--- a/src/HOL/Probability/Lebesgue_Integration.thy Wed Feb 02 10:35:41 2011 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Wed Feb 02 12:34:45 2011 +0100
@@ -7,6 +7,7 @@
begin
lemma sums_If_finite:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
assumes finite: "finite {r. P r}"
shows "(\<lambda>r. if P r then f r else 0) sums (\<Sum>r\<in>{r. P r}. f r)" (is "?F sums _")
proof cases
@@ -24,7 +25,8 @@
qed
lemma sums_single:
- "(\<lambda>r. if r = i then f r else 0) sums f i"
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ shows "(\<lambda>r. if r = i then f r else 0) sums f i"
using sums_If_finite[of "\<lambda>r. r = i" f] by simp
section "Simple function"
@@ -37,12 +39,12 @@
*}
-definition (in sigma_algebra) "simple_function g \<longleftrightarrow>
+definition "simple_function M g \<longleftrightarrow>
finite (g ` space M) \<and>
(\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
lemma (in sigma_algebra) simple_functionD:
- assumes "simple_function g"
+ assumes "simple_function M g"
shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
proof -
show "finite (g ` space M)"
@@ -55,7 +57,7 @@
lemma (in sigma_algebra) simple_function_indicator_representation:
fixes f ::"'a \<Rightarrow> pextreal"
- assumes f: "simple_function f" and x: "x \<in> space M"
+ assumes f: "simple_function M f" and x: "x \<in> space M"
shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
(is "?l = ?r")
proof -
@@ -69,7 +71,7 @@
qed
lemma (in measure_space) simple_function_notspace:
- "simple_function (\<lambda>x. h x * indicator (- space M) x::pextreal)" (is "simple_function ?h")
+ "simple_function M (\<lambda>x. h x * indicator (- space M) x::pextreal)" (is "simple_function M ?h")
proof -
have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
@@ -79,7 +81,7 @@
lemma (in sigma_algebra) simple_function_cong:
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
- shows "simple_function f \<longleftrightarrow> simple_function g"
+ shows "simple_function M f \<longleftrightarrow> simple_function M g"
proof -
have "f ` space M = g ` space M"
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
@@ -87,15 +89,21 @@
thus ?thesis unfolding simple_function_def using assms by simp
qed
+lemma (in sigma_algebra) simple_function_cong_algebra:
+ assumes "sets N = sets M" "space N = space M"
+ shows "simple_function M f \<longleftrightarrow> simple_function N f"
+ unfolding simple_function_def assms ..
+
lemma (in sigma_algebra) borel_measurable_simple_function:
- assumes "simple_function f"
+ assumes "simple_function M f"
shows "f \<in> borel_measurable M"
proof (rule borel_measurableI)
fix S
let ?I = "f ` (f -` S \<inter> space M)"
have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
have "finite ?I"
- using assms unfolding simple_function_def by (auto intro: finite_subset)
+ using assms unfolding simple_function_def
+ using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto
hence "?U \<in> sets M"
apply (rule finite_UN)
using assms unfolding simple_function_def by auto
@@ -105,17 +113,17 @@
lemma (in sigma_algebra) simple_function_borel_measurable:
fixes f :: "'a \<Rightarrow> 'x::t2_space"
assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
- shows "simple_function f"
+ shows "simple_function M f"
using assms unfolding simple_function_def
by (auto intro: borel_measurable_vimage)
lemma (in sigma_algebra) simple_function_const[intro, simp]:
- "simple_function (\<lambda>x. c)"
+ "simple_function M (\<lambda>x. c)"
by (auto intro: finite_subset simp: simple_function_def)
lemma (in sigma_algebra) simple_function_compose[intro, simp]:
- assumes "simple_function f"
- shows "simple_function (g \<circ> f)"
+ assumes "simple_function M f"
+ shows "simple_function M (g \<circ> f)"
unfolding simple_function_def
proof safe
show "finite ((g \<circ> f) ` space M)"
@@ -132,7 +140,7 @@
lemma (in sigma_algebra) simple_function_indicator[intro, simp]:
assumes "A \<in> sets M"
- shows "simple_function (indicator A)"
+ shows "simple_function M (indicator A)"
proof -
have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
by (auto simp: indicator_def)
@@ -143,9 +151,9 @@
qed
lemma (in sigma_algebra) simple_function_Pair[intro, simp]:
- assumes "simple_function f"
- assumes "simple_function g"
- shows "simple_function (\<lambda>x. (f x, g x))" (is "simple_function ?p")
+ assumes "simple_function M f"
+ assumes "simple_function M g"
+ shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
unfolding simple_function_def
proof safe
show "finite (?p ` space M)"
@@ -161,16 +169,16 @@
qed
lemma (in sigma_algebra) simple_function_compose1:
- assumes "simple_function f"
- shows "simple_function (\<lambda>x. g (f x))"
+ assumes "simple_function M f"
+ shows "simple_function M (\<lambda>x. g (f x))"
using simple_function_compose[OF assms, of g]
by (simp add: comp_def)
lemma (in sigma_algebra) simple_function_compose2:
- assumes "simple_function f" and "simple_function g"
- shows "simple_function (\<lambda>x. h (f x) (g x))"
+ assumes "simple_function M f" and "simple_function M g"
+ shows "simple_function M (\<lambda>x. h (f x) (g x))"
proof -
- have "simple_function ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
+ have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
using assms by auto
thus ?thesis by (simp_all add: comp_def)
qed
@@ -183,14 +191,14 @@
and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
- assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
- shows "simple_function (\<lambda>x. \<Sum>i\<in>P. f i x)"
+ assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
+ shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
proof cases
assume "finite P" from this assms show ?thesis by induct auto
qed auto
lemma (in sigma_algebra) simple_function_le_measurable:
- assumes "simple_function f" "simple_function g"
+ assumes "simple_function M f" "simple_function M g"
shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
proof -
have *: "{x \<in> space M. f x \<le> g x} =
@@ -214,7 +222,7 @@
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
fixes u :: "'a \<Rightarrow> pextreal"
assumes u: "u \<in> borel_measurable M"
- shows "\<exists>f. (\<forall>i. simple_function (f i) \<and> (\<forall>x\<in>space M. f i x \<noteq> \<omega>)) \<and> f \<up> u"
+ shows "\<exists>f. (\<forall>i. simple_function M (f i) \<and> (\<forall>x\<in>space M. f i x \<noteq> \<omega>)) \<and> f \<up> u"
proof -
have "\<exists>f. \<forall>x j. (of_nat j \<le> u x \<longrightarrow> f x j = j*2^j) \<and>
(u x < of_nat j \<longrightarrow> of_nat (f x j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f x j)))"
@@ -406,10 +414,10 @@
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
fixes u :: "'a \<Rightarrow> pextreal"
assumes "u \<in> borel_measurable M"
- obtains (x) f where "f \<up> u" "\<And>i. simple_function (f i)" "\<And>i. \<omega>\<notin>f i`space M"
+ obtains (x) f where "f \<up> u" "\<And>i. simple_function M (f i)" "\<And>i. \<omega>\<notin>f i`space M"
proof -
from borel_measurable_implies_simple_function_sequence[OF assms]
- obtain f where x: "\<And>i. simple_function (f i)" "f \<up> u"
+ obtain f where x: "\<And>i. simple_function M (f i)" "f \<up> u"
and fin: "\<And>i. \<And>x. x\<in>space M \<Longrightarrow> f i x \<noteq> \<omega>" by auto
{ fix i from fin[of _ i] have "\<omega> \<notin> f i`space M" by fastsimp }
with x show thesis by (auto intro!: that[of f])
@@ -417,7 +425,7 @@
lemma (in sigma_algebra) simple_function_eq_borel_measurable:
fixes f :: "'a \<Rightarrow> pextreal"
- shows "simple_function f \<longleftrightarrow>
+ shows "simple_function M f \<longleftrightarrow>
finite (f`space M) \<and> f \<in> borel_measurable M"
using simple_function_borel_measurable[of f]
borel_measurable_simple_function[of f]
@@ -425,8 +433,8 @@
lemma (in measure_space) simple_function_restricted:
fixes f :: "'a \<Rightarrow> pextreal" assumes "A \<in> sets M"
- shows "sigma_algebra.simple_function (restricted_space A) f \<longleftrightarrow> simple_function (\<lambda>x. f x * indicator A x)"
- (is "sigma_algebra.simple_function ?R f \<longleftrightarrow> simple_function ?f")
+ shows "simple_function (restricted_space A) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator A x)"
+ (is "simple_function ?R f \<longleftrightarrow> simple_function M ?f")
proof -
interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
have "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
@@ -463,29 +471,26 @@
qed
lemma (in sigma_algebra) simple_function_subalgebra:
- assumes "sigma_algebra.simple_function N f"
- and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M" "sigma_algebra N"
- shows "simple_function f"
- using assms
- unfolding simple_function_def
- unfolding sigma_algebra.simple_function_def[OF N_subalgebra(3)]
- by auto
+ assumes "simple_function N f"
+ and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
+ shows "simple_function M f"
+ using assms unfolding simple_function_def by auto
lemma (in measure_space) simple_function_vimage:
assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
- and f: "sigma_algebra.simple_function M' f"
- shows "simple_function (\<lambda>x. f (T x))"
+ and f: "simple_function M' f"
+ shows "simple_function M (\<lambda>x. f (T x))"
proof (intro simple_function_def[THEN iffD2] conjI ballI)
interpret T: sigma_algebra M' by fact
have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
using T unfolding measurable_def by auto
then show "finite ((\<lambda>x. f (T x)) ` space M)"
- using f unfolding T.simple_function_def by (auto intro: finite_subset)
+ using f unfolding simple_function_def by (auto intro: finite_subset)
fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
then have "i \<in> f ` space M'"
using T unfolding measurable_def by auto
then have "f -` {i} \<inter> space M' \<in> sets M'"
- using f unfolding T.simple_function_def by auto
+ using f unfolding simple_function_def by auto
then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
using T unfolding measurable_def by auto
also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
@@ -495,12 +500,18 @@
section "Simple integral"
-definition (in measure_space) simple_integral (binder "\<integral>\<^isup>S " 10) where
- "simple_integral f = (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M))"
+definition simple_integral_def:
+ "integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * measure M (f -` {x} \<inter> space M))"
+
+syntax
+ "_simple_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> pextreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> pextreal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
+
+translations
+ "\<integral>\<^isup>S x. f \<partial>M" == "CONST integral\<^isup>S M (%x. f)"
lemma (in measure_space) simple_integral_cong:
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
- shows "simple_integral f = simple_integral g"
+ shows "integral\<^isup>S M f = integral\<^isup>S M g"
proof -
have "f ` space M = g ` space M"
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
@@ -509,18 +520,18 @@
qed
lemma (in measure_space) simple_integral_cong_measure:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A" and "simple_function f"
- shows "measure_space.simple_integral M \<nu> f = simple_integral f"
+ assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
+ and "simple_function M f"
+ shows "integral\<^isup>S N f = integral\<^isup>S M f"
proof -
- interpret v: measure_space M \<nu>
- by (rule measure_space_cong) fact
- from simple_functionD[OF `simple_function f`] assms show ?thesis
- unfolding simple_integral_def v.simple_integral_def
- by (auto intro!: setsum_cong)
+ interpret v: measure_space N
+ by (rule measure_space_cong) fact+
+ from simple_functionD[OF `simple_function M f`] assms show ?thesis
+ by (auto intro!: setsum_cong simp: simple_integral_def)
qed
lemma (in measure_space) simple_integral_const[simp]:
- "(\<integral>\<^isup>Sx. c) = c * \<mu> (space M)"
+ "(\<integral>\<^isup>Sx. c \<partial>M) = c * \<mu> (space M)"
proof (cases "space M = {}")
case True thus ?thesis unfolding simple_integral_def by simp
next
@@ -529,8 +540,8 @@
qed
lemma (in measure_space) simple_function_partition:
- assumes "simple_function f" and "simple_function g"
- shows "simple_integral f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
+ assumes "simple_function M f" and "simple_function M g"
+ shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
(is "_ = setsum _ (?p ` space M)")
proof-
let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
@@ -561,7 +572,7 @@
ultimately
have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
by (subst measure_finitely_additive) auto }
- hence "simple_integral f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
+ hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
unfolding simple_integral_def
by (subst setsum_Sigma[symmetric],
auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
@@ -584,8 +595,8 @@
qed
lemma (in measure_space) simple_integral_add[simp]:
- assumes "simple_function f" and "simple_function g"
- shows "(\<integral>\<^isup>Sx. f x + g x) = simple_integral f + simple_integral g"
+ assumes "simple_function M f" and "simple_function M g"
+ shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g"
proof -
{ fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
assume "x \<in> space M"
@@ -595,15 +606,15 @@
thus ?thesis
unfolding
simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
- simple_function_partition[OF `simple_function f` `simple_function g`]
- simple_function_partition[OF `simple_function g` `simple_function f`]
