src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63886 685fb01256af child 63940 0d82c4c94014 permissions -rw-r--r--
tuned proofs;
```     1 theory Equivalence_Lebesgue_Henstock_Integration
```
```     2   imports Lebesgue_Measure Henstock_Kurzweil_Integration
```
```     3 begin
```
```     4
```
```     5 subsection \<open>Equivalence Lebesgue integral on @{const lborel} and HK-integral\<close>
```
```     6
```
```     7 lemma has_integral_measure_lborel:
```
```     8   fixes A :: "'a::euclidean_space set"
```
```     9   assumes A[measurable]: "A \<in> sets borel" and finite: "emeasure lborel A < \<infinity>"
```
```    10   shows "((\<lambda>x. 1) has_integral measure lborel A) A"
```
```    11 proof -
```
```    12   { fix l u :: 'a
```
```    13     have "((\<lambda>x. 1) has_integral measure lborel (box l u)) (box l u)"
```
```    14     proof cases
```
```    15       assume "\<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b"
```
```    16       then show ?thesis
```
```    17         apply simp
```
```    18         apply (subst has_integral_restrict[symmetric, OF box_subset_cbox])
```
```    19         apply (subst has_integral_spike_interior_eq[where g="\<lambda>_. 1"])
```
```    20         using has_integral_const[of "1::real" l u]
```
```    21         apply (simp_all add: inner_diff_left[symmetric] content_cbox_cases)
```
```    22         done
```
```    23     next
```
```    24       assume "\<not> (\<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b)"
```
```    25       then have "box l u = {}"
```
```    26         unfolding box_eq_empty by (auto simp: not_le intro: less_imp_le)
```
```    27       then show ?thesis
```
```    28         by simp
```
```    29     qed }
```
```    30   note has_integral_box = this
```
```    31
```
```    32   { fix a b :: 'a let ?M = "\<lambda>A. measure lborel (A \<inter> box a b)"
```
```    33     have "Int_stable  (range (\<lambda>(a, b). box a b))"
```
```    34       by (auto simp: Int_stable_def box_Int_box)
```
```    35     moreover have "(range (\<lambda>(a, b). box a b)) \<subseteq> Pow UNIV"
```
```    36       by auto
```
```    37     moreover have "A \<in> sigma_sets UNIV (range (\<lambda>(a, b). box a b))"
```
```    38        using A unfolding borel_eq_box by simp
```
```    39     ultimately have "((\<lambda>x. 1) has_integral ?M A) (A \<inter> box a b)"
```
```    40     proof (induction rule: sigma_sets_induct_disjoint)
```
```    41       case (basic A) then show ?case
```
```    42         by (auto simp: box_Int_box has_integral_box)
```
```    43     next
```
```    44       case empty then show ?case
```
```    45         by simp
```
```    46     next
```
```    47       case (compl A)
```
```    48       then have [measurable]: "A \<in> sets borel"
```
```    49         by (simp add: borel_eq_box)
```
```    50
```
```    51       have "((\<lambda>x. 1) has_integral ?M (box a b)) (box a b)"
```
```    52         by (simp add: has_integral_box)
```
```    53       moreover have "((\<lambda>x. if x \<in> A \<inter> box a b then 1 else 0) has_integral ?M A) (box a b)"
```
```    54         by (subst has_integral_restrict) (auto intro: compl)
```
```    55       ultimately have "((\<lambda>x. 1 - (if x \<in> A \<inter> box a b then 1 else 0)) has_integral ?M (box a b) - ?M A) (box a b)"
```
```    56         by (rule has_integral_sub)
```
```    57       then have "((\<lambda>x. (if x \<in> (UNIV - A) \<inter> box a b then 1 else 0)) has_integral ?M (box a b) - ?M A) (box a b)"
```
```    58         by (rule has_integral_cong[THEN iffD1, rotated 1]) auto
```
```    59       then have "((\<lambda>x. 1) has_integral ?M (box a b) - ?M A) ((UNIV - A) \<inter> box a b)"
```
```    60         by (subst (asm) has_integral_restrict) auto
```
```    61       also have "?M (box a b) - ?M A = ?M (UNIV - A)"
```
```    62         by (subst measure_Diff[symmetric]) (auto simp: emeasure_lborel_box_eq Diff_Int_distrib2)
```
```    63       finally show ?case .
