--- a/src/HOL/Analysis/Set_Integral.thy Mon Apr 09 17:21:10 2018 +0100
+++ b/src/HOL/Analysis/Set_Integral.thy Wed Apr 11 16:34:44 2018 +0100
@@ -15,18 +15,18 @@
Notation
*)
-abbreviation "set_borel_measurable M A f \<equiv> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable M"
+definition "set_borel_measurable M A f \<equiv> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable M"
-abbreviation "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
+definition "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
-abbreviation "set_lebesgue_integral M A f \<equiv> lebesgue_integral M (\<lambda>x. indicator A x *\<^sub>R f x)"
+definition "set_lebesgue_integral M A f \<equiv> lebesgue_integral M (\<lambda>x. indicator A x *\<^sub>R f x)"
syntax
-"_ascii_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
-("(4LINT (_):(_)/|(_)./ _)" [0,60,110,61] 60)
+ "_ascii_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
+ ("(4LINT (_):(_)/|(_)./ _)" [0,60,110,61] 60)
translations
-"LINT x:A|M. f" == "CONST set_lebesgue_integral M A (\<lambda>x. f)"
+ "LINT x:A|M. f" == "CONST set_lebesgue_integral M A (\<lambda>x. f)"
(*
Notation for integration wrt lebesgue measure on the reals:
@@ -38,12 +38,12 @@
*)
syntax
-"_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real"
-("(2LBINT _./ _)" [0,60] 60)
+ "_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real"
+ ("(2LBINT _./ _)" [0,60] 60)
syntax
-"_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> real"
-("(3LBINT _:_./ _)" [0,60,61] 60)
+ "_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> real"
+ ("(3LBINT _:_./ _)" [0,60,61] 60)
(*
Basic properties
@@ -60,7 +60,7 @@
shows "f -` B \<inter> X \<in> sets M"
proof -
have "f \<in> borel_measurable (restrict_space M X)"
- using assms by (subst borel_measurable_restrict_space_iff) auto
+ using assms unfolding set_borel_measurable_def by (subst borel_measurable_restrict_space_iff) auto
then have "f -` B \<inter> space (restrict_space M X) \<in> sets (restrict_space M X)"
by (rule measurable_sets) fact
with \<open>X \<in> sets M\<close> show ?thesis
@@ -70,7 +70,9 @@
lemma set_lebesgue_integral_cong:
assumes "A \<in> sets M" and "\<forall>x. x \<in> A \<longrightarrow> f x = g x"
shows "(LINT x:A|M. f x) = (LINT x:A|M. g x)"
- using assms by (auto intro!: Bochner_Integration.integral_cong split: split_indicator simp add: sets.sets_into_space)
+ unfolding set_lebesgue_integral_def
+ using assms
+ by (metis indicator_simps(2) real_vector.scale_zero_left)
lemma set_lebesgue_integral_cong_AE:
assumes [measurable]: "A \<in> sets M" "f \<in> borel_measurable M" "g \<in> borel_measurable M"
@@ -79,13 +81,15 @@
proof-
have "AE x in M. indicator A x *\<^sub>R f x = indicator A x *\<^sub>R g x"
using assms by auto
- thus ?thesis by (intro integral_cong_AE) auto
+ thus ?thesis
+ unfolding set_lebesgue_integral_def by (intro integral_cong_AE) auto
qed
lemma set_integrable_cong_AE:
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
AE x \<in> A in M. f x = g x \<Longrightarrow> A \<in> sets M \<Longrightarrow>
set_integrable M A f = set_integrable M A g"
+ unfolding set_integrable_def
by (rule integrable_cong_AE) auto
lemma set_integrable_subset:
@@ -94,8 +98,9 @@
shows "set_integrable M B f"
proof -
have "set_integrable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
- by (rule integrable_mult_indicator) fact+
+ using assms integrable_mult_indicator set_integrable_def by blast