+ simple_function_partition[OF `simple_function M f` `simple_function M g`]
+ simple_function_partition[OF `simple_function M g` `simple_function M f`]
apply (subst (3) Int_commute)
by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
qed
lemma (in measure_space) simple_integral_setsum[simp]:
- assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
- shows "(\<integral>\<^isup>Sx. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
+ assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
+ shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))"
proof cases
assume "finite P"
from this assms show ?thesis
@@ -611,8 +622,8 @@
qed auto
lemma (in measure_space) simple_integral_mult[simp]:
- assumes "simple_function f"
- shows "(\<integral>\<^isup>Sx. c * f x) = c * simple_integral f"
+ assumes "simple_function M f"
+ shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f"
proof -
note mult = simple_function_mult[OF simple_function_const[of c] assms]
{ fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
@@ -626,8 +637,8 @@
qed
lemma (in sigma_algebra) simple_function_If:
- assumes sf: "simple_function f" "simple_function g" and A: "A \<in> sets M"
- shows "simple_function (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function ?IF")
+ assumes sf: "simple_function M f" "simple_function M g" and A: "A \<in> sets M"
+ shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
proof -
def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
show ?thesis unfolding simple_function_def
@@ -648,17 +659,17 @@
qed
lemma (in measure_space) simple_integral_mono_AE:
- assumes "simple_function f" and "simple_function g"
+ assumes "simple_function M f" and "simple_function M g"
and mono: "AE x. f x \<le> g x"
- shows "simple_integral f \<le> simple_integral g"
+ shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
proof -
let "?S x" = "(g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
"\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
show ?thesis
unfolding *
- simple_function_partition[OF `simple_function f` `simple_function g`]
- simple_function_partition[OF `simple_function g` `simple_function f`]
+ simple_function_partition[OF `simple_function M f` `simple_function M g`]
+ simple_function_partition[OF `simple_function M g` `simple_function M f`]
proof (safe intro!: setsum_mono)
fix x assume "x \<in> space M"
then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
@@ -680,23 +691,23 @@
qed
lemma (in measure_space) simple_integral_mono:
- assumes "simple_function f" and "simple_function g"
+ assumes "simple_function M f" and "simple_function M g"
and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
- shows "simple_integral f \<le> simple_integral g"
+ shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
proof (rule simple_integral_mono_AE[OF assms(1, 2)])
show "AE x. f x \<le> g x"
using mono by (rule AE_cong) auto
qed
lemma (in measure_space) simple_integral_cong_AE:
- assumes "simple_function f" "simple_function g" and "AE x. f x = g x"
- shows "simple_integral f = simple_integral g"
+ assumes "simple_function M f" "simple_function M g" and "AE x. f x = g x"
+ shows "integral\<^isup>S M f = integral\<^isup>S M g"
using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
lemma (in measure_space) simple_integral_cong':
- assumes sf: "simple_function f" "simple_function g"
+ assumes sf: "simple_function M f" "simple_function M g"
and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
- shows "simple_integral f = simple_integral g"
+ shows "integral\<^isup>S M f = integral\<^isup>S M g"
proof (intro simple_integral_cong_AE sf AE_I)
show "\<mu> {x\<in>space M. f x \<noteq> g x} = 0" by fact
show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
@@ -705,12 +716,12 @@
lemma (in measure_space) simple_integral_indicator:
assumes "A \<in> sets M"
- assumes "simple_function f"
- shows "(\<integral>\<^isup>Sx. f x * indicator A x) =
+ assumes "simple_function M f"
+ shows "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
(\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
proof cases
assume "A = space M"
- moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x) = simple_integral f"
+ moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = integral\<^isup>S M f"
by (auto intro!: simple_integral_cong)
moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
ultimately show ?thesis by (simp add: simple_integral_def)
@@ -726,7 +737,7 @@
next
show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
qed
- have *: "(\<integral>\<^isup>Sx. f x * indicator A x) =
+ have *: "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
(\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
unfolding simple_integral_def I
proof (rule setsum_mono_zero_cong_left)
@@ -752,7 +763,7 @@
lemma (in measure_space) simple_integral_indicator_only[simp]:
assumes "A \<in> sets M"
- shows "simple_integral (indicator A) = \<mu> A"
+ shows "integral\<^isup>S M (indicator A) = \<mu> A"
proof cases
assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
@@ -765,22 +776,22 @@
qed
lemma (in measure_space) simple_integral_null_set:
- assumes "simple_function u" "N \<in> null_sets"
- shows "(\<integral>\<^isup>Sx. u x * indicator N x) = 0"
+ assumes "simple_function M u" "N \<in> null_sets"
+ shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0"
proof -
have "AE x. indicator N x = (0 :: pextreal)"
using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
- then have "(\<integral>\<^isup>Sx. u x * indicator N x) = (\<integral>\<^isup>Sx. 0)"
+ then have "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^isup>Sx. 0 \<partial>M)"
using assms by (intro simple_integral_cong_AE) (auto intro!: AE_disjI2)
then show ?thesis by simp
qed
lemma (in measure_space) simple_integral_cong_AE_mult_indicator:
- assumes sf: "simple_function f" and eq: "AE x. x \<in> S" and "S \<in> sets M"
- shows "simple_integral f = (\<integral>\<^isup>Sx. f x * indicator S x)"
+ assumes sf: "simple_function M f" and eq: "AE x. x \<in> S" and "S \<in> sets M"
+ shows "integral\<^isup>S M f = (\<integral>\<^isup>Sx. f x * indicator S x \<partial>M)"
proof (rule simple_integral_cong_AE)
- show "simple_function f" by fact
- show "simple_function (\<lambda>x. f x * indicator S x)"
+ show "simple_function M f" by fact
+ show "simple_function M (\<lambda>x. f x * indicator S x)"
using sf `S \<in> sets M` by auto
from eq show "AE x. f x = f x * indicator S x"
by (rule AE_mp) simp
@@ -788,10 +799,9 @@
lemma (in measure_space) simple_integral_restricted:
assumes "A \<in> sets M"
- assumes sf: "simple_function (\<lambda>x. f x * indicator A x)"
- shows "measure_space.simple_integral (restricted_space A) \<mu> f = (\<integral>\<^isup>Sx. f x * indicator A x)"
- (is "_ = simple_integral ?f")
- unfolding measure_space.simple_integral_def[OF restricted_measure_space[OF `A \<in> sets M`]]
+ assumes sf: "simple_function M (\<lambda>x. f x * indicator A x)"
+ shows "integral\<^isup>S (restricted_space A) f = (\<integral>\<^isup>Sx. f x * indicator A x \<partial>M)"
+ (is "_ = integral\<^isup>S M ?f")
unfolding simple_integral_def
proof (simp, safe intro!: setsum_mono_zero_cong_left)
from sf show "finite (?f ` space M)"
@@ -816,64 +826,71 @@
qed
lemma (in measure_space) simple_integral_subalgebra:
- assumes N: "measure_space N \<mu>" and [simp]: "space N = space M"
- shows "measure_space.simple_integral N \<mu> = simple_integral"
- unfolding simple_integral_def_raw
- unfolding measure_space.simple_integral_def_raw[OF N] by simp
+ assumes N: "measure_space N" and [simp]: "space N = space M" "measure N = measure M"
+ shows "integral\<^isup>S N = integral\<^isup>S M"
+ unfolding simple_integral_def_raw by simp
lemma (in measure_space) simple_integral_vimage:
assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
- and f: "sigma_algebra.simple_function M' f"
- shows "measure_space.simple_integral M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f = (\<integral>\<^isup>S x. f (T x))"
- (is "measure_space.simple_integral M' ?nu f = _")
+ and f: "simple_function M' f"
+ and \<nu>: "\<And>A. A \<in> sets M' \<Longrightarrow> measure M' A = \<mu> (T -` A \<inter> space M)"
+ shows "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
proof -
- interpret T: measure_space M' ?nu using T by (rule measure_space_vimage) auto
- show "T.simple_integral f = (\<integral>\<^isup>S x. f (T x))"
- unfolding simple_integral_def T.simple_integral_def
+ interpret T: measure_space M' using \<nu> by (rule measure_space_vimage[OF T])
+ show "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
+ unfolding simple_integral_def
proof (intro setsum_mono_zero_cong_right ballI)
show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
using T unfolding measurable_def by auto
show "finite (f ` space M')"
- using f unfolding T.simple_function_def by auto
+ using f unfolding simple_function_def by auto
next
fix i assume "i \<in> f ` space M' - (\<lambda>x. f (T x)) ` space M"
then have "T -` (f -` {i} \<inter> space M') \<inter> space M = {}" by (auto simp: image_iff)
- then show "i * \<mu> (T -` (f -` {i} \<inter> space M') \<inter> space M) = 0" by simp
+ with f[THEN T.simple_functionD(2), THEN \<nu>]
+ show "i * T.\<mu> (f -` {i} \<inter> space M') = 0" by simp
next
fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M"
then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
using T unfolding measurable_def by auto
- then show "i * \<mu> (T -` (f -` {i} \<inter> space M') \<inter> space M) = i * \<mu> ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
+ with f[THEN T.simple_functionD(2), THEN \<nu>]
+ show "i * T.\<mu> (f -` {i} \<inter> space M') = i * \<mu> ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
by auto
qed
qed
-section "Continuous posititve integration"
+section "Continuous positive integration"
+
+definition positive_integral_def:
+ "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> f}. integral\<^isup>S M g)"
-definition (in measure_space) positive_integral (binder "\<integral>\<^isup>+ " 10) where
- "positive_integral f = SUPR {g. simple_function g \<and> g \<le> f} simple_integral"
+syntax
+ "_positive_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> pextreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> pextreal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
+
+translations
+ "\<integral>\<^isup>+ x. f \<partial>M" == "CONST integral\<^isup>P M (%x. f)"
-lemma (in measure_space) positive_integral_alt:
- "positive_integral f =
- (SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M} simple_integral)" (is "_ = ?alt")
+lemma (in measure_space) positive_integral_alt: "integral\<^isup>P M f =
+ (SUP g : {g. simple_function M g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}. integral\<^isup>S M g)"
+ (is "_ = ?alt")
proof (rule antisym SUP_leI)
- show "positive_integral f \<le> ?alt" unfolding positive_integral_def
+ show "integral\<^isup>P M f \<le> ?alt" unfolding positive_integral_def
proof (safe intro!: SUP_leI)
- fix g assume g: "simple_function g" "g \<le> f"
+ fix g assume g: "simple_function M g" "g \<le> f"
let ?G = "g -` {\<omega>} \<inter> space M"
- show "simple_integral g \<le>
- SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g ` space M} simple_integral"
- (is "simple_integral g \<le> SUPR ?A simple_integral")
+ show "integral\<^isup>S M g \<le>
+ (SUP h : {i. simple_function M i \<and> i \<le> f \<and> \<omega> \<notin> i ` space M}. integral\<^isup>S M h)"
+ (is "integral\<^isup>S M g \<le> SUPR ?A _")
proof cases
let ?g = "\<lambda>x. indicator (space M - ?G) x * g x"
- have g': "simple_function ?g"
+ have g': "simple_function M ?g"
using g by (auto intro: simple_functionD)
moreover
assume "\<mu> ?G = 0"
then have "AE x. g x = ?g x" using g
by (intro AE_I[where N="?G"])
(auto intro: simple_functionD simp: indicator_def)
- with g(1) g' have "simple_integral g = simple_integral ?g"
+ with g(1) g' have "integral\<^isup>S M g = integral\<^isup>S M ?g"
by (rule simple_integral_cong_AE)
moreover have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
from this `g \<le> f` have "?g \<le> f" by (rule order_trans)
@@ -885,15 +902,15 @@
then have "?G \<noteq> {}" by auto
then have "\<omega> \<in> g`space M" by force
then have "space M \<noteq> {}" by auto
- have "SUPR ?A simple_integral = \<omega>"
+ have "SUPR ?A (integral\<^isup>S M) = \<omega>"
proof (intro SUP_\<omega>[THEN iffD2] allI impI)
fix x assume "x < \<omega>"
then guess n unfolding less_\<omega>_Ex_of_nat .. note n = this
then have "0 < n" by (intro neq0_conv[THEN iffD1] notI) simp
let ?g = "\<lambda>x. (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * indicator ?G x"
- show "\<exists>i\<in>?A. x < simple_integral i"
+ show "\<exists>i\<in>?A. x < integral\<^isup>S M i"
proof (intro bexI impI CollectI conjI)
- show "simple_function ?g" using g
+ show "simple_function M ?g" using g
by (auto intro!: simple_functionD simple_function_add)
have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
from this g(2) show "?g \<le> f" by (rule order_trans)
@@ -902,10 +919,10 @@
have "x < (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * \<mu> ?G"
using n `\<mu> ?G \<noteq> 0` `0 < n`
by (auto simp: pextreal_noteq_omega_Ex field_simps)
- also have "\<dots> = simple_integral ?g" using g `space M \<noteq> {}`
+ also have "\<dots> = integral\<^isup>S M ?g" using g `space M \<noteq> {}`
by (subst simple_integral_indicator)
(auto simp: image_constant ac_simps dest: simple_functionD)
- finally show "x < simple_integral ?g" .
