] .
   321   show "0 \<le> m" using assms by (rule has_integral_nonneg) auto
   322   have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) ----> integral UNIV (?I s)"
   323   proof (intro dominated_convergence(2) ballI)
   324     show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto
   325     fix n show "?I (s \<inter> cube n) integrable_on UNIV"
   326       unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD)
   327     fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def)
   328   next
   329     fix x :: 'a
   330     from mem_big_cube obtain k where k: "x \<in> cube k" .
   331     { fix n have "?I (s \<inter> cube (n + k)) x = ?I s x"
   332       using k cube_subset[of k "n + k"] by (auto simp: indicator_def) }
   333     note * = this
   334     show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x"
   335       by (rule LIMSEQ_offset[where k=k]) (auto simp: *)
   336   qed
   337   then show "(\<lambda>n. integral (cube n) (?I s)) ----> m"
   338     unfolding integral_unique[OF assms] integral_indicator_UNIV by simp
   339 qed
   340 
   341 lemma has_integral_iff_lmeasure:
   342   "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m)"
   343 proof
   344   assume "(indicator A has_integral m) UNIV"
   345   with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this]
   346   show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m"
   347     by (auto intro: has_integral_nonneg)
   348 next
   349   assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m"
   350   then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto
   351 qed
   352 
   353 lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV"
   354   shows "lmeasure s = Real (integral UNIV (indicator s))"
   355   using assms unfolding integrable_on_def
   356 proof safe
   357   fix y :: real assume "(indicator s has_integral y) UNIV"
   358   from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this]
   359   show "lmeasure s = Real (integral UNIV (indicator s))" by simp
   360 qed
   361 
   362 lemma lebesgue_simple_function_indicator:
   363   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
   364   assumes f:"lebesgue.simple_function f"
   365   shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
   366   apply(rule,subst lebesgue.simple_function_indicator_representation[OF f]) by auto
   367 
   368 lemma integral_eq_lmeasure:
   369   "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lmeasure s)"
   370   by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg)
   371 
   372 lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lmeasure s \<noteq> \<omega>"
   373   using lmeasure_eq_integral[OF assms] by auto
   374 
   375 lemma negligible_iff_lebesgue_null_sets:
   376   "negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets"
   377 proof
   378   assume "negligible A"
   379   from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
   380   show "A \<in> lebesgue.null_sets" by auto
   381 next
   382   assume A: "A \<in> lebesgue.null_sets"
   383   then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] by auto
   384   show "negligible A" unfolding negligible_def
   385   proof (intro allI)
   386     fix a b :: 'a
   387     have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}"
   388       by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *)
   389     then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)"
   390       using * by (auto intro!: integral_subset_le has_integral_integrable)
   391     moreover have "(0::real) \<le> integral {a..b} (indicator A)"
   392       using integrable by (auto intro!: integral_nonneg)
   393     ultimately have "integral {a..b} (indicator A) = (0::real)"
   394       using integral_unique[OF *] by auto
   395     then show "(indicator A has_integral (0::real)) {a..b}"
   396       using integrable_integral[OF integrable] by simp
   397   qed
   398 qed
   399 
   400 lemma integral_const[simp]:
   401   fixes a b :: "'a::ordered_euclidean_space"
   402   shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
   403   by (rule integral_unique) (rule has_integral_const)
   404 
   405 lemma lmeasure_UNIV[intro]: "lmeasure (UNIV::'a::ordered_euclidean_space set) = \<omega>"
   406   unfolding lmeasure_def SUP_\<omega>
   407 proof (intro allI impI)
   408   fix x assume "x < \<omega>"
   409   then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto
   410   then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto
   411   show "\<exists>i\<in>UNIV. x < Real (integral (cube i) (indicator UNIV::'a\<Rightarrow>real))"
   412   proof (intro bexI[of _ n])
   413     have [simp]: "indicator UNIV = (\<lambda>x. 1)" by (auto simp: fun_eq_iff)
   414     { fix m :: nat assume "0 < m" then have "real n \<le> (\<Prod>x<m. 2 * real n)"
   415       proof (induct m)
   416         case (Suc m)
   417         show ?case
   418         proof cases
   419           assume "m = 0" then show ?thesis by (simp add: lessThan_Suc)
   420         next
   421           assume "m \<noteq> 0" then have "real n \<le> (\<Prod>x<m. 2 * real n)" using Suc by auto
   422           then show ?thesis
   423             by (auto simp: lessThan_Suc field_simps mult_le_cancel_left1)
   424         qed
   425       qed auto } note this[OF DIM_positive[where 'a='a], simp]
   426     then have [simp]: "0 \<le> (\<Prod>x<DIM('a). 2 * real n)" using real_of_nat_ge_zero by arith
   427     have "x < Real (of_nat n)" using n r by auto
   428     also have "Real (of_nat n) \<le> Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
   429       by (auto simp: real_eq_of_nat[symmetric] cube_def content_closed_interval_cases)
   430     finally show "x < Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" .