with \<open>B \<subseteq> A\<close> show ?thesis
+ unfolding set_integrable_def
by (simp add: indicator_inter_arith[symmetric] Int_absorb2)
qed
@@ -104,11 +109,12 @@
assumes f: "set_integrable M S f" and T: "T \<in> sets (restrict_space M S)"
shows "set_integrable M T f"
proof -
- obtain T' where T_eq: "T = S \<inter> T'" and "T' \<in> sets M" using T by (auto simp: sets_restrict_space)
-
+ obtain T' where T_eq: "T = S \<inter> T'" and "T' \<in> sets M"
+ using T by (auto simp: sets_restrict_space)
have \<open>integrable M (\<lambda>x. indicator T' x *\<^sub>R (indicator S x *\<^sub>R f x))\<close>
- by (rule integrable_mult_indicator; fact)
+ using \<open>T' \<in> sets M\<close> f integrable_mult_indicator set_integrable_def by blast
then show ?thesis
+ unfolding set_integrable_def
unfolding T_eq indicator_inter_arith by (simp add: ac_simps)
qed
@@ -116,41 +122,49 @@
(* TODO: borel_integrable_atLeastAtMost should be named something like set_integrable_Icc_isCont *)
lemma set_integral_scaleR_right [simp]: "LINT t:A|M. a *\<^sub>R f t = a *\<^sub>R (LINT t:A|M. f t)"
+ unfolding set_lebesgue_integral_def
by (subst integral_scaleR_right[symmetric]) (auto intro!: Bochner_Integration.integral_cong)
lemma set_integral_mult_right [simp]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "LINT t:A|M. a * f t = a * (LINT t:A|M. f t)"
+ unfolding set_lebesgue_integral_def
by (subst integral_mult_right_zero[symmetric]) auto
lemma set_integral_mult_left [simp]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "LINT t:A|M. f t * a = (LINT t:A|M. f t) * a"
+ unfolding set_lebesgue_integral_def
by (subst integral_mult_left_zero[symmetric]) auto
lemma set_integral_divide_zero [simp]:
fixes a :: "'a::{real_normed_field, field, second_countable_topology}"
shows "LINT t:A|M. f t / a = (LINT t:A|M. f t) / a"
+ unfolding set_lebesgue_integral_def
by (subst integral_divide_zero[symmetric], intro Bochner_Integration.integral_cong)
(auto split: split_indicator)
lemma set_integrable_scaleR_right [simp, intro]:
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a *\<^sub>R f t)"
+ unfolding set_integrable_def
unfolding scaleR_left_commute by (rule integrable_scaleR_right)
lemma set_integrable_scaleR_left [simp, intro]:
fixes a :: "_ :: {banach, second_countable_topology}"
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t *\<^sub>R a)"
+ unfolding set_integrable_def
using integrable_scaleR_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
lemma set_integrable_mult_right [simp, intro]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a * f t)"
+ unfolding set_integrable_def
using integrable_mult_right[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
lemma set_integrable_mult_left [simp, intro]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t * a)"
+ unfolding set_integrable_def
using integrable_mult_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
lemma set_integrable_divide [simp, intro]:
@@ -159,10 +173,11 @@
shows "set_integrable M A (\<lambda>t. f t / a)"
proof -
have "integrable M (\<lambda>x. indicator A x *\<^sub>R f x / a)"
- using assms by (rule integrable_divide_zero)
+ using assms unfolding set_integrable_def by (rule integrable_divide_zero)
also have "(\<lambda>x. indicator A x *\<^sub>R f x / a) = (\<lambda>x. indicator A x *\<^sub>R (f x / a))"
by (auto split: split_indicator)
- finally show ?thesis .
+ finally show ?thesis
+ unfolding set_integrable_def .
