--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Fri Sep 16 13:56:51 2016 +0200
@@ -0,0 +1,606 @@
+theory Equivalence_Lebesgue_Henstock_Integration
+ imports Lebesgue_Measure Henstock_Kurzweil_Integration
+begin
+
+subsection \<open>Equivalence Lebesgue integral on @{const lborel} and HK-integral\<close>
+
+lemma has_integral_measure_lborel:
+ fixes A :: "'a::euclidean_space set"
+ assumes A[measurable]: "A \<in> sets borel" and finite: "emeasure lborel A < \<infinity>"
+ shows "((\<lambda>x. 1) has_integral measure lborel A) A"
+proof -
+ { fix l u :: 'a
+ have "((\<lambda>x. 1) has_integral measure lborel (box l u)) (box l u)"
+ proof cases
+ assume "\<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b"
+ then show ?thesis
+ apply simp
+ apply (subst has_integral_restrict[symmetric, OF box_subset_cbox])
+ apply (subst has_integral_spike_interior_eq[where g="\<lambda>_. 1"])
+ using has_integral_const[of "1::real" l u]
+ apply (simp_all add: inner_diff_left[symmetric] content_cbox_cases)
+ done
+ next
+ assume "\<not> (\<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b)"
+ then have "box l u = {}"
+ unfolding box_eq_empty by (auto simp: not_le intro: less_imp_le)
+ then show ?thesis
+ by simp
+ qed }
+ note has_integral_box = this
+
+ { fix a b :: 'a let ?M = "\<lambda>A. measure lborel (A \<inter> box a b)"
+ have "Int_stable (range (\<lambda>(a, b). box a b))"
+ by (auto simp: Int_stable_def box_Int_box)
+ moreover have "(range (\<lambda>(a, b). box a b)) \<subseteq> Pow UNIV"
+ by auto
+ moreover have "A \<in> sigma_sets UNIV (range (\<lambda>(a, b). box a b))"
+ using A unfolding borel_eq_box by simp
+ ultimately have "((\<lambda>x. 1) has_integral ?M A) (A \<inter> box a b)"
+ proof (induction rule: sigma_sets_induct_disjoint)
+ case (basic A) then show ?case
+ by (auto simp: box_Int_box has_integral_box)
+ next
+ case empty then show ?case
+ by simp
+ next
+ case (compl A)
+ then have [measurable]: "A \<in> sets borel"
+ by (simp add: borel_eq_box)
+
+ have "((\<lambda>x. 1) has_integral ?M (box a b)) (box a b)"
+ by (simp add: has_integral_box)
+ moreover have "((\<lambda>x. if x \<in> A \<inter> box a b then 1 else 0) has_integral ?M A) (box a b)"
+ by (subst has_integral_restrict) (auto intro: compl)
+ 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)"
+ by (rule has_integral_sub)
+ 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)"
+ by (rule has_integral_cong[THEN iffD1, rotated 1]) auto
+ then have "((\<lambda>x. 1) has_integral ?M (box a b) - ?M A) ((UNIV - A) \<inter> box a b)"
+ by (subst (asm) has_integral_restrict) auto
+ also have "?M (box a b) - ?M A = ?M (UNIV - A)"
+ by (subst measure_Diff[symmetric]) (auto simp: emeasure_lborel_box_eq Diff_Int_distrib2)
+ finally show ?case .
