--- a/src/HOL/Probability/Lebesgue_Integration.thy Fri Jan 14 14:21:26 2011 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Fri Jan 14 14:21:48 2011 +0100
@@ -489,7 +489,7 @@
section "Simple integral"
-definition (in measure_space)
+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))"
lemma (in measure_space) simple_integral_cong:
@@ -514,7 +514,7 @@
qed
lemma (in measure_space) simple_integral_const[simp]:
- "simple_integral (\<lambda>x. c) = c * \<mu> (space M)"
+ "(\<integral>\<^isup>Sx. c) = c * \<mu> (space M)"
proof (cases "space M = {}")
case True thus ?thesis unfolding simple_integral_def by simp
next
@@ -579,7 +579,7 @@
lemma (in measure_space) simple_integral_add[simp]:
assumes "simple_function f" and "simple_function g"
- shows "simple_integral (\<lambda>x. f x + g x) = simple_integral f + simple_integral g"
+ shows "(\<integral>\<^isup>Sx. f x + g x) = simple_integral f + simple_integral g"
proof -
{ fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
assume "x \<in> space M"
@@ -597,7 +597,7 @@
lemma (in measure_space) simple_integral_setsum[simp]:
assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
- shows "simple_integral (\<lambda>x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
+ shows "(\<integral>\<^isup>Sx. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
proof cases
assume "finite P"
from this assms show ?thesis
@@ -606,7 +606,7 @@
lemma (in measure_space) simple_integral_mult[simp]:
assumes "simple_function f"
- shows "simple_integral (\<lambda>x. c * f x) = c * simple_integral f"
+ shows "(\<integral>\<^isup>Sx. c * f x) = c * simple_integral 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"
@@ -700,11 +700,11 @@
lemma (in measure_space) simple_integral_indicator:
assumes "A \<in> sets M"
assumes "simple_function f"
- shows "simple_integral (\<lambda>x. f x * indicator A x) =
+ shows "(\<integral>\<^isup>Sx. f x * indicator A x) =
(\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
proof cases
assume "A = space M"
- moreover hence "simple_integral (\<lambda>x. f x * indicator A x) = simple_integral f"
+ moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x) = simple_integral 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)
@@ -720,7 +720,7 @@
next
show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
qed
- have *: "simple_integral (\<lambda>x. f x * indicator A x) =
+ have *: "(\<integral>\<^isup>Sx. f x * indicator A x) =
(\<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)
@@ -760,18 +760,18 @@
lemma (in measure_space) simple_integral_null_set:
assumes "simple_function u" "N \<in> null_sets"
- shows "simple_integral (\<lambda>x. u x * indicator N x) = 0"
+ shows "(\<integral>\<^isup>Sx. u x * indicator N x) = 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 "simple_integral (\<lambda>x. u x * indicator N x) = simple_integral (\<lambda>x. 0)"
+ then have "(\<integral>\<^isup>Sx. u x * indicator N x) = (\<integral>\<^isup>Sx. 0)"
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 = simple_integral (\<lambda>x. f x * indicator S x)"
+ shows "simple_integral f = (\<integral>\<^isup>Sx. f x * indicator S x)"
proof (rule simple_integral_cong_AE)
show "simple_function f" by fact
show "simple_function (\<lambda>x. f x * indicator S x)"
@@ -783,7 +783,7 @@
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 = simple_integral (\<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`]]
unfolding simple_integral_def
@@ -842,7 +842,7 @@
section "Continuous posititve integration"
-definition (in measure_space)
+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"
lemma (in measure_space) positive_integral_alt:
@@ -1071,7 +1071,7 @@
fixes g :: "'d \<Rightarrow> pextreal" and f :: "'d \<Rightarrow> 'a"
assumes f: "bij_inv S (space M) f h"
shows "measure_space.positive_integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) g =
- positive_integral (\<lambda>x. g (h x))"
+ (\<integral>\<^isup>+x. g (h x))"
proof -
interpret v: measure_space "vimage_algebra S f" "\<lambda>A. \<mu> (f ` A)"
using f by (rule measure_space_isomorphic[OF bij_inv_bij_betw(1)])
@@ -1149,7 +1149,7 @@
unfolding pextreal_SUP_cmult[symmetric]