```
```    64     next
```
```    65       case (union F)
```
```    66       then have [measurable]: "\<And>i. F i \<in> sets borel"
```
```    67         by (simp add: borel_eq_box subset_eq)
```
```    68       have "((\<lambda>x. if x \<in> UNION UNIV F \<inter> box a b then 1 else 0) has_integral ?M (\<Union>i. F i)) (box a b)"
```
```    69       proof (rule has_integral_monotone_convergence_increasing)
```
```    70         let ?f = "\<lambda>k x. \<Sum>i<k. if x \<in> F i \<inter> box a b then 1 else 0 :: real"
```
```    71         show "\<And>k. (?f k has_integral (\<Sum>i<k. ?M (F i))) (box a b)"
```
```    72           using union.IH by (auto intro!: has_integral_setsum simp del: Int_iff)
```
```    73         show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
```
```    74           by (intro setsum_mono2) auto
```
```    75         from union(1) have *: "\<And>x i j. x \<in> F i \<Longrightarrow> x \<in> F j \<longleftrightarrow> j = i"
```
```    76           by (auto simp add: disjoint_family_on_def)
```
```    77         show "\<And>x. (\<lambda>k. ?f k x) \<longlonglongrightarrow> (if x \<in> UNION UNIV F \<inter> box a b then 1 else 0)"
```
```    78           apply (auto simp: * setsum.If_cases Iio_Int_singleton)
```
```    79           apply (rule_tac k="Suc xa" in LIMSEQ_offset)
```
```    80           apply simp
```
```    81           done
```
```    82         have *: "emeasure lborel ((\<Union>x. F x) \<inter> box a b) \<le> emeasure lborel (box a b)"
```
```    83           by (intro emeasure_mono) auto
```
```    84
```
```    85         with union(1) show "(\<lambda>k. \<Sum>i<k. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"
```
```    86           unfolding sums_def[symmetric] UN_extend_simps
```
```    87           by (intro measure_UNION) (auto simp: disjoint_family_on_def emeasure_lborel_box_eq top_unique)
```
```    88       qed
```
```    89       then show ?case
```
```    90         by (subst (asm) has_integral_restrict) auto
```
```    91     qed }
```
```    92   note * = this
```
```    93
```
```    94   show ?thesis
```
```    95   proof (rule has_integral_monotone_convergence_increasing)
```
```    96     let ?B = "\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One) :: 'a set"
```
```    97     let ?f = "\<lambda>n::nat. \<lambda>x. if x \<in> A \<inter> ?B n then 1 else 0 :: real"
```
```    98     let ?M = "\<lambda>n. measure lborel (A \<inter> ?B n)"
```
```    99
```
```   100     show "\<And>n::nat. (?f n has_integral ?M n) A"
```
```   101       using * by (subst has_integral_restrict) simp_all
```
```   102     show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
```
```   103       by (auto simp: box_def)
```
```   104     { fix x assume "x \<in> A"
```
```   105       moreover have "(\<lambda>k. indicator (A \<inter> ?B k) x :: real) \<longlonglongrightarrow> indicator (\<Union>k::nat. A \<inter> ?B k) x"
```
```   106         by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def box_def)
```
```   107       ultimately show "(\<lambda>k. if x \<in> A \<inter> ?B k then 1 else 0::real) \<longlonglongrightarrow> 1"
```
```   108         by (simp add: indicator_def UN_box_eq_UNIV) }
```
```   109
```
```   110     have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> emeasure lborel (\<Union>n::nat. A \<inter> ?B n)"
```
```   111       by (intro Lim_emeasure_incseq) (auto simp: incseq_def box_def)
```
```   112     also have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) = (\<lambda>n. measure lborel (A \<inter> ?B n))"
```
```   113     proof (intro ext emeasure_eq_ennreal_measure)
```
```   114       fix n have "emeasure lborel (A \<inter> ?B n) \<le> emeasure lborel (?B n)"
```
```   115         by (intro emeasure_mono) auto
```
```   116       then show "emeasure lborel (A \<inter> ?B n) \<noteq> top"
```
```   117         by (auto simp: top_unique)
```
```   118     qed
```
```   119     finally show "(\<lambda>n. measure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> measure lborel A"
```
```   120       using emeasure_eq_ennreal_measure[of lborel A] finite
```
```   121       by (simp add: UN_box_eq_UNIV less_top)
```
```   122   qed
```
```   123 qed
```
```   124
```
```   125 lemma nn_integral_has_integral:
```
```   126   fixes f::"'a::euclidean_space \<Rightarrow> real"
```
```   127   assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) = ennreal r" "0 \<le> r"
```
```   128   shows "(f has_integral r) UNIV"
```
```   129 using f proof (induct f arbitrary: r rule: borel_measurable_induct_real)
```
```   130   case (set A)
```
```   131   then have "((\<lambda>x. 1) has_integral measure lborel A) A"
```
```   132     by (intro has_integral_measure_lborel) (auto simp: ennreal_indicator)
```
```   133   with set show ?case
```
```   134     by (simp add: ennreal_indicator measure_def) (simp add: indicator_def)
```
```   135 next
```
```   136   case (mult g c)
```
```   137   then have "ennreal c * (\<integral>\<^sup>+ x. g x \<partial>lborel) = ennreal r"
```