+ finally show "x < integral\<^isup>S M ?g" .
qed
qed
then show ?thesis by simp
@@ -914,40 +931,41 @@
qed (auto intro!: SUP_subset simp: positive_integral_def)
lemma (in measure_space) positive_integral_cong_measure:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A"
- shows "measure_space.positive_integral M \<nu> f = positive_integral f"
+ assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
+ shows "integral\<^isup>P N f = integral\<^isup>P M f"
proof -
- interpret v: measure_space M \<nu>
- by (rule measure_space_cong) fact
+ interpret v: measure_space N
+ by (rule measure_space_cong) fact+
with assms show ?thesis
- unfolding positive_integral_def v.positive_integral_def SUPR_def
+ unfolding positive_integral_def SUPR_def
by (auto intro!: arg_cong[where f=Sup] image_cong
- simp: simple_integral_cong_measure[of \<nu>])
+ simp: simple_integral_cong_measure[OF assms]
+ simple_function_cong_algebra[OF assms(2,3)])
qed
lemma (in measure_space) positive_integral_alt1:
- "positive_integral f =
- (SUP g : {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}. simple_integral g)"
+ "integral\<^isup>P M f =
+ (SUP g : {g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}. integral\<^isup>S M g)"
unfolding positive_integral_alt SUPR_def
proof (safe intro!: arg_cong[where f=Sup])
fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
- assume "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
- hence "?g \<le> f" "simple_function ?g" "simple_integral g = simple_integral ?g"
+ assume "simple_function M g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
+ hence "?g \<le> f" "simple_function M ?g" "integral\<^isup>S M g = integral\<^isup>S M ?g"
"\<omega> \<notin> g`space M"
unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
- thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
+ thus "integral\<^isup>S M g \<in> integral\<^isup>S M ` {g. simple_function M g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
by auto
next
- fix g assume "simple_function g" "g \<le> f" "\<omega> \<notin> g`space M"
- hence "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
+ fix g assume "simple_function M g" "g \<le> f" "\<omega> \<notin> g`space M"
+ hence "simple_function M g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
by (auto simp add: le_fun_def image_iff)
- thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}"
+ thus "integral\<^isup>S M g \<in> integral\<^isup>S M ` {g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}"
by auto
qed
lemma (in measure_space) positive_integral_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
- shows "positive_integral f = positive_integral g"
+ shows "integral\<^isup>P M f = integral\<^isup>P M g"
proof -
have "\<And>h. (\<forall>x\<in>space M. h x \<le> f x \<and> h x \<noteq> \<omega>) = (\<forall>x\<in>space M. h x \<le> g x \<and> h x \<noteq> \<omega>)"
using assms by auto
@@ -955,30 +973,30 @@
qed
lemma (in measure_space) positive_integral_eq_simple_integral:
- assumes "simple_function f"
- shows "positive_integral f = simple_integral f"
+ assumes "simple_function M f"
+ shows "integral\<^isup>P M f = integral\<^isup>S M f"
unfolding positive_integral_def
proof (safe intro!: pextreal_SUPI)
- fix g assume "simple_function g" "g \<le> f"
- with assms show "simple_integral g \<le> simple_integral f"
+ fix g assume "simple_function M g" "g \<le> f"
+ with assms show "integral\<^isup>S M g \<le> integral\<^isup>S M f"
by (auto intro!: simple_integral_mono simp: le_fun_def)
next
- fix y assume "\<forall>x. x\<in>{g. simple_function g \<and> g \<le> f} \<longrightarrow> simple_integral x \<le> y"
- with assms show "simple_integral f \<le> y" by auto
+ fix y assume "\<forall>x. x\<in>{g. simple_function M g \<and> g \<le> f} \<longrightarrow> integral\<^isup>S M x \<le> y"
+ with assms show "integral\<^isup>S M f \<le> y" by auto
qed
lemma (in measure_space) positive_integral_mono_AE:
assumes ae: "AE x. u x \<le> v x"
- shows "positive_integral u \<le> positive_integral v"
+ shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
unfolding positive_integral_alt1
proof (safe intro!: SUPR_mono)
- fix a assume a: "simple_function a" and mono: "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
+ fix a assume a: "simple_function M a" and mono: "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
from ae obtain N where N: "{x\<in>space M. \<not> u x \<le> v x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
by (auto elim!: AE_E)
- have "simple_function (\<lambda>x. a x * indicator (space M - N) x)"
+ have "simple_function M (\<lambda>x. a x * indicator (space M - N) x)"
using `N \<in> sets M` a by auto
- with a show "\<exists>b\<in>{g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> v x \<and> g x \<noteq> \<omega>)}.
- simple_integral a \<le> simple_integral b"
+ with a show "\<exists>b\<in>{g. simple_function M g \<and> (\<forall>x\<in>space M. g x \<le> v x \<and> g x \<noteq> \<omega>)}.
+ integral\<^isup>S M a \<le> integral\<^isup>S M b"
proof (safe intro!: bexI[of _ "\<lambda>x. a x * indicator (space M - N) x"]
simple_integral_mono_AE)
show "AE x. a x \<le> a x * indicator (space M - N) x"
@@ -987,7 +1005,7 @@
N \<inter> {x \<in> space M. a x \<noteq> 0}" (is "?N = _")
using `N \<in> sets M`[THEN sets_into_space] by (auto simp: indicator_def)
then show "?N \<in> sets M"
- using `N \<in> sets M` `simple_function a`[THEN borel_measurable_simple_function]
+ using `N \<in> sets M` `simple_function M a`[THEN borel_measurable_simple_function]
by (auto intro!: measure_mono Int)
then have "\<mu> ?N \<le> \<mu> N"
unfolding * using `N \<in> sets M` by (auto intro!: measure_mono)
@@ -1010,12 +1028,12 @@
qed
lemma (in measure_space) positive_integral_cong_AE:
- "AE x. u x = v x \<Longrightarrow> positive_integral u = positive_integral v"
+ "AE x. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
by (auto simp: eq_iff intro!: positive_integral_mono_AE)
lemma (in measure_space) positive_integral_mono:
assumes mono: "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x"
- shows "positive_integral u \<le> positive_integral v"
+ shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
using mono by (auto intro!: AE_cong positive_integral_mono_AE)
lemma image_set_cong:
@@ -1027,15 +1045,15 @@
lemma (in measure_space) positive_integral_SUP_approx:
assumes "f \<up> s"
and f: "\<And>i. f i \<in> borel_measurable M"
- and "simple_function u"
+ and "simple_function M u"
and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
- shows "simple_integral u \<le> (SUP i. positive_integral (f i))" (is "_ \<le> ?S")
+ shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S")
proof (rule pextreal_le_mult_one_interval)
fix a :: pextreal assume "0 < a" "a < 1"
hence "a \<noteq> 0" by auto
let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
have B: "\<And>i. ?B i \<in> sets M"
- using f `simple_function u` by (auto simp: borel_measurable_simple_function)
+ using f `simple_function M u` by (auto simp: borel_measurable_simple_function)
let "?uB i x" = "u x * indicator (?B i) x"
@@ -1049,7 +1067,7 @@
note B_mono = this
have u: "\<And>x. x \<in> space M \<Longrightarrow> u -` {u x} \<inter> space M \<in> sets M"
- using `simple_function u` by (auto simp add: simple_function_def)
+ using `simple_function M u` by (auto simp add: simple_function_def)
have "\<And>i. (\<lambda>n. (u -` {i} \<inter> space M) \<inter> ?B n) \<up> (u -` {i} \<inter> space M)" using B_mono unfolding isoton_def
proof safe
@@ -1071,8 +1089,8 @@
qed auto
note measure_conv = measure_up[OF Int[OF u B] this]
- have "simple_integral u = (SUP i. simple_integral (?uB i))"
- unfolding simple_integral_indicator[OF B `simple_function u`]
+ have "integral\<^isup>S M u = (SUP i. integral\<^isup>S M (?uB i))"
+ unfolding simple_integral_indicator[OF B `simple_function M u`]
proof (subst SUPR_pextreal_setsum, safe)
fix x n assume "x \<in> space M"
have "\<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f n x})
@@ -1082,52 +1100,51 @@
\<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
by (auto intro: mult_left_mono)
next
- show "simple_integral u =
+ show "integral\<^isup>S M u =
(\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (u -` {i} \<inter> space M \<inter> ?B n))"
using measure_conv unfolding simple_integral_def isoton_def
by (auto intro!: setsum_cong simp: pextreal_SUP_cmult)
qed
moreover
- have "a * (SUP i. simple_integral (?uB i)) \<le> ?S"
+ have "a * (SUP i. integral\<^isup>S M (?uB i)) \<le> ?S"
unfolding pextreal_SUP_cmult[symmetric]
proof (safe intro!: SUP_mono bexI)
fix i
- have "a * simple_integral (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x)"
- using B `simple_function u`
+ have "a * integral\<^isup>S M (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x \<partial>M)"
+ using B `simple_function M u`
by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
- also have "\<dots> \<le> positive_integral (f i)"
+ also have "\<dots> \<le> integral\<^isup>P M (f i)"
proof -
have "\<And>x. a * ?uB i x \<le> f i x" unfolding indicator_def by auto
- hence *: "simple_function (\<lambda>x. a * ?uB i x)" using B assms(3)
+ hence *: "simple_function M (\<lambda>x. a * ?uB i x)" using B assms(3)
by (auto intro!: simple_integral_mono)
show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
by (auto intro!: positive_integral_mono simp: indicator_def)
qed
- finally show "a * simple_integral (?uB i) \<le> positive_integral (f i)"
+ finally show "a * integral\<^isup>S M (?uB i) \<le> integral\<^isup>P M (f i)"
by auto
qed simp
- ultimately show "a * simple_integral u \<le> ?S" by simp
+ ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp
qed
text {* Beppo-Levi monotone convergence theorem *}
lemma (in measure_space) positive_integral_isoton:
assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
- shows "(\<lambda>i. positive_integral (f i)) \<up> positive_integral u"
+ shows "(\<lambda>i. integral\<^isup>P M (f i)) \<up> integral\<^isup>P M u"
unfolding isoton_def
proof safe
- fix i show "positive_integral (f i) \<le> positive_integral (f (Suc i))"
+ fix i show "integral\<^isup>P M (f i) \<le> integral\<^isup>P M (f (Suc i))"
apply (rule positive_integral_mono)
using `f \<up> u` unfolding isoton_def le_fun_def by auto
next
have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
-
- show "(SUP i. positive_integral (f i)) = positive_integral u"
+ show "(SUP i. integral\<^isup>P M (f i)) = integral\<^isup>P M u"
proof (rule antisym)
from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
- show "(SUP j. positive_integral (f j)) \<le> positive_integral u"
+ show "(SUP j. integral\<^isup>P M (f j)) \<le> integral\<^isup>P M u"
by (auto intro!: SUP_leI positive_integral_mono)
next
- show "positive_integral u \<le> (SUP i. positive_integral (f i))"
+ show "integral\<^isup>P M u \<le> (SUP i. integral\<^isup>P M (f i))"
unfolding positive_integral_alt[of u]
by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
qed
@@ -1136,12 +1153,12 @@
lemma (in measure_space) positive_integral_monotone_convergence_SUP:
assumes "\<And>i x. x \<in> space M \<Longrightarrow> f i x \<le> f (Suc i) x"
assumes "\<And>i. f i \<in> borel_measurable M"
- shows "(SUP i. positive_integral (f i)) = (\<integral>\<^isup>+ x. SUP i. f i x)"
- (is "_ = positive_integral ?u")
+ shows "(SUP i. integral\<^isup>P M (f i)) = (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
+ (is "_ = integral\<^isup>P M ?u")
proof -
show ?thesis
proof (rule antisym)
- show "(SUP j. positive_integral (f j)) \<le> positive_integral ?u"
+ show "(SUP j. integral\<^isup>P M (f j)) \<le> integral\<^isup>P M ?u"
by (auto intro!: SUP_leI positive_integral_mono le_SUPI)
next
def rf \<equiv> "\<lambda>i. \<lambda>x\<in>space M. f i x" and ru \<equiv> "\<lambda>x\<in>space M. ?u x"
@@ -1151,26 +1168,26 @@
unfolding isoton_def le_fun_def fun_eq_iff SUPR_apply
using SUP_const[OF UNIV_not_empty]
by (auto simp: restrict_def le_fun_def fun_eq_iff)
- ultimately have "positive_integral ru \<le> (SUP i. positive_integral (rf i))"
+ ultimately have "integral\<^isup>P M ru \<le> (SUP i. integral\<^isup>P M (rf i))"
unfolding positive_integral_alt[of ru]
by (auto simp: le_fun_def intro!: SUP_leI positive_integral_SUP_approx)
- then show "positive_integral ?u \<le> (SUP i. positive_integral (f i))"
+ then show "integral\<^isup>P M ?u \<le> (SUP i. integral\<^isup>P M (f i))"
unfolding ru_def rf_def by (simp cong: positive_integral_cong)
qed
qed
lemma (in measure_space) SUP_simple_integral_sequences:
- assumes f: "f \<up> u" "\<And>i. simple_function (f i)"
- and g: "g \<up> u" "\<And>i. simple_function (g i)"
- shows "(SUP i. simple_integral (f i)) = (SUP i. simple_integral (g i))"
+ assumes f: "f \<up> u" "\<And>i. simple_function M (f i)"
+ and g: "g \<up> u" "\<And>i. simple_function M (g i)"
+ shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))"
(is "SUPR _ ?F = SUPR _ ?G")
proof -
- have "(SUP i. ?F i) = (SUP i. positive_integral (f i))"
+ have "(SUP i. ?F i) = (SUP i. integral\<^isup>P M (f i))"
using assms by (simp add: positive_integral_eq_simple_integral)
- also have "\<dots> = positive_integral u"
+ also have "\<dots> = integral\<^isup>P M u"
using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
unfolding isoton_def by simp
- also have "\<dots> = (SUP i. positive_integral (g i))"
+ also have "\<dots> = (SUP i. integral\<^isup>P M (g i))"
using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
unfolding isoton_def by simp
also have "\<dots> = (SUP i. ?G i)"
@@ -1179,38 +1196,36 @@
qed
lemma (in measure_space) positive_integral_const[simp]:
- "(\<integral>\<^isup>+ x. c) = c * \<mu> (space M)"
+ "(\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
by (subst positive_integral_eq_simple_integral) auto
lemma (in measure_space) positive_integral_isoton_simple:
- assumes "f \<up> u" and e: "\<And>i. simple_function (f i)"
- shows "(\<lambda>i. simple_integral (f i)) \<up> positive_integral u"
+ assumes "f \<up> u" and e: "\<And>i. simple_function M (f i)"
+ shows "(\<lambda>i. integral\<^isup>S M (f i)) \<up> integral\<^isup>P M u"
using positive_integral_isoton[OF `f \<up> u` e(1)[THEN borel_measurable_simple_function]]
unfolding positive_integral_eq_simple_integral[OF e] .