   431   qed auto
   432 qed
   433 
   434 lemma
   435   fixes a b ::"'a::ordered_euclidean_space"
   436   shows lmeasure_atLeastAtMost[simp]: "lmeasure {a..b} = Real (content {a..b})"
   437 proof -
   438   have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV"
   439     unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw)
   440   from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV
   441     by (simp add: indicator_def_raw)
   442 qed
   443 
   444 lemma atLeastAtMost_singleton_euclidean[simp]:
   445   fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
   446   by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a])
   447 
   448 lemma content_singleton[simp]: "content {a} = 0"
   449 proof -
   450   have "content {a .. a} = 0"
   451     by (subst content_closed_interval) auto
   452   then show ?thesis by simp
   453 qed
   454 
   455 lemma lmeasure_singleton[simp]:
   456   fixes a :: "'a::ordered_euclidean_space" shows "lmeasure {a} = 0"
   457   using lmeasure_atLeastAtMost[of a a] by simp
   458 
   459 declare content_real[simp]
   460 
   461 lemma
   462   fixes a b :: real
   463   shows lmeasure_real_greaterThanAtMost[simp]:
   464     "lmeasure {a <.. b} = Real (if a \<le> b then b - a else 0)"
   465 proof cases
   466   assume "a < b"
   467   then have "lmeasure {a <.. b} = lmeasure {a .. b} - lmeasure {a}"
   468     by (subst lebesgue.measure_Diff[symmetric])
   469        (auto intro!: arg_cong[where f=lmeasure])
   470   then show ?thesis by auto
   471 qed auto
   472 
   473 lemma
   474   fixes a b :: real
   475   shows lmeasure_real_atLeastLessThan[simp]:
   476     "lmeasure {a ..< b} = Real (if a \<le> b then b - a else 0)"
   477 proof cases
   478   assume "a < b"
   479   then have "lmeasure {a ..< b} = lmeasure {a .. b} - lmeasure {b}"
   480     by (subst lebesgue.measure_Diff[symmetric])
   481        (auto intro!: arg_cong[where f=lmeasure])
   482   then show ?thesis by auto
   483 qed auto
   484 
   485 lemma
   486   fixes a b :: real
   487   shows lmeasure_real_greaterThanLessThan[simp]:
   488     "lmeasure {a <..< b} = Real (if a \<le> b then b - a else 0)"
   489 proof cases
   490   assume "a < b"
   491   then have "lmeasure {a <..< b} = lmeasure {a <.. b} - lmeasure {b}"
   492     by (subst lebesgue.measure_Diff[symmetric])
   493        (auto intro!: arg_cong[where f=lmeasure])
   494   then show ?thesis by auto
   495 qed auto
   496 
   497 interpretation borel: measure_space borel lmeasure
   498 proof
   499   show "countably_additive borel lmeasure"
   500     using lebesgue.ca unfolding countably_additive_def
   501     apply safe apply (erule_tac x=A in allE) by auto
   502 qed auto
   503 
   504 interpretation borel: sigma_finite_measure borel lmeasure
   505 proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
   506   show "range cube \<subseteq> sets borel" by (auto intro: borel_closed)
   507   { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
   508   thus "(\<Union>i. cube i) = space borel" by auto
   509   show "\<forall>i. lmeasure (cube i) \<noteq> \<omega>" unfolding cube_def by auto
   510 qed
   511 
   512 interpretation lebesgue: sigma_finite_measure lebesgue lmeasure
   513 proof
   514   from borel.sigma_finite guess A ..