```   138     by (subst nn_integral_cmult[symmetric]) (auto simp: ennreal_mult)
```
```   139   with \<open>0 \<le> r\<close> \<open>0 \<le> c\<close>
```
```   140   obtain r' where "(c = 0 \<and> r = 0) \<or> (0 \<le> r' \<and> (\<integral>\<^sup>+ x. ennreal (g x) \<partial>lborel) = ennreal r' \<and> r = c * r')"
```
```   141     by (cases "\<integral>\<^sup>+ x. ennreal (g x) \<partial>lborel" rule: ennreal_cases)
```
```   142        (auto split: if_split_asm simp: ennreal_mult_top ennreal_mult[symmetric])
```
```   143   with mult show ?case
```
```   144     by (auto intro!: has_integral_cmult_real)
```
```   145 next
```
```   146   case (add g h)
```
```   147   then have "(\<integral>\<^sup>+ x. h x + g x \<partial>lborel) = (\<integral>\<^sup>+ x. h x \<partial>lborel) + (\<integral>\<^sup>+ x. g x \<partial>lborel)"
```
```   148     by (simp add: nn_integral_add)
```
```   149   with add obtain a b where "0 \<le> a" "0 \<le> b" "(\<integral>\<^sup>+ x. h x \<partial>lborel) = ennreal a" "(\<integral>\<^sup>+ x. g x \<partial>lborel) = ennreal b" "r = a + b"
```
```   150     by (cases "\<integral>\<^sup>+ x. h x \<partial>lborel" "\<integral>\<^sup>+ x. g x \<partial>lborel" rule: ennreal2_cases)
```
```   151        (auto simp: add_top nn_integral_add top_add ennreal_plus[symmetric] simp del: ennreal_plus)
```
```   152   with add show ?case
```
```   153     by (auto intro!: has_integral_add)
```
```   154 next
```
```   155   case (seq U)
```
```   156   note seq(1)[measurable] and f[measurable]
```
```   157
```
```   158   { fix i x
```
```   159     have "U i x \<le> f x"
```
```   160       using seq(5)
```
```   161       apply (rule LIMSEQ_le_const)
```
```   162       using seq(4)
```
```   163       apply (auto intro!: exI[of _ i] simp: incseq_def le_fun_def)
```
```   164       done }
```
```   165   note U_le_f = this
```
```   166
```
```   167   { fix i
```
```   168     have "(\<integral>\<^sup>+x. U i x \<partial>lborel) \<le> (\<integral>\<^sup>+x. f x \<partial>lborel)"
```
```   169       using seq(2) f(2) U_le_f by (intro nn_integral_mono) simp
```
```   170     then obtain p where "(\<integral>\<^sup>+x. U i x \<partial>lborel) = ennreal p" "p \<le> r" "0 \<le> p"
```
```   171       using seq(6) \<open>0\<le>r\<close> by (cases "\<integral>\<^sup>+x. U i x \<partial>lborel" rule: ennreal_cases) (auto simp: top_unique)
```
```   172     moreover note seq
```
```   173     ultimately have "\<exists>p. (\<integral>\<^sup>+x. U i x \<partial>lborel) = ennreal p \<and> 0 \<le> p \<and> p \<le> r \<and> (U i has_integral p) UNIV"
```
```   174       by auto }
```
```   175   then obtain p where p: "\<And>i. (\<integral>\<^sup>+x. ennreal (U i x) \<partial>lborel) = ennreal (p i)"
```
```   176     and bnd: "\<And>i. p i \<le> r" "\<And>i. 0 \<le> p i"
```
```   177     and U_int: "\<And>i.(U i has_integral (p i)) UNIV" by metis
```
```   178
```
```   179   have int_eq: "\<And>i. integral UNIV (U i) = p i" using U_int by (rule integral_unique)
```
```   180
```
```   181   have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) \<longlonglongrightarrow> integral UNIV f"
```
```   182   proof (rule monotone_convergence_increasing)
```
```   183     show "\<forall>k. U k integrable_on UNIV" using U_int by auto
```
```   184     show "\<forall>k. \<forall>x\<in>UNIV. U k x \<le> U (Suc k) x" using \<open>incseq U\<close> by (auto simp: incseq_def le_fun_def)
```
```   185     then show "bounded {integral UNIV (U k) |k. True}"
```
```   186       using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
```
```   187     show "\<forall>x\<in>UNIV. (\<lambda>k. U k x) \<longlonglongrightarrow> f x"
```
```   188       using seq by auto
```
```   189   qed
```
```   190   moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lborel)) \<longlonglongrightarrow> (\<integral>\<^sup>+x. f x \<partial>lborel)"
```
```   191     using seq f(2) U_le_f by (intro nn_integral_dominated_convergence[where w=f]) auto
```
```   192   ultimately have "integral UNIV f = r"
```
```   193     by (auto simp add: bnd int_eq p seq intro: LIMSEQ_unique)
```
```   194   with * show ?case
```
```   195     by (simp add: has_integral_integral)
```
```   196 qed
```
```   197
```
```   198 lemma nn_integral_lborel_eq_integral:
```
```   199   fixes f::"'a::euclidean_space \<Rightarrow> real"
```
```   200   assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) < \<infinity>"
```
```   201   shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = integral UNIV f"
```
```   202 proof -
```
```   203   from f(3) obtain r where r: "(\<integral>\<^sup>+x. f x \<partial>lborel) = ennreal r" "0 \<le> r"
```
```   204     by (cases "\<integral>\<^sup>+x. f x \<partial>lborel" rule: ennreal_cases) auto
```
```   205   then show ?thesis
```
```   206     using nn_integral_has_integral[OF f(1,2) r] by (simp add: integral_unique)
```
```   207 qed
```
```   208
```
```   209 lemma nn_integral_integrable_on:
```
```   210   fixes f::"'a::euclidean_space \<Rightarrow> real"
```
```   211   assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) < \<infinity>"