lemma (in measure_space) positive_integral_vimage:
assumes T: "sigma_algebra M'" "T \<in> measurable M M'" and f: "f \<in> borel_measurable M'"
- shows "measure_space.positive_integral M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f = (\<integral>\<^isup>+ x. f (T x))"
- (is "measure_space.positive_integral M' ?nu f = _")
+ and \<nu>: "\<And>A. A \<in> sets M' \<Longrightarrow> measure M' A = \<mu> (T -` A \<inter> space M)"
+ shows "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
proof -
- interpret T: measure_space M' ?nu using T by (rule measure_space_vimage) auto
- obtain f' where f': "f' \<up> f" "\<And>i. T.simple_function (f' i)"
+ interpret T: measure_space M' using \<nu> by (rule measure_space_vimage[OF T])
+ obtain f' where f': "f' \<up> f" "\<And>i. simple_function M' (f' i)"
using T.borel_measurable_implies_simple_function_sequence[OF f] by blast
- then have f: "(\<lambda>i x. f' i (T x)) \<up> (\<lambda>x. f (T x))" "\<And>i. simple_function (\<lambda>x. f' i (T x))"
+ then have f: "(\<lambda>i x. f' i (T x)) \<up> (\<lambda>x. f (T x))" "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
using simple_function_vimage[OF T] unfolding isoton_fun_expand by auto
- show "T.positive_integral f = (\<integral>\<^isup>+ x. f (T x))"
+ show "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
using positive_integral_isoton_simple[OF f]
using T.positive_integral_isoton_simple[OF f']
- unfolding simple_integral_vimage[OF T f'(2)] isoton_def
- by simp
+ by (simp add: simple_integral_vimage[OF T f'(2) \<nu>] isoton_def)
qed
lemma (in measure_space) positive_integral_linear:
assumes f: "f \<in> borel_measurable M"
and g: "g \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. a * f x + g x) =
- a * positive_integral f + positive_integral g"
- (is "positive_integral ?L = _")
+ shows "(\<integral>\<^isup>+ x. a * f x + g x \<partial>M) = a * integral\<^isup>P M f + integral\<^isup>P M g"
+ (is "integral\<^isup>P M ?L = _")
proof -
from borel_measurable_implies_simple_function_sequence'[OF f] guess u .
note u = this positive_integral_isoton_simple[OF this(1-2)]
@@ -1222,46 +1237,45 @@
using assms by simp
from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
note positive_integral_isoton_simple[OF this(1-2)] and l = this
- moreover have
- "(SUP i. simple_integral (l i)) = (SUP i. simple_integral (?L' i))"
+ moreover have "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
proof (rule SUP_simple_integral_sequences[OF l(1-2)])
- show "?L' \<up> ?L" "\<And>i. simple_function (?L' i)"
+ show "?L' \<up> ?L" "\<And>i. simple_function M (?L' i)"
using u v by (auto simp: isoton_fun_expand isoton_add isoton_cmult_right)
qed
moreover from u v have L'_isoton:
- "(\<lambda>i. simple_integral (?L' i)) \<up> a * positive_integral f + positive_integral g"
+ "(\<lambda>i. integral\<^isup>S M (?L' i)) \<up> a * integral\<^isup>P M f + integral\<^isup>P M g"
by (simp add: isoton_add isoton_cmult_right)
ultimately show ?thesis by (simp add: isoton_def)
qed
lemma (in measure_space) positive_integral_cmult:
assumes "f \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. c * f x) = c * positive_integral f"
+ shows "(\<integral>\<^isup>+ x. c * f x \<partial>M) = c * integral\<^isup>P M f"
using positive_integral_linear[OF assms, of "\<lambda>x. 0" c] by auto
lemma (in measure_space) positive_integral_multc:
assumes "f \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. f x * c) = positive_integral f * c"
+ shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c"
unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
lemma (in measure_space) positive_integral_indicator[simp]:
- "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x) = \<mu> A"
+ "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x\<partial>M) = \<mu> A"
by (subst positive_integral_eq_simple_integral)
(auto simp: simple_function_indicator simple_integral_indicator)
lemma (in measure_space) positive_integral_cmult_indicator:
- "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x) = c * \<mu> A"
+ "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
by (subst positive_integral_eq_simple_integral)
(auto simp: simple_function_indicator simple_integral_indicator)
lemma (in measure_space) positive_integral_add:
assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. f x + g x) = positive_integral f + positive_integral g"
+ shows "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = integral\<^isup>P M f + integral\<^isup>P M g"
using positive_integral_linear[OF assms, of 1] by simp
lemma (in measure_space) positive_integral_setsum:
assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. positive_integral (f i))"
+ shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))"
proof cases
assume "finite P"
from this assms show ?thesis
@@ -1277,14 +1291,13 @@
lemma (in measure_space) positive_integral_diff:
assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
- and fin: "positive_integral g \<noteq> \<omega>"
+ and fin: "integral\<^isup>P M g \<noteq> \<omega>"
and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
- shows "(\<integral>\<^isup>+ x. f x - g x) = positive_integral f - positive_integral g"
+ shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g"
proof -
have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
using f g by (rule borel_measurable_pextreal_diff)
- have "(\<integral>\<^isup>+x. f x - g x) + positive_integral g =
- positive_integral f"
+ have "(\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g = integral\<^isup>P M f"
unfolding positive_integral_add[OF borel g, symmetric]
proof (rule positive_integral_cong)
fix x assume "x \<in> space M"
@@ -1297,9 +1310,9 @@
lemma (in measure_space) positive_integral_psuminf:
assumes "\<And>i. f i \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. \<Sum>\<^isub>\<infinity> i. f i x) = (\<Sum>\<^isub>\<infinity> i. positive_integral (f i))"
+ shows "(\<integral>\<^isup>+ x. (\<Sum>\<^isub>\<infinity> i. f i x) \<partial>M) = (\<Sum>\<^isub>\<infinity> i. integral\<^isup>P M (f i))"
proof -
- have "(\<lambda>i. (\<integral>\<^isup>+x. \<Sum>i<i. f i x)) \<up> (\<integral>\<^isup>+x. \<Sum>\<^isub>\<infinity>i. f i x)"
+ have "(\<lambda>i. (\<integral>\<^isup>+x. (\<Sum>i<i. f i x) \<partial>M)) \<up> (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>i. f i x) \<partial>M)"
by (rule positive_integral_isoton)
(auto intro!: borel_measurable_pextreal_setsum assms positive_integral_mono
arg_cong[where f=Sup]
@@ -1312,79 +1325,86 @@
lemma (in measure_space) positive_integral_lim_INF:
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pextreal"
assumes "\<And>i. u i \<in> borel_measurable M"
- shows "(\<integral>\<^isup>+ x. SUP n. INF m. u (m + n) x) \<le>
- (SUP n. INF m. positive_integral (u (m + n)))"
+ shows "(\<integral>\<^isup>+ x. (SUP n. INF m. u (m + n) x) \<partial>M) \<le>
+ (SUP n. INF m. integral\<^isup>P M (u (m + n)))"
proof -
- have "(\<integral>\<^isup>+x. SUP n. INF m. u (m + n) x)
- = (SUP n. (\<integral>\<^isup>+x. INF m. u (m + n) x))"
+ have "(\<integral>\<^isup>+x. (SUP n. INF m. u (m + n) x) \<partial>M)
+ = (SUP n. (\<integral>\<^isup>+x. (INF m. u (m + n) x) \<partial>M))"
using assms
by (intro positive_integral_monotone_convergence_SUP[symmetric] INF_mono)
(auto simp del: add_Suc simp add: add_Suc[symmetric])
- also have "\<dots> \<le> (SUP n. INF m. positive_integral (u (m + n)))"
+ also have "\<dots> \<le> (SUP n. INF m. integral\<^isup>P M (u (m + n)))"
by (auto intro!: SUP_mono bexI le_INFI positive_integral_mono INF_leI)
finally show ?thesis .
qed
lemma (in measure_space) measure_space_density:
assumes borel: "u \<in> borel_measurable M"
- shows "measure_space M (\<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x))" (is "measure_space M ?v")
-proof
- show "?v {} = 0" by simp
- show "countably_additive M ?v"
- unfolding countably_additive_def
- proof safe
- fix A :: "nat \<Rightarrow> 'a set"
- assume "range A \<subseteq> sets M"
- hence "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
- using borel by (auto intro: borel_measurable_indicator)
- moreover assume "disjoint_family A"
- note psuminf_indicator[OF this]
- ultimately show "(\<Sum>\<^isub>\<infinity>n. ?v (A n)) = ?v (\<Union>x. A x)"
- by (simp add: positive_integral_psuminf[symmetric])
+ and M'[simp]: "M' = (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)\<rparr>)"
+ shows "measure_space M'"
+proof -
+ interpret M': sigma_algebra M' by (intro sigma_algebra_cong) auto
+ show ?thesis
+ proof
+ show "measure M' {} = 0" unfolding M' by simp
+ show "countably_additive M' (measure M')"
+ proof (intro countably_additiveI)
+ fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M'"
+ then have "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
+ using borel by (auto intro: borel_measurable_indicator)
+ moreover assume "disjoint_family A"
+ note psuminf_indicator[OF this]
+ ultimately show "(\<Sum>\<^isub>\<infinity>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
+ by (simp add: positive_integral_psuminf[symmetric])
+ qed
qed
qed
lemma (in measure_space) positive_integral_translated_density:
assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- shows "measure_space.positive_integral M (\<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x)) g =
- (\<integral>\<^isup>+ x. f x * g x)" (is "measure_space.positive_integral M ?T _ = _")
+ and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
+ shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
proof -
- from measure_space_density[OF assms(1)]
- interpret T: measure_space M ?T .
+ from measure_space_density[OF assms(1) M']
+ interpret T: measure_space M' .
+ have borel[simp]:
+ "borel_measurable M' = borel_measurable M"
+ "simple_function M' = simple_function M"
+ unfolding measurable_def simple_function_def_raw by (auto simp: M')
from borel_measurable_implies_simple_function_sequence[OF assms(2)]
- obtain G where G: "\<And>i. simple_function (G i)" "G \<up> g" by blast
+ obtain G where G: "\<And>i. simple_function M (G i)" "G \<up> g" by blast
note G_borel = borel_measurable_simple_function[OF this(1)]
- from T.positive_integral_isoton[OF `G \<up> g` G_borel]
- have *: "(\<lambda>i. T.positive_integral (G i)) \<up> T.positive_integral g" .
+ from T.positive_integral_isoton[unfolded borel, OF `G \<up> g` G_borel]
+ have *: "(\<lambda>i. integral\<^isup>P M' (G i)) \<up> integral\<^isup>P M' g" .