+ next
+ case (union F)
+ then have [measurable]: "\<And>i. F i \<in> sets borel"
+ by (simp add: borel_eq_box subset_eq)
+ 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)"
+ proof (rule has_integral_monotone_convergence_increasing)
+ let ?f = "\<lambda>k x. \<Sum>i<k. if x \<in> F i \<inter> box a b then 1 else 0 :: real"
+ show "\<And>k. (?f k has_integral (\<Sum>i<k. ?M (F i))) (box a b)"
+ using union.IH by (auto intro!: has_integral_setsum simp del: Int_iff)
+ show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
+ by (intro setsum_mono2) auto
+ from union(1) have *: "\<And>x i j. x \<in> F i \<Longrightarrow> x \<in> F j \<longleftrightarrow> j = i"
+ by (auto simp add: disjoint_family_on_def)
+ show "\<And>x. (\<lambda>k. ?f k x) \<longlonglongrightarrow> (if x \<in> UNION UNIV F \<inter> box a b then 1 else 0)"
+ apply (auto simp: * setsum.If_cases Iio_Int_singleton)
+ apply (rule_tac k="Suc xa" in LIMSEQ_offset)
+ apply simp
+ done
+ have *: "emeasure lborel ((\<Union>x. F x) \<inter> box a b) \<le> emeasure lborel (box a b)"
+ by (intro emeasure_mono) auto
+
+ with union(1) show "(\<lambda>k. \<Sum>i<k. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"
+ unfolding sums_def[symmetric] UN_extend_simps
+ by (intro measure_UNION) (auto simp: disjoint_family_on_def emeasure_lborel_box_eq top_unique)
+ qed
+ then show ?case
+ by (subst (asm) has_integral_restrict) auto
+ qed }
+ note * = this
+
+ show ?thesis
+ proof (rule has_integral_monotone_convergence_increasing)
+ let ?B = "\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One) :: 'a set"
+ let ?f = "\<lambda>n::nat. \<lambda>x. if x \<in> A \<inter> ?B n then 1 else 0 :: real"
+ let ?M = "\<lambda>n. measure lborel (A \<inter> ?B n)"
+
+ show "\<And>n::nat. (?f n has_integral ?M n) A"
+ using * by (subst has_integral_restrict) simp_all
+ show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
+ by (auto simp: box_def)
+ { fix x assume "x \<in> A"
+ moreover have "(\<lambda>k. indicator (A \<inter> ?B k) x :: real) \<longlonglongrightarrow> indicator (\<Union>k::nat. A \<inter> ?B k) x"
+ by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def box_def)
+ ultimately show "(\<lambda>k. if x \<in> A \<inter> ?B k then 1 else 0::real) \<longlonglongrightarrow> 1"
+ by (simp add: indicator_def UN_box_eq_UNIV) }
+
+ have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> emeasure lborel (\<Union>n::nat. A \<inter> ?B n)"
+ by (intro Lim_emeasure_incseq) (auto simp: incseq_def box_def)
+ also have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) = (\<lambda>n. measure lborel (A \<inter> ?B n))"
+ proof (intro ext emeasure_eq_ennreal_measure)
+ fix n have "emeasure lborel (A \<inter> ?B n) \<le> emeasure lborel (?B n)"
+ by (intro emeasure_mono) auto
+ then show "emeasure lborel (A \<inter> ?B n) \<noteq> top"
+ by (auto simp: top_unique)
+ qed
+ finally show "(\<lambda>n. measure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> measure lborel A"
+ using emeasure_eq_ennreal_measure[of lborel A] finite
+ by (simp add: UN_box_eq_UNIV less_top)
+ qed
+qed
+
+lemma nn_integral_has_integral:
+ fixes f::"'a::euclidean_space \<Rightarrow> real"
+ 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"
+ shows "(f has_integral r) UNIV"
+using f proof (induct f arbitrary: r rule: borel_measurable_induct_real)
+ case (set A)
+ then have "((\<lambda>x. 1) has_integral measure lborel A) A"
+ by (intro has_integral_measure_lborel) (auto simp: ennreal_indicator)
+ with set show ?case
+ by (simp add: ennreal_indicator measure_def) (simp add: indicator_def)
+next
+ case (mult g c)
+ then have "ennreal c * (\<integral>\<^sup>+ x. g x \<partial>lborel) = ennreal r"