```
```   212   shows "f integrable_on UNIV"
```
```   213 proof -
```
```   214   from f(3) obtain r where r: "(\<integral>\<^sup>+x. f x \<partial>lborel) = ennreal r" "0 \<le> r"
```
```   215     by (cases "\<integral>\<^sup>+x. f x \<partial>lborel" rule: ennreal_cases) auto
```
```   216   then show ?thesis
```
```   217     by (intro has_integral_integrable[where i=r] nn_integral_has_integral[where r=r] f)
```
```   218 qed
```
```   219
```
```   220 lemma nn_integral_has_integral_lborel:
```
```   221   fixes f :: "'a::euclidean_space \<Rightarrow> real"
```
```   222   assumes f_borel: "f \<in> borel_measurable borel" and nonneg: "\<And>x. 0 \<le> f x"
```
```   223   assumes I: "(f has_integral I) UNIV"
```
```   224   shows "integral\<^sup>N lborel f = I"
```
```   225 proof -
```
```   226   from f_borel have "(\<lambda>x. ennreal (f x)) \<in> borel_measurable lborel" by auto
```
```   227   from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this
```
```   228   let ?B = "\<lambda>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One) :: 'a set"
```
```   229
```
```   230   note F(1)[THEN borel_measurable_simple_function, measurable]
```
```   231
```
```   232   have "0 \<le> I"
```
```   233     using I by (rule has_integral_nonneg) (simp add: nonneg)
```
```   234
```
```   235   have F_le_f: "enn2real (F i x) \<le> f x" for i x
```
```   236     using F(3,4)[where x=x] nonneg SUP_upper[of i UNIV "\<lambda>i. F i x"]
```
```   237     by (cases "F i x" rule: ennreal_cases) auto
```
```   238   let ?F = "\<lambda>i x. F i x * indicator (?B i) x"
```
```   239   have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) = (SUP i. integral\<^sup>N lborel (\<lambda>x. ?F i x))"
```
```   240   proof (subst nn_integral_monotone_convergence_SUP[symmetric])
```
```   241     { fix x
```
```   242       obtain j where j: "x \<in> ?B j"
```
```   243         using UN_box_eq_UNIV by auto
```
```   244
```
```   245       have "ennreal (f x) = (SUP i. F i x)"
```
```   246         using F(4)[of x] nonneg[of x] by (simp add: max_def)
```
```   247       also have "\<dots> = (SUP i. ?F i x)"
```
```   248       proof (rule SUP_eq)
```
```   249         fix i show "\<exists>j\<in>UNIV. F i x \<le> ?F j x"
```
```   250           using j F(2)
```
```   251           by (intro bexI[of _ "max i j"])
```
```   252              (auto split: split_max split_indicator simp: incseq_def le_fun_def box_def)
```
```   253       qed (auto intro!: F split: split_indicator)
```
```   254       finally have "ennreal (f x) =  (SUP i. ?F i x)" . }
```
```   255     then show "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) = (\<integral>\<^sup>+ x. (SUP i. ?F i x) \<partial>lborel)"
```
```   256       by simp
```
```   257   qed (insert F, auto simp: incseq_def le_fun_def box_def split: split_indicator)
```
```   258   also have "\<dots> \<le> ennreal I"
```
```   259   proof (rule SUP_least)
```
```   260     fix i :: nat
```
```   261     have finite_F: "(\<integral>\<^sup>+ x. ennreal (enn2real (F i x) * indicator (?B i) x) \<partial>lborel) < \<infinity>"
```
```   262     proof (rule nn_integral_bound_simple_function)
```
```   263       have "emeasure lborel {x \<in> space lborel. ennreal (enn2real (F i x) * indicator (?B i) x) \<noteq> 0} \<le>
```
```   264         emeasure lborel (?B i)"
```
```   265         by (intro emeasure_mono)  (auto split: split_indicator)
```
```   266       then show "emeasure lborel {x \<in> space lborel. ennreal (enn2real (F i x) * indicator (?B i) x) \<noteq> 0} < \<infinity>"
```
```   267         by (auto simp: less_top[symmetric] top_unique)
```
```   268     qed (auto split: split_indicator
```
```   269               intro!: F simple_function_compose1[where g="enn2real"] simple_function_ennreal)
```
```   270
```
```   271     have int_F: "(\<lambda>x. enn2real (F i x) * indicator (?B i) x) integrable_on UNIV"
```
```   272       using F(4) finite_F
```
```   273       by (intro nn_integral_integrable_on) (auto split: split_indicator simp: enn2real_nonneg)
```
```   274
```
```   275     have "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) =
```
```   276       (\<integral>\<^sup>+ x. ennreal (enn2real (F i x) * indicator (?B i) x) \<partial>lborel)"
```
```   277       using F(3,4)
```
```   278       by (intro nn_integral_cong) (auto simp: image_iff eq_commute split: split_indicator)
```
```   279     also have "\<dots> = ennreal (integral UNIV (\<lambda>x. enn2real (F i x) * indicator (?B i) x))"
```
```   280       using F
```
```   281       by (intro nn_integral_lborel_eq_integral[OF _ _ finite_F])
```
```   282          (auto split: split_indicator intro: enn2real_nonneg)
```
```   283     also have "\<dots> \<le> ennreal I"
```
```   284       by (auto intro!: has_integral_le[OF integrable_integral[OF int_F] I] nonneg F_le_f
```
```   285                simp: \<open>0 \<le> I\<close> split: split_indicator )
```
```   286     finally show "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) \<le> ennreal I" .