   515   moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
   516   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>)"
   517     by auto
   518 qed
   519 
   520 lemma simple_function_has_integral:
   521   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
   522   assumes f:"lebesgue.simple_function f"
   523   and f':"\<forall>x. f x \<noteq> \<omega>"
   524   and om:"\<forall>x\<in>range f. lmeasure (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
   525   shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
   526   unfolding lebesgue.simple_integral_def
   527   apply(subst lebesgue_simple_function_indicator[OF f])
   528 proof -
   529   case goal1
   530   have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
   531     "\<forall>x\<in>range f. x * lmeasure (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
   532     using f' om unfolding indicator_def by auto
   533   show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym]
   534     unfolding real_of_pextreal_setsum'[OF *(2),THEN sym]
   535     unfolding real_of_pextreal_setsum space_lebesgue
   536     apply(rule has_integral_setsum)
   537   proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD)
   538     fix y::'a show "((\<lambda>x. real (f y * indicator (f -` {f y}) x)) has_integral
   539       real (f y * lmeasure (f -` {f y} \<inter> UNIV))) UNIV"
   540     proof(cases "f y = 0") case False
   541       have mea:"(indicator (f -` {f y}) ::_\<Rightarrow>real) integrable_on UNIV"
   542         apply(rule lmeasure_finite_integrable)
   543         using assms unfolding lebesgue.simple_function_def using False by auto
   544       have *:"\<And>x. real (indicator (f -` {f y}) x::pextreal) = (indicator (f -` {f y}) x)"
   545         by (auto simp: indicator_def)
   546       show ?thesis unfolding real_of_pextreal_mult[THEN sym]
   547         apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def])
   548         unfolding Int_UNIV_right lmeasure_eq_integral[OF mea,THEN sym]
   549         unfolding integral_eq_lmeasure[OF mea, symmetric] *
   550         apply(rule integrable_integral) using mea .
   551     qed auto
   552   qed
   553 qed
   554 
   555 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
   556   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
   557   using assms by auto
   558 
   559 lemma simple_function_has_integral':
   560   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
   561   assumes f:"lebesgue.simple_function f"
   562   and i: "lebesgue.simple_integral f \<noteq> \<omega>"
   563   shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
   564 proof- let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x"
   565   { fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this
   566   have **:"{x. f x \<noteq> ?f x} = f -` {\<omega>}" by auto
   567   have **:"lmeasure {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
   568     using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**)
   569   show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **])
   570     apply(rule lebesgue.simple_function_compose1[OF f])
   571     unfolding * defer apply(rule simple_function_has_integral)
   572   proof-
   573     show "lebesgue.simple_function ?f"
   574       using lebesgue.simple_function_compose1[OF f] .
   575     show "\<forall>x. ?f x \<noteq> \<omega>" by auto
   576     show "\<forall>x\<in>range ?f. lmeasure (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
   577     proof (safe, simp, safe, rule ccontr)
   578       fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0"
   579       hence "(\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y} = f -` {f y}"
   580         by (auto split: split_if_asm)
   581       moreover assume "lmeasure ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
   582       ultimately have "lmeasure (f -` {f y}) = \<omega>" by simp
   583       moreover
   584       have "f y * lmeasure (f -` {f y}) \<noteq> \<omega>" using i f
   585         unfolding lebesgue.simple_integral_def setsum_\<omega> lebesgue.simple_function_def
   586         by auto
   587       ultimately have "f y = 0" by (auto split: split_if_asm)
   588       then show False using `f y \<noteq> 0` by simp
   589     qed
   590   qed
   591 qed
   592 
   593 lemma (in measure_space) positive_integral_monotone_convergence:
   594   fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pextreal"
   595   assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)"
   596   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
   597   shows "u \<in> borel_measurable M"
   598   and "(\<lambda>i. positive_integral (f i)) ----> positive_integral u" (is ?ilim)
   599 proof -
   600   from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
   601   show ?ilim using mono lim i by auto
   602   have "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal
   603     unfolding fun_eq_iff mono_def by auto
   604   moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
   605     using i by auto
   606   ultimately show "u \<in> borel_measurable M" by simp
   607 qed
   608 
   609 lemma positive_integral_has_integral:
   610   fixes f::"'a::ordered_euclidean_space => pextreal"
   611   assumes f:"f \<in> borel_measurable lebesgue"
   612   and int_om:"lebesgue.positive_integral f \<noteq> \<omega>"
   613   and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
   614   shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.positive_integral f))) UNIV"
   615 proof- let ?i = "lebesgue.positive_integral f"
   616   from lebesgue.borel_measurable_implies_simple_function_sequence[OF f]
   617   guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2)
   618   let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)"
   619   have u_simple:"\<And>k. lebesgue.simple_integral (u k) = lebesgue.positive_integral (u k)"
   620     apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) ..
