src/HOL/Probability/Lebesgue_Measure.thy
author hoelzl
Wed Feb 02 12:34:45 2011 +0100 (2011-02-02)
changeset 41689 3e39b0e730d6
parent 41661 baf1964bc468
child 41704 8c539202f854
permissions -rw-r--r--
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
changed syntax for simple_function, simple_integral, positive_integral, integral and RN_deriv.
introduced binder variants for simple_integral, positive_integral and integral.
     1 (*  Author: Robert Himmelmann, TU Muenchen *)
     2 header {* Lebsegue measure *}
     3 theory Lebesgue_Measure
     4   imports Product_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 cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
    46   unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
    47 
    48 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
    49   unfolding Pi_def by auto
    50 
    51 definition lebesgue :: "'a::ordered_euclidean_space measure_space" where
    52   "lebesgue = \<lparr> space = UNIV,
    53     sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n},
    54     measure = \<lambda>A. SUP n. Real (integral (cube n) (indicator A)) \<rparr>"
    55 
    56 lemma space_lebesgue[simp]: "space lebesgue = UNIV"
    57   unfolding lebesgue_def by simp
    58 
    59 lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n"
    60   unfolding lebesgue_def by simp
    61 
    62 lemma lebesgueI: "(\<And>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n) \<Longrightarrow> A \<in> sets lebesgue"
    63   unfolding lebesgue_def by simp
    64 
    65 lemma absolutely_integrable_on_indicator[simp]:
    66   fixes A :: "'a::ordered_euclidean_space set"
    67   shows "((indicator A :: _ \<Rightarrow> real) absolutely_integrable_on X) \<longleftrightarrow>
    68     (indicator A :: _ \<Rightarrow> real) integrable_on X"
    69   unfolding absolutely_integrable_on_def by simp
    70 
    71 lemma LIMSEQ_indicator_UN:
    72   "(\<lambda>k. indicator (\<Union> i<k. A i) x) ----> (indicator (\<Union>i. A i) x :: real)"
    73 proof cases
    74   assume "\<exists>i. x \<in> A i" then guess i .. note i = this
    75   then have *: "\<And>n. (indicator (\<Union> i<n + Suc i. A i) x :: real) = 1"
    76     "(indicator (\<Union> i. A i) x :: real) = 1" by (auto simp: indicator_def)
    77   show ?thesis
    78     apply (rule LIMSEQ_offset[of _ "Suc i"]) unfolding * by auto
    79 qed (auto simp: indicator_def)
    80 
    81 lemma indicator_add:
    82   "A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x"
    83   unfolding indicator_def by auto
    84 
    85 interpretation lebesgue: sigma_algebra lebesgue
    86 proof (intro sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI lebesgueI)
    87   fix A n assume A: "A \<in> sets lebesgue"
    88   have "indicator (space lebesgue - A) = (\<lambda>x. 1 - indicator A x :: real)"
    89     by (auto simp: fun_eq_iff indicator_def)
    90   then show "(indicator (space lebesgue - A) :: _ \<Rightarrow> real) integrable_on cube n"
    91     using A by (auto intro!: integrable_sub dest: lebesgueD simp: cube_def)
    92 next
    93   fix n show "(indicator {} :: _\<Rightarrow>real) integrable_on cube n"
    94     by (auto simp: cube_def indicator_def_raw)
    95 next
    96   fix A :: "nat \<Rightarrow> 'a set" and n ::nat assume "range A \<subseteq> sets lebesgue"
    97   then have A: "\<And>i. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
    98     by (auto dest: lebesgueD)
    99   show "(indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "?g integrable_on _")
   100   proof (intro dominated_convergence[where g="?g"] ballI)
   101     fix k show "(indicator (\<Union>i<k. A i) :: _ \<Rightarrow> real) integrable_on cube n"
   102     proof (induct k)
   103       case (Suc k)
   104       have *: "(\<Union> i<Suc k. A i) = (\<Union> i<k. A i) \<union> A k"
   105         unfolding lessThan_Suc UN_insert by auto
   106       have *: "(\<lambda>x. max (indicator (\<Union> i<k. A i) x) (indicator (A k) x) :: real) =
   107           indicator (\<Union> i<Suc k. A i)" (is "(\<lambda>x. max (?f x) (?g x)) = _")
   108         by (auto simp: fun_eq_iff * indicator_def)
   109       show ?case
   110         using absolutely_integrable_max[of ?f "cube n" ?g] A Suc by (simp add: *)
   111     qed auto
   112   qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
   113 qed simp
   114 
   115 interpretation lebesgue: measure_space lebesgue
   116 proof
   117   have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff)
   118   show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
   119 next
   120   show "countably_additive lebesgue (measure lebesgue)"
   121   proof (intro countably_additive_def[THEN iffD2] allI impI)
   122     fix A :: "nat \<Rightarrow> 'b set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A"
   123     then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
   124       by (auto dest: lebesgueD)
   125     let "?m n i" = "integral (cube n) (indicator (A i) :: _\<Rightarrow>real)"
   126     let "?M n I" = "integral (cube n) (indicator (\<Union>i\<in>I. A i) :: _\<Rightarrow>real)"
   127     have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: integral_nonneg)
   128     assume "(\<Union>i. A i) \<in> sets lebesgue"
   129     then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n"
   130       by (auto dest: lebesgueD)
   131     show "(\<Sum>\<^isub>\<infinity>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
   132     proof (simp add: lebesgue_def, subst psuminf_SUP_eq)
   133       fix n i show "Real (?m n i) \<le> Real (?m (Suc n) i)"
   134         using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le)
   135     next
   136       show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (?m n i)) = (SUP n. Real (?M n UNIV))"
   137         unfolding psuminf_def
   138       proof (subst setsum_Real, (intro arg_cong[where f="SUPR UNIV"] ext ballI nn SUP_eq_LIMSEQ[THEN iffD2])+)
   139         fix n :: nat show "mono (\<lambda>m. \<Sum>x<m. ?m n x)"
   140         proof (intro mono_iff_le_Suc[THEN iffD2] allI)
   141           fix m show "(\<Sum>x<m. ?m n x) \<le> (\<Sum>x<Suc m. ?m n x)"
   142             using nn[of n m] by auto
   143         qed
   144         show "0 \<le> ?M n UNIV"
   145           using UN_A by (auto intro!: integral_nonneg)
   146         fix m show "0 \<le> (\<Sum>x<m. ?m n x)" by (auto intro!: setsum_nonneg)
   147       next
   148         fix n