```
```   287   qed
```
```   288   finally have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) < \<infinity>"
```
```   289     by (auto simp: less_top[symmetric] top_unique)
```
```   290   from nn_integral_lborel_eq_integral[OF assms(1,2) this] I show ?thesis
```
```   291     by (simp add: integral_unique)
```
```   292 qed
```
```   293
```
```   294 lemma has_integral_iff_emeasure_lborel:
```
```   295   fixes A :: "'a::euclidean_space set"
```
```   296   assumes A[measurable]: "A \<in> sets borel" and [simp]: "0 \<le> r"
```
```   297   shows "((\<lambda>x. 1) has_integral r) A \<longleftrightarrow> emeasure lborel A = ennreal r"
```
```   298 proof (cases "emeasure lborel A = \<infinity>")
```
```   299   case emeasure_A: True
```
```   300   have "\<not> (\<lambda>x. 1::real) integrable_on A"
```
```   301   proof
```
```   302     assume int: "(\<lambda>x. 1::real) integrable_on A"
```
```   303     then have "(indicator A::'a \<Rightarrow> real) integrable_on UNIV"
```
```   304       unfolding indicator_def[abs_def] integrable_restrict_univ .
```
```   305     then obtain r where "((indicator A::'a\<Rightarrow>real) has_integral r) UNIV"
```
```   306       by auto
```
```   307     from nn_integral_has_integral_lborel[OF _ _ this] emeasure_A show False
```
```   308       by (simp add: ennreal_indicator)
```
```   309   qed
```
```   310   with emeasure_A show ?thesis
```
```   311     by auto
```
```   312 next
```
```   313   case False
```
```   314   then have "((\<lambda>x. 1) has_integral measure lborel A) A"
```
```   315     by (simp add: has_integral_measure_lborel less_top)
```
```   316   with False show ?thesis
```
```   317     by (auto simp: emeasure_eq_ennreal_measure has_integral_unique)
```
```   318 qed
```
```   319
```
```   320 lemma has_integral_integral_real:
```
```   321   fixes f::"'a::euclidean_space \<Rightarrow> real"
```
```   322   assumes f: "integrable lborel f"
```
```   323   shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
```
```   324 using f proof induct
```
```   325   case (base A c) then show ?case
```
```   326     by (auto intro!: has_integral_mult_left simp: )
```
```   327        (simp add: emeasure_eq_ennreal_measure indicator_def has_integral_measure_lborel)
```
```   328 next
```
```   329   case (add f g) then show ?case
```
```   330     by (auto intro!: has_integral_add)
```
```   331 next
```
```   332   case (lim f s)
```
```   333   show ?case
```
```   334   proof (rule has_integral_dominated_convergence)
```
```   335     show "\<And>i. (s i has_integral integral\<^sup>L lborel (s i)) UNIV" by fact
```
```   336     show "(\<lambda>x. norm (2 * f x)) integrable_on UNIV"
```
```   337       using \<open>integrable lborel f\<close>
```
```   338       by (intro nn_integral_integrable_on)
```
```   339          (auto simp: integrable_iff_bounded abs_mult  nn_integral_cmult ennreal_mult ennreal_mult_less_top)
```
```   340     show "\<And>k. \<forall>x\<in>UNIV. norm (s k x) \<le> norm (2 * f x)"
```
```   341       using lim by (auto simp add: abs_mult)
```
```   342     show "\<forall>x\<in>UNIV. (\<lambda>k. s k x) \<longlonglongrightarrow> f x"
```
```   343       using lim by auto
```
```   344     show "(\<lambda>k. integral\<^sup>L lborel (s k)) \<longlonglongrightarrow> integral\<^sup>L lborel f"
```
```   345       using lim lim(1)[THEN borel_measurable_integrable]
```
```   346       by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) auto
```
```   347   qed
```
```   348 qed
```
```   349
```
```   350 context
```
```   351   fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```   352 begin
```
```   353
```
```   354 lemma has_integral_integral_lborel:
```
```   355   assumes f: "integrable lborel f"
```
```   356   shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
```
```   357 proof -
```
```   358   have "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b)) UNIV"
```
```   359     using f by (intro has_integral_setsum finite_Basis ballI has_integral_scaleR_left has_integral_integral_real) auto
```
```   360   also have eq_f: "(\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) = f"
```
```   361     by (simp add: fun_eq_iff euclidean_representation)
```
```   362   also have "(\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b) = integral\<^sup>L lborel f"
```
```   363     using f by (subst (2) eq_f[symmetric]) simp
```
```   364   finally show ?thesis .