qed
lemma set_integral_add [simp, intro]:
@@ -170,21 +185,23 @@
assumes "set_integrable M A f" "set_integrable M A g"
shows "set_integrable M A (\<lambda>x. f x + g x)"
and "LINT x:A|M. f x + g x = (LINT x:A|M. f x) + (LINT x:A|M. g x)"
- using assms by (simp_all add: scaleR_add_right)
+ using assms unfolding set_integrable_def set_lebesgue_integral_def by (simp_all add: scaleR_add_right)
lemma set_integral_diff [simp, intro]:
assumes "set_integrable M A f" "set_integrable M A g"
shows "set_integrable M A (\<lambda>x. f x - g x)" and "LINT x:A|M. f x - g x =
(LINT x:A|M. f x) - (LINT x:A|M. g x)"
- using assms by (simp_all add: scaleR_diff_right)
+ using assms unfolding set_integrable_def set_lebesgue_integral_def by (simp_all add: scaleR_diff_right)
(* question: why do we have this for negation, but multiplication by a constant
requires an integrability assumption? *)
lemma set_integral_uminus: "set_integrable M A f \<Longrightarrow> LINT x:A|M. - f x = - (LINT x:A|M. f x)"
+ unfolding set_integrable_def set_lebesgue_integral_def
by (subst integral_minus[symmetric]) simp_all
lemma set_integral_complex_of_real:
"LINT x:A|M. complex_of_real (f x) = of_real (LINT x:A|M. f x)"
+ unfolding set_lebesgue_integral_def
by (subst integral_complex_of_real[symmetric])
(auto intro!: Bochner_Integration.integral_cong split: split_indicator)
@@ -193,28 +210,31 @@
assumes "set_integrable M A f" "set_integrable M A g"
"\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)"
-using assms by (auto intro: integral_mono split: split_indicator)
+ using assms unfolding set_integrable_def set_lebesgue_integral_def
+ by (auto intro: integral_mono split: split_indicator)
lemma set_integral_mono_AE:
fixes f g :: "_ \<Rightarrow> real"
assumes "set_integrable M A f" "set_integrable M A g"
"AE x \<in> A in M. f x \<le> g x"
shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)"
-using assms by (auto intro: integral_mono_AE split: split_indicator)
+ using assms unfolding set_integrable_def set_lebesgue_integral_def
+ by (auto intro: integral_mono_AE split: split_indicator)
lemma set_integrable_abs: "set_integrable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar> :: real)"
- using integrable_abs[of M "\<lambda>x. f x * indicator A x"] by (simp add: abs_mult ac_simps)
+ using integrable_abs[of M "\<lambda>x. f x * indicator A x"]unfolding set_integrable_def by (simp add: abs_mult ac_simps)
lemma set_integrable_abs_iff:
fixes f :: "_ \<Rightarrow> real"
shows "set_borel_measurable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
+ unfolding set_integrable_def set_borel_measurable_def
by (subst (2) integrable_abs_iff[symmetric]) (simp_all add: abs_mult ac_simps)
lemma set_integrable_abs_iff':
fixes f :: "_ \<Rightarrow> real"
shows "f \<in> borel_measurable M \<Longrightarrow> A \<in> sets M \<Longrightarrow>
set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
-by (intro set_integrable_abs_iff) auto
+ by (simp add: set_borel_measurable_def set_integrable_abs_iff)
lemma set_integrable_discrete_difference:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
@@ -222,6 +242,7 @@
assumes diff: "(A - B) \<union> (B - A) \<subseteq> X"
assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
shows "set_integrable M A f \<longleftrightarrow> set_integrable M B f"
+ unfolding set_integrable_def
proof (rule integrable_discrete_difference[where X=X])
show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x"
using diff by (auto split: split_indicator)
@@ -233,6 +254,7 @@
assumes diff: "(A - B) \<union> (B - A) \<subseteq> X"
assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
shows "set_lebesgue_integral M A f = set_lebesgue_integral M B f"
+ unfolding set_lebesgue_integral_def
proof (rule integral_discrete_difference[where X=X])