   621   have int_u_le:"\<And>k. lebesgue.simple_integral (u k) \<le> lebesgue.positive_integral f"
   622     unfolding u_simple apply(rule lebesgue.positive_integral_mono)
   623     using isoton_Sup[OF u(3)] unfolding le_fun_def by auto
   624   have u_int_om:"\<And>i. lebesgue.simple_integral (u i) \<noteq> \<omega>"
   625   proof- case goal1 thus ?case using int_u_le[of i] int_om by auto qed
   626 
   627   note u_int = simple_function_has_integral'[OF u(1) this]
   628   have "(\<lambda>x. real (f x)) integrable_on UNIV \<and>
   629     (\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) ----> Integration.integral UNIV (\<lambda>x. real (f x))"
   630     apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int)
   631   proof safe case goal1 show ?case apply(rule real_of_pextreal_mono) using u(2,3) by auto
   632   next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym])
   633       prefer 3 apply(subst Real_real') defer apply(subst Real_real')
   634       using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto
   635   next case goal3
   636     show ?case apply(rule bounded_realI[where B="real (lebesgue.positive_integral f)"])
   637       apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int)
   638       unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le])
   639       using u int_om by auto
   640   qed note int = conjunctD2[OF this]
   641 
   642   have "(\<lambda>i. lebesgue.simple_integral (u i)) ----> ?i" unfolding u_simple
   643     apply(rule lebesgue.positive_integral_monotone_convergence(2))
   644     apply(rule lebesgue.borel_measurable_simple_function[OF u(1)])
   645     using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto
   646   hence "(\<lambda>i. real (lebesgue.simple_integral (u i))) ----> real ?i" apply-
   647     apply(subst lim_Real[THEN sym]) prefer 3
   648     apply(subst Real_real') defer apply(subst Real_real')
   649     using u f_om int_om u_int_om by auto
   650   note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]]
   651   show ?thesis unfolding * by(rule integrable_integral[OF int(1)])
   652 qed
   653 
   654 lemma lebesgue_integral_has_integral:
   655   fixes f::"'a::ordered_euclidean_space => real"
   656   assumes f:"lebesgue.integrable f"
   657   shows "(f has_integral (lebesgue.integral f)) UNIV"
   658 proof- let ?n = "\<lambda>x. - min (f x) 0" and ?p = "\<lambda>x. max (f x) 0"
   659   have *:"f = (\<lambda>x. ?p x - ?n x)" apply rule by auto
   660   note f = lebesgue.integrableD[OF f]
   661   show ?thesis unfolding lebesgue.integral_def apply(subst *)
   662   proof(rule has_integral_sub) case goal1
   663     have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto
   664     note lebesgue.borel_measurable_Real[OF f(1)]
   665     from positive_integral_has_integral[OF this f(2) *]
   666     show ?case unfolding real_Real_max .
   667   next case goal2
   668     have *:"\<forall>x. Real (- f x) \<noteq> \<omega>" by auto
   669     note lebesgue.borel_measurable_uminus[OF f(1)]
   670     note lebesgue.borel_measurable_Real[OF this]
   671     from positive_integral_has_integral[OF this f(3) *]
   672     show ?case unfolding real_Real_max minus_min_eq_max by auto
   673   qed
   674 qed
   675 
   676 lemma lebesgue_positive_integral_eq_borel:
   677   "f \<in> borel_measurable borel \<Longrightarrow> lebesgue.positive_integral f = borel.positive_integral f "
   678   by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
   679 
   680 lemma lebesgue_integral_eq_borel:
   681   assumes "f \<in> borel_measurable borel"
   682   shows "lebesgue.integrable f = borel.integrable f" (is ?P)
   683     and "lebesgue.integral f = borel.integral f" (is ?I)
   684 proof -
   685   have *: "sigma_algebra borel" by default
   686   have "sets borel \<subseteq> sets lebesgue" by auto
   687   from lebesgue.integral_subalgebra[OF assms this _ *]
   688   show ?P ?I by auto
   689 qed
   690 
   691 lemma borel_integral_has_integral:
   692   fixes f::"'a::ordered_euclidean_space => real"
   693   assumes f:"borel.integrable f"
   694   shows "(f has_integral (borel.integral f)) UNIV"
   695 proof -
   696   have borel: "f \<in> borel_measurable borel"
   697     using f unfolding borel.integrable_def by auto
   698   from f show ?thesis
   699     using lebesgue_integral_has_integral[of f]
   700     unfolding lebesgue_integral_eq_borel[OF borel] by simp
   701 qed
   702 