+ by (subst nn_integral_cmult[symmetric]) (auto simp: ennreal_mult)
+ with \<open>0 \<le> r\<close> \<open>0 \<le> c\<close>
+ 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')"
+ by (cases "\<integral>\<^sup>+ x. ennreal (g x) \<partial>lborel" rule: ennreal_cases)
+ (auto split: if_split_asm simp: ennreal_mult_top ennreal_mult[symmetric])
+ with mult show ?case
+ by (auto intro!: has_integral_cmult_real)
+next
+ case (add g h)
+ 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)"
+ by (simp add: nn_integral_add)
+ 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"
+ by (cases "\<integral>\<^sup>+ x. h x \<partial>lborel" "\<integral>\<^sup>+ x. g x \<partial>lborel" rule: ennreal2_cases)
+ (auto simp: add_top nn_integral_add top_add ennreal_plus[symmetric] simp del: ennreal_plus)
+ with add show ?case
+ by (auto intro!: has_integral_add)
+next
+ case (seq U)
+ note seq(1)[measurable] and f[measurable]
+
+ { fix i x
+ have "U i x \<le> f x"
+ using seq(5)
+ apply (rule LIMSEQ_le_const)
+ using seq(4)
+ apply (auto intro!: exI[of _ i] simp: incseq_def le_fun_def)
+ done }
+ note U_le_f = this
+
+ { fix i
+ have "(\<integral>\<^sup>+x. U i x \<partial>lborel) \<le> (\<integral>\<^sup>+x. f x \<partial>lborel)"
+ using seq(2) f(2) U_le_f by (intro nn_integral_mono) simp
+ then obtain p where "(\<integral>\<^sup>+x. U i x \<partial>lborel) = ennreal p" "p \<le> r" "0 \<le> p"
+ 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)
+ moreover note seq
+ 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"
+ by auto }
+ then obtain p where p: "\<And>i. (\<integral>\<^sup>+x. ennreal (U i x) \<partial>lborel) = ennreal (p i)"
+ and bnd: "\<And>i. p i \<le> r" "\<And>i. 0 \<le> p i"
+ and U_int: "\<And>i.(U i has_integral (p i)) UNIV" by metis
+
+ have int_eq: "\<And>i. integral UNIV (U i) = p i" using U_int by (rule integral_unique)
+
+ have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) \<longlonglongrightarrow> integral UNIV f"
+ proof (rule monotone_convergence_increasing)
+ show "\<forall>k. U k integrable_on UNIV" using U_int by auto
+ 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)
+ then show "bounded {integral UNIV (U k) |k. True}"
+ using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
+ show "\<forall>x\<in>UNIV. (\<lambda>k. U k x) \<longlonglongrightarrow> f x"
+ using seq by auto
+ qed
+ moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lborel)) \<longlonglongrightarrow> (\<integral>\<^sup>+x. f x \<partial>lborel)"
+ using seq f(2) U_le_f by (intro nn_integral_dominated_convergence[where w=f]) auto
+ ultimately have "integral UNIV f = r"
+ by (auto simp add: bnd int_eq p seq intro: LIMSEQ_unique)
+ with * show ?case
+ by (simp add: has_integral_integral)
+qed
+
+lemma nn_integral_lborel_eq_integral:
+ fixes f::"'a::euclidean_space \<Rightarrow> real"
+ assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) < \<infinity>"
+ shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = integral UNIV f"
+proof -
+ from f(3) obtain r where r: "(\<integral>\<^sup>+x. f x \<partial>lborel) = ennreal r" "0 \<le> r"
+ by (cases "\<integral>\<^sup>+x. f x \<partial>lborel" rule: ennreal_cases) auto
+ then show ?thesis
+ using nn_integral_has_integral[OF f(1,2) r] by (simp add: integral_unique)
+qed
+
+lemma nn_integral_integrable_on:
+ fixes f::"'a::euclidean_space \<Rightarrow> real"
+ assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) < \<infinity>"
+ shows "f integrable_on UNIV"
+proof -
+ from f(3) obtain r where r: "(\<integral>\<^sup>+x. f x \<partial>lborel) = ennreal r" "0 \<le> r"