proof (safe intro!: SUP_mono bexI)
fix i
- have "a * simple_integral (?uB i) = simple_integral (\<lambda>x. a * ?uB i x)"
+ have "a * simple_integral (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x)"
using B `simple_function u`
by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
also have "\<dots> \<le> positive_integral (f i)"
@@ -1193,7 +1193,7 @@
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)) = positive_integral (\<lambda>x. SUP i. f i x)"
+ shows "(SUP i. positive_integral (f i)) = (\<integral>\<^isup>+ x. SUP i. f i x)"
(is "_ = positive_integral ?u")
proof -
show ?thesis
@@ -1236,7 +1236,7 @@
qed
lemma (in measure_space) positive_integral_const[simp]:
- "positive_integral (\<lambda>x. c) = c * \<mu> (space M)"
+ "(\<integral>\<^isup>+ x. c) = c * \<mu> (space M)"
by (subst positive_integral_eq_simple_integral) auto
lemma (in measure_space) positive_integral_isoton_simple:
@@ -1248,7 +1248,7 @@
lemma (in measure_space) positive_integral_linear:
assumes f: "f \<in> borel_measurable M"
and g: "g \<in> borel_measurable M"
- shows "positive_integral (\<lambda>x. a * f x + g x) =
+ shows "(\<integral>\<^isup>+ x. a * f x + g x) =
a * positive_integral f + positive_integral g"
(is "positive_integral ?L = _")
proof -
@@ -1276,30 +1276,32 @@
lemma (in measure_space) positive_integral_cmult:
assumes "f \<in> borel_measurable M"
- shows "positive_integral (\<lambda>x. c * f x) = c * positive_integral f"
+ shows "(\<integral>\<^isup>+ x. c * f x) = c * positive_integral 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 "positive_integral (\<lambda>x. f x * c) = positive_integral f * c"
+ shows "(\<integral>\<^isup>+ x. f x * c) = positive_integral 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> positive_integral (\<lambda>x. indicator A x) = \<mu> A"
-by (subst positive_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator)
+ "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x) = \<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> positive_integral (\<lambda>x. c * indicator A x) = c * \<mu> A"
-by (subst positive_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator)
+ "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x) = 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 "positive_integral (\<lambda>x. f x + g x) = positive_integral f + positive_integral g"
+ shows "(\<integral>\<^isup>+ x. f x + g x) = positive_integral f + positive_integral 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 "positive_integral (\<lambda>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) = (\<Sum>i\<in>P. positive_integral (f i))"
proof cases
assume "finite P"
from this assms show ?thesis
@@ -1317,11 +1319,11 @@
assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
and fin: "positive_integral g \<noteq> \<omega>"
and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
- shows "positive_integral (\<lambda>x. f x - g x) = positive_integral f - positive_integral g"
+ shows "(\<integral>\<^isup>+ x. f x - g x) = positive_integral f - positive_integral g"
proof -
have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
using f g by (rule borel_measurable_pextreal_diff)
- have "positive_integral (\<lambda>x. f x - g x) + positive_integral g =
+ have "(\<integral>\<^isup>+x. f x - g x) + positive_integral g =
positive_integral f"
unfolding positive_integral_add[OF borel g, symmetric]
proof (rule positive_integral_cong)
@@ -1335,9 +1337,9 @@
lemma (in measure_space) positive_integral_psuminf:
assumes "\<And>i. f i \<in> borel_measurable M"
- shows "positive_integral (\<lambda>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) = (\<Sum>\<^isub>\<infinity> i. positive_integral (f i))"
proof -
- have "(\<lambda>i. positive_integral (\<lambda>x. \<Sum>i<i. f i x)) \<up> positive_integral (\<lambda>x. \<Sum>\<^isub>\<infinity>i. f i x)"
+ have "(\<lambda>i. (\<integral>\<^isup>+x. \<Sum>i<i. f i x)) \<up> (\<integral>\<^isup>+x. \<Sum>\<^isub>\<infinity>i. f i x)"
by (rule positive_integral_isoton)
(auto intro!: borel_measurable_pextreal_setsum assms positive_integral_mono
arg_cong[where f=Sup]
@@ -1350,11 +1352,11 @@