{ fix i
have [simp]: "finite (G i ` space M)"
using G(1) unfolding simple_function_def by auto
- have "T.positive_integral (G i) = T.simple_integral (G i)"
+ have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
using G T.positive_integral_eq_simple_integral by simp
- also have "\<dots> = (\<integral>\<^isup>+x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x))"
- apply (simp add: T.simple_integral_def)
+ also have "\<dots> = (\<integral>\<^isup>+x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x) \<partial>M)"
+ apply (simp add: simple_integral_def M')
apply (subst positive_integral_cmult[symmetric])
- using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
+ using G_borel assms(1) apply (fastsimp intro: borel_measurable_vimage)
apply (subst positive_integral_setsum[symmetric])
- using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
+ using G_borel assms(1) apply (fastsimp intro: borel_measurable_vimage)
by (simp add: setsum_right_distrib field_simps)
- also have "\<dots> = (\<integral>\<^isup>+x. f x * G i x)"
+ also have "\<dots> = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
by (auto intro!: positive_integral_cong
simp: indicator_def if_distrib setsum_cases)
- finally have "T.positive_integral (G i) = (\<integral>\<^isup>+x. f x * G i x)" . }
- with * have eq_Tg: "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x)) \<up> T.positive_integral g" by simp
+ finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)" . }
+ with * have eq_Tg: "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x \<partial>M)) \<up> integral\<^isup>P M' g" by simp
from G(2) have "(\<lambda>i x. f x * G i x) \<up> (\<lambda>x. f x * g x)"
unfolding isoton_fun_expand by (auto intro!: isoton_cmult_right)
- then have "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x)) \<up> (\<integral>\<^isup>+x. f x * g x)"
+ then have "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x \<partial>M)) \<up> (\<integral>\<^isup>+x. f x * g x \<partial>M)"
using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pextreal_times)
- with eq_Tg show "T.positive_integral g = (\<integral>\<^isup>+x. f x * g x)"
+ with eq_Tg show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)"
unfolding isoton_def by simp
qed
lemma (in measure_space) positive_integral_null_set:
- assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x) = 0"
+ assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
proof -
- have "(\<integral>\<^isup>+ x. u x * indicator N x) = (\<integral>\<^isup>+ x. 0)"
+ have "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
proof (intro positive_integral_cong_AE AE_I)
show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
by (auto simp: indicator_def)
@@ -1396,20 +1416,20 @@
lemma (in measure_space) positive_integral_Markov_inequality:
assumes borel: "u \<in> borel_measurable M" and "A \<in> sets M" and c: "c \<noteq> \<omega>"
- shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x)"
+ shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
(is "\<mu> ?A \<le> _ * ?PI")
proof -
have "?A \<in> sets M"
using `A \<in> sets M` borel by auto
- hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x)"
+ hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
using positive_integral_indicator by simp
- also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x))"
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)"
proof (rule positive_integral_mono)
fix x assume "x \<in> space M"
show "indicator ?A x \<le> c * (u x * indicator A x)"
by (cases "x \<in> ?A") auto
qed
- also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x)"
+ also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
using assms
by (auto intro!: positive_integral_cmult borel_measurable_indicator)
finally show ?thesis .
@@ -1417,11 +1437,11 @@
lemma (in measure_space) positive_integral_0_iff:
assumes borel: "u \<in> borel_measurable M"
- shows "positive_integral u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
+ shows "integral\<^isup>P M u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
(is "_ \<longleftrightarrow> \<mu> ?A = 0")
proof -
have A: "?A \<in> sets M" using borel by auto
- have u: "(\<integral>\<^isup>+ x. u x * indicator ?A x) = positive_integral u"
+ have u: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
by (auto intro!: positive_integral_cong simp: indicator_def)
show ?thesis
@@ -1429,10 +1449,10 @@
assume "\<mu> ?A = 0"
hence "?A \<in> null_sets" using `?A \<in> sets M` by auto
from positive_integral_null_set[OF this]
- have "0 = (\<integral>\<^isup>+ x. u x * indicator ?A x)" by simp
- thus "positive_integral u = 0" unfolding u by simp
+ have "0 = (\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M)" by simp
+ thus "integral\<^isup>P M u = 0" unfolding u by simp
next
- assume *: "positive_integral u = 0"
+ assume *: "integral\<^isup>P M u = 0"
let "?M n" = "{x \<in> space M. 1 \<le> of_nat n * u x}"
have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
proof -
@@ -1469,34 +1489,34 @@
lemma (in measure_space) positive_integral_restricted:
assumes "A \<in> sets M"
- shows "measure_space.positive_integral (restricted_space A) \<mu> f = (\<integral>\<^isup>+ x. f x * indicator A x)"
- (is "measure_space.positive_integral ?R \<mu> f = positive_integral ?f")
+ shows "integral\<^isup>P (restricted_space A) f = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
+ (is "integral\<^isup>P ?R f = integral\<^isup>P M ?f")
proof -
- have msR: "measure_space ?R \<mu>" by (rule restricted_measure_space[OF `A \<in> sets M`])
- then interpret R: measure_space ?R \<mu> .
+ have msR: "measure_space ?R" by (rule restricted_measure_space[OF `A \<in> sets M`])
+ then interpret R: measure_space ?R .
have saR: "sigma_algebra ?R" by fact
- have *: "R.positive_integral f = R.positive_integral ?f"
+ have *: "integral\<^isup>P ?R f = integral\<^isup>P ?R ?f"
by (intro R.positive_integral_cong) auto
show ?thesis
- unfolding * R.positive_integral_def positive_integral_def
+ unfolding * positive_integral_def
unfolding simple_function_restricted[OF `A \<in> sets M`]
apply (simp add: SUPR_def)
apply (rule arg_cong[where f=Sup])
proof (auto simp add: image_iff simple_integral_restricted[OF `A \<in> sets M`])
- fix g assume "simple_function (\<lambda>x. g x * indicator A x)"
+ fix g assume "simple_function M (\<lambda>x. g x * indicator A x)"
"g \<le> f"
- then show "\<exists>x. simple_function x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and>
- (\<integral>\<^isup>Sx. g x * indicator A x) = simple_integral x"
+ then show "\<exists>x. simple_function M x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and>
+ (\<integral>\<^isup>Sx. g x * indicator A x \<partial>M) = integral\<^isup>S M x"
apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
by (auto simp: indicator_def le_fun_def)
next
- fix g assume g: "simple_function g" "g \<le> (\<lambda>x. f x * indicator A x)"
+ fix g assume g: "simple_function M g" "g \<le> (\<lambda>x. f x * indicator A x)"
then have *: "(\<lambda>x. g x * indicator A x) = g"
"\<And>x. g x * indicator A x = g x"
"\<And>x. g x \<le> f x"
by (auto simp: le_fun_def fun_eq_iff indicator_def split: split_if_asm)
- from g show "\<exists>x. simple_function (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and>
- simple_integral g = simple_integral (\<lambda>xa. x xa * indicator A xa)"
+ from g show "\<exists>x. simple_function M (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and>
+ integral\<^isup>S M g = integral\<^isup>S M (\<lambda>xa. x xa * indicator A xa)"
using `A \<in> sets M`[THEN sets_into_space]
apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
by (fastsimp simp: le_fun_def *)
@@ -1505,103 +1525,113 @@
lemma (in measure_space) positive_integral_subalgebra:
assumes borel: "f \<in> borel_measurable N"
- and N: "sets N \<subseteq> sets M" "space N = space M" and sa: "sigma_algebra N"
- shows "measure_space.positive_integral N \<mu> f = positive_integral f"
+ and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
+ and sa: "sigma_algebra N"
+ shows "integral\<^isup>P N f = integral\<^isup>P M f"
proof -
- interpret N: measure_space N \<mu> using measure_space_subalgebra[OF sa N] .
+ interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
from N.borel_measurable_implies_simple_function_sequence[OF borel]
- obtain fs where Nsf: "\<And>i. N.simple_function (fs i)" and "fs \<up> f" by blast
- then have sf: "\<And>i. simple_function (fs i)"
- using simple_function_subalgebra[OF _ N sa] by blast
- from positive_integral_isoton_simple[OF `fs \<up> f` sf] N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
- show ?thesis unfolding isoton_def simple_integral_def N.simple_integral_def `space N = space M` by simp
+ obtain fs where Nsf: "\<And>i. simple_function N (fs i)" and "fs \<up> f" by blast
+ then have sf: "\<And>i. simple_function M (fs i)"
+ using simple_function_subalgebra[OF _ N(1,2)] by blast
+ from N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
+ have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * N.\<mu> (fs i -` {x} \<inter> space M))"
+ unfolding isoton_def simple_integral_def `space N = space M` by simp
+ also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * \<mu> (fs i -` {x} \<inter> space M))"
+ using N N.simple_functionD(2)[OF Nsf] unfolding `space N = space M` by auto
+ also have "\<dots> = integral\<^isup>P M f"
+ using positive_integral_isoton_simple[OF `fs \<up> f` sf]
+ unfolding isoton_def simple_integral_def `space N = space M` by simp
+ finally show ?thesis .
qed
section "Lebesgue Integral"
-definition (in measure_space) integrable where
- "integrable f \<longleftrightarrow> f \<in> borel_measurable M \<and>
- (\<integral>\<^isup>+ x. Real (f x)) \<noteq> \<omega> \<and>
- (\<integral>\<^isup>+ x. Real (- f x)) \<noteq> \<omega>"
+definition integrable where
+ "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
+ (\<integral>\<^isup>+ x. Real (f x) \<partial>M) \<noteq> \<omega> \<and>
+ (\<integral>\<^isup>+ x. Real (- f x) \<partial>M) \<noteq> \<omega>"
-lemma (in measure_space) integrableD[dest]:
- assumes "integrable f"
- shows "f \<in> borel_measurable M"
- "(\<integral>\<^isup>+ x. Real (f x)) \<noteq> \<omega>"
- "(\<integral>\<^isup>+ x. Real (- f x)) \<noteq> \<omega>"
+lemma integrableD[dest]:
+ assumes "integrable M f"
+ shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. Real (f x) \<partial>M) \<noteq> \<omega>" "(\<integral>\<^isup>+ x. Real (- f x) \<partial>M) \<noteq> \<omega>"
using assms unfolding integrable_def by auto
-definition (in measure_space) integral (binder "\<integral> " 10) where
- "integral f = real ((\<integral>\<^isup>+ x. Real (f x))) - real ((\<integral>\<^isup>+ x. Real (- f x)))"
+definition lebesgue_integral_def:
+ "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. Real (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. Real (- f x) \<partial>M))"
+
+syntax
+ "_lebesgue_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
+
+translations
+ "\<integral> x. f \<partial>M" == "CONST integral\<^isup>L M (%x. f)"
lemma (in measure_space) integral_cong:
- assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
- shows "integral f = integral g"
- using assms by (simp cong: positive_integral_cong add: integral_def)
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
+ shows "integral\<^isup>L M f = integral\<^isup>L M g"
+ using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
lemma (in measure_space) integral_cong_measure:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A"
- shows "measure_space.integral M \<nu> f = integral f"
+ assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
+ shows "integral\<^isup>L N f = integral\<^isup>L M f"
proof -
- interpret v: measure_space M \<nu>
- by (rule measure_space_cong) fact
+ interpret v: measure_space N
+ by (rule measure_space_cong) fact+
show ?thesis
- unfolding integral_def v.integral_def
- by (simp add: positive_integral_cong_measure[OF assms])
+ by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
qed
lemma (in measure_space) integral_cong_AE:
assumes cong: "AE x. f x = g x"
- shows "integral f = integral g"
+ shows "integral\<^isup>L M f = integral\<^isup>L M g"
proof -
have "AE x. Real (f x) = Real (g x)"
using cong by (rule AE_mp) simp
moreover have "AE x. Real (- f x) = Real (- g x)"
using cong by (rule AE_mp) simp
ultimately show ?thesis
- by (simp cong: positive_integral_cong_AE add: integral_def)
+ by (simp cong: positive_integral_cong_AE add: lebesgue_integral_def)
qed
lemma (in measure_space) integrable_cong:
- "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable f \<longleftrightarrow> integrable g"
+ "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
lemma (in measure_space) integral_eq_positive_integral:
assumes "\<And>x. 0 \<le> f x"
- shows "integral f = real ((\<integral>\<^isup>+ x. Real (f x)))"
+ shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
proof -
have "\<And>x. Real (- f x) = 0" using assms by simp
- thus ?thesis by (simp del: Real_eq_0 add: integral_def)
+ thus ?thesis by (simp del: Real_eq_0 add: lebesgue_integral_def)
qed
lemma (in measure_space) integral_vimage:
assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
- assumes f: "measure_space.integrable M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f"
- shows "integrable (\<lambda>x. f (T x))" (is ?P)
- and "measure_space.integral M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f = (\<integral>x. f (T x))" (is ?I)
+ and \<nu>: "\<And>A. A \<in> sets M' \<Longrightarrow> measure M' A = \<mu> (T -` A \<inter> space M)"
+ assumes f: "integrable M' f"
+ shows "integrable M (\<lambda>x. f (T x))" (is ?P)
+ and "integral\<^isup>L M' f = (\<integral>x. f (T x) \<partial>M)" (is ?I)
proof -
- interpret T: measure_space M' "\<lambda>A. \<mu> (T -` A \<inter> space M)"
- using T by (rule measure_space_vimage) auto
+ interpret T: measure_space M' using \<nu> by (rule measure_space_vimage[OF T])
from measurable_comp[OF T(2), of f borel]
have borel: "(\<lambda>x. Real (f x)) \<in> borel_measurable M'" "(\<lambda>x. Real (- f x)) \<in> borel_measurable M'"
and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
- using f unfolding T.integrable_def by (auto simp: comp_def)
+ using f unfolding integrable_def by (auto simp: comp_def)
then show ?P ?I
- using f unfolding T.integral_def integral_def T.integrable_def integrable_def
- unfolding borel[THEN positive_integral_vimage[OF T]] by auto
+ using f unfolding lebesgue_integral_def integrable_def
+ by (auto simp: borel[THEN positive_integral_vimage[OF T], OF \<nu>])
qed
lemma (in measure_space) integral_minus[intro, simp]:
- assumes "integrable f"
- shows "integrable (\<lambda>x. - f x)" "(\<integral>x. - f x) = - integral f"
- using assms by (auto simp: integrable_def integral_def)
+ assumes "integrable M f"
+ shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
+ using assms by (auto simp: integrable_def lebesgue_integral_def)
lemma (in measure_space) integral_of_positive_diff:
- assumes integrable: "integrable u" "integrable v"
+ assumes integrable: "integrable M u" "integrable M v"
and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
- shows "integrable f" and "integral f = integral u - integral v"
+ shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
proof -
- let ?PI = positive_integral
let "?f x" = "Real (f x)"
let "?mf x" = "Real (- f x)"
let "?u x" = "Real (u x)"
@@ -1615,38 +1645,39 @@
"f \<in> borel_measurable M"
by (auto simp: f_def[symmetric] integrable_def)
- have "?PI (\<lambda>x. Real (u x - v x)) \<le> ?PI ?u"
+ have "(\<integral>\<^isup>+ x. Real (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
using pos by (auto intro!: positive_integral_mono)
- moreover have "?PI (\<lambda>x. Real (v x - u x)) \<le> ?PI ?v"
+ moreover have "(\<integral>\<^isup>+ x. Real (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
using pos by (auto intro!: positive_integral_mono)
- ultimately show f: "integrable f"
- using `integrable u` `integrable v` `f \<in> borel_measurable M`
+ ultimately show f: "integrable M f"
+ using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
by (auto simp: integrable_def f_def)
- hence mf: "integrable (\<lambda>x. - f x)" ..