   149         have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto
   150         from lebesgueD[OF this]
   151         have "(\<lambda>m. ?M n {..< m}) ----> ?M n UNIV"
   152           (is "(\<lambda>m. integral _ (?A m)) ----> ?I")
   153           by (intro dominated_convergence(2)[where f="?A" and h="\<lambda>x. 1::real"])
   154              (auto intro: LIMSEQ_indicator_UN simp: cube_def)
   155         moreover
   156         { fix m have *: "(\<Sum>x<m. ?m n x) = ?M n {..< m}"
   157           proof (induct m)
   158             case (Suc m)
   159             have "(\<Union>i<m. A i) \<in> sets lebesgue" using rA by auto
   160             then have "(indicator (\<Union>i<m. A i) :: _\<Rightarrow>real) integrable_on (cube n)"
   161               by (auto dest!: lebesgueD)
   162             moreover
   163             have "(\<Union>i<m. A i) \<inter> A m = {}"
   164               using rA(2)[unfolded disjoint_family_on_def, THEN bspec, of m]
   165               by auto
   166             then have "\<And>x. indicator (\<Union>i<Suc m. A i) x =
   167               indicator (\<Union>i<m. A i) x + (indicator (A m) x :: real)"
   168               by (auto simp: indicator_add lessThan_Suc ac_simps)
   169             ultimately show ?case
   170               using Suc A by (simp add: integral_add[symmetric])
   171           qed auto }
   172         ultimately show "(\<lambda>m. \<Sum>x<m. ?m n x) ----> ?M n UNIV"
   173           by simp
   174       qed
   175     qed
   176   qed
   177 qed
   178 
   179 lemma has_integral_interval_cube:
   180   fixes a b :: "'a::ordered_euclidean_space"
   181   shows "(indicator {a .. b} has_integral
   182     content ({\<chi>\<chi> i. max (- real n) (a $$ i) .. \<chi>\<chi> i. min (real n) (b $$ i)} :: 'a set)) (cube n)"
   183     (is "(?I has_integral content ?R) (cube n)")
   184 proof -
   185   let "{?N .. ?P}" = ?R
   186   have [simp]: "(\<lambda>x. if x \<in> cube n then ?I x else 0) = indicator ?R"
   187     by (auto simp: indicator_def cube_def fun_eq_iff eucl_le[where 'a='a])
   188   have "(?I has_integral content ?R) (cube n) \<longleftrightarrow> (indicator ?R has_integral content ?R) UNIV"
   189     unfolding has_integral_restrict_univ[where s="cube n", symmetric] by simp
   190   also have "\<dots> \<longleftrightarrow> ((\<lambda>x. 1) has_integral content ?R) ?R"
   191     unfolding indicator_def_raw has_integral_restrict_univ ..
   192   finally show ?thesis
   193     using has_integral_const[of "1::real" "?N" "?P"] by simp
   194 qed
   195 
   196 lemma lebesgueI_borel[intro, simp]:
   197   fixes s::"'a::ordered_euclidean_space set"
   198   assumes "s \<in> sets borel" shows "s \<in> sets lebesgue"
   199 proof -
   200   let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})"
   201   have *:"?S \<subseteq> sets lebesgue"
   202   proof (safe intro!: lebesgueI)
   203     fix n :: nat and a b :: 'a
   204     let ?N = "\<chi>\<chi> i. max (- real n) (a $$ i)"
   205     let ?P = "\<chi>\<chi> i. min (real n) (b $$ i)"
   206     show "(indicator {a..b} :: 'a\<Rightarrow>real) integrable_on cube n"
   207       unfolding integrable_on_def
   208       using has_integral_interval_cube[of a b] by auto
   209   qed
   210   have "s \<in> sigma_sets UNIV ?S" using assms
   211     unfolding borel_eq_atLeastAtMost by (simp add: sigma_def)
   212   thus ?thesis
   213     using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *]
   214     by (auto simp: sigma_def)
   215 qed
   216 
   217 lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
   218   assumes "negligible s" shows "s \<in> sets lebesgue"
   219   using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI)
   220 
   221 lemma lmeasure_eq_0:
   222   fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "lebesgue.\<mu> S = 0"
   223 proof -
   224   have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0"
   225     unfolding lebesgue_integral_def using assms
   226     by (intro integral_unique some1_equality ex_ex1I)
   227        (auto simp: cube_def negligible_def)
   228   then show ?thesis by (auto simp: lebesgue_def)
   229 qed
   230 
   231 lemma lmeasure_iff_LIMSEQ:
   232   assumes "A \<in> sets lebesgue" "0 \<le> m"
   233   shows "lebesgue.\<mu> A = Real m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
   234 proof (simp add: lebesgue_def, intro SUP_eq_LIMSEQ)
   235   show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))"
   236     using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD)
   237   fix n show "0 \<le> integral (cube n) (indicator A::_=>real)"
   238     using assms by (auto dest!: lebesgueD intro!: integral_nonneg)
   239 qed fact
   240 
   241 lemma has_integral_indicator_UNIV:
   242   fixes s A :: "'a::ordered_euclidean_space set" and x :: real
   243   shows "((indicator (s \<inter> A) :: 'a\<Rightarrow>real) has_integral x) UNIV = ((indicator s :: _\<Rightarrow>real) has_integral x) A"
   244 proof -
   245   have "(\<lambda>x. if x \<in> A then indicator s x else 0) = (indicator (s \<inter> A) :: _\<Rightarrow>real)"
   246     by (auto simp: fun_eq_iff indicator_def)
   247   then show ?thesis
   248     unfolding has_integral_restrict_univ[where s=A, symmetric] by simp
   249 qed
   250 
   251 lemma
   252   fixes s a :: "'a::ordered_euclidean_space set"
   253   shows integral_indicator_UNIV:
   254     "integral UNIV (indicator (s \<inter> A) :: 'a\<Rightarrow>real) = integral A (indicator s :: _\<Rightarrow>real)"
   255   and integrable_indicator_UNIV:
   256     "(indicator (s \<inter> A) :: 'a\<Rightarrow>real) integrable_on UNIV \<longleftrightarrow> (indicator s :: 'a\<Rightarrow>real) integrable_on A"
   257   unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto
   258 
   259 lemma lmeasure_finite_has_integral:
   260   fixes s :: "'a::ordered_euclidean_space set"
   261   assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = Real m" "0 \<le> m"
   262   shows "(indicator s has_integral m) UNIV"
   263 proof -
   264   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
   265   have **: "(?I s) integrable_on UNIV \<and> (\<lambda>k. integral UNIV (?I (s \<inter> cube k))) ----> integral UNIV (?I s)"
   266   proof (intro monotone_convergence_increasing allI ballI)
   267     have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) ----> m"
   268       using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1, 3)] .