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 "positive_integral (\<lambda>x. SUP n. INF m. u (m + n) x) \<le>
+ shows "(\<integral>\<^isup>+ x. SUP n. INF m. u (m + n) x) \<le>
(SUP n. INF m. positive_integral (u (m + n)))"
proof -
- have "positive_integral (\<lambda>x. SUP n. INF m. u (m + n) x)
- = (SUP n. positive_integral (\<lambda>x. INF m. u (m + n) x))"
+ have "(\<integral>\<^isup>+x. SUP n. INF m. u (m + n) x)
+ = (SUP n. (\<integral>\<^isup>+x. INF m. u (m + n) x))"
using assms
by (intro positive_integral_monotone_convergence_SUP[symmetric] INF_mono)
(auto simp del: add_Suc simp add: add_Suc[symmetric])
@@ -1365,7 +1367,7 @@
lemma (in measure_space) measure_space_density:
assumes borel: "u \<in> borel_measurable M"
- shows "measure_space M (\<lambda>A. positive_integral (\<lambda>x. u x * indicator A x))" (is "measure_space M ?v")
+ 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"
@@ -1384,8 +1386,8 @@
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. positive_integral (\<lambda>x. f x * indicator A x)) g =
- positive_integral (\<lambda>x. f x * g x)" (is "measure_space.positive_integral M ?T _ = _")
+ 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 _ = _")
proof -
from measure_space_density[OF assms(1)]
interpret T: measure_space M ?T .
@@ -1399,30 +1401,30 @@
using G(1) unfolding simple_function_def by auto
have "T.positive_integral (G i) = T.simple_integral (G i)"
using G T.positive_integral_eq_simple_integral by simp
- also have "\<dots> = positive_integral (\<lambda>x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x))"
+ 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)
apply (subst positive_integral_cmult[symmetric])
using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
apply (subst positive_integral_setsum[symmetric])
using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
by (simp add: setsum_right_distrib field_simps)
- also have "\<dots> = positive_integral (\<lambda>x. f x * G i x)"
+ also have "\<dots> = (\<integral>\<^isup>+x. f x * G i x)"
by (auto intro!: positive_integral_cong
simp: indicator_def if_distrib setsum_cases)
- finally have "T.positive_integral (G i) = positive_integral (\<lambda>x. f x * G i x)" . }
- with * have eq_Tg: "(\<lambda>i. positive_integral (\<lambda>x. f x * G i x)) \<up> T.positive_integral g" by simp
+ 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
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. positive_integral (\<lambda>x. f x * G i x)) \<up> positive_integral (\<lambda>x. f x * g x)"
+ then have "(\<lambda>i. (\<integral>\<^isup>+x. f x * G i x)) \<up> (\<integral>\<^isup>+x. f x * g x)"
using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pextreal_times)
- with eq_Tg show "T.positive_integral g = positive_integral (\<lambda>x. f x * g x)"
+ with eq_Tg show "T.positive_integral g = (\<integral>\<^isup>+x. f x * g x)"
unfolding isoton_def by simp
qed
lemma (in measure_space) positive_integral_null_set:
- assumes "N \<in> null_sets" shows "positive_integral (\<lambda>x. u x * indicator N x) = 0"
+ assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x) = 0"
proof -
- have "positive_integral (\<lambda>x. u x * indicator N x) = positive_integral (\<lambda>x. 0)"
+ have "(\<integral>\<^isup>+ x. u x * indicator N x) = (\<integral>\<^isup>+ x. 0)"
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)
@@ -1434,20 +1436,20 @@
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 * positive_integral (\<lambda>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)"
(is "\<mu> ?A \<le> _ * ?PI")
proof -
have "?A \<in> sets M"
using `A \<in> sets M` borel by auto
- hence "\<mu> ?A = positive_integral (\<lambda>x. indicator ?A x)"
+ hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x)"
using positive_integral_indicator by simp
- also have "\<dots> \<le> positive_integral (\<lambda>x. c * (u x * indicator A x))"
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x))"
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 * positive_integral (\<lambda>x. u x * indicator A x)"
+ also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x)"
using assms
by (auto intro!: positive_integral_cmult borel_measurable_indicator)
finally show ?thesis .