+ hence mf: "integrable M (\<lambda>x. - f x)" ..
have *: "\<And>x. Real (- v x) = 0" "\<And>x. Real (- u x) = 0"
using pos by auto
have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
unfolding f_def using pos by simp
- hence "?PI (\<lambda>x. ?u x + ?mf x) = ?PI (\<lambda>x. ?v x + ?f x)" by simp
- hence "real (?PI ?u + ?PI ?mf) = real (?PI ?v + ?PI ?f)"
+ hence "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
+ hence "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
+ real (integral\<^isup>P M ?v + integral\<^isup>P M ?f)"
using positive_integral_add[OF u_borel mf_borel]
using positive_integral_add[OF v_borel f_borel]
by auto
- then show "integral f = integral u - integral v"
- using f mf `integrable u` `integrable v`
- unfolding integral_def integrable_def *
- by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?v", cases "?PI ?u")
+ then show "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
+ using f mf `integrable M u` `integrable M v`
+ unfolding lebesgue_integral_def integrable_def *
+ by (cases "integral\<^isup>P M ?f", cases "integral\<^isup>P M ?mf", cases "integral\<^isup>P M ?v", cases "integral\<^isup>P M ?u")
(auto simp add: field_simps)
qed
lemma (in measure_space) integral_linear:
- assumes "integrable f" "integrable g" and "0 \<le> a"
- shows "integrable (\<lambda>t. a * f t + g t)"
- and "(\<integral> t. a * f t + g t) = a * integral f + integral g"
+ assumes "integrable M f" "integrable M g" and "0 \<le> a"
+ shows "integrable M (\<lambda>t. a * f t + g t)"
+ and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g"
proof -
- let ?PI = positive_integral
+ let ?PI = "integral\<^isup>P M"
let "?f x" = "Real (f x)"
let "?g x" = "Real (g x)"
let "?mf x" = "Real (- f x)"
@@ -1670,37 +1701,36 @@
positive_integral_linear[OF pos]
positive_integral_linear[OF neg]
- have "integrable ?p" "integrable ?n"
+ have "integrable M ?p" "integrable M ?n"
"\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
using assms p n unfolding integrable_def * linear by auto
note diff = integral_of_positive_diff[OF this]
- show "integrable (\<lambda>t. a * f t + g t)" by (rule diff)
+ show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
- from assms show "(\<integral> t. a * f t + g t) = a * integral f + integral g"
- unfolding diff(2) unfolding integral_def * linear integrable_def
+ from assms show "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g"
+ unfolding diff(2) unfolding lebesgue_integral_def * linear integrable_def
by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?g", cases "?PI ?mg")
(auto simp add: field_simps zero_le_mult_iff)
qed
lemma (in measure_space) integral_add[simp, intro]:
- assumes "integrable f" "integrable g"
- shows "integrable (\<lambda>t. f t + g t)"
- and "(\<integral> t. f t + g t) = integral f + integral g"
+ assumes "integrable M f" "integrable M g"
+ shows "integrable M (\<lambda>t. f t + g t)"
+ and "(\<integral> t. f t + g t \<partial>M) = integral\<^isup>L M f + integral\<^isup>L M g"
using assms integral_linear[where a=1] by auto
lemma (in measure_space) integral_zero[simp, intro]:
- shows "integrable (\<lambda>x. 0)"
- and "(\<integral> x.0) = 0"
- unfolding integrable_def integral_def
+ shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
+ unfolding integrable_def lebesgue_integral_def
by (auto simp add: borel_measurable_const)
lemma (in measure_space) integral_cmult[simp, intro]:
- assumes "integrable f"
- shows "integrable (\<lambda>t. a * f t)" (is ?P)
- and "(\<integral> t. a * f t) = a * integral f" (is ?I)
+ assumes "integrable M f"
+ shows "integrable M (\<lambda>t. a * f t)" (is ?P)
+ and "(\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f" (is ?I)
proof -
- have "integrable (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t) = a * integral f"
+ have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f"
proof (cases rule: le_cases)
assume "0 \<le> a" show ?thesis
using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
@@ -1716,56 +1746,56 @@
qed
lemma (in measure_space) integral_multc:
- assumes "integrable f"
- shows "(\<integral> x. f x * c) = integral f * c"
+ assumes "integrable M f"
+ shows "(\<integral> x. f x * c \<partial>M) = integral\<^isup>L M f * c"
unfolding mult_commute[of _ c] integral_cmult[OF assms] ..
lemma (in measure_space) integral_mono_AE:
- assumes fg: "integrable f" "integrable g"
+ assumes fg: "integrable M f" "integrable M g"
and mono: "AE t. f t \<le> g t"
- shows "integral f \<le> integral g"
+ shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
proof -
have "AE x. Real (f x) \<le> Real (g x)"
using mono by (rule AE_mp) (auto intro!: AE_cong)
- moreover have "AE x. Real (- g x) \<le> Real (- f x)"
+ moreover have "AE x. Real (- g x) \<le> Real (- f x)"
using mono by (rule AE_mp) (auto intro!: AE_cong)
ultimately show ?thesis using fg
- by (auto simp: integral_def integrable_def diff_minus
+ by (auto simp: lebesgue_integral_def integrable_def diff_minus
intro!: add_mono real_of_pextreal_mono positive_integral_mono_AE)
qed
lemma (in measure_space) integral_mono:
- assumes fg: "integrable f" "integrable g"
+ assumes fg: "integrable M f" "integrable M g"
and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
- shows "integral f \<le> integral g"
+ shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
apply (rule integral_mono_AE[OF fg])
using mono by (rule AE_cong) auto
lemma (in measure_space) integral_diff[simp, intro]:
- assumes f: "integrable f" and g: "integrable g"
- shows "integrable (\<lambda>t. f t - g t)"
- and "(\<integral> t. f t - g t) = integral f - integral g"
+ assumes f: "integrable M f" and g: "integrable M g"
+ shows "integrable M (\<lambda>t. f t - g t)"
+ and "(\<integral> t. f t - g t \<partial>M) = integral\<^isup>L M f - integral\<^isup>L M g"
using integral_add[OF f integral_minus(1)[OF g]]
unfolding diff_minus integral_minus(2)[OF g]
by auto
lemma (in measure_space) integral_indicator[simp, intro]:
assumes "a \<in> sets M" and "\<mu> a \<noteq> \<omega>"
- shows "integral (indicator a) = real (\<mu> a)" (is ?int)
- and "integrable (indicator a)" (is ?able)
+ shows "integral\<^isup>L M (indicator a) = real (\<mu> a)" (is ?int)
+ and "integrable M (indicator a)" (is ?able)
proof -
have *:
"\<And>A x. Real (indicator A x) = indicator A x"
"\<And>A x. Real (- indicator A x) = 0" unfolding indicator_def by auto
show ?int ?able
- using assms unfolding integral_def integrable_def
+ using assms unfolding lebesgue_integral_def integrable_def
by (auto simp: * positive_integral_indicator borel_measurable_indicator)
qed
lemma (in measure_space) integral_cmul_indicator:
assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<omega>"
- shows "integrable (\<lambda>x. c * indicator A x)" (is ?P)
- and "(\<integral>x. c * indicator A x) = c * real (\<mu> A)" (is ?I)
+ shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
+ and "(\<integral>x. c * indicator A x \<partial>M) = c * real (\<mu> A)" (is ?I)
proof -
show ?P
proof (cases "c = 0")
@@ -1779,9 +1809,9 @@
qed
lemma (in measure_space) integral_setsum[simp, intro]:
- assumes "\<And>n. n \<in> S \<Longrightarrow> integrable (f n)"
- shows "(\<integral>x. \<Sum> i \<in> S. f i x) = (\<Sum> i \<in> S. integral (f i))" (is "?int S")
- and "integrable (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
+ assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f n)"
+ shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^isup>L M (f i))" (is "?int S")
+ and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
proof -
have "?int S \<and> ?I S"
proof (cases "finite S")
@@ -1792,8 +1822,8 @@
qed
lemma (in measure_space) integrable_abs:
- assumes "integrable f"
- shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
+ assumes "integrable M f"
+ shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
proof -
have *:
"\<And>x. Real \<bar>f x\<bar> = Real (f x) + Real (- f x)"
@@ -1808,56 +1838,55 @@
lemma (in measure_space) integral_subalgebra:
assumes borel: "f \<in> borel_measurable N"
- and N: "sets N \<subseteq> sets M" "space N = space M" and sa: "sigma_algebra N"
- shows "measure_space.integrable N \<mu> f \<longleftrightarrow> integrable f" (is ?P)
- and "measure_space.integral N \<mu> f = integral f" (is ?I)
+ and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A" and sa: "sigma_algebra N"
+ shows "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
+ and "integral\<^isup>L N f = integral\<^isup>L M f" (is ?I)
proof -
- interpret N: measure_space N \<mu> using measure_space_subalgebra[OF sa N] .
+ interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
have "(\<lambda>x. Real (f x)) \<in> borel_measurable N" "(\<lambda>x. Real (- f x)) \<in> borel_measurable N"
using borel by auto
note * = this[THEN positive_integral_subalgebra[OF _ N sa]]
have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
using assms unfolding measurable_def by auto
- then show ?P ?I unfolding integrable_def N.integrable_def integral_def N.integral_def
- unfolding * by auto
+ then show ?P ?I by (auto simp: * integrable_def lebesgue_integral_def)
qed
lemma (in measure_space) integrable_bound:
- assumes "integrable f"
+ assumes "integrable M f"
and f: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
"\<And>x. x \<in> space M \<Longrightarrow> \<bar>g x\<bar> \<le> f x"
assumes borel: "g \<in> borel_measurable M"
- shows "integrable g"
+ shows "integrable M g"
proof -
- have "(\<integral>\<^isup>+ x. Real (g x)) \<le> (\<integral>\<^isup>+ x. Real \<bar>g x\<bar>)"
+ have "(\<integral>\<^isup>+ x. Real (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. Real \<bar>g x\<bar> \<partial>M)"
by (auto intro!: positive_integral_mono)
- also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x))"
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
using f by (auto intro!: positive_integral_mono)
also have "\<dots> < \<omega>"
- using `integrable f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
- finally have pos: "(\<integral>\<^isup>+ x. Real (g x)) < \<omega>" .
+ using `integrable M f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
+ finally have pos: "(\<integral>\<^isup>+ x. Real (g x) \<partial>M) < \<omega>" .
- have "(\<integral>\<^isup>+ x. Real (- g x)) \<le> (\<integral>\<^isup>+ x. Real (\<bar>g x\<bar>))"
+ have "(\<integral>\<^isup>+ x. Real (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. Real (\<bar>g x\<bar>) \<partial>M)"
by (auto intro!: positive_integral_mono)
- also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x))"
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x) \<partial>M)"
using f by (auto intro!: positive_integral_mono)
also have "\<dots> < \<omega>"
- using `integrable f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
- finally have neg: "(\<integral>\<^isup>+ x. Real (- g x)) < \<omega>" .