+ by (cases "\<integral>\<^sup>+x. f x \<partial>lborel" rule: ennreal_cases) auto
+ then show ?thesis
+ by (intro has_integral_integrable[where i=r] nn_integral_has_integral[where r=r] f)
+qed
+
+lemma nn_integral_has_integral_lborel:
+ fixes f :: "'a::euclidean_space \<Rightarrow> real"
+ assumes f_borel: "f \<in> borel_measurable borel" and nonneg: "\<And>x. 0 \<le> f x"
+ assumes I: "(f has_integral I) UNIV"
+ shows "integral\<^sup>N lborel f = I"
+proof -
+ from f_borel have "(\<lambda>x. ennreal (f x)) \<in> borel_measurable lborel" by auto
+ from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this
+ let ?B = "\<lambda>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One) :: 'a set"
+
+ note F(1)[THEN borel_measurable_simple_function, measurable]
+
+ have "0 \<le> I"
+ using I by (rule has_integral_nonneg) (simp add: nonneg)
+
+ have F_le_f: "enn2real (F i x) \<le> f x" for i x
+ using F(3,4)[where x=x] nonneg SUP_upper[of i UNIV "\<lambda>i. F i x"]
+ by (cases "F i x" rule: ennreal_cases) auto
+ let ?F = "\<lambda>i x. F i x * indicator (?B i) x"
+ have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) = (SUP i. integral\<^sup>N lborel (\<lambda>x. ?F i x))"
+ proof (subst nn_integral_monotone_convergence_SUP[symmetric])
+ { fix x
+ obtain j where j: "x \<in> ?B j"
+ using UN_box_eq_UNIV by auto
+
+ have "ennreal (f x) = (SUP i. F i x)"
+ using F(4)[of x] nonneg[of x] by (simp add: max_def)
+ also have "\<dots> = (SUP i. ?F i x)"
+ proof (rule SUP_eq)
+ fix i show "\<exists>j\<in>UNIV. F i x \<le> ?F j x"
+ using j F(2)
+ by (intro bexI[of _ "max i j"])
+ (auto split: split_max split_indicator simp: incseq_def le_fun_def box_def)
+ qed (auto intro!: F split: split_indicator)
+ finally have "ennreal (f x) = (SUP i. ?F i x)" . }
+ then show "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) = (\<integral>\<^sup>+ x. (SUP i. ?F i x) \<partial>lborel)"
+ by simp
+ qed (insert F, auto simp: incseq_def le_fun_def box_def split: split_indicator)
+ also have "\<dots> \<le> ennreal I"
+ proof (rule SUP_least)
+ fix i :: nat
+ have finite_F: "(\<integral>\<^sup>+ x. ennreal (enn2real (F i x) * indicator (?B i) x) \<partial>lborel) < \<infinity>"
+ proof (rule nn_integral_bound_simple_function)
+ have "emeasure lborel {x \<in> space lborel. ennreal (enn2real (F i x) * indicator (?B i) x) \<noteq> 0} \<le>
+ emeasure lborel (?B i)"
+ by (intro emeasure_mono) (auto split: split_indicator)
+ then show "emeasure lborel {x \<in> space lborel. ennreal (enn2real (F i x) * indicator (?B i) x) \<noteq> 0} < \<infinity>"
+ by (auto simp: less_top[symmetric] top_unique)
+ qed (auto split: split_indicator
+ intro!: F simple_function_compose1[where g="enn2real"] simple_function_ennreal)
+
+ have int_F: "(\<lambda>x. enn2real (F i x) * indicator (?B i) x) integrable_on UNIV"
+ using F(4) finite_F
+ by (intro nn_integral_integrable_on) (auto split: split_indicator simp: enn2real_nonneg)
+
+ have "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) =
+ (\<integral>\<^sup>+ x. ennreal (enn2real (F i x) * indicator (?B i) x) \<partial>lborel)"
+ using F(3,4)
+ by (intro nn_integral_cong) (auto simp: image_iff eq_commute split: split_indicator)
+ also have "\<dots> = ennreal (integral UNIV (\<lambda>x. enn2real (F i x) * indicator (?B i) x))"
+ using F
+ by (intro nn_integral_lborel_eq_integral[OF _ _ finite_F])
+ (auto split: split_indicator intro: enn2real_nonneg)
+ also have "\<dots> \<le> ennreal I"
+ by (auto intro!: has_integral_le[OF integrable_integral[OF int_F] I] nonneg F_le_f
+ simp: \<open>0 \<le> I\<close> split: split_indicator )
+ finally show "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) \<le> ennreal I" .