@@ -1459,7 +1461,7 @@
(is "_ \<longleftrightarrow> \<mu> ?A = 0")
proof -
have A: "?A \<in> sets M" using borel by auto
- have u: "positive_integral (\<lambda>x. u x * indicator ?A x) = positive_integral u"
+ have u: "(\<integral>\<^isup>+ x. u x * indicator ?A x) = positive_integral u"
by (auto intro!: positive_integral_cong simp: indicator_def)
show ?thesis
@@ -1467,7 +1469,7 @@
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 = positive_integral (\<lambda>x. u x * indicator ?A x)" by simp
+ have "0 = (\<integral>\<^isup>+ x. u x * indicator ?A x)" by simp
thus "positive_integral u = 0" unfolding u by simp
next
assume *: "positive_integral u = 0"
@@ -1507,7 +1509,7 @@
lemma (in measure_space) positive_integral_restricted:
assumes "A \<in> sets M"
- shows "measure_space.positive_integral (restricted_space A) \<mu> f = positive_integral (\<lambda>x. f x * indicator A x)"
+ 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")
proof -
have msR: "measure_space ?R \<mu>" by (rule restricted_measure_space[OF `A \<in> sets M`])
@@ -1524,7 +1526,7 @@
fix g assume "simple_function (\<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>
- simple_integral (\<lambda>x. g x * indicator A x) = simple_integral x"
+ (\<integral>\<^isup>Sx. g x * indicator A x) = simple_integral x"
apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
by (auto simp: indicator_def le_fun_def)
next
@@ -1560,20 +1562,18 @@
definition (in measure_space) integrable where
"integrable f \<longleftrightarrow> f \<in> borel_measurable M \<and>
- positive_integral (\<lambda>x. Real (f x)) \<noteq> \<omega> \<and>
- positive_integral (\<lambda>x. Real (- f x)) \<noteq> \<omega>"
+ (\<integral>\<^isup>+ x. Real (f x)) \<noteq> \<omega> \<and>
+ (\<integral>\<^isup>+ x. Real (- f x)) \<noteq> \<omega>"
lemma (in measure_space) integrableD[dest]:
assumes "integrable f"
shows "f \<in> borel_measurable M"
- "positive_integral (\<lambda>x. Real (f x)) \<noteq> \<omega>"
- "positive_integral (\<lambda>x. Real (- f x)) \<noteq> \<omega>"
+ "(\<integral>\<^isup>+ x. Real (f x)) \<noteq> \<omega>"
+ "(\<integral>\<^isup>+ x. Real (- f x)) \<noteq> \<omega>"
using assms unfolding integrable_def by auto
-definition (in measure_space) integral where
- "integral f =
- real (positive_integral (\<lambda>x. Real (f x))) -
- real (positive_integral (\<lambda>x. Real (- f x)))"
+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)))"
lemma (in measure_space) integral_cong:
assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
@@ -1609,7 +1609,7 @@
lemma (in measure_space) integral_eq_positive_integral:
assumes "\<And>x. 0 \<le> f x"
- shows "integral f = real (positive_integral (\<lambda>x. Real (f x)))"
+ shows "integral f = real ((\<integral>\<^isup>+ x. Real (f x)))"
proof -
have "\<And>x. Real (- f x) = 0" using assms by simp
thus ?thesis by (simp del: Real_eq_0 add: integral_def)
@@ -1617,7 +1617,7 @@
lemma (in measure_space) integral_vimage_inv:
assumes f: "bij_betw f S (space M)"
- shows "measure_space.integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>x. g x) = integral (\<lambda>x. g (the_inv_into S f x))"
+ shows "measure_space.integral (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>x. g x) = (\<integral>x. g (the_inv_into S f x))"
proof -
interpret v: measure_space "vimage_algebra S f" "\<lambda>A. \<mu> (f ` A)"
using f by (rule measure_space_isomorphic)
@@ -1634,7 +1634,7 @@
lemma (in measure_space) integral_minus[intro, simp]:
assumes "integrable f"