```
```   365 qed
```
```   366
```
```   367 lemma integrable_on_lborel: "integrable lborel f \<Longrightarrow> f integrable_on UNIV"
```
```   368   using has_integral_integral_lborel by auto
```
```   369
```
```   370 lemma integral_lborel: "integrable lborel f \<Longrightarrow> integral UNIV f = (\<integral>x. f x \<partial>lborel)"
```
```   371   using has_integral_integral_lborel by auto
```
```   372
```
```   373 end
```
```   374
```
```   375 subsection \<open>Fundamental Theorem of Calculus for the Lebesgue integral\<close>
```
```   376
```
```   377 text \<open>
```
```   378
```
```   379 For the positive integral we replace continuity with Borel-measurability.
```
```   380
```
```   381 \<close>
```
```   382
```
```   383 lemma
```
```   384   fixes f :: "real \<Rightarrow> real"
```
```   385   assumes [measurable]: "f \<in> borel_measurable borel"
```
```   386   assumes f: "\<And>x. x \<in> {a..b} \<Longrightarrow> DERIV F x :> f x" "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x" and "a \<le> b"
```
```   387   shows nn_integral_FTC_Icc: "(\<integral>\<^sup>+x. ennreal (f x) * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?nn)
```
```   388     and has_bochner_integral_FTC_Icc_nonneg:
```
```   389       "has_bochner_integral lborel (\<lambda>x. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
```
```   390     and integral_FTC_Icc_nonneg: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
```
```   391     and integrable_FTC_Icc_nonneg: "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is ?int)
```
```   392 proof -
```
```   393   have *: "(\<lambda>x. f x * indicator {a..b} x) \<in> borel_measurable borel" "\<And>x. 0 \<le> f x * indicator {a..b} x"
```
```   394     using f(2) by (auto split: split_indicator)
```
```   395
```
```   396   have F_mono: "a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b\<Longrightarrow> F x \<le> F y" for x y
```
```   397     using f by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
```
```   398
```
```   399   have "(f has_integral F b - F a) {a..b}"
```
```   400     by (intro fundamental_theorem_of_calculus)
```
```   401        (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric]
```
```   402              intro: has_field_derivative_subset[OF f(1)] \<open>a \<le> b\<close>)
```
```   403   then have i: "((\<lambda>x. f x * indicator {a .. b} x) has_integral F b - F a) UNIV"
```
```   404     unfolding indicator_def if_distrib[where f="\<lambda>x. a * x" for a]
```
```   405     by (simp cong del: if_weak_cong del: atLeastAtMost_iff)
```
```   406   then have nn: "(\<integral>\<^sup>+x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a"
```
```   407     by (rule nn_integral_has_integral_lborel[OF *])
```
```   408   then show ?has
```
```   409     by (rule has_bochner_integral_nn_integral[rotated 3]) (simp_all add: * F_mono \<open>a \<le> b\<close>)
```
```   410   then show ?eq ?int
```
```   411     unfolding has_bochner_integral_iff by auto
```
```   412   show ?nn
```
```   413     by (subst nn[symmetric])
```
```   414        (auto intro!: nn_integral_cong simp add: ennreal_mult f split: split_indicator)
```
```   415 qed
```
```   416
```
```   417 lemma
```
```   418   fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
```
```   419   assumes "a \<le> b"
```
```   420   assumes "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
```
```   421   assumes cont: "continuous_on {a .. b} f"
```
```   422   shows has_bochner_integral_FTC_Icc:
```
```   423       "has_bochner_integral lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x) (F b - F a)" (is ?has)
```
```   424     and integral_FTC_Icc: "(\<integral>x. indicator {a .. b} x *\<^sub>R f x \<partial>lborel) = F b - F a" (is ?eq)
```
```   425 proof -
```
```   426   let ?f = "\<lambda>x. indicator {a .. b} x *\<^sub>R f x"
```
```   427   have int: "integrable lborel ?f"
```
```   428     using borel_integrable_compact[OF _ cont] by auto
```
```   429   have "(f has_integral F b - F a) {a..b}"
```
```   430     using assms(1,2) by (intro fundamental_theorem_of_calculus) auto
```
```   431   moreover
```
```   432   have "(f has_integral integral\<^sup>L lborel ?f) {a..b}"
```
```   433     using has_integral_integral_lborel[OF int]
```
```   434     unfolding indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a]
```
```   435     by (simp cong del: if_weak_cong del: atLeastAtMost_iff)
```
```   436   ultimately show ?eq
```
```   437     by (auto dest: has_integral_unique)
