--- a/src/HOL/Analysis/Lebesgue_Measure.thy Thu Sep 15 22:41:05 2016 +0200
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy Fri Sep 16 13:56:51 2016 +0200
@@ -469,6 +469,10 @@
"emeasure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
+lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
+ using emeasure_lborel_cbox[of x x] nonempty_Basis
+ by (auto simp del: emeasure_lborel_cbox nonempty_Basis simp add: cbox_sing setprod_constant)
+
lemma
fixes l u :: real
assumes [simp]: "l \<le> u"
@@ -484,6 +488,17 @@
and measure_lborel_cbox[simp]: "measure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
by (simp_all add: measure_def inner_diff_left setprod_nonneg)
+lemma measure_lborel_cbox_eq:
+ "measure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
+ using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)
+
+lemma measure_lborel_box_eq:
+ "measure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
+ using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
+
+lemma measure_lborel_singleton[simp]: "measure lborel {x} = 0"
+ by (simp add: measure_def)
+
lemma sigma_finite_lborel: "sigma_finite_measure lborel"
proof
show "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets lborel \<and> \<Union>A = space lborel \<and> (\<forall>a\<in>A. emeasure lborel a \<noteq> \<infinity>)"
@@ -516,10 +531,6 @@
done
qed
-lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
- using emeasure_lborel_cbox[of x x] nonempty_Basis
- by (auto simp del: emeasure_lborel_cbox nonempty_Basis simp add: cbox_sing setprod_constant)
-
lemma emeasure_lborel_countable:
fixes A :: "'a::euclidean_space set"
assumes "countable A"
@@ -572,46 +583,38 @@
{ fix i show "emeasure lborel (?A i) \<noteq> \<infinity>" by auto }
{ fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X"
- apply (auto simp: emeasure_eq emeasure_lborel_box_eq )
+ apply (auto simp: emeasure_eq emeasure_lborel_box_eq)
apply (subst box_eq_empty(1)[THEN iffD2])
apply (auto intro: less_imp_le simp: not_le)
done }
qed
-lemma lborel_affine:
- fixes t :: "'a::euclidean_space" assumes "c \<noteq> 0"
- shows "lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>_. \<bar>c\<bar>^DIM('a))" (is "_ = ?D")
+lemma lborel_affine_euclidean:
+ fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
+ defines "T x \<equiv> t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)"
+ assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
+ shows "lborel = density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
proof (rule lborel_eqI)
let ?B = "Basis :: 'a set"
fix l u assume le: "\<And>b. b \<in> ?B \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
- show "emeasure ?D (box l u) = (\<Prod>b\<in>?B. (u - l) \<bullet> b)"
- proof cases
- assume "0 < c"
- then have "(\<lambda>x. t + c *\<^sub>R x) -` box l u = box ((l - t) /\<^sub>R c) ((u - t) /\<^sub>R c)"
- by (auto simp: field_simps box_def inner_simps)
- with \<open>0 < c\<close> show ?thesis
- using le
- by (auto simp: field_simps inner_simps setprod_dividef setprod_constant setprod_nonneg
- ennreal_mult[symmetric] emeasure_density nn_integral_distr emeasure_distr
- nn_integral_cmult emeasure_lborel_box_eq borel_measurable_indicator')
- next
- assume "\<not> 0 < c" with \<open>c \<noteq> 0\<close> have "c < 0" by auto
- then have "box ((u - t) /\<^sub>R c) ((l - t) /\<^sub>R c) = (\<lambda>x. t + c *\<^sub>R x) -` box l u"
- by (auto simp: field_simps box_def inner_simps)
- then have *: "\<And>x. indicator (box l u) (t + c *\<^sub>R x) = (indicator (box ((u - t) /\<^sub>R c) ((l - t) /\<^sub>R c)) x :: ennreal)"
- by (auto split: split_indicator)
- have **: "(\<Prod>x\<in>Basis. (l \<bullet> x - u \<bullet> x) / c) = (\<Prod>x\<in>Basis. u \<bullet> x - l \<bullet> x) / (-c) ^ card (Basis::'a set)"
- using \<open>c < 0\<close>
- by (auto simp add: field_simps setprod_dividef[symmetric] setprod_constant[symmetric]
- intro!: setprod.cong)
- show ?thesis
- using \<open>c < 0\<close> le
- by (auto simp: * ** field_simps emeasure_density nn_integral_distr nn_integral_cmult
- emeasure_lborel_box_eq inner_simps setprod_nonneg ennreal_mult[symmetric]
- borel_measurable_indicator')
- qed
+ have [measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
+ by (simp add: T_def[abs_def])
+ have eq: "T -` box l u = box
+ (\<Sum>j\<in>Basis. (((if 0 < c j then l - t else u - t) \<bullet> j) / c j) *\<^sub>R j)
+ (\<Sum>j\<in>Basis. (((if 0 < c j then u - t else l - t) \<bullet> j) / c j) *\<^sub>R j)"
+ using c by (auto simp: box_def T_def field_simps inner_simps divide_less_eq)
+ with le c show "emeasure ?D (box l u) = (\<Prod>b\<in>?B. (u - l) \<bullet> b)"
+ by (auto simp: emeasure_density emeasure_distr nn_integral_multc emeasure_lborel_box_eq inner_simps
+ field_simps divide_simps ennreal_mult'[symmetric] setprod_nonneg setprod.distrib[symmetric]
+ intro!: setprod.cong)
qed simp
+lemma lborel_affine:
+ fixes t :: "'a::euclidean_space"
+ shows "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>_. \<bar>c\<bar>^DIM('a))"
+ using lborel_affine_euclidean[where c="\<lambda>_::'a. c" and t=t]
+ unfolding scaleR_scaleR[symmetric] scaleR_setsum_right[symmetric] euclidean_representation setprod_constant by simp
+
lemma lborel_real_affine:
"c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. ennreal (abs c))"
using lborel_affine[of c t] by simp
@@ -726,378 +729,6 @@
lemma lborelD_Collect[measurable (raw)]: "{x\<in>space borel. P x} \<in> sets borel \<Longrightarrow> {x\<in>space lborel. P x} \<in> sets lborel" by simp
lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel" by simp
-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>
-
lemma emeasure_bounded_finite:
assumes "bounded A" shows "emeasure lborel A < \<infinity>"
proof -
@@ -1149,233 +780,4 @@
by (auto simp: mult.commute)
qed
-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 integral_add[symmetric])
- apply (auto intro!: borel_integrable_atLeastAtMost continuous_intros)
- by (auto intro!: DERIV_isCont 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