+ qed
+ finally have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) < \<infinity>"
+ by (auto simp: less_top[symmetric] top_unique)
+ from nn_integral_lborel_eq_integral[OF assms(1,2) this] I show ?thesis
+ by (simp add: integral_unique)
+qed
+
+lemma has_integral_iff_emeasure_lborel:
+ fixes A :: "'a::euclidean_space set"
+ assumes A[measurable]: "A \<in> sets borel" and [simp]: "0 \<le> r"
+ shows "((\<lambda>x. 1) has_integral r) A \<longleftrightarrow> emeasure lborel A = ennreal r"
+proof (cases "emeasure lborel A = \<infinity>")
+ case emeasure_A: True
+ have "\<not> (\<lambda>x. 1::real) integrable_on A"
+ proof
+ assume int: "(\<lambda>x. 1::real) integrable_on A"
+ then have "(indicator A::'a \<Rightarrow> real) integrable_on UNIV"
+ unfolding indicator_def[abs_def] integrable_restrict_univ .
+ then obtain r where "((indicator A::'a\<Rightarrow>real) has_integral r) UNIV"
+ by auto
+ from nn_integral_has_integral_lborel[OF _ _ this] emeasure_A show False
+ by (simp add: ennreal_indicator)
+ qed
+ with emeasure_A show ?thesis
+ by auto
+next
+ case False
+ then have "((\<lambda>x. 1) has_integral measure lborel A) A"
+ by (simp add: has_integral_measure_lborel less_top)
+ with False show ?thesis
+ by (auto simp: emeasure_eq_ennreal_measure has_integral_unique)
+qed
+
+lemma has_integral_integral_real:
+ fixes f::"'a::euclidean_space \<Rightarrow> real"
+ assumes f: "integrable lborel f"
+ shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
+using f proof induct
+ case (base A c) then show ?case
+ by (auto intro!: has_integral_mult_left simp: )
+ (simp add: emeasure_eq_ennreal_measure indicator_def has_integral_measure_lborel)
+next
+ case (add f g) then show ?case
+ by (auto intro!: has_integral_add)
+next
+ case (lim f s)
+ show ?case
+ proof (rule has_integral_dominated_convergence)
+ show "\<And>i. (s i has_integral integral\<^sup>L lborel (s i)) UNIV" by fact
+ show "(\<lambda>x. norm (2 * f x)) integrable_on UNIV"
+ using \<open>integrable lborel f\<close>
+ by (intro nn_integral_integrable_on)
+ (auto simp: integrable_iff_bounded abs_mult nn_integral_cmult ennreal_mult ennreal_mult_less_top)
+ show "\<And>k. \<forall>x\<in>UNIV. norm (s k x) \<le> norm (2 * f x)"
+ using lim by (auto simp add: abs_mult)
+ show "\<forall>x\<in>UNIV. (\<lambda>k. s k x) \<longlonglongrightarrow> f x"
+ using lim by auto
+ show "(\<lambda>k. integral\<^sup>L lborel (s k)) \<longlonglongrightarrow> integral\<^sup>L lborel f"
+ using lim lim(1)[THEN borel_measurable_integrable]
+ by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) auto
+ qed
+qed
+
+context
+ fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+begin
+
+lemma has_integral_integral_lborel:
+ assumes f: "integrable lborel f"
+ shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
+proof -
+ 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"
+ using f by (intro has_integral_setsum finite_Basis ballI has_integral_scaleR_left has_integral_integral_real) auto
+ also have eq_f: "(\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) = f"
+ by (simp add: fun_eq_iff euclidean_representation)
+ also have "(\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b) = integral\<^sup>L lborel f"
+ using f by (subst (2) eq_f[symmetric]) simp
+ finally show ?thesis .
+qed
+
+lemma integrable_on_lborel: "integrable lborel f \<Longrightarrow> f integrable_on UNIV"
+ using has_integral_integral_lborel by auto
+
+lemma integral_lborel: "integrable lborel f \<Longrightarrow> integral UNIV f = (\<integral>x. f x \<partial>lborel)"
+ using has_integral_integral_lborel by auto
+
+end
+
+subsection \<open>Fundamental Theorem of Calculus for the Lebesgue integral\<close>
+
+text \<open>
+
+For the positive integral we replace continuity with Borel-measurability.