   703 lemma continuous_on_imp_borel_measurable:
   704   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
   705   assumes "continuous_on UNIV f"
   706   shows "f \<in> borel_measurable borel"
   707   apply(rule borel.borel_measurableI)
   708   using continuous_open_preimage[OF assms] unfolding vimage_def by auto
   709 
   710 lemma (in measure_space) integral_monotone_convergence_pos':
   711   assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
   712   and pos: "\<And>x i. 0 \<le> f i x"
   713   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
   714   and ilim: "(\<lambda>i. integral (f i)) ----> x"
   715   shows "integrable u \<and> integral u = x"
   716   using integral_monotone_convergence_pos[OF assms] by auto
   717 
   718 definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where
   719   "e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)"
   720 
   721 definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where
   722   "p2e x = (\<chi>\<chi> i. x i)"
   723 
   724 lemma e2p_p2e[simp]:
   725   "x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x"
   726   by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def)
   727 
   728 lemma p2e_e2p[simp]:
   729   "p2e (e2p x) = (x::'a::ordered_euclidean_space)"
   730   by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def)
   731 
   732 lemma bij_inv_p2e_e2p:
   733   shows "bij_inv ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) (UNIV :: 'a::ordered_euclidean_space set)
   734      p2e e2p" (is "bij_inv ?P ?U _ _")
   735 proof (rule bij_invI)
   736   show "p2e \<in> ?P \<rightarrow> ?U" "e2p \<in> ?U \<rightarrow> ?P" by (auto simp: e2p_def)
   737 qed auto
   738 
   739 interpretation borel_product: product_sigma_finite "\<lambda>x. borel::real algebra" "\<lambda>x. lmeasure"
   740   by default
   741 
   742 lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
   743   unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
   744 
   745 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
   746   unfolding Pi_def by auto
   747 
   748 lemma measurable_e2p_on_generator:
   749   "e2p \<in> measurable \<lparr> space = UNIV::'a set, sets = range lessThan \<rparr>
   750   (product_algebra
   751     (\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>)
   752     {..<DIM('a::ordered_euclidean_space)})"
   753   (is "e2p \<in> measurable ?E ?P")
   754 proof (unfold measurable_def, intro CollectI conjI ballI)
   755   show "e2p \<in> space ?E \<rightarrow> space ?P" by (auto simp: e2p_def)
   756   fix A assume "A \<in> sets ?P"
   757   then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)"
   758     and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)"
   759     by (auto elim!: product_algebraE)
   760   then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto
   761   from this[THEN bchoice] guess xs ..
   762   then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})"
   763     using A by auto
   764   have "e2p -` A = {..< (\<chi>\<chi> i. xs i) :: 'a}"
   765     using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq
   766       euclidean_eq[where 'a='a] eucl_less[where 'a='a])
   767   then show "e2p -` A \<inter> space ?E \<in> sets ?E" by simp
   768 qed
   769 
   770 lemma measurable_p2e_on_generator:
   771   "p2e \<in> measurable
   772     (product_algebra
   773       (\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>)
   774       {..<DIM('a::ordered_euclidean_space)})
   775     \<lparr> space = UNIV::'a set, sets = range lessThan \<rparr>"
   776   (is "p2e \<in> measurable ?P ?E")
   777 proof (unfold measurable_def, intro CollectI conjI ballI)
   778   show "p2e \<in> space ?P \<rightarrow> space ?E" by simp
   779   fix A assume "A \<in> sets ?E"
   780   then obtain x where "A = {..<x}" by auto
   781   then have "p2e -` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x $$ i})"
   782     using DIM_positive
   783     by (auto simp: Pi_iff set_eq_iff p2e_def
   784                    euclidean_eq[where 'a='a] eucl_less[where 'a='a])
   785   then show "p2e -` A \<inter> space ?P \<in> sets ?P" by auto
   786 qed
   787 
   788 lemma borel_vimage_algebra_eq:
   789   defines "F \<equiv> product_algebra (\<lambda>x. \<lparr> space = (UNIV::real set), sets = range lessThan \<rparr>) {..<DIM('a::ordered_euclidean_space)}"
   790   shows "sigma_algebra.vimage_algebra (borel::'a::ordered_euclidean_space algebra) (space (sigma F)) p2e = sigma F"