show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x"
using diff by (auto split: split_indicator)
@@ -246,35 +268,44 @@
proof -
have "set_integrable M (A - B) f"
using f_A by (rule set_integrable_subset) auto
- from Bochner_Integration.integrable_add[OF this f_B] show ?thesis
+ with f_B have "integrable M (\<lambda>x. indicator (A - B) x *\<^sub>R f x + indicator B x *\<^sub>R f x)"
+ unfolding set_integrable_def using integrable_add by blast
+ then show ?thesis
+ unfolding set_integrable_def
by (rule integrable_cong[THEN iffD1, rotated 2]) (auto split: split_indicator)
qed
+lemma set_integrable_empty [simp]: "set_integrable M {} f"
+ by (auto simp: set_integrable_def)
+
lemma set_integrable_UN:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> set_integrable M (A i) f"
"\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M"
shows "set_integrable M (\<Union>i\<in>I. A i) f"
-using assms by (induct I) (auto intro!: set_integrable_Un)
+ using assms
+ by (induct I) (auto simp: set_integrable_Un sets.finite_UN)
lemma set_integral_Un:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "A \<inter> B = {}"
and "set_integrable M A f"
and "set_integrable M B f"
- shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
-by (auto simp add: indicator_union_arith indicator_inter_arith[symmetric]
- scaleR_add_left assms)
+shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
+ using assms
+ unfolding set_integrable_def set_lebesgue_integral_def
+ by (auto simp add: indicator_union_arith indicator_inter_arith[symmetric] scaleR_add_left)
lemma set_integral_cong_set:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
- assumes [measurable]: "set_borel_measurable M A f" "set_borel_measurable M B f"
+ assumes "set_borel_measurable M A f" "set_borel_measurable M B f"
and ae: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
shows "LINT x:B|M. f x = LINT x:A|M. f x"
+ unfolding set_lebesgue_integral_def
proof (rule integral_cong_AE)
show "AE x in M. indicator B x *\<^sub>R f x = indicator A x *\<^sub>R f x"
using ae by (auto simp: subset_eq split: split_indicator)
-qed fact+
+qed (use assms in \<open>auto simp: set_borel_measurable_def\<close>)
lemma set_borel_measurable_subset:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
@@ -282,10 +313,12 @@
shows "set_borel_measurable M B f"
proof -
have "set_borel_measurable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
- by measurable
- also have "(\<lambda>x. indicator B x *\<^sub>R indicator A x *\<^sub>R f x) = (\<lambda>x. indicator B x *\<^sub>R f x)"
+ using assms unfolding set_borel_measurable_def
+ using borel_measurable_indicator borel_measurable_scaleR by blast
+ moreover have "(\<lambda>x. indicator B x *\<^sub>R indicator A x *\<^sub>R f x) = (\<lambda>x. indicator B x *\<^sub>R f x)"
using \<open>B \<subseteq> A\<close> by (auto simp: fun_eq_iff split: split_indicator)
- finally show ?thesis .
+ ultimately show ?thesis
+ unfolding set_borel_measurable_def by simp
qed
lemma set_integral_Un_AE:
@@ -298,7 +331,7 @@
have f: "set_integrable M (A \<union> B) f"
by (intro set_integrable_Un assms)
then have f': "set_borel_measurable M (A \<union> B) f"
- by (rule borel_measurable_integrable)
+ using integrable_iff_bounded set_borel_measurable_def set_integrable_def by blast
have "LINT x:A\<union>B|M. f x = LINT x:(A - A \<inter> B) \<union> (B - A \<inter> B)|M. f x"
proof (rule set_integral_cong_set)
show "AE x in M. (x \<in> A - A \<inter> B \<union> (B - A \<inter> B)) = (x \<in> A \<union> B)"
@@ -321,9 +354,13 @@
and "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
shows "(LINT x:(\<Union>i\<in>I. A i)|M. f x) = (\<Sum>i\<in>I. LINT x:A i|M. f x)"
using assms
- apply induct
- apply (auto intro!: set_integral_Un set_integrable_Un set_integrable_UN simp: disjoint_family_on_def)