+ using `integrable M f` unfolding integrable_def by (auto simp: pextreal_less_\<omega>)
+ finally have neg: "(\<integral>\<^isup>+ x. Real (- g x) \<partial>M) < \<omega>" .
from neg pos borel show ?thesis
unfolding integrable_def by auto
qed
lemma (in measure_space) integrable_abs_iff:
- "f \<in> borel_measurable M \<Longrightarrow> integrable (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable f"
+ "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
by (auto intro!: integrable_bound[where g=f] integrable_abs)
lemma (in measure_space) integrable_max:
- assumes int: "integrable f" "integrable g"
- shows "integrable (\<lambda> x. max (f x) (g x))"
+ assumes int: "integrable M f" "integrable M g"
+ shows "integrable M (\<lambda> x. max (f x) (g x))"
proof (rule integrable_bound)
- show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
+ show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
using int by (simp add: integrable_abs)
show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
using int unfolding integrable_def by auto
@@ -1868,10 +1897,10 @@
qed
lemma (in measure_space) integrable_min:
- assumes int: "integrable f" "integrable g"
- shows "integrable (\<lambda> x. min (f x) (g x))"
+ assumes int: "integrable M f" "integrable M g"
+ shows "integrable M (\<lambda> x. min (f x) (g x))"
proof (rule integrable_bound)
- show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
+ show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
using int by (simp add: integrable_abs)
show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
using int unfolding integrable_def by auto
@@ -1882,33 +1911,33 @@
qed
lemma (in measure_space) integral_triangle_inequality:
- assumes "integrable f"
- shows "\<bar>integral f\<bar> \<le> (\<integral>x. \<bar>f x\<bar>)"
+ assumes "integrable M f"
+ shows "\<bar>integral\<^isup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
proof -
- have "\<bar>integral f\<bar> = max (integral f) (- integral f)" by auto
- also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar>)"
+ have "\<bar>integral\<^isup>L M f\<bar> = max (integral\<^isup>L M f) (- integral\<^isup>L M f)" by auto
+ also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
using assms integral_minus(2)[of f, symmetric]
by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
finally show ?thesis .
qed
lemma (in measure_space) integral_positive:
- assumes "integrable f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
- shows "0 \<le> integral f"
+ assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
+ shows "0 \<le> integral\<^isup>L M f"
proof -
- have "0 = (\<integral>x. 0)" by (auto simp: integral_zero)
- also have "\<dots> \<le> integral f"
+ have "0 = (\<integral>x. 0 \<partial>M)" by (auto simp: integral_zero)
+ also have "\<dots> \<le> integral\<^isup>L M f"
using assms by (rule integral_mono[OF integral_zero(1)])
finally show ?thesis .
qed
lemma (in measure_space) integral_monotone_convergence_pos:
- assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
+ assumes i: "\<And>i. integrable M (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
and pos: "\<And>x i. 0 \<le> f i x"
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
- and ilim: "(\<lambda>i. integral (f i)) ----> x"
- shows "integrable u"
- and "integral u = x"
+ and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
+ shows "integrable M u"
+ and "integral\<^isup>L M u = x"
proof -
{ fix x have "0 \<le> u x"
using mono pos[of 0 x] incseq_le[OF _ lim, of x 0]
@@ -1928,43 +1957,42 @@
hence borel_u: "u \<in> borel_measurable M"
using pos_u by (auto simp: borel_measurable_Real_eq SUP_F)
- have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. Real (f n x)) = Real (integral (f n))"
- using i unfolding integral_def integrable_def by (auto simp: Real_real)
+ have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. Real (f n x) \<partial>M) = Real (integral\<^isup>L M (f n))"
+ using i unfolding lebesgue_integral_def integrable_def by (auto simp: Real_real)
- have pos_integral: "\<And>n. 0 \<le> integral (f n)"
+ have pos_integral: "\<And>n. 0 \<le> integral\<^isup>L M (f n)"
using pos i by (auto simp: integral_positive)
hence "0 \<le> x"
using LIMSEQ_le_const[OF ilim, of 0] by auto
- have "(\<lambda>i. (\<integral>\<^isup>+ x. Real (f i x))) \<up> (\<integral>\<^isup>+ x. Real (u x))"
+ have "(\<lambda>i. (\<integral>\<^isup>+ x. Real (f i x) \<partial>M)) \<up> (\<integral>\<^isup>+ x. Real (u x) \<partial>M)"
proof (rule positive_integral_isoton)
from SUP_F mono pos
show "(\<lambda>i x. Real (f i x)) \<up> (\<lambda>x. Real (u x))"
unfolding isoton_fun_expand by (auto simp: isoton_def mono_def)
qed (rule borel_f)
- hence pI: "(\<integral>\<^isup>+ x. Real (u x)) =
- (SUP n. (\<integral>\<^isup>+ x. Real (f n x)))"
+ hence pI: "(\<integral>\<^isup>+ x. Real (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. Real (f n x) \<partial>M))"
unfolding isoton_def by simp
also have "\<dots> = Real x" unfolding integral_eq
proof (rule SUP_eq_LIMSEQ[THEN iffD2])
- show "mono (\<lambda>n. integral (f n))"
+ show "mono (\<lambda>n. integral\<^isup>L M (f n))"
using mono i by (auto simp: mono_def intro!: integral_mono)
- show "\<And>n. 0 \<le> integral (f n)" using pos_integral .
+ show "\<And>n. 0 \<le> integral\<^isup>L M (f n)" using pos_integral .
show "0 \<le> x" using `0 \<le> x` .
- show "(\<lambda>n. integral (f n)) ----> x" using ilim .
+ show "(\<lambda>n. integral\<^isup>L M (f n)) ----> x" using ilim .
qed
- finally show "integrable u" "integral u = x" using borel_u `0 \<le> x`
- unfolding integrable_def integral_def by auto
+ finally show "integrable M u" "integral\<^isup>L M u = x" using borel_u `0 \<le> x`
+ unfolding integrable_def lebesgue_integral_def by auto
qed
lemma (in measure_space) integral_monotone_convergence:
- assumes f: "\<And>i. integrable (f i)" and "mono f"
+ assumes f: "\<And>i. integrable M (f i)" and "mono f"
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
- and ilim: "(\<lambda>i. integral (f i)) ----> x"
- shows "integrable u"
- and "integral u = x"
+ and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
+ shows "integrable M u"
+ and "integral\<^isup>L M u = x"
proof -
- have 1: "\<And>i. integrable (\<lambda>x. f i x - f 0 x)"
+ have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
using f by (auto intro!: integral_diff)
have 2: "\<And>x. mono (\<lambda>n. f n x - f 0 x)" using `mono f`
unfolding mono_def le_fun_def by auto
@@ -1972,43 +2000,43 @@
unfolding mono_def le_fun_def by (auto simp: field_simps)
have 4: "\<And>x. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
using lim by (auto intro!: LIMSEQ_diff)
- have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x)) ----> x - integral (f 0)"
+ have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^isup>L M (f 0)"
using f ilim by (auto intro!: LIMSEQ_diff simp: integral_diff)
note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5]
- have "integrable (\<lambda>x. (u x - f 0 x) + f 0 x)"
+ have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
using diff(1) f by (rule integral_add(1))
- with diff(2) f show "integrable u" "integral u = x"
+ with diff(2) f show "integrable M u" "integral\<^isup>L M u = x"
by (auto simp: integral_diff)
qed
lemma (in measure_space) integral_0_iff:
- assumes "integrable f"
- shows "(\<integral>x. \<bar>f x\<bar>) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
+ assumes "integrable M f"
+ shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
proof -
have *: "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
- have "integrable (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
+ have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
hence "(\<lambda>x. Real (\<bar>f x\<bar>)) \<in> borel_measurable M"
- "(\<integral>\<^isup>+ x. Real \<bar>f x\<bar>) \<noteq> \<omega>" unfolding integrable_def by auto
+ "(\<integral>\<^isup>+ x. Real \<bar>f x\<bar> \<partial>M) \<noteq> \<omega>" unfolding integrable_def by auto
from positive_integral_0_iff[OF this(1)] this(2)
- show ?thesis unfolding integral_def *
+ show ?thesis unfolding lebesgue_integral_def *
by (simp add: real_of_pextreal_eq_0)
qed
lemma (in measure_space) positive_integral_omega:
assumes "f \<in> borel_measurable M"
- and "positive_integral f \<noteq> \<omega>"
+ and "integral\<^isup>P M f \<noteq> \<omega>"
shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
proof -
- have "\<omega> * \<mu> (f -` {\<omega>} \<inter> space M) = (\<integral>\<^isup>+ x. \<omega> * indicator (f -` {\<omega>} \<inter> space M) x)"
+ have "\<omega> * \<mu> (f -` {\<omega>} \<inter> space M) = (\<integral>\<^isup>+ x. \<omega> * indicator (f -` {\<omega>} \<inter> space M) x \<partial>M)"
using positive_integral_cmult_indicator[OF borel_measurable_vimage, OF assms(1), of \<omega> \<omega>] by simp
- also have "\<dots> \<le> positive_integral f"
+ also have "\<dots> \<le> integral\<^isup>P M f"
by (auto intro!: positive_integral_mono simp: indicator_def)
finally show ?thesis
using assms(2) by (cases ?thesis) auto
qed
lemma (in measure_space) positive_integral_omega_AE:
- assumes "f \<in> borel_measurable M" "positive_integral f \<noteq> \<omega>" shows "AE x. f x \<noteq> \<omega>"
+ assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<omega>" shows "AE x. f x \<noteq> \<omega>"
proof (rule AE_I)
show "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
by (rule positive_integral_omega[OF assms])
@@ -2017,39 +2045,39 @@
qed auto
lemma (in measure_space) simple_integral_omega:
- assumes "simple_function f"
- and "simple_integral f \<noteq> \<omega>"
+ assumes "simple_function M f"
+ and "integral\<^isup>S M f \<noteq> \<omega>"
shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
proof (rule positive_integral_omega)
show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
- show "positive_integral f \<noteq> \<omega>"
+ show "integral\<^isup>P M f \<noteq> \<omega>"
using assms by (simp add: positive_integral_eq_simple_integral)
qed
lemma (in measure_space) integral_real:
fixes f :: "'a \<Rightarrow> pextreal"
assumes "AE x. f x \<noteq> \<omega>"
- shows "(\<integral>x. real (f x)) = real (positive_integral f)" (is ?plus)
- and "(\<integral>x. - real (f x)) = - real (positive_integral f)" (is ?minus)
+ shows "(\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f)" (is ?plus)
+ and "(\<integral>x. - real (f x) \<partial>M) = - real (integral\<^isup>P M f)" (is ?minus)
proof -
- have "(\<integral>\<^isup>+ x. Real (real (f x))) = positive_integral f"
+ have "(\<integral>\<^isup>+ x. Real (real (f x)) \<partial>M) = integral\<^isup>P M f"
apply (rule positive_integral_cong_AE)
apply (rule AE_mp[OF assms(1)])
by (auto intro!: AE_cong simp: Real_real)
moreover
- have "(\<integral>\<^isup>+ x. Real (- real (f x))) = (\<integral>\<^isup>+ x. 0)"
+ have "(\<integral>\<^isup>+ x. Real (- real (f x)) \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
by (intro positive_integral_cong) auto
ultimately show ?plus ?minus
- by (auto simp: integral_def integrable_def)
+ by (auto simp: lebesgue_integral_def integrable_def)
qed
lemma (in measure_space) integral_dominated_convergence:
- assumes u: "\<And>i. integrable (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
- and w: "integrable w" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> w x"
+ assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
+ and w: "integrable M w" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> w x"
and u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
- shows "integrable u'"
- and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar>)) ----> 0" (is "?lim_diff")
- and "(\<lambda>i. integral (u i)) ----> integral u'" (is ?lim)
+ shows "integrable M u'"
+ and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar> \<partial>M)) ----> 0" (is "?lim_diff")
+ and "(\<lambda>i. integral\<^isup>L M (u i)) ----> integral\<^isup>L M u'" (is ?lim)
proof -
{ fix x j assume x: "x \<in> space M"
from u'[OF x] have "(\<lambda>i. \<bar>u i x\<bar>) ----> \<bar>u' x\<bar>" by (rule LIMSEQ_imp_rabs)
@@ -2061,9 +2089,9 @@
have u'_borel: "u' \<in> borel_measurable M"
using u' by (blast intro: borel_measurable_LIMSEQ[of u])
- show "integrable u'"
+ show "integrable M u'"
proof (rule integrable_bound)
- show "integrable w" by fact
+ show "integrable M w" by fact
show "u' \<in> borel_measurable M" by fact
next
fix x assume x: "x \<in> space M"
@@ -2072,8 +2100,8 @@
qed
let "?diff n x" = "2 * w x - \<bar>u n x - u' x\<bar>"
- have diff: "\<And>n. integrable (\<lambda>x. \<bar>u n x - u' x\<bar>)"
- using w u `integrable u'`
+ have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)"
+ using w u `integrable M u'`
by (auto intro!: integral_add integral_diff integral_cmult integrable_abs)
{ fix j x assume x: "x \<in> space M"
@@ -2083,31 +2111,31 @@
finally have "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp }
note diff_less_2w = this
- have PI_diff: "\<And>m n. (\<integral>\<^isup>+ x. Real (?diff (m + n) x)) =
- (\<integral>\<^isup>+ x. Real (2 * w x)) - (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)"
+ have PI_diff: "\<And>m n. (\<integral>\<^isup>+ x. Real (?diff (m + n) x) \<partial>M) =
+ (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)"
using diff w diff_less_2w
by (subst positive_integral_diff[symmetric])
(auto simp: integrable_def intro!: positive_integral_cong)
- have "integrable (\<lambda>x. 2 * w x)"
+ have "integrable M (\<lambda>x. 2 * w x)"
using w by (auto intro: integral_cmult)
- hence I2w_fin: "(\<integral>\<^isup>+ x. Real (2 * w x)) \<noteq> \<omega>" and
+ hence I2w_fin: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) \<noteq> \<omega>" and
borel_2w: "(\<lambda>x. Real (2 * w x)) \<in> borel_measurable M"
unfolding integrable_def by auto
- have "(INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)) = 0" (is "?lim_SUP = 0")
+ have "(INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)) = 0" (is "?lim_SUP = 0")
proof cases
- assume eq_0: "(\<integral>\<^isup>+ x. Real (2 * w x)) = 0"
- have "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar>) \<le> (\<integral>\<^isup>+ x. Real (2 * w x))"
+ assume eq_0: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) = 0"
+ have "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar> \<partial>M) \<le> (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M)"
proof (rule positive_integral_mono)
fix i x assume "x \<in> space M" from diff_less_2w[OF this, of i]
show "Real \<bar>u i x - u' x\<bar> \<le> Real (2 * w x)" by auto
qed
- hence "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar>) = 0" using eq_0 by auto
+ hence "\<And>i. (\<integral>\<^isup>+ x. Real \<bar>u i x - u' x\<bar> \<partial>M) = 0" using eq_0 by auto
thus ?thesis by simp
next
- assume neq_0: "(\<integral>\<^isup>+ x. Real (2 * w x)) \<noteq> 0"
- have "(\<integral>\<^isup>+ x. Real (2 * w x)) = (\<integral>\<^isup>+ x. SUP n. INF m. Real (?diff (m + n) x))"
+ assume neq_0: "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) \<noteq> 0"
+ have "(\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP n. INF m. Real (?diff (m + n) x)) \<partial>M)"
proof (rule positive_integral_cong, subst add_commute)
fix x assume x: "x \<in> space M"
show "Real (2 * w x) = (SUP n. INF m. Real (?diff (n + m) x))"
@@ -2119,22 +2147,22 @@
thus "(\<lambda>i. ?diff i x) ----> 2 * w x" by simp
qed
qed
- also have "\<dots> \<le> (SUP n. INF m. (\<integral>\<^isup>+ x. Real (?diff (m + n) x)))"
+ also have "\<dots> \<le> (SUP n. INF m. (\<integral>\<^isup>+ x. Real (?diff (m + n) x) \<partial>M))"
using u'_borel w u unfolding integrable_def
by (auto intro!: positive_integral_lim_INF)
- also have "\<dots> = (\<integral>\<^isup>+ x. Real (2 * w x)) -
- (INF n. SUP m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>))"
+ also have "\<dots> = (\<integral>\<^isup>+ x. Real (2 * w x) \<partial>M) -
+ (INF n. SUP m. \<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)"
unfolding PI_diff pextreal_INF_minus[OF I2w_fin] pextreal_SUP_minus ..