- shows "integrable (\<lambda>x. - f x)" "integral (\<lambda>x. - f x) = - integral f"
+ shows "integrable (\<lambda>x. - f x)" "(\<integral>x. - f x) = - integral f"
using assms by (auto simp: integrable_def integral_def)
lemma (in measure_space) integral_of_positive_diff:
@@ -1685,7 +1685,7 @@
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 (\<lambda>t. a * f t + g t) = a * integral f + integral g"
+ and "(\<integral> t. a * f t + g t) = a * integral f + integral g"
proof -
let ?PI = positive_integral
let "?f x" = "Real (f x)"
@@ -1718,7 +1718,7 @@
show "integrable (\<lambda>t. a * f t + g t)" by (rule diff)
- from assms show "integral (\<lambda>t. a * f t + g t) = a * integral f + integral g"
+ from assms show "(\<integral> t. a * f t + g t) = a * integral f + integral g"
unfolding diff(2) unfolding 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)
@@ -1727,21 +1727,21 @@
lemma (in measure_space) integral_add[simp, intro]:
assumes "integrable f" "integrable g"
shows "integrable (\<lambda>t. f t + g t)"
- and "integral (\<lambda>t. f t + g t) = integral f + integral g"
+ and "(\<integral> t. f t + g t) = integral f + integral 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 (\<lambda>x. 0) = 0"
+ and "(\<integral> x.0) = 0"
unfolding integrable_def 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 (\<lambda>t. a * f t) = a * integral f" (is ?I)
+ and "(\<integral> t. a * f t) = a * integral f" (is ?I)
proof -
- have "integrable (\<lambda>t. a * f t) \<and> integral (\<lambda>t. a * f t) = a * integral f"
+ have "integrable (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t) = a * integral f"
proof (cases rule: le_cases)
assume "0 \<le> a" show ?thesis
using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
@@ -1758,7 +1758,7 @@
lemma (in measure_space) integral_multc:
assumes "integrable f"
- shows "integral (\<lambda>x. f x * c) = integral f * c"
+ shows "(\<integral> x. f x * c) = integral f * c"
unfolding mult_commute[of _ c] integral_cmult[OF assms] ..
lemma (in measure_space) integral_mono_AE:
@@ -1785,7 +1785,7 @@
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 (\<lambda>t. f t - g t) = integral f - integral g"
+ and "(\<integral> t. f t - g t) = integral f - integral g"
using integral_add[OF f integral_minus(1)[OF g]]
unfolding diff_minus integral_minus(2)[OF g]
by auto
@@ -1806,7 +1806,7 @@
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 (\<lambda>x. c * indicator A x) = c * real (\<mu> A)" (is ?I)
+ and "(\<integral>x. c * indicator A x) = c * real (\<mu> A)" (is ?I)
proof -
show ?P
proof (cases "c = 0")
@@ -1821,7 +1821,7 @@
lemma (in measure_space) integral_setsum[simp, intro]:
assumes "\<And>n. n \<in> S \<Longrightarrow> integrable (f n)"
- shows "integral (\<lambda>x. \<Sum> i \<in> S. f i x) = (\<Sum> i \<in> S. integral (f i))" (is "?int S")
+ 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")
proof -
have "?int S \<and> ?I S"
@@ -1854,21 +1854,21 @@
assumes borel: "g \<in> borel_measurable M"
shows "integrable g"
proof -
- have "positive_integral (\<lambda>x. Real (g x)) \<le> positive_integral (\<lambda>x. Real \<bar>g x\<bar>)"
+ have "(\<integral>\<^isup>+ x. Real (g x)) \<le> (\<integral>\<^isup>+ x. Real \<bar>g x\<bar>)"
by (auto intro!: positive_integral_mono)
- also have "\<dots> \<le> positive_integral (\<lambda>x. Real (f x))"
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x))"
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: "positive_integral (\<lambda>x. Real (g x)) < \<omega>" .