   269     { fix n have "integral (cube n) (?I s) \<le> m"
   270         using cube_subset assms
   271         by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI)
   272            (auto dest!: lebesgueD) }
   273     moreover
   274     { fix n have "0 \<le> integral (cube n) (?I s)"
   275       using assms by (auto dest!: lebesgueD intro!: integral_nonneg) }
   276     ultimately
   277     show "bounded {integral UNIV (?I (s \<inter> cube k)) |k. True}"
   278       unfolding bounded_def
   279       apply (rule_tac exI[of _ 0])
   280       apply (rule_tac exI[of _ m])
   281       by (auto simp: dist_real_def integral_indicator_UNIV)
   282     fix k show "?I (s \<inter> cube k) integrable_on UNIV"
   283       unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD)
   284     fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x"
   285       using cube_subset[of k "Suc k"] by (auto simp: indicator_def)
   286   next
   287     fix x :: 'a
   288     from mem_big_cube obtain k where k: "x \<in> cube k" .
   289     { fix n have "?I (s \<inter> cube (n + k)) x = ?I s x"
   290       using k cube_subset[of k "n + k"] by (auto simp: indicator_def) }
   291     note * = this
   292     show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x"
   293       by (rule LIMSEQ_offset[where k=k]) (auto simp: *)
   294   qed
   295   note ** = conjunctD2[OF this]
   296   have m: "m = integral UNIV (?I s)"
   297     apply (intro LIMSEQ_unique[OF _ **(2)])
   298     using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1,3)] integral_indicator_UNIV .
   299   show ?thesis
   300     unfolding m by (intro integrable_integral **)
   301 qed
   302 
   303 lemma lmeasure_finite_integrable: assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s \<noteq> \<omega>"
   304   shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
   305 proof (cases "lebesgue.\<mu> s")
   306   case (preal m) from lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
   307   show ?thesis unfolding integrable_on_def by auto
   308 qed (insert assms, auto)
   309 
   310 lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
   311   shows "s \<in> sets lebesgue"
   312 proof (intro lebesgueI)
   313   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
   314   fix n show "(?I s) integrable_on cube n" unfolding cube_def
   315   proof (intro integrable_on_subinterval)
   316     show "(?I s) integrable_on UNIV"
   317       unfolding integrable_on_def using assms by auto
   318   qed auto
   319 qed
   320 
   321 lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
   322   shows "lebesgue.\<mu> s = Real m"
   323 proof (intro lmeasure_iff_LIMSEQ[THEN iffD2])
   324   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
   325   show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] .
   326   show "0 \<le> m" using assms by (rule has_integral_nonneg) auto
   327   have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) ----> integral UNIV (?I s)"
   328   proof (intro dominated_convergence(2) ballI)
   329     show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto
   330     fix n show "?I (s \<inter> cube n) integrable_on UNIV"
   331       unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD)
   332     fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def)
   333   next
   334     fix x :: 'a
   335     from mem_big_cube obtain k where k: "x \<in> cube k" .