finally show ?thesis using neq_0 I2w_fin by (rule pextreal_le_minus_imp_0)
qed
-
+
have [simp]: "\<And>n m x. Real (- \<bar>u (m + n) x - u' x\<bar>) = 0" by auto
- have [simp]: "\<And>n m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>) =
- Real ((\<integral>x. \<bar>u (n + m) x - u' x\<bar>))"
- using diff by (subst add_commute) (simp add: integral_def integrable_def Real_real)
+ have [simp]: "\<And>n m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M) =
+ Real ((\<integral>x. \<bar>u (n + m) x - u' x\<bar> \<partial>M))"
+ using diff by (subst add_commute) (simp add: lebesgue_integral_def integrable_def Real_real)
- have "(SUP n. INF m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar>)) \<le> ?lim_SUP"
+ have "(SUP n. INF m. (\<integral>\<^isup>+ x. Real \<bar>u (m + n) x - u' x\<bar> \<partial>M)) \<le> ?lim_SUP"
(is "?lim_INF \<le> _") by (subst (1 2) add_commute) (rule lim_INF_le_lim_SUP)
hence "?lim_INF = Real 0" "?lim_SUP = Real 0" using `?lim_SUP = 0` by auto
thus ?lim_diff using diff by (auto intro!: integral_positive lim_INF_eq_lim_SUP)
@@ -2143,28 +2171,28 @@
proof (rule LIMSEQ_I)
fix r :: real assume "0 < r"
from LIMSEQ_D[OF `?lim_diff` this]
- obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar>) < r"
+ obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r"
using diff by (auto simp: integral_positive)
- show "\<exists>N. \<forall>n\<ge>N. norm (integral (u n) - integral u') < r"
+ show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r"
proof (safe intro!: exI[of _ N])
fix n assume "N \<le> n"
- have "\<bar>integral (u n) - integral u'\<bar> = \<bar>(\<integral>x. u n x - u' x)\<bar>"
- using u `integrable u'` by (auto simp: integral_diff)
- also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar>)" using u `integrable u'`
+ have "\<bar>integral\<^isup>L M (u n) - integral\<^isup>L M u'\<bar> = \<bar>(\<integral>x. u n x - u' x \<partial>M)\<bar>"
+ using u `integrable M u'` by (auto simp: integral_diff)
+ also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'`
by (rule_tac integral_triangle_inequality) (auto intro!: integral_diff)
also note N[OF `N \<le> n`]
- finally show "norm (integral (u n) - integral u') < r" by simp
+ finally show "norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r" by simp
qed
qed
qed
lemma (in measure_space) integral_sums:
- assumes borel: "\<And>i. integrable (f i)"
+ assumes borel: "\<And>i. integrable M (f i)"
and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
- and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar>))"
- shows "integrable (\<lambda>x. (\<Sum>i. f i x))" (is "integrable ?S")
- and "(\<lambda>i. integral (f i)) sums (\<integral>x. (\<Sum>i. f i x))" (is ?integral)
+ and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
+ shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
+ and "(\<lambda>i. integral\<^isup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
proof -
have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
using summable unfolding summable_def by auto
@@ -2173,10 +2201,10 @@
let "?w y" = "if y \<in> space M then w y else 0"
- obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar>)) sums x"
+ obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
using sums unfolding summable_def ..
- have 1: "\<And>n. integrable (\<lambda>x. \<Sum>i = 0..<n. f i x)"
+ have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i = 0..<n. f i x)"
using borel by (auto intro!: integral_setsum)
{ fix j x assume [simp]: "x \<in> space M"
@@ -2185,21 +2213,21 @@
finally have "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp }
note 2 = this
- have 3: "integrable ?w"
+ have 3: "integrable M ?w"
proof (rule integral_monotone_convergence(1))
let "?F n y" = "(\<Sum>i = 0..<n. \<bar>f i y\<bar>)"
let "?w' n y" = "if y \<in> space M then ?F n y else 0"
- have "\<And>n. integrable (?F n)"
+ have "\<And>n. integrable M (?F n)"
using borel by (auto intro!: integral_setsum integrable_abs)
- thus "\<And>n. integrable (?w' n)" by (simp cong: integrable_cong)
+ thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
show "mono ?w'"
by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
{ fix x show "(\<lambda>n. ?w' n x) ----> ?w x"
using w by (cases "x \<in> space M") (simp_all add: LIMSEQ_const sums_def) }
- have *: "\<And>n. integral (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar>))"
+ have *: "\<And>n. integral\<^isup>L M (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
using borel by (simp add: integral_setsum integrable_abs cong: integral_cong)
from abs_sum
- show "(\<lambda>i. integral (?w' i)) ----> x" unfolding * sums_def .
+ show "(\<lambda>i. integral\<^isup>L M (?w' i)) ----> x" unfolding * sums_def .
qed
have 4: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> ?w x" using 2[of _ 0] by simp
@@ -2210,7 +2238,7 @@
note int = integral_dominated_convergence(1,3)[OF 1 2 3 4 5]
- from int show "integrable ?S" by simp
+ from int show "integrable M ?S" by simp
show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF borel]
using int(2) by simp
@@ -2224,12 +2252,12 @@
and enum_zero: "enum ` (-S) \<subseteq> {0}"
and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
and abs_summable: "summable (\<lambda>r. \<bar>enum r * real (\<mu> (f -` {enum r} \<inter> space M))\<bar>)"
- shows "integrable f"
- and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral f" (is ?sums)
+ shows "integrable M f"
+ and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums)
proof -
let "?A r" = "f -` {enum r} \<inter> space M"
let "?F r x" = "enum r * indicator (?A r) x"
- have enum_eq: "\<And>r. enum r * real (\<mu> (?A r)) = integral (?F r)"
+ have enum_eq: "\<And>r. enum r * real (\<mu> (?A r)) = integral\<^isup>L M (?F r)"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
{ fix x assume "x \<in> space M"
@@ -2250,19 +2278,19 @@
by (auto intro!: sums_single simp: F F_abs) }
note F_sums_f = this(1) and F_abs_sums_f = this(2)
- have int_f: "integral f = (\<integral>x. \<Sum>r. ?F r x)" "integrable f = integrable (\<lambda>x. \<Sum>r. ?F r x)"
+ have int_f: "integral\<^isup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
{ fix r
- have "(\<integral>x. \<bar>?F r x\<bar>) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x)"
+ have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
by (auto simp: indicator_def intro!: integral_cong)
also have "\<dots> = \<bar>enum r\<bar> * real (\<mu> (?A r))"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
- finally have "(\<integral>x. \<bar>?F r x\<bar>) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
+ finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
by (simp add: abs_mult_pos real_pextreal_pos) }
note int_abs_F = this
- have 1: "\<And>i. integrable (\<lambda>x. ?F i x)"
+ have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
@@ -2272,7 +2300,7 @@
show ?sums unfolding enum_eq int_f by simp
from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
- show "integrable f" unfolding int_f by simp
+ show "integrable M f" unfolding int_f by simp
qed
section "Lebesgue integration on finite space"
@@ -2280,8 +2308,8 @@
lemma (in measure_space) integral_on_finite:
assumes f: "f \<in> borel_measurable M" and finite: "finite (f`space M)"
and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<omega>"
- shows "integrable f"
- and "(\<integral>x. f x) =
+ shows "integrable M f"
+ and "(\<integral>x. f x \<partial>M) =
(\<Sum> r \<in> f`space M. r * real (\<mu> (f -` {r} \<inter> space M)))" (is "?integral")
proof -
let "?A r" = "f -` {r} \<inter> space M"
@@ -2295,40 +2323,40 @@
finally have "f x = ?S x" . }
note f_eq = this
- have f_eq_S: "integrable f \<longleftrightarrow> integrable ?S" "integral f = integral ?S"
+ have f_eq_S: "integrable M f \<longleftrightarrow> integrable M ?S" "integral\<^isup>L M f = integral\<^isup>L M ?S"
by (auto intro!: integrable_cong integral_cong simp only: f_eq)
- show "integrable f" ?integral using fin f f_eq_S
+ show "integrable M f" ?integral using fin f f_eq_S
by (simp_all add: integral_cmul_indicator borel_measurable_vimage)
qed
-lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function f"
+lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function M f"
unfolding simple_function_def using finite_space by auto
lemma (in finite_measure_space) borel_measurable_finite[intro, simp]: "f \<in> borel_measurable M"
by (auto intro: borel_measurable_simple_function)
lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
- "positive_integral f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
+ "integral\<^isup>P M f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
proof -
- have *: "positive_integral f = (\<integral>\<^isup>+ x. \<Sum>y\<in>space M. f y * indicator {y} x)"
+ have *: "integral\<^isup>P M f = (\<integral>\<^isup>+ x. (\<Sum>y\<in>space M. f y * indicator {y} x) \<partial>M)"
by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
show ?thesis unfolding * using borel_measurable_finite[of f]
by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
qed
lemma (in finite_measure_space) integral_finite_singleton:
- shows "integrable f"
- and "integral f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
+ shows "integrable M f"
+ and "integral\<^isup>L M f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
proof -
have [simp]:
- "(\<integral>\<^isup>+ x. Real (f x)) = (\<Sum>x \<in> space M. Real (f x) * \<mu> {x})"
- "(\<integral>\<^isup>+ x. Real (- f x)) = (\<Sum>x \<in> space M. Real (- f x) * \<mu> {x})"
+ "(\<integral>\<^isup>+ x. Real (f x) \<partial>M) = (\<Sum>x \<in> space M. Real (f x) * \<mu> {x})"
+ "(\<integral>\<^isup>+ x. Real (- f x) \<partial>M) = (\<Sum>x \<in> space M. Real (- f x) * \<mu> {x})"
unfolding positive_integral_finite_eq_setsum by auto
- show "integrable f" using finite_space finite_measure
+ show "integrable M f" using finite_space finite_measure
by (simp add: setsum_\<omega> integrable_def)
show ?I using finite_measure
- apply (simp add: integral_def real_of_pextreal_setsum[symmetric]
+ apply (simp add: lebesgue_integral_def real_of_pextreal_setsum[symmetric]
real_of_pextreal_mult[symmetric] setsum_subtractf[symmetric])
by (rule setsum_cong) (simp_all split: split_if)
qed