-by (subst set_integral_Un, auto intro: set_integrable_UN)
+proof induction
+ case (insert x F)
+ then have "A x \<inter> UNION F A = {}"
+ by (meson disjoint_family_on_insert)
+ with insert show ?case
+ by (simp add: set_integral_Un set_integrable_Un set_integrable_UN disjoint_family_on_insert)
+qed (simp add: set_lebesgue_integral_def)
(* TODO: find a better name? *)
lemma pos_integrable_to_top:
@@ -332,10 +369,11 @@
assumes nneg: "\<And>x i. x \<in> A i \<Longrightarrow> 0 \<le> f x"
and intgbl: "\<And>i::nat. set_integrable M (A i) f"
and lim: "(\<lambda>i::nat. LINT x:A i|M. f x) \<longlonglongrightarrow> l"
- shows "set_integrable M (\<Union>i. A i) f"
+shows "set_integrable M (\<Union>i. A i) f"
+ unfolding set_integrable_def
apply (rule integrable_monotone_convergence[where f = "\<lambda>i::nat. \<lambda>x. indicator (A i) x *\<^sub>R f x" and x = l])
- apply (rule intgbl)
- prefer 3 apply (rule lim)
+ apply (rule intgbl [unfolded set_integrable_def])
+ prefer 3 apply (rule lim [unfolded set_lebesgue_integral_def])
apply (rule AE_I2)
using \<open>mono A\<close> apply (auto simp: mono_def nneg split: split_indicator) []
proof (rule AE_I2)
@@ -356,8 +394,7 @@
then show "(\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R f x) \<in> borel_measurable M"
apply (rule borel_measurable_LIMSEQ_real)
apply assumption
- apply (intro borel_measurable_integrable intgbl)
- done
+ using intgbl set_integrable_def by blast
qed
(* Proof from Royden Real Analysis, p. 91. *)
@@ -367,16 +404,18 @@
and disj: "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
and intgbl: "set_integrable M (\<Union>i. A i) f"
shows "LINT x:(\<Union>i. A i)|M. f x = (\<Sum>i. (LINT x:(A i)|M. f x))"
+ unfolding set_lebesgue_integral_def
proof (subst integral_suminf[symmetric])
- show int_A: "\<And>i. set_integrable M (A i) f"
- using intgbl by (rule set_integrable_subset) auto
+ show int_A: "integrable M (\<lambda>x. indicat_real (A i) x *\<^sub>R f x)" for i
+ using intgbl unfolding set_integrable_def [symmetric]
+ by (rule set_integrable_subset) auto
{ fix x assume "x \<in> space M"
have "(\<lambda>i. indicator (A i) x *\<^sub>R f x) sums (indicator (\<Union>i. A i) x *\<^sub>R f x)"
by (intro sums_scaleR_left indicator_sums) fact }
note sums = this
have norm_f: "\<And>i. set_integrable M (A i) (\<lambda>x. norm (f x))"
- using int_A[THEN integrable_norm] by auto
+ using int_A[THEN integrable_norm] unfolding set_integrable_def by auto
show "AE x in M. summable (\<lambda>i. norm (indicator (A i) x *\<^sub>R f x))"
using disj by (intro AE_I2) (auto intro!: summable_mult2 sums_summable[OF indicator_sums])
@@ -387,21 +426,22 @@
show "0 \<le> LINT x|M. norm (indicator (A n) x *\<^sub>R f x)"
using norm_f by (auto intro!: integral_nonneg_AE)
- have "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) =
- (\<Sum>i<n. set_lebesgue_integral M (A i) (\<lambda>x. norm (f x)))"
- by (simp add: abs_mult)
+ have "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) = (\<Sum>i<n. LINT x:A i|M. norm (f x))"
+ by (simp add: abs_mult set_lebesgue_integral_def)
also have "\<dots> = set_lebesgue_integral M (\<Union>i<n. A i) (\<lambda>x. norm (f x))"
using norm_f
by (subst set_integral_finite_Union) (auto simp: disjoint_family_on_def disj)
also have "\<dots> \<le> set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))"
- using intgbl[THEN integrable_norm]
- by (intro integral_mono set_integrable_UN[of "{..<n}"] norm_f)
- (auto split: split_indicator)
+ using intgbl[unfolded set_integrable_def, THEN integrable_norm] norm_f
+ unfolding set_lebesgue_integral_def set_integrable_def
+ apply (intro integral_mono set_integrable_UN[of "{..<n}", unfolded set_integrable_def])
+ apply (auto split: split_indicator)
+ done
finally show "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) \<le>