+
+\<close>
+
+lemma
+ fixes f :: "real \<Rightarrow> real"
+ assumes [measurable]: "f \<in> borel_measurable borel"
+ 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"
+ shows nn_integral_FTC_Icc: "(\<integral>\<^sup>+x. ennreal (f x) * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?nn)
+ and has_bochner_integral_FTC_Icc_nonneg:
+ "has_bochner_integral lborel (\<lambda>x. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
+ and integral_FTC_Icc_nonneg: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
+ and integrable_FTC_Icc_nonneg: "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is ?int)
+proof -
+ have *: "(\<lambda>x. f x * indicator {a..b} x) \<in> borel_measurable borel" "\<And>x. 0 \<le> f x * indicator {a..b} x"
+ using f(2) by (auto split: split_indicator)
+
+ have F_mono: "a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b\<Longrightarrow> F x \<le> F y" for x y
+ using f by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
+
+ have "(f has_integral F b - F a) {a..b}"
+ by (intro fundamental_theorem_of_calculus)
+ (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric]
+ intro: has_field_derivative_subset[OF f(1)] \<open>a \<le> b\<close>)
+ then have i: "((\<lambda>x. f x * indicator {a .. b} x) has_integral F b - F a) UNIV"
+ unfolding indicator_def if_distrib[where f="\<lambda>x. a * x" for a]
+ by (simp cong del: if_weak_cong del: atLeastAtMost_iff)
+ then have nn: "(\<integral>\<^sup>+x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a"
+ by (rule nn_integral_has_integral_lborel[OF *])
+ then show ?has
+ by (rule has_bochner_integral_nn_integral[rotated 3]) (simp_all add: * F_mono \<open>a \<le> b\<close>)
+ then show ?eq ?int
+ unfolding has_bochner_integral_iff by auto
+ show ?nn
+ by (subst nn[symmetric])
+ (auto intro!: nn_integral_cong simp add: ennreal_mult f split: split_indicator)
+qed
+
+lemma
+ fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
+ assumes "a \<le> b"
+ assumes "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
+ assumes cont: "continuous_on {a .. b} f"
+ shows has_bochner_integral_FTC_Icc:
+ "has_bochner_integral lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x) (F b - F a)" (is ?has)
+ and integral_FTC_Icc: "(\<integral>x. indicator {a .. b} x *\<^sub>R f x \<partial>lborel) = F b - F a" (is ?eq)
+proof -
+ let ?f = "\<lambda>x. indicator {a .. b} x *\<^sub>R f x"
+ have int: "integrable lborel ?f"
+ using borel_integrable_compact[OF _ cont] by auto
+ have "(f has_integral F b - F a) {a..b}"
+ using assms(1,2) by (intro fundamental_theorem_of_calculus) auto
+ moreover
+ have "(f has_integral integral\<^sup>L lborel ?f) {a..b}"
+ using has_integral_integral_lborel[OF int]
+ unfolding indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a]
+ by (simp cong del: if_weak_cong del: atLeastAtMost_iff)
+ ultimately show ?eq
+ by (auto dest: has_integral_unique)
+ then show ?has
+ using int by (auto simp: has_bochner_integral_iff)
+qed
+
+lemma
+ fixes f :: "real \<Rightarrow> real"
+ assumes "a \<le> b"
+ assumes deriv: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> DERIV F x :> f x"