   336     { fix n have "?I (s \<inter> cube (n + k)) x = ?I s x"
   337       using k cube_subset[of k "n + k"] by (auto simp: indicator_def) }
   338     note * = this
   339     show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x"
   340       by (rule LIMSEQ_offset[where k=k]) (auto simp: *)
   341   qed
   342   then show "(\<lambda>n. integral (cube n) (?I s)) ----> m"
   343     unfolding integral_unique[OF assms] integral_indicator_UNIV by simp
   344 qed
   345 
   346 lemma has_integral_iff_lmeasure:
   347   "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m)"
   348 proof
   349   assume "(indicator A has_integral m) UNIV"
   350   with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this]
   351   show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m"
   352     by (auto intro: has_integral_nonneg)
   353 next
   354   assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = Real m"
   355   then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto
   356 qed
   357 
   358 lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV"
   359   shows "lebesgue.\<mu> s = Real (integral UNIV (indicator s))"
   360   using assms unfolding integrable_on_def
   361 proof safe
   362   fix y :: real assume "(indicator s has_integral y) UNIV"
   363   from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this]
   364   show "lebesgue.\<mu> s = Real (integral UNIV (indicator s))" by simp
   365 qed
   366 
   367 lemma lebesgue_simple_function_indicator:
   368   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
   369   assumes f:"simple_function lebesgue f"
   370   shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
   371   by (rule, subst lebesgue.simple_function_indicator_representation[OF f]) auto
   372 
   373 lemma integral_eq_lmeasure:
   374   "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lebesgue.\<mu> s)"
   375   by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg)
   376 
   377 lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<omega>"
   378   using lmeasure_eq_integral[OF assms] by auto
   379 
   380 lemma negligible_iff_lebesgue_null_sets:
   381   "negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets"
   382 proof
   383   assume "negligible A"
   384   from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
   385   show "A \<in> lebesgue.null_sets" by auto
   386 next
   387   assume A: "A \<in> lebesgue.null_sets"
   388   then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] by auto
   389   show "negligible A" unfolding negligible_def
   390   proof (intro allI)
   391     fix a b :: 'a
   392     have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}"
   393       by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *)
   394     then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)"
   395       using * by (auto intro!: integral_subset_le has_integral_integrable)
   396     moreover have "(0::real) \<le> integral {a..b} (indicator A)"
   397       using integrable by (auto intro!: integral_nonneg)
   398     ultimately have "integral {a..b} (indicator A) = (0::real)"
   399       using integral_unique[OF *] by auto
   400     then show "(indicator A has_integral (0::real)) {a..b}"
   401       using integrable_integral[OF integrable] by simp
   402   qed
   403 qed
   404 
   405 lemma integral_const[simp]:
   406   fixes a b :: "'a::ordered_euclidean_space"
   407   shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
   408   by (rule integral_unique) (rule has_integral_const)
   409 
   410 lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<omega>"
   411 proof (simp add: lebesgue_def SUP_\<omega>, intro allI impI)
   412   fix x assume "x < \<omega>"
   413   then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto
   414   then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto
   415   show "\<exists>i. x < Real (integral (cube i) (indicator UNIV::'a\<Rightarrow>real))"
   416   proof (intro exI[of _ n])
   417     have [simp]: "indicator UNIV = (\<lambda>x. 1)" by (auto simp: fun_eq_iff)
   418     { fix m :: nat assume "0 < m" then have "real n \<le> (\<Prod>x<m. 2 * real n)"
   419       proof (induct m)
   420         case (Suc m)
   421         show ?case
   422         proof cases
   423           assume "m = 0" then show ?thesis by (simp add: lessThan_Suc)
   424         next
   425           assume "m \<noteq> 0" then have "real n \<le> (\<Prod>x<m. 2 * real n)" using Suc by auto
   426           then show ?thesis
   427             by (auto simp: lessThan_Suc field_simps mult_le_cancel_left1)
   428         qed
   429       qed auto } note this[OF DIM_positive[where 'a='a], simp]
   430     then have [simp]: "0 \<le> (\<Prod>x<DIM('a). 2 * real n)" using real_of_nat_ge_zero by arith
   431     have "x < Real (of_nat n)" using n r by auto
   432     also have "Real (of_nat n) \<le> Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
   433       by (auto simp: real_eq_of_nat[symmetric] cube_def content_closed_interval_cases)
   434     finally show "x < Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" .
   435   qed
   436 qed
   437 
   438 lemma
   439   fixes a b ::"'a::ordered_euclidean_space"
   440   shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = Real (content {a..b})"
   441 proof -
   442   have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV"
   443     unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw)
   444   from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV
   445     by (simp add: indicator_def_raw)
   446 qed
   447 
   448 lemma atLeastAtMost_singleton_euclidean[simp]:
   449   fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
   450   by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a])
   451 
   452 lemma content_singleton[simp]: "content {a} = 0"
   453 proof -
   454   have "content {a .. a} = 0"
   455     by (subst content_closed_interval) auto
   456   then show ?thesis by simp
   457 qed
   458 
   459 lemma lmeasure_singleton[simp]:
   460   fixes a :: "'a::ordered_euclidean_space" shows "lebesgue.\<mu> {a} = 0"
   461   using lmeasure_atLeastAtMost[of a a] by simp
   462 
   463 declare content_real[simp]
   464 
   465 lemma
   466   fixes a b :: real
   467   shows lmeasure_real_greaterThanAtMost[simp]:
   468     "lebesgue.\<mu> {a <.. b} = Real (if a \<le> b then b - a else 0)"
   469 proof cases
   470   assume "a < b"
   471   then have "lebesgue.\<mu> {a <.. b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {a}"
   472     by (subst lebesgue.measure_Diff[symmetric])
   473        (auto intro!: arg_cong[where f=lebesgue.\<mu>])
   474   then show ?thesis by auto
   475 qed auto
   476 
   477 lemma
   478   fixes a b :: real
   479   shows lmeasure_real_atLeastLessThan[simp]:
   480     "lebesgue.\<mu> {a ..< b} = Real (if a \<le> b then b - a else 0)"
   481 proof cases
   482   assume "a < b"
   483   then have "lebesgue.\<mu> {a ..< b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {b}"
   484     by (subst lebesgue.measure_Diff[symmetric])
   485        (auto intro!: arg_cong[where f=lebesgue.\<mu>])
   486   then show ?thesis by auto
   487 qed auto
   488 
   489 lemma
   490   fixes a b :: real
   491   shows lmeasure_real_greaterThanLessThan[simp]:
   492     "lebesgue.\<mu> {a <..< b} = Real (if a \<le> b then b - a else 0)"
   493 proof cases
   494   assume "a < b"
   495   then have "lebesgue.\<mu> {a <..< b} = lebesgue.\<mu> {a <.. b} - lebesgue.\<mu> {b}"
   496     by (subst lebesgue.measure_Diff[symmetric])
   497        (auto intro!: arg_cong[where f=lebesgue.\<mu>])
   498   then show ?thesis by auto
   499 qed auto
   500 
   501 definition "lborel = lebesgue \<lparr> sets := sets borel \<rparr>"
   502 
   503 lemma
   504   shows space_lborel[simp]: "space lborel = UNIV"
   505   and sets_lborel[simp]: "sets lborel = sets borel"
   506   and measure_lborel[simp]: "measure lborel = lebesgue.\<mu>"
   507   and measurable_lborel[simp]: "measurable lborel = measurable borel"
   508   by (simp_all add: measurable_def_raw lborel_def)
   509 
   510 interpretation lborel: measure_space lborel
   511   where "space lborel = UNIV"
   512   and "sets lborel = sets borel"
   513   and "measure lborel = lebesgue.\<mu>"
   514   and "measurable lborel = measurable borel"
   515 proof -
   516   show "measure_space lborel"
   517   proof
   518     show "countably_additive lborel (measure lborel)"
   519       using lebesgue.ca unfolding countably_additive_def lborel_def
   520       apply safe apply (erule_tac x=A in allE) by auto
   521   qed (auto simp: lborel_def)
   522 qed simp_all
   523 
   524 interpretation lborel: sigma_finite_measure lborel
   525   where "space lborel = UNIV"
   526   and "sets lborel = sets borel"
   527   and "measure lborel = lebesgue.\<mu>"
   528   and "measurable lborel = measurable borel"
   529 proof -
   530   show "sigma_finite_measure lborel"
   531   proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
   532     show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
   533     { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
   534     thus "(\<Union>i. cube i) = space lborel" by auto
   535     show "\<forall>i. measure lborel (cube i) \<noteq> \<omega>" by (simp add: cube_def)
   536   qed
   537 qed simp_all
   538 
   539 interpretation lebesgue: sigma_finite_measure lebesgue
   540 proof
   541   from lborel.sigma_finite guess A ..