set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))"
by simp
qed
- show "set_lebesgue_integral M (UNION UNIV A) f = LINT x|M. (\<Sum>i. indicator (A i) x *\<^sub>R f x)"
+ show "LINT x|M. indicator (UNION UNIV A) x *\<^sub>R f x = LINT x|M. (\<Sum>i. indicator (A i) x *\<^sub>R f x)"
apply (rule Bochner_Integration.integral_cong[OF refl])
apply (subst suminf_scaleR_left[OF sums_summable[OF indicator_sums, OF disj], symmetric])
using sums_unique[OF indicator_sums[OF disj]]
@@ -413,14 +453,18 @@
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "incseq A"
and intgbl: "set_integrable M (\<Union>i. A i) f"
- shows "(\<lambda>i. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Union>i. A i)|M. f x"
+shows "(\<lambda>i. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Union>i. A i)|M. f x"
+ unfolding set_lebesgue_integral_def
proof (intro integral_dominated_convergence[where w="\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R norm (f x)"])
have int_A: "\<And>i. set_integrable M (A i) f"
using intgbl by (rule set_integrable_subset) auto
- then show "\<And>i. set_borel_measurable M (A i) f" "set_borel_measurable M (\<Union>i. A i) f"
- "set_integrable M (\<Union>i. A i) (\<lambda>x. norm (f x))"
- using intgbl integrable_norm[OF intgbl] by auto
-
+ show "\<And>i. (\<lambda>x. indicator (A i) x *\<^sub>R f x) \<in> borel_measurable M"
+ using int_A integrable_iff_bounded set_integrable_def by blast
+ show "(\<lambda>x. indicator (UNION UNIV A) x *\<^sub>R f x) \<in> borel_measurable M"
+ using integrable_iff_bounded intgbl set_integrable_def by blast
+ show "integrable M (\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R norm (f x))"
+ using int_A intgbl integrable_norm unfolding set_integrable_def
+ by fastforce
{ fix x i assume "x \<in> A i"
with A have "(\<lambda>xa. indicator (A xa) x::real) \<longlonglongrightarrow> 1 \<longleftrightarrow> (\<lambda>xa. 1::real) \<longlonglongrightarrow> 1"
by (intro filterlim_cong refl)
@@ -435,15 +479,19 @@
assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "decseq A"
and int0: "set_integrable M (A 0) f"
shows "(\<lambda>i::nat. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Inter>i. A i)|M. f x"
+ unfolding set_lebesgue_integral_def
proof (rule integral_dominated_convergence)
have int_A: "\<And>i. set_integrable M (A i) f"
using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
- show "set_integrable M (A 0) (\<lambda>x. norm (f x))"
- using int0[THEN integrable_norm] by simp
+ have "integrable M (\<lambda>c. norm (indicat_real (A 0) c *\<^sub>R f c))"
+ by (metis (no_types) int0 integrable_norm set_integrable_def)
+ then show "integrable M (\<lambda>x. indicator (A 0) x *\<^sub>R norm (f x))"
+ by force
have "set_integrable M (\<Inter>i. A i) f"
using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
- with int_A show "set_borel_measurable M (\<Inter>i. A i) f" "\<And>i. set_borel_measurable M (A i) f"
- by auto
+ with int_A show "(\<lambda>x. indicat_real (INTER UNIV A) x *\<^sub>R f x) \<in> borel_measurable M"
+ "\<And>i. (\<lambda>x. indicat_real (A i) x *\<^sub>R f x) \<in> borel_measurable M"
+ by (auto simp: set_integrable_def)
show "\<And>i. AE x in M. norm (indicator (A i) x *\<^sub>R f x) \<le> indicator (A 0) x *\<^sub>R norm (f x)"
using A by (auto split: split_indicator simp: decseq_def)
{ fix x i assume "x \<in> space M" "x \<notin> A i"
@@ -462,7 +510,8 @@
proof-
have "set_lebesgue_integral M {a} f = set_lebesgue_integral M {a} (%x. f a)"
by (intro set_lebesgue_integral_cong) simp_all
- then show ?thesis using assms by simp
+ then show ?thesis using assms
+ unfolding set_lebesgue_integral_def by simp
qed
@@ -523,6 +572,7 @@
lemma set_measurable_continuous_on_ivl:
assumes "continuous_on {a..b} (f :: real \<Rightarrow> real)"