+ assumes cont: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
+ shows has_bochner_integral_FTC_Icc_real:
+ "has_bochner_integral lborel (\<lambda>x. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
+ and integral_FTC_Icc_real: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
+proof -
+ have 1: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
+ unfolding has_field_derivative_iff_has_vector_derivative[symmetric]
+ using deriv by (auto intro: DERIV_subset)
+ have 2: "continuous_on {a .. b} f"
+ using cont by (intro continuous_at_imp_continuous_on) auto
+ show ?has ?eq
+ 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]
+ by (auto simp: mult.commute)
+qed
+
+lemma nn_integral_FTC_atLeast:
+ fixes f :: "real \<Rightarrow> real"
+ assumes f_borel: "f \<in> borel_measurable borel"
+ assumes f: "\<And>x. a \<le> x \<Longrightarrow> DERIV F x :> f x"
+ assumes nonneg: "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> f x"
+ assumes lim: "(F \<longlongrightarrow> T) at_top"
+ shows "(\<integral>\<^sup>+x. ennreal (f x) * indicator {a ..} x \<partial>lborel) = T - F a"
+proof -
+ let ?f = "\<lambda>(i::nat) (x::real). ennreal (f x) * indicator {a..a + real i} x"
+ let ?fR = "\<lambda>x. ennreal (f x) * indicator {a ..} x"
+
+ have F_mono: "a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> F x \<le> F y" for x y
+ using f nonneg by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
+ then have F_le_T: "a \<le> x \<Longrightarrow> F x \<le> T" for x
+ by (intro tendsto_le_const[OF _ lim])
+ (auto simp: trivial_limit_at_top_linorder eventually_at_top_linorder)
+
+ have "(SUP i::nat. ?f i x) = ?fR x" for x
+ proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
+ from reals_Archimedean2[of "x - a"] guess n ..
+ then have "eventually (\<lambda>n. ?f n x = ?fR x) sequentially"
+ by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator)
+ then show "(\<lambda>n. ?f n x) \<longlonglongrightarrow> ?fR x"
+ by (rule Lim_eventually)
+ qed (auto simp: nonneg incseq_def le_fun_def split: split_indicator)
+ then have "integral\<^sup>N lborel ?fR = (\<integral>\<^sup>+ x. (SUP i::nat. ?f i x) \<partial>lborel)"
+ by simp
+ also have "\<dots> = (SUP i::nat. (\<integral>\<^sup>+ x. ?f i x \<partial>lborel))"
+ proof (rule nn_integral_monotone_convergence_SUP)
+ show "incseq ?f"
+ using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator)
+ show "\<And>i. (?f i) \<in> borel_measurable lborel"
+ using f_borel by auto
+ qed
+ also have "\<dots> = (SUP i::nat. ennreal (F (a + real i) - F a))"
+ by (subst nn_integral_FTC_Icc[OF f_borel f nonneg]) auto
+ also have "\<dots> = T - F a"
+ proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
+ have "(\<lambda>x. F (a + real x)) \<longlonglongrightarrow> T"
+ apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top])
+ apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl])
+ apply (rule filterlim_real_sequentially)
+ done
+ then show "(\<lambda>n. ennreal (F (a + real n) - F a)) \<longlonglongrightarrow> ennreal (T - F a)"
+ by (simp add: F_mono F_le_T tendsto_diff)
+ qed (auto simp: incseq_def intro!: ennreal_le_iff[THEN iffD2] F_mono)
+ finally show ?thesis .