```
```   438   then show ?has
```
```   439     using int by (auto simp: has_bochner_integral_iff)
```
```   440 qed
```
```   441
```
```   442 lemma
```
```   443   fixes f :: "real \<Rightarrow> real"
```
```   444   assumes "a \<le> b"
```
```   445   assumes deriv: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> DERIV F x :> f x"
```
```   446   assumes cont: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
```
```   447   shows has_bochner_integral_FTC_Icc_real:
```
```   448       "has_bochner_integral lborel (\<lambda>x. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
```
```   449     and integral_FTC_Icc_real: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
```
```   450 proof -
```
```   451   have 1: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
```
```   452     unfolding has_field_derivative_iff_has_vector_derivative[symmetric]
```
```   453     using deriv by (auto intro: DERIV_subset)
```
```   454   have 2: "continuous_on {a .. b} f"
```
```   455     using cont by (intro continuous_at_imp_continuous_on) auto
```
```   456   show ?has ?eq
```
```   457     using has_bochner_integral_FTC_Icc[OF \<open>a \<le> b\<close> 1 2] integral_FTC_Icc[OF \<open>a \<le> b\<close> 1 2]
```
```   458     by (auto simp: mult.commute)
```
```   459 qed
```
```   460
```
```   461 lemma nn_integral_FTC_atLeast:
```
```   462   fixes f :: "real \<Rightarrow> real"
```
```   463   assumes f_borel: "f \<in> borel_measurable borel"
```
```   464   assumes f: "\<And>x. a \<le> x \<Longrightarrow> DERIV F x :> f x"
```
```   465   assumes nonneg: "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> f x"
```
```   466   assumes lim: "(F \<longlongrightarrow> T) at_top"
```
```   467   shows "(\<integral>\<^sup>+x. ennreal (f x) * indicator {a ..} x \<partial>lborel) = T - F a"
```
```   468 proof -
```
```   469   let ?f = "\<lambda>(i::nat) (x::real). ennreal (f x) * indicator {a..a + real i} x"
```
```   470   let ?fR = "\<lambda>x. ennreal (f x) * indicator {a ..} x"
```
```   471
```
```   472   have F_mono: "a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> F x \<le> F y" for x y
```
```   473     using f nonneg by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
```
```   474   then have F_le_T: "a \<le> x \<Longrightarrow> F x \<le> T" for x
```
```   475     by (intro tendsto_le_const[OF _ lim])
```
```   476        (auto simp: trivial_limit_at_top_linorder eventually_at_top_linorder)
```
```   477
```
```   478   have "(SUP i::nat. ?f i x) = ?fR x" for x
```
```   479   proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
```
```   480     from reals_Archimedean2[of "x - a"] guess n ..
```
```   481     then have "eventually (\<lambda>n. ?f n x = ?fR x) sequentially"
```
```   482       by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator)
```
```   483     then show "(\<lambda>n. ?f n x) \<longlonglongrightarrow> ?fR x"
```
```   484       by (rule Lim_eventually)
```
```   485   qed (auto simp: nonneg incseq_def le_fun_def split: split_indicator)
```
```   486   then have "integral\<^sup>N lborel ?fR = (\<integral>\<^sup>+ x. (SUP i::nat. ?f i x) \<partial>lborel)"
```
```   487     by simp
```
```   488   also have "\<dots> = (SUP i::nat. (\<integral>\<^sup>+ x. ?f i x \<partial>lborel))"
```
```   489   proof (rule nn_integral_monotone_convergence_SUP)
```
```   490     show "incseq ?f"
```
```   491       using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator)
```
```   492     show "\<And>i. (?f i) \<in> borel_measurable lborel"
```
```   493       using f_borel by auto
```
```   494   qed
```
```   495   also have "\<dots> = (SUP i::nat. ennreal (F (a + real i) - F a))"
```
```   496     by (subst nn_integral_FTC_Icc[OF f_borel f nonneg]) auto
```
```   497   also have "\<dots> = T - F a"
```
```   498   proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
```
```   499     have "(\<lambda>x. F (a + real x)) \<longlonglongrightarrow> T"
```
```   500       apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top])
```
```   501       apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl])
```
```   502       apply (rule filterlim_real_sequentially)
```
```   503       done
```
```   504     then show "(\<lambda>n. ennreal (F (a + real n) - F a)) \<longlonglongrightarrow> ennreal (T - F a)"
```
```   505       by (simp add: F_mono F_le_T tendsto_diff)
```
```   506   qed (auto simp: incseq_def intro!: ennreal_le_iff[THEN iffD2] F_mono)
```
```   507   finally show ?thesis .