   791   unfolding borel_eq_lessThan
   792 proof (intro vimage_algebra_sigma)
   793   let ?E = "\<lparr>space = (UNIV::'a set), sets = range lessThan\<rparr>"
   794   show "bij_inv (space (sigma F)) (space (sigma ?E)) p2e e2p"
   795     using bij_inv_p2e_e2p unfolding F_def by simp
   796   show "sets F \<subseteq> Pow (space F)" "sets ?E \<subseteq> Pow (space ?E)" unfolding F_def
   797     by (intro product_algebra_sets_into_space) auto
   798   show "p2e \<in> measurable F ?E"
   799     "e2p \<in> measurable ?E F"
   800     unfolding F_def using measurable_p2e_on_generator measurable_e2p_on_generator by auto
   801 qed
   802 
   803 lemma product_borel_eq_vimage:
   804   "sigma (product_algebra (\<lambda>x. borel) {..<DIM('a::ordered_euclidean_space)}) =
   805   sigma_algebra.vimage_algebra borel (extensional {..<DIM('a)})
   806   (p2e:: _ \<Rightarrow> 'a::ordered_euclidean_space)"
   807   unfolding borel_vimage_algebra_eq[simplified]
   808   unfolding borel_eq_lessThan
   809   apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>i. \<lambda>n. lessThan (real n)"])
   810   unfolding lessThan_iff
   811 proof- fix i assume i:"i<DIM('a)"
   812   show "(\<lambda>n. {..<real n}) \<up> space \<lparr>space = UNIV, sets = range lessThan\<rparr>"
   813     by(auto intro!:real_arch_lt isotoneI)
   814 qed auto
   815 
   816 lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
   817   apply(rule image_Int[THEN sym])
   818   using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)]
   819   unfolding bij_betw_def by auto
   820 
   821 lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space"
   822   shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>"
   823   unfolding Int_stable_def algebra.select_convs
   824 proof safe fix a b x y::'a
   825   have *:"e2p ` {a..b} \<inter> e2p ` {x..y} =
   826     (\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a $$ i) (x $$ i), \<chi>\<chi> i. min (b $$ i) (y $$ i)::'a)"
   827     unfolding e2p_Int inter_interval by auto
   828   show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding *
   829     apply(rule range_eqI) ..
   830 qed
   831 
   832 lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space"
   833   shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>"
   834   unfolding Int_stable_def algebra.select_convs
   835   apply safe unfolding inter_interval by auto
   836 
   837 lemma inj_on_disjoint_family_on: assumes "disjoint_family_on A S" "inj f"
   838   shows "disjoint_family_on (\<lambda>x. f ` A x) S"
   839   unfolding disjoint_family_on_def
   840 proof(rule,rule,rule)
   841   fix x1 x2 assume x:"x1 \<in> S" "x2 \<in> S" "x1 \<noteq> x2"
   842   show "f ` A x1 \<inter> f ` A x2 = {}"
   843   proof(rule ccontr) case goal1
   844     then obtain z where z:"z \<in> f ` A x1 \<inter> f ` A x2" by auto
   845     then obtain z1 z2 where z12:"z1 \<in> A x1" "z2 \<in> A x2" "f z1 = z" "f z2 = z" by auto
   846     hence "z1 = z2" using assms(2) unfolding inj_on_def by blast
   847     hence "x1 = x2" using z12(1-2) using assms[unfolded disjoint_family_on_def] using x by auto
   848     thus False using x(3) by auto
   849   qed
   850 qed
   851 
   852 declare restrict_extensional[intro]
   853 
   854 lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}"
   855   unfolding e2p_def by auto
   856 
   857 lemma e2p_image_vimage: fixes A::"'a::ordered_euclidean_space set"
   858   shows "e2p ` A = p2e -` A \<inter> extensional {..<DIM('a)}"
   859 proof(rule set_eqI,rule)
   860   fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this
   861   show "x \<in> p2e -` A \<inter> extensional {..<DIM('a)}"
   862     apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto
   863 next fix x assume "x \<in> p2e -` A \<inter> extensional {..<DIM('a)}"
   864   thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto
   865 qed
   866 
   867 lemma lmeasure_measure_eq_borel_prod:
   868   fixes A :: "('a::ordered_euclidean_space) set"
   869   assumes "A \<in> sets borel"
   870   shows "lmeasure A = borel_product.product_measure {..<DIM('a)} (e2p ` A :: (nat \<Rightarrow> real) set)"
   871 proof (rule measure_unique_Int_stable[where X=A and A=cube])
   872   interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
   873   show "Int_stable \<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
   874     (is "Int_stable ?E" ) using Int_stable_cuboids' .