+qed
+
+lemma integral_power:
+ "a \<le> b \<Longrightarrow> (\<integral>x. x^k * indicator {a..b} x \<partial>lborel) = (b^Suc k - a^Suc k) / Suc k"
+proof (subst integral_FTC_Icc_real)
+ fix x show "DERIV (\<lambda>x. x^Suc k / Suc k) x :> x^k"
+ by (intro derivative_eq_intros) auto
+qed (auto simp: field_simps simp del: of_nat_Suc)
+
+subsection \<open>Integration by parts\<close>
+
+lemma integral_by_parts_integrable:
+ fixes f g F G::"real \<Rightarrow> real"
+ assumes "a \<le> b"
+ assumes cont_f[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
+ assumes cont_g[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
+ assumes [intro]: "!!x. DERIV F x :> f x"
+ assumes [intro]: "!!x. DERIV G x :> g x"
+ shows "integrable lborel (\<lambda>x.((F x) * (g x) + (f x) * (G x)) * indicator {a .. b} x)"
+ by (auto intro!: borel_integrable_atLeastAtMost continuous_intros) (auto intro!: DERIV_isCont)
+
+lemma integral_by_parts:
+ fixes f g F G::"real \<Rightarrow> real"
+ assumes [arith]: "a \<le> b"
+ assumes cont_f[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
+ assumes cont_g[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
+ assumes [intro]: "!!x. DERIV F x :> f x"
+ assumes [intro]: "!!x. DERIV G x :> g x"
+ shows "(\<integral>x. (F x * g x) * indicator {a .. b} x \<partial>lborel)
+ = F b * G b - F a * G a - \<integral>x. (f x * G x) * indicator {a .. b} x \<partial>lborel"
+proof-
+ 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"
+ by (rule integral_FTC_Icc_real, auto intro!: derivative_eq_intros continuous_intros)
+ (auto intro!: DERIV_isCont)
+
+ have "(\<integral>x. (F x * g x + f x * G x) * indicator {a .. b} x \<partial>lborel) =
+ (\<integral>x. (F x * g x) * indicator {a .. b} x \<partial>lborel) + \<integral>x. (f x * G x) * indicator {a .. b} x \<partial>lborel"
+ apply (subst Bochner_Integration.integral_add[symmetric])
+ apply (auto intro!: borel_integrable_atLeastAtMost continuous_intros)
+ by (auto intro!: DERIV_isCont Bochner_Integration.integral_cong split: split_indicator)
+
+ thus ?thesis using 0 by auto
+qed
+
+lemma integral_by_parts':
+ fixes f g F G::"real \<Rightarrow> real"
+ assumes "a \<le> b"
+ assumes "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
+ assumes "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
+ assumes "!!x. DERIV F x :> f x"
+ assumes "!!x. DERIV G x :> g x"
+ shows "(\<integral>x. indicator {a .. b} x *\<^sub>R (F x * g x) \<partial>lborel)
+ = F b * G b - F a * G a - \<integral>x. indicator {a .. b} x *\<^sub>R (f x * G x) \<partial>lborel"
+ using integral_by_parts[OF assms] by (simp add: ac_simps)
+
+lemma has_bochner_integral_even_function:
+ fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
+ assumes f: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x) x"
+ assumes even: "\<And>x. f (- x) = f x"
+ shows "has_bochner_integral lborel f (2 *\<^sub>R x)"
+proof -
+ have indicator: "\<And>x::real. indicator {..0} (- x) = indicator {0..} x"
+ by (auto split: split_indicator)
+ have "has_bochner_integral lborel (\<lambda>x. indicator {.. 0} x *\<^sub>R f x) x"
+ by (subst lborel_has_bochner_integral_real_affine_iff[where c="-1" and t=0])
+ (auto simp: indicator even f)
+ 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)"
+ by (rule has_bochner_integral_add)
+ then have "has_bochner_integral lborel f (x + x)"
+ by (rule has_bochner_integral_discrete_difference[where X="{0}", THEN iffD1, rotated 4])
+ (auto split: split_indicator)
+ then show ?thesis
+ by (simp add: scaleR_2)
+qed
+
+lemma has_bochner_integral_odd_function:
+ fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
+ assumes f: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x) x"
+ assumes odd: "\<And>x. f (- x) = - f x"
+ shows "has_bochner_integral lborel f 0"
+proof -
+ have indicator: "\<And>x::real. indicator {..0} (- x) = indicator {0..} x"
+ by (auto split: split_indicator)
+ have "has_bochner_integral lborel (\<lambda>x. - indicator {.. 0} x *\<^sub>R f x) x"
+ by (subst lborel_has_bochner_integral_real_affine_iff[where c="-1" and t=0])
+ (auto simp: indicator odd f)
+ from has_bochner_integral_minus[OF this]
+ have "has_bochner_integral lborel (\<lambda>x. indicator {.. 0} x *\<^sub>R f x) (- x)"
+ by simp
+ 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)"
+ by (rule has_bochner_integral_add)
+ then have "has_bochner_integral lborel f (x + - x)"
+ by (rule has_bochner_integral_discrete_difference[where X="{0}", THEN iffD1, rotated 4])
+ (auto split: split_indicator)
+ then show ?thesis
+ by simp
+qed
+
+end