+ finally have pos: "(\<integral>\<^isup>+ x. Real (g x)) < \<omega>" .
- have "positive_integral (\<lambda>x. Real (- g x)) \<le> positive_integral (\<lambda>x. Real (\<bar>g x\<bar>))"
+ have "(\<integral>\<^isup>+ x. Real (- g x)) \<le> (\<integral>\<^isup>+ x. Real (\<bar>g x\<bar>))"
by (auto intro!: positive_integral_mono)
- also have "\<dots> \<le> positive_integral (\<lambda>x. Real (f x))"
+ also have "\<dots> \<le> (\<integral>\<^isup>+ x. Real (f x))"
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: "positive_integral (\<lambda>x. Real (- g x)) < \<omega>" .
+ finally have neg: "(\<integral>\<^isup>+ x. Real (- g x)) < \<omega>" .
from neg pos borel show ?thesis
unfolding integrable_def by auto
@@ -1908,10 +1908,10 @@
lemma (in measure_space) integral_triangle_inequality:
assumes "integrable f"
- shows "\<bar>integral f\<bar> \<le> integral (\<lambda>x. \<bar>f x\<bar>)"
+ shows "\<bar>integral f\<bar> \<le> (\<integral>x. \<bar>f x\<bar>)"
proof -
have "\<bar>integral f\<bar> = max (integral f) (- integral f)" by auto
- also have "\<dots> \<le> integral (\<lambda>x. \<bar>f x\<bar>)"
+ also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar>)"
using assms integral_minus(2)[of f, symmetric]
by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
finally show ?thesis .
@@ -1921,7 +1921,7 @@
assumes "integrable f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
shows "0 \<le> integral f"
proof -
- have "0 = integral (\<lambda>x. 0)" by (auto simp: integral_zero)
+ have "0 = (\<integral>x. 0)" by (auto simp: integral_zero)
also have "\<dots> \<le> integral f"
using assms by (rule integral_mono[OF integral_zero(1)])
finally show ?thesis .
@@ -1953,7 +1953,7 @@
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. positive_integral (\<lambda>x. Real (f n x)) = Real (integral (f n))"
+ 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 pos_integral: "\<And>n. 0 \<le> integral (f n)"
@@ -1961,14 +1961,14 @@
hence "0 \<le> x"
using LIMSEQ_le_const[OF ilim, of 0] by auto
- have "(\<lambda>i. positive_integral (\<lambda>x. Real (f i x))) \<up> positive_integral (\<lambda>x. Real (u x))"
+ have "(\<lambda>i. (\<integral>\<^isup>+ x. Real (f i x))) \<up> (\<integral>\<^isup>+ x. Real (u x))"
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: "positive_integral (\<lambda>x. Real (u x)) =
- (SUP n. positive_integral (\<lambda>x. Real (f n x)))"
+ hence pI: "(\<integral>\<^isup>+ x. Real (u x)) =
+ (SUP n. (\<integral>\<^isup>+ x. Real (f n x)))"
unfolding isoton_def by simp
also have "\<dots> = Real x" unfolding integral_eq
proof (rule SUP_eq_LIMSEQ[THEN iffD2])
@@ -1997,7 +1997,7 @@
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 (\<lambda>x. f i x - f 0 x)) ----> x - integral (f 0)"
+ have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x)) ----> x - integral (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)"
@@ -2008,12 +2008,12 @@
lemma (in measure_space) integral_0_iff:
assumes "integrable f"
- shows "integral (\<lambda>x. \<bar>f x\<bar>) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
+ shows "(\<integral>x. \<bar>f x\<bar>) = 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)
hence "(\<lambda>x. Real (\<bar>f x\<bar>)) \<in> borel_measurable M"
- "positive_integral (\<lambda>x. Real \<bar>f x\<bar>) \<noteq> \<omega>" unfolding integrable_def by auto
+ "(\<integral>\<^isup>+ x. Real \<bar>f x\<bar>) \<noteq> \<omega>" unfolding integrable_def by auto
from positive_integral_0_iff[OF this(1)] this(2)