   875   show "borel = sigma ?E" using borel_eq_atLeastAtMost .
   876   show "\<And>i. lmeasure (cube i) \<noteq> \<omega>" unfolding cube_def by auto
   877   show "\<And>X. X \<in> sets ?E \<Longrightarrow>
   878     lmeasure X = borel_product.product_measure {..<DIM('a)} (e2p ` X :: (nat \<Rightarrow> real) set)"
   879   proof- case goal1 then obtain a b where X:"X = {a..b}" by auto
   880     { presume *:"X \<noteq> {} \<Longrightarrow> ?case"
   881       show ?case apply(cases,rule *,assumption) by auto }
   882     def XX \<equiv> "\<lambda>i. {a $$ i .. b $$ i}" assume  "X \<noteq> {}"  note X' = this[unfolded X interval_ne_empty]
   883     have *:"Pi\<^isub>E {..<DIM('a)} XX = e2p ` X" apply(rule set_eqI)
   884     proof fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} XX"
   885       thus "x \<in> e2p ` X" unfolding image_iff apply(rule_tac x="\<chi>\<chi> i. x i" in bexI)
   886         unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by rule auto
   887     next fix x assume "x \<in> e2p ` X" then guess y unfolding image_iff .. note y = this
   888       show "x \<in> Pi\<^isub>E {..<DIM('a)} XX" unfolding y using y(1)
   889         unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by auto
   890     qed
   891     have "lmeasure X = (\<Prod>x<DIM('a). Real (b $$ x - a $$ x))"  using X' apply- unfolding X
   892       unfolding lmeasure_atLeastAtMost content_closed_interval apply(subst Real_setprod) by auto
   893     also have "... = (\<Prod>i<DIM('a). lmeasure (XX i))" apply(rule setprod_cong2)
   894       unfolding XX_def lmeasure_atLeastAtMost apply(subst content_real) using X' by auto
   895     also have "... = borel_product.product_measure {..<DIM('a)} (e2p ` X)" unfolding *[THEN sym]
   896       apply(rule fprod.measure_times[THEN sym]) unfolding XX_def by auto
   897     finally show ?case .
   898   qed
   899 
   900   show "range cube \<subseteq> sets \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
   901     unfolding cube_def_raw by auto
   902   have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp
   903   thus "cube \<up> space \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
   904     apply-apply(rule isotoneI) apply(rule cube_subset_Suc) by auto
   905   show "A \<in> sets borel " by fact
   906   show "measure_space borel lmeasure" by default
   907   show "measure_space borel
   908      (\<lambda>a::'a set. finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` a))"
   909     apply default unfolding countably_additive_def
   910   proof safe fix A::"nat \<Rightarrow> 'a set" assume A:"range A \<subseteq> sets borel" "disjoint_family A"
   911       "(\<Union>i. A i) \<in> sets borel"
   912     note fprod.ca[unfolded countably_additive_def,rule_format]
   913     note ca = this[of "\<lambda> n. e2p ` (A n)"]
   914     show "(\<Sum>\<^isub>\<infinity>n. finite_product_sigma_finite.measure
   915         (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` A n)) =
   916            finite_product_sigma_finite.measure (\<lambda>x. borel)
   917             (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` (\<Union>i. A i))" unfolding image_UN
   918     proof(rule ca) show "range (\<lambda>n. e2p ` A n) \<subseteq> sets
   919        (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
   920         unfolding product_borel_eq_vimage
   921       proof case goal1
   922         then guess y unfolding image_iff .. note y=this(2)
   923         show ?case unfolding borel.in_vimage_algebra y apply-
   924           apply(rule_tac x="A y" in bexI,rule e2p_image_vimage)
   925           using A(1) by auto
   926       qed
   927 
   928       show "disjoint_family (\<lambda>n. e2p ` A n)" apply(rule inj_on_disjoint_family_on)
   929         using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)] using A(2) unfolding bij_betw_def by auto
   930       show "(\<Union>n. e2p ` A n) \<in> sets (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
   931         unfolding product_borel_eq_vimage borel.in_vimage_algebra
   932       proof(rule bexI[OF _ A(3)],rule set_eqI,rule)
   933         fix x assume x:"x \<in> (\<Union>n. e2p ` A n)" hence "p2e x \<in> (\<Union>i. A i)" by auto