shows "set_borel_measurable borel {a..b} f"
+ unfolding set_borel_measurable_def
by (rule borel_measurable_continuous_on_indicator[OF _ assms]) simp
@@ -670,19 +720,19 @@
lemma set_integral_null_delta:
fixes f::"_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes [measurable]: "integrable M f" "A \<in> sets M" "B \<in> sets M"
- and "(A - B) \<union> (B - A) \<in> null_sets M"
+ and null: "(A - B) \<union> (B - A) \<in> null_sets M"
shows "(\<integral>x \<in> A. f x \<partial>M) = (\<integral>x \<in> B. f x \<partial>M)"
-proof (rule set_integral_cong_set, auto)
+proof (rule set_integral_cong_set)
have *: "AE a in M. a \<notin> (A - B) \<union> (B - A)"
- using assms(4) AE_not_in by blast
+ using null AE_not_in by blast
then show "AE x in M. (x \<in> B) = (x \<in> A)"
by auto
-qed
+qed (simp_all add: set_borel_measurable_def)
lemma set_integral_space:
assumes "integrable M f"
shows "(\<integral>x \<in> space M. f x \<partial>M) = (\<integral>x. f x \<partial>M)"
-by (metis (mono_tags, lifting) indicator_simps(1) Bochner_Integration.integral_cong real_vector.scale_one)
+ by (metis (no_types, lifting) indicator_simps(1) integral_cong scaleR_one set_lebesgue_integral_def)
lemma null_if_pos_func_has_zero_nn_int:
fixes f::"'a \<Rightarrow> ennreal"
@@ -703,7 +753,8 @@
proof -
have "AE x in M. indicator A x * f x = 0"
apply (subst integral_nonneg_eq_0_iff_AE[symmetric])
- using assms integrable_mult_indicator[OF \<open>A \<in> sets M\<close> assms(1)] by auto
+ using assms integrable_mult_indicator[OF \<open>A \<in> sets M\<close> assms(1)]
+ by (auto simp: set_lebesgue_integral_def)
then have "AE x\<in>A in M. f x = 0" by auto
then have "AE x\<in>A in M. False" using assms(3) by auto
then show "A \<in> null_sets M" using assms(2) by (simp add: AE_iff_null_sets)
@@ -728,31 +779,29 @@
lemma density_unique_real:
fixes f f'::"_ \<Rightarrow> real"
- assumes [measurable]: "integrable M f" "integrable M f'"
+ assumes M[measurable]: "integrable M f" "integrable M f'"
assumes density_eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>x \<in> A. f x \<partial>M) = (\<integral>x \<in> A. f' x \<partial>M)"
shows "AE x in M. f x = f' x"
proof -
define A where "A = {x \<in> space M. f x < f' x}"
then have [measurable]: "A \<in> sets M" by simp
have "(\<integral>x \<in> A. (f' x - f x) \<partial>M) = (\<integral>x \<in> A. f' x \<partial>M) - (\<integral>x \<in> A. f x \<partial>M)"
- using \<open>A \<in> sets M\<close> assms(1) assms(2) integrable_mult_indicator by blast
+ using \<open>A \<in> sets M\<close> M integrable_mult_indicator set_integrable_def by blast
then have "(\<integral>x \<in> A. (f' x - f x) \<partial>M) = 0" using assms(3) by simp
then have "A \<in> null_sets M"
using A_def null_if_pos_func_has_zero_int[where ?f = "\<lambda>x. f' x - f x" and ?A = A] assms by auto
then have "AE x in M. x \<notin> A" by (simp add: AE_not_in)
then have *: "AE x in M. f' x \<le> f x" unfolding A_def by auto
-
define B where "B = {x \<in> space M. f' x < f x}"
then have [measurable]: "B \<in> sets M" by simp
have "(\<integral>x \<in> B. (f x - f' x) \<partial>M) = (\<integral>x \<in> B. f x \<partial>M) - (\<integral>x \<in> B. f' x \<partial>M)"
- using \<open>B \<in> sets M\<close> assms(1) assms(2) integrable_mult_indicator by blast
+ using \<open>B \<in> sets M\<close> M integrable_mult_indicator set_integrable_def by blast
then have "(\<integral>x \<in> B. (f x - f' x) \<partial>M) = 0" using assms(3) by simp
then have "B \<in> null_sets M"
using B_def null_if_pos_func_has_zero_int[where ?f = "\<lambda>x. f x - f' x" and ?A = B] assms by auto
then have "AE x in M. x \<notin> B" by (simp add: AE_not_in)
then have "AE x in M. f' x \<ge> f x" unfolding B_def by auto
-
then show ?thesis using * by auto
qed