   542   moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
   543   ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<omega>)"
   544     by auto
   545 qed
   546 
   547 lemma simple_function_has_integral:
   548   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
   549   assumes f:"simple_function lebesgue f"
   550   and f':"\<forall>x. f x \<noteq> \<omega>"
   551   and om:"\<forall>x\<in>range f. lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
   552   shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
   553   unfolding simple_integral_def
   554   apply(subst lebesgue_simple_function_indicator[OF f])
   555 proof -
   556   case goal1
   557   have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
   558     "\<forall>x\<in>range f. x * lebesgue.\<mu> (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
   559     using f' om unfolding indicator_def by auto
   560   show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym]
   561     unfolding real_of_pextreal_setsum'[OF *(2),THEN sym]
   562     unfolding real_of_pextreal_setsum space_lebesgue
   563     apply(rule has_integral_setsum)
   564   proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD)
   565     fix y::'a show "((\<lambda>x. real (f y * indicator (f -` {f y}) x)) has_integral
   566       real (f y * lebesgue.\<mu> (f -` {f y} \<inter> UNIV))) UNIV"
   567     proof(cases "f y = 0") case False
   568       have mea:"(indicator (f -` {f y}) ::_\<Rightarrow>real) integrable_on UNIV"
   569         apply(rule lmeasure_finite_integrable)
   570         using assms unfolding simple_function_def using False by auto
   571       have *:"\<And>x. real (indicator (f -` {f y}) x::pextreal) = (indicator (f -` {f y}) x)"
   572         by (auto simp: indicator_def)
   573       show ?thesis unfolding real_of_pextreal_mult[THEN sym]
   574         apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def])
   575         unfolding Int_UNIV_right lmeasure_eq_integral[OF mea,THEN sym]
   576         unfolding integral_eq_lmeasure[OF mea, symmetric] *
   577         apply(rule integrable_integral) using mea .
   578     qed auto
   579   qed
   580 qed
   581 
   582 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
   583   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
   584   using assms by auto
   585 
   586 lemma simple_function_has_integral':
   587   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal"
   588   assumes f:"simple_function lebesgue f"
   589   and i: "integral\<^isup>S lebesgue f \<noteq> \<omega>"
   590   shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
   591 proof- let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x"
   592   { fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this
   593   have **:"{x. f x \<noteq> ?f x} = f -` {\<omega>}" by auto
   594   have **:"lebesgue.\<mu> {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
   595     using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**)
   596   show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **])
   597     apply(rule lebesgue.simple_function_compose1[OF f])
   598     unfolding * defer apply(rule simple_function_has_integral)
   599   proof-
   600     show "simple_function lebesgue ?f"
   601       using lebesgue.simple_function_compose1[OF f] .
   602     show "\<forall>x. ?f x \<noteq> \<omega>" by auto
   603     show "\<forall>x\<in>range ?f. lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
   604     proof (safe, simp, safe, rule ccontr)
   605       fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0"
   606       hence "(\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y} = f -` {f y}"
   607         by (auto split: split_if_asm)
   608       moreover assume "lebesgue.\<mu> ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
   609       ultimately have "lebesgue.\<mu> (f -` {f y}) = \<omega>" by simp
   610       moreover
   611       have "f y * lebesgue.\<mu> (f -` {f y}) \<noteq> \<omega>" using i f
   612         unfolding simple_integral_def setsum_\<omega> simple_function_def
   613         by auto
   614       ultimately have "f y = 0" by (auto split: split_if_asm)
   615       then show False using `f y \<noteq> 0` by simp
   616     qed
   617   qed
   618 qed
   619 
   620 lemma (in measure_space) positive_integral_monotone_convergence:
   621   fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pextreal"
   622   assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)"
   623   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
   624   shows "u \<in> borel_measurable M"
   625   and "(\<lambda>i. integral\<^isup>P M (f i)) ----> integral\<^isup>P M u" (is ?ilim)
   626 proof -
   627   from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
   628   show ?ilim using mono lim i by auto
   629   have "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal
   630     unfolding fun_eq_iff mono_def by auto
   631   moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
   632     using i by auto
   633   ultimately show "u \<in> borel_measurable M" by simp
   634 qed
   635 
   636 lemma positive_integral_has_integral:
   637   fixes f::"'a::ordered_euclidean_space => pextreal"
   638   assumes f:"f \<in> borel_measurable lebesgue"
   639   and int_om:"integral\<^isup>P lebesgue f \<noteq> \<omega>"
   640   and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
   641   shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV"
   642 proof- let ?i = "integral\<^isup>P lebesgue f"
   643   from lebesgue.borel_measurable_implies_simple_function_sequence[OF f]
   644   guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2)
   645   let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)"
   646   have u_simple:"\<And>k. integral\<^isup>S lebesgue (u k) = integral\<^isup>P lebesgue (u k)"
   647     apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) ..