   934         moreover have "x \<in> extensional {..<DIM('a)}"
   935           using x unfolding extensional_def e2p_def_raw by auto
   936         ultimately show "x \<in> p2e -` (\<Union>i. A i) \<inter> extensional {..<DIM('a)}" by auto
   937       next fix x assume x:"x \<in> p2e -` (\<Union>i. A i) \<inter> extensional {..<DIM('a)}"
   938         hence "p2e x \<in> (\<Union>i. A i)" by auto
   939         hence "\<exists>n. x \<in> e2p ` A n" apply safe apply(rule_tac x=i in exI)
   940           unfolding image_iff apply(rule_tac x="p2e x" in bexI)
   941           apply(subst e2p_p2e) using x by auto
   942         thus "x \<in> (\<Union>n. e2p ` A n)" by auto
   943       qed
   944     qed
   945   qed auto
   946 qed
   947 
   948 lemma e2p_p2e'[simp]: fixes x::"'a::ordered_euclidean_space"
   949   assumes "A \<subseteq> extensional {..<DIM('a)}"
   950   shows "e2p ` (p2e ` A ::'a set) = A"
   951   apply(rule set_eqI) unfolding image_iff Bex_def apply safe defer
   952   apply(rule_tac x="p2e x" in exI,safe) using assms by auto
   953 
   954 lemma range_p2e:"range (p2e::_\<Rightarrow>'a::ordered_euclidean_space) = UNIV"
   955   apply safe defer unfolding image_iff apply(rule_tac x="\<lambda>i. x $$ i" in bexI)
   956   unfolding p2e_def by auto
   957 
   958 lemma p2e_inv_extensional:"(A::'a::ordered_euclidean_space set)
   959   = p2e ` (p2e -` A \<inter> extensional {..<DIM('a)})"
   960   unfolding p2e_def_raw apply safe unfolding image_iff
   961 proof- fix x assume "x\<in>A"
   962   let ?y = "\<lambda>i. if i<DIM('a) then x$$i else undefined"
   963   have *:"Chi ?y = x" apply(subst euclidean_eq) by auto
   964   show "\<exists>xa\<in>Chi -` A \<inter> extensional {..<DIM('a)}. x = Chi xa" apply(rule_tac x="?y" in bexI)
   965     apply(subst euclidean_eq) unfolding extensional_def using `x\<in>A` by(auto simp: *)
   966 qed
   967 
   968 lemma borel_fubini_positiv_integral:
   969   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal"
   970   assumes f: "f \<in> borel_measurable borel"
   971   shows "borel.positive_integral f =
   972           borel_product.product_positive_integral {..<DIM('a)} (f \<circ> p2e)"
   973 proof- def U \<equiv> "extensional {..<DIM('a)} :: (nat \<Rightarrow> real) set"
   974   interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
   975   have *:"sigma_algebra.vimage_algebra borel U (p2e:: _ \<Rightarrow> 'a)
   976     = sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)})"
   977     unfolding U_def product_borel_eq_vimage[symmetric] ..
   978   show ?thesis
   979     unfolding borel.positive_integral_vimage[unfolded space_borel, OF bij_inv_p2e_e2p[THEN bij_inv_bij_betw(1)]]
   980     apply(subst fprod.positive_integral_cong_measure[THEN sym, of "\<lambda>A. lmeasure (p2e ` A)"])
   981     unfolding U_def[symmetric] *[THEN sym] o_def
   982   proof- fix A assume A:"A \<in> sets (sigma_algebra.vimage_algebra borel U (p2e ::_ \<Rightarrow> 'a))"
   983     hence *:"A \<subseteq> extensional {..<DIM('a)}" unfolding U_def by auto
   984     from A guess B unfolding borel.in_vimage_algebra U_def ..
   985     then have "(p2e ` A::'a set) \<in> sets borel"
   986       by (simp add: p2e_inv_extensional[of B, symmetric])
   987     from lmeasure_measure_eq_borel_prod[OF this] show "lmeasure (p2e ` A::'a set) =
   988       finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} A"
   989       unfolding e2p_p2e'[OF *] .
   990   qed auto
   991 qed
   992 
   993 lemma borel_fubini:
   994   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
   995   assumes f: "f \<in> borel_measurable borel"
   996   shows "borel.integral f = borel_product.product_integral {..<DIM('a)} (f \<circ> p2e)"
   997 proof- interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
   998   have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto
   999   have 2:"(\<lambda>x. Real (- f x)) \<in> borel_measurable borel" using f by auto
  1000   show ?thesis unfolding fprod.integral_def borel.integral_def
  1001     unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2]
  1002     unfolding o_def ..
  1003 qed
  1004 
  1005 end