```
```   508 qed
```
```   509
```
```   510 lemma integral_power:
```
```   511   "a \<le> b \<Longrightarrow> (\<integral>x. x^k * indicator {a..b} x \<partial>lborel) = (b^Suc k - a^Suc k) / Suc k"
```
```   512 proof (subst integral_FTC_Icc_real)
```
```   513   fix x show "DERIV (\<lambda>x. x^Suc k / Suc k) x :> x^k"
```
```   514     by (intro derivative_eq_intros) auto
```
```   515 qed (auto simp: field_simps simp del: of_nat_Suc)
```
```   516
```
```   517 subsection \<open>Integration by parts\<close>
```
```   518
```
```   519 lemma integral_by_parts_integrable:
```
```   520   fixes f g F G::"real \<Rightarrow> real"
```
```   521   assumes "a \<le> b"
```
```   522   assumes cont_f[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
```
```   523   assumes cont_g[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
```
```   524   assumes [intro]: "!!x. DERIV F x :> f x"
```
```   525   assumes [intro]: "!!x. DERIV G x :> g x"
```
```   526   shows  "integrable lborel (\<lambda>x.((F x) * (g x) + (f x) * (G x)) * indicator {a .. b} x)"
```
```   527   by (auto intro!: borel_integrable_atLeastAtMost continuous_intros) (auto intro!: DERIV_isCont)
```
```   528
```
```   529 lemma integral_by_parts:
```
```   530   fixes f g F G::"real \<Rightarrow> real"
```
```   531   assumes [arith]: "a \<le> b"
```
```   532   assumes cont_f[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
```
```   533   assumes cont_g[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
```
```   534   assumes [intro]: "!!x. DERIV F x :> f x"
```
```   535   assumes [intro]: "!!x. DERIV G x :> g x"
```
```   536   shows "(\<integral>x. (F x * g x) * indicator {a .. b} x \<partial>lborel)
```
```   537             =  F b * G b - F a * G a - \<integral>x. (f x * G x) * indicator {a .. b} x \<partial>lborel"
```
```   538 proof-
```
```   539   have 0: "(\<integral>x. (F x * g x + f x * G x) * indicator {a .. b} x \<partial>lborel) = F b * G b - F a * G a"
```
```   540     by (rule integral_FTC_Icc_real, auto intro!: derivative_eq_intros continuous_intros)
```
```   541       (auto intro!: DERIV_isCont)
```
```   542
```
```   543   have "(\<integral>x. (F x * g x + f x * G x) * indicator {a .. b} x \<partial>lborel) =
```
```   544     (\<integral>x. (F x * g x) * indicator {a .. b} x \<partial>lborel) + \<integral>x. (f x * G x) * indicator {a .. b} x \<partial>lborel"
```
```   545     apply (subst Bochner_Integration.integral_add[symmetric])
```
```   546     apply (auto intro!: borel_integrable_atLeastAtMost continuous_intros)
```
```   547     by (auto intro!: DERIV_isCont Bochner_Integration.integral_cong split: split_indicator)
```
```   548
```
```   549   thus ?thesis using 0 by auto
```
```   550 qed
```
```   551
```
```   552 lemma integral_by_parts':
```
```   553   fixes f g F G::"real \<Rightarrow> real"
```
```   554   assumes "a \<le> b"
```
```   555   assumes "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
```
```   556   assumes "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
```
```   557   assumes "!!x. DERIV F x :> f x"
```
```   558   assumes "!!x. DERIV G x :> g x"
```
```   559   shows "(\<integral>x. indicator {a .. b} x *\<^sub>R (F x * g x) \<partial>lborel)
```
```   560             =  F b * G b - F a * G a - \<integral>x. indicator {a .. b} x *\<^sub>R (f x * G x) \<partial>lborel"
```
```   561   using integral_by_parts[OF assms] by (simp add: ac_simps)
```
```   562
```
```   563 lemma has_bochner_integral_even_function:
```
```   564   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```   565   assumes f: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x) x"
```
```   566   assumes even: "\<And>x. f (- x) = f x"
```
```   567   shows "has_bochner_integral lborel f (2 *\<^sub>R x)"
```
```   568 proof -
```
```   569   have indicator: "\<And>x::real. indicator {..0} (- x) = indicator {0..} x"
```
```   570     by (auto split: split_indicator)
```
```   571   have "has_bochner_integral lborel (\<lambda>x. indicator {.. 0} x *\<^sub>R f x) x"
```
```   572     by (subst lborel_has_bochner_integral_real_affine_iff[where c="-1" and t=0])
```
```   573        (auto simp: indicator even f)
```
```   574   with f have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x + indicator {.. 0} x *\<^sub>R f x) (x + x)"
```
```   575     by (rule has_bochner_integral_add)
```
```   576   then have "has_bochner_integral lborel f (x + x)"
```
```   577     by (rule has_bochner_integral_discrete_difference[where X="{0}", THEN iffD1, rotated 4])
```
```   578        (auto split: split_indicator)
```
```   579   then show ?thesis
```
```   580     by (simp add: scaleR_2)
```
```   581 qed
```
```   582
```
```   583 lemma has_bochner_integral_odd_function:
```
```   584   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```   585   assumes f: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x) x"
```
```   586   assumes odd: "\<And>x. f (- x) = - f x"
```
```   587   shows "has_bochner_integral lborel f 0"
```
```   588 proof -
```
```   589   have indicator: "\<And>x::real. indicator {..0} (- x) = indicator {0..} x"
```
```   590     by (auto split: split_indicator)
```
```   591   have "has_bochner_integral lborel (\<lambda>x. - indicator {.. 0} x *\<^sub>R f x) x"
```
```   592     by (subst lborel_has_bochner_integral_real_affine_iff[where c="-1" and t=0])
```
```   593        (auto simp: indicator odd f)
```
```   594   from has_bochner_integral_minus[OF this]
```
```   595   have "has_bochner_integral lborel (\<lambda>x. indicator {.. 0} x *\<^sub>R f x) (- x)"
```
```   596     by simp
```
```   597   with f have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x + indicator {.. 0} x *\<^sub>R f x) (x + - x)"
```
```   598     by (rule has_bochner_integral_add)
```
```   599   then have "has_bochner_integral lborel f (x + - x)"
```
```   600     by (rule has_bochner_integral_discrete_difference[where X="{0}", THEN iffD1, rotated 4])
```
```   601        (auto split: split_indicator)
```
```   602   then show ?thesis
```
```   603     by simp
```
```   604 qed
```
```   605
```
```   606 end
```