   648   have int_u_le:"\<And>k. integral\<^isup>S lebesgue (u k) \<le> integral\<^isup>P lebesgue f"
   649     unfolding u_simple apply(rule lebesgue.positive_integral_mono)
   650     using isoton_Sup[OF u(3)] unfolding le_fun_def by auto
   651   have u_int_om:"\<And>i. integral\<^isup>S lebesgue (u i) \<noteq> \<omega>"
   652   proof- case goal1 thus ?case using int_u_le[of i] int_om by auto qed
   653 
   654   note u_int = simple_function_has_integral'[OF u(1) this]
   655   have "(\<lambda>x. real (f x)) integrable_on UNIV \<and>
   656     (\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) ----> Integration.integral UNIV (\<lambda>x. real (f x))"
   657     apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int)
   658   proof safe case goal1 show ?case apply(rule real_of_pextreal_mono) using u(2,3) by auto
   659   next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym])
   660       prefer 3 apply(subst Real_real') defer apply(subst Real_real')
   661       using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto
   662   next case goal3
   663     show ?case apply(rule bounded_realI[where B="real (integral\<^isup>P lebesgue f)"])
   664       apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int)
   665       unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le])
   666       using u int_om by auto
   667   qed note int = conjunctD2[OF this]
   668 
   669   have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> ?i" unfolding u_simple
   670     apply(rule lebesgue.positive_integral_monotone_convergence(2))
   671     apply(rule lebesgue.borel_measurable_simple_function[OF u(1)])
   672     using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto
   673   hence "(\<lambda>i. real (integral\<^isup>S lebesgue (u i))) ----> real ?i" apply-
   674     apply(subst lim_Real[THEN sym]) prefer 3
   675     apply(subst Real_real') defer apply(subst Real_real')
   676     using u f_om int_om u_int_om by auto
   677   note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]]
   678   show ?thesis unfolding * by(rule integrable_integral[OF int(1)])
   679 qed
   680 
   681 lemma lebesgue_integral_has_integral:
   682   fixes f::"'a::ordered_euclidean_space => real"
   683   assumes f:"integrable lebesgue f"
   684   shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV"
   685 proof- let ?n = "\<lambda>x. - min (f x) 0" and ?p = "\<lambda>x. max (f x) 0"
   686   have *:"f = (\<lambda>x. ?p x - ?n x)" apply rule by auto
   687   note f = integrableD[OF f]
   688   show ?thesis unfolding lebesgue_integral_def apply(subst *)
   689   proof(rule has_integral_sub) case goal1
   690     have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto
   691     note lebesgue.borel_measurable_Real[OF f(1)]
   692     from positive_integral_has_integral[OF this f(2) *]
   693     show ?case unfolding real_Real_max .
   694   next case goal2
   695     have *:"\<forall>x. Real (- f x) \<noteq> \<omega>" by auto
   696     note lebesgue.borel_measurable_uminus[OF f(1)]
   697     note lebesgue.borel_measurable_Real[OF this]
   698     from positive_integral_has_integral[OF this f(3) *]
   699     show ?case unfolding real_Real_max minus_min_eq_max by auto
   700   qed
   701 qed
   702 
   703 lemma lebesgue_positive_integral_eq_borel:
   704   "f \<in> borel_measurable borel \<Longrightarrow> integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
   705   by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
   706 
   707 lemma lebesgue_integral_eq_borel:
   708   assumes "f \<in> borel_measurable borel"
   709   shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P)
   710     and "integral\<^isup>L lebesgue f = integral\<^isup>L lborel f" (is ?I)
   711 proof -
   712   have *: "sigma_algebra lborel" by default
   713   have "sets lborel \<subseteq> sets lebesgue" by auto
   714   from lebesgue.integral_subalgebra[of f lborel, OF _ this _ _ *] assms
   715   show ?P ?I by auto
   716 qed
   717 
   718 lemma borel_integral_has_integral:
   719   fixes f::"'a::ordered_euclidean_space => real"
   720   assumes f:"integrable lborel f"
   721   shows "(f has_integral (integral\<^isup>L lborel f)) UNIV"
   722 proof -
   723   have borel: "f \<in> borel_measurable borel"
   724     using f unfolding integrable_def by auto
   725   from f show ?thesis
   726     using lebesgue_integral_has_integral[of f]
   727     unfolding lebesgue_integral_eq_borel[OF borel] by simp
   728 qed
   729 
   730 lemma continuous_on_imp_borel_measurable:
   731   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
   732   assumes "continuous_on UNIV f"
   733   shows "f \<in> borel_measurable borel"
   734   apply(rule borel.borel_measurableI)
   735   using continuous_open_preimage[OF assms] unfolding vimage_def by auto
   736 
   737 lemma (in measure_space) integral_monotone_convergence_pos':
   738   assumes i: "\<And>i. integrable M (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
   739   and pos: "\<And>x i. 0 \<le> f i x"
   740   and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
   741   and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
   742   shows "integrable M u \<and> integral\<^isup>L M u = x"
   743   using integral_monotone_convergence_pos[OF assms] by auto
   744 
   745 definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where
   746   "e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)"
   747 
   748 definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where
   749   "p2e x = (\<chi>\<chi> i. x i)"
   750 
   751 lemma e2p_p2e[simp]:
   752   "x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x"
   753   by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def)
   754 
   755 lemma p2e_e2p[simp]:
   756   "p2e (e2p x) = (x::'a::ordered_euclidean_space)"
   757   by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def)
   758 
   759 lemma bij_inv_p2e_e2p:
   760   shows "bij_inv ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) (UNIV :: 'a::ordered_euclidean_space set)
   761      p2e e2p" (is "bij_inv ?P ?U _ _")
   762 proof (rule bij_invI)
   763   show "p2e \<in> ?P \<rightarrow> ?U" "e2p \<in> ?U \<rightarrow> ?P" by (auto simp: e2p_def)
   764 qed auto
   765 
   766 declare restrict_extensional[intro]
   767 
   768 lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}"
   769   unfolding e2p_def by auto
   770 
   771 lemma e2p_image_vimage: fixes A::"'a::ordered_euclidean_space set"
   772   shows "e2p ` A = p2e -` A \<inter> extensional {..<DIM('a)}"
   773 proof(rule set_eqI,rule)
   774   fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this
   775   show "x \<in> p2e -` A \<inter> extensional {..<DIM('a)}"
   776     apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto
   777 next fix x assume "x \<in> p2e -` A \<inter> extensional {..<DIM('a)}"
   778   thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto
   779 qed
   780 
   781 interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure_space"
   782   by default
   783 
   784 interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure_space" "{..<DIM('a::ordered_euclidean_space)}"
   785   where "space lborel = UNIV"
   786   and "sets lborel = sets borel"
   787   and "measure lborel = lebesgue.\<mu>"
   788   and "measurable lborel = measurable borel"
   789 proof -
   790   show "finite_product_sigma_finite (\<lambda>x. lborel::real measure_space) {..<DIM('a::ordered_euclidean_space)}"
   791     by default simp
   792 qed simp_all
   793 
   794 lemma sets_product_borel:
   795   assumes [intro]: "finite I"
   796   shows "sets (\<Pi>\<^isub>M i\<in>I.
   797      \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>) =
   798    sets (\<Pi>\<^isub>M i\<in>I. lborel)" (is "sets ?G = _")
   799 proof -
   800   have "sets ?G = sets (\<Pi>\<^isub>M i\<in>I.
   801        sigma \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>)"
   802     by (subst sigma_product_algebra_sigma_eq[of I "\<lambda>_ i. {..<real i}" ])
   803        (auto intro!: measurable_sigma_sigma isotoneI real_arch_lt
   804              simp: product_algebra_def)
   805   then show ?thesis
   806     unfolding lborel_def borel_eq_lessThan lebesgue_def sigma_def by simp
   807 qed
   808 
   809 lemma measurable_e2p:
   810   "e2p \<in> measurable (borel::'a::ordered_euclidean_space algebra)
   811                     (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))"
   812     (is "_ \<in> measurable ?E ?P")
   813 proof -
   814   let ?B = "\<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>"
   815   let ?G = "product_algebra_generator {..<DIM('a)} (\<lambda>_. ?B)"
   816   have "e2p \<in> measurable ?E (sigma ?G)"
   817   proof (rule borel.measurable_sigma)
   818     show "e2p \<in> space ?E \<rightarrow> space ?G" by (auto simp: e2p_def)
   819     fix A assume "A \<in> sets ?G"
   820     then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)"
   821       and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)"
   822       by (auto elim!: product_algebraE simp: )
   823     then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto
   824     from this[THEN bchoice] guess xs ..
   825     then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})"
   826       using A by auto
   827     have "e2p -` A = {..< (\<chi>\<chi> i. xs i) :: 'a}"
   828       using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq
   829         euclidean_eq[where 'a='a] eucl_less[where 'a='a])
   830     then show "e2p -` A \<inter> space ?E \<in> sets ?E" by simp
   831   qed (auto simp: product_algebra_generator_def)
   832   with sets_product_borel[of "{..<DIM('a)}"] show ?thesis
   833     unfolding measurable_def product_algebra_def by simp
   834 qed
   835 
   836 lemma measurable_p2e:
   837   "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
   838     (borel :: 'a::ordered_euclidean_space algebra)"
   839   (is "p2e \<in> measurable ?P _")
   840   unfolding borel_eq_lessThan
   841 proof (intro lborel_space.measurable_sigma)
   842   let ?E = "\<lparr> space = UNIV :: 'a set, sets = range lessThan \<rparr>"
   843   show "p2e \<in> space ?P \<rightarrow> space ?E" by simp
   844   fix A assume "A \<in> sets ?E"
   845   then obtain x where "A = {..<x}" by auto
   846   then have "p2e -` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x $$ i})"
   847     using DIM_positive
   848     by (auto simp: Pi_iff set_eq_iff p2e_def
   849                    euclidean_eq[where 'a='a] eucl_less[where 'a='a])
   850   then show "p2e -` A \<inter> space ?P \<in> sets ?P" by auto
   851 qed simp
   852 
   853 lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
   854   apply(rule image_Int[THEN sym])
   855   using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)]
   856   unfolding bij_betw_def by auto
   857 
   858 lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space"
   859   shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>"
   860   unfolding Int_stable_def algebra.select_convs
   861 proof safe fix a b x y::'a
   862   have *:"e2p ` {a..b} \<inter> e2p ` {x..y} =
   863     (\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a $$ i) (x $$ i), \<chi>\<chi> i. min (b $$ i) (y $$ i)::'a)"
   864     unfolding e2p_Int inter_interval by auto
   865   show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding *
   866     apply(rule range_eqI) ..
   867 qed
   868 
   869 lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space"
   870   shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>"
   871   unfolding Int_stable_def algebra.select_convs
   872   apply safe unfolding inter_interval by auto
   873 
   874 lemma lmeasure_measure_eq_borel_prod:
   875   fixes A :: "('a::ordered_euclidean_space) set"
   876   assumes "A \<in> sets borel"
   877   shows "lebesgue.\<mu> A = lborel_space.\<mu> TYPE('a) (e2p ` A)" (is "_ = ?m A")
   878 proof (rule measure_unique_Int_stable[where X=A and A=cube])
   879   show "Int_stable \<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
   880     (is "Int_stable ?E" ) using Int_stable_cuboids' .
   881   have [simp]: "sigma ?E = borel" using borel_eq_atLeastAtMost ..
   882   show "\<And>i. lebesgue.\<mu> (cube i) \<noteq> \<omega>" unfolding cube_def by auto
   883   show "\<And>X. X \<in> sets ?E \<Longrightarrow> lebesgue.\<mu> X = ?m X"
   884   proof- case goal1 then obtain a b where X:"X = {a..b}" by auto
   885     { presume *:"X \<noteq> {} \<Longrightarrow> ?case"
   886       show ?case apply(cases,rule *,assumption) by auto }
   887     def XX \<equiv> "\<lambda>i. {a $$ i .. b $$ i}" assume  "X \<noteq> {}"  note X' = this[unfolded X interval_ne_empty]
   888     have *:"Pi\<^isub>E {..<DIM('a)} XX = e2p ` X" apply(rule set_eqI)
   889     proof fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} XX"
   890       thus "x \<in> e2p ` X" unfolding image_iff apply(rule_tac x="\<chi>\<chi> i. x i" in bexI)
   891         unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by rule auto
   892     next fix x assume "x \<in> e2p ` X" then guess y unfolding image_iff .. note y = this
   893       show "x \<in> Pi\<^isub>E {..<DIM('a)} XX" unfolding y using y(1)
   894         unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by auto
   895     qed
   896     have "lebesgue.\<mu> X = (\<Prod>x<DIM('a). Real (b $$ x - a $$ x))"  using X' apply- unfolding X
   897       unfolding lmeasure_atLeastAtMost content_closed_interval apply(subst Real_setprod) by auto
   898     also have "... = (\<Prod>i<DIM('a). lebesgue.\<mu> (XX i))" apply(rule setprod_cong2)
   899       unfolding XX_def lmeasure_atLeastAtMost apply(subst content_real) using X' by auto
   900     also have "... = ?m X" unfolding *[THEN sym]
   901       apply(rule lborel_space.measure_times[symmetric]) unfolding XX_def by auto
   902     finally show ?case .
   903   qed
   904 
   905   show "range cube \<subseteq> sets \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
   906     unfolding cube_def_raw by auto
   907   have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp
   908   thus "cube \<up> space \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
   909     apply-apply(rule isotoneI) apply(rule cube_subset_Suc) by auto
   910   show "A \<in> sets (sigma ?E)" using assms by simp
   911   have "measure_space lborel" by default
   912   then show "measure_space \<lparr> space = space ?E, sets = sets (sigma ?E), measure = measure lebesgue\<rparr>"
   913     unfolding lebesgue_def lborel_def by simp
   914   let ?M = "\<lparr> space = space ?E, sets = sets (sigma ?E), measure = ?m \<rparr>"
   915   show "measure_space ?M"
   916   proof (rule lborel_space.measure_space_vimage)
   917     show "sigma_algebra ?M" by (rule lborel.sigma_algebra_cong) auto
   918     show "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel) ?M"
   919       using measurable_p2e unfolding measurable_def by auto
   920     fix A :: "'a set" assume "A \<in> sets ?M"
   921     show "measure ?M A = lborel_space.\<mu> TYPE('a) (p2e -` A \<inter> space (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel))"
   922       by (simp add: e2p_image_vimage)
   923   qed
   924 qed simp
   925 
   926 lemma range_e2p:"range (e2p::'a::ordered_euclidean_space \<Rightarrow> _) = extensional {..<DIM('a)}"
   927   unfolding e2p_def_raw
   928   apply auto
   929   by (rule_tac x="\<chi>\<chi> i. x i" in image_eqI) (auto simp: fun_eq_iff extensional_def)
   930 
   931 lemma borel_fubini_positiv_integral:
   932   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal"
   933   assumes f: "f \<in> borel_measurable borel"
   934   shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(lborel_space.P TYPE('a))"
   935 proof (rule lborel.positive_integral_vimage[symmetric, of _ "e2p :: 'a \<Rightarrow> _" "(\<lambda>x. f (p2e x))", unfolded p2e_e2p])
   936   show "(e2p :: 'a \<Rightarrow> _) \<in> measurable borel (lborel_space.P TYPE('a))" by (rule measurable_e2p)
   937   show "sigma_algebra (lborel_space.P TYPE('a))" by default
   938   from measurable_comp[OF measurable_p2e f]
   939   show "(\<lambda>x. f (p2e x)) \<in> borel_measurable (lborel_space.P TYPE('a))" by (simp add: comp_def)
   940   let "?L A" = "lebesgue.\<mu> ((e2p::'a \<Rightarrow> (nat \<Rightarrow> real)) -` A \<inter> UNIV)"
   941   fix A :: "(nat \<Rightarrow> real) set" assume "A \<in> sets (lborel_space.P TYPE('a))"
   942   then have A: "(e2p::'a \<Rightarrow> (nat \<Rightarrow> real)) -` A \<inter> space borel \<in> sets borel"
   943     by (rule measurable_sets[OF measurable_e2p])
   944   have [simp]: "A \<inter> extensional {..<DIM('a)} = A"
   945     using `A \<in> sets (lborel_space.P TYPE('a))`[THEN lborel_space.sets_into_space] by auto
   946   show "lborel_space.\<mu> TYPE('a) A = ?L A"
   947     using lmeasure_measure_eq_borel_prod[OF A] by (simp add: range_e2p)
   948 qed
   949 
   950 lemma borel_fubini:
   951   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
   952   assumes f: "f \<in> borel_measurable borel"
   953   shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>(lborel_space.P TYPE('a))"
   954 proof -
   955   have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto
   956   have 2:"(\<lambda>x. Real (- f x)) \<in> borel_measurable borel" using f by auto
   957   show ?thesis unfolding lebesgue_integral_def
   958     unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2]
   959     unfolding o_def ..
   960 qed
   961 
   962 end