show ?thesis unfolding integral_def *
by (simp add: real_of_pextreal_eq_0)
@@ -2024,7 +2024,7 @@
and "positive_integral f \<noteq> \<omega>"
shows "\<mu> (f -` {\<omega>} \<inter> space M) = 0"
proof -
- have "\<omega> * \<mu> (f -` {\<omega>} \<inter> space M) = positive_integral (\<lambda>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)"
using positive_integral_cmult_indicator[OF borel_measurable_vimage, OF assms(1), of \<omega> \<omega>] by simp
also have "\<dots> \<le> positive_integral f"
by (auto intro!: positive_integral_mono simp: indicator_def)
@@ -2054,15 +2054,15 @@
lemma (in measure_space) integral_real:
fixes f :: "'a \<Rightarrow> pextreal"
assumes "AE x. f x \<noteq> \<omega>"
- shows "integral (\<lambda>x. real (f x)) = real (positive_integral f)" (is ?plus)
- and "integral (\<lambda>x. - real (f x)) = - real (positive_integral f)" (is ?minus)
+ shows "(\<integral>x. real (f x)) = real (positive_integral f)" (is ?plus)
+ and "(\<integral>x. - real (f x)) = - real (positive_integral f)" (is ?minus)
proof -
- have "positive_integral (\<lambda>x. Real (real (f x))) = positive_integral f"
+ have "(\<integral>\<^isup>+ x. Real (real (f x))) = positive_integral f"
apply (rule positive_integral_cong_AE)
apply (rule AE_mp[OF assms(1)])
by (auto intro!: AE_cong simp: Real_real)
moreover
- have "positive_integral (\<lambda>x. Real (- real (f x))) = positive_integral (\<lambda>x. 0)"
+ have "(\<integral>\<^isup>+ x. Real (- real (f x))) = (\<integral>\<^isup>+ x. 0)"
by (intro positive_integral_cong) auto
ultimately show ?plus ?minus
by (auto simp: integral_def integrable_def)
@@ -2073,7 +2073,7 @@
and w: "integrable 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 (\<lambda>x. \<bar>u i x - u' x\<bar>)) ----> 0" (is "?lim_diff")
+ 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)
proof -
{ fix x j assume x: "x \<in> space M"
@@ -2108,31 +2108,31 @@
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. positive_integral (\<lambda>x. Real (?diff (m + n) x)) =
- positive_integral (\<lambda>x. Real (2 * w x)) - positive_integral (\<lambda>x. Real \<bar>u (m + n) x - u' x\<bar>)"
+ 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>)"
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)"
using w by (auto intro: integral_cmult)
- hence I2w_fin: "positive_integral (\<lambda>x. Real (2 * w x)) \<noteq> \<omega>" and
+ hence I2w_fin: "(\<integral>\<^isup>+ x. Real (2 * w x)) \<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. positive_integral (\<lambda>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>)) = 0" (is "?lim_SUP = 0")
proof cases
- assume eq_0: "positive_integral (\<lambda>x. Real (2 * w x)) = 0"
- have "\<And>i. positive_integral (\<lambda>x. Real \<bar>u i x - u' x\<bar>) \<le> positive_integral (\<lambda>x. Real (2 * w x))"
+ 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))"
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. positive_integral (\<lambda>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>) = 0" using eq_0 by auto
thus ?thesis by simp
next
- assume neq_0: "positive_integral (\<lambda>x. Real (2 * w x)) \<noteq> 0"
- have "positive_integral (\<lambda>x. Real (2 * w x)) = positive_integral (\<lambda>x. SUP n. INF m. Real (?diff (m + n) x))"
+ 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))"
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))"
@@ -2144,22 +2144,22 @@
thus "(\<lambda>i. ?diff i x) ----> 2 * w x" by simp
qed
qed
- also have "\<dots> \<le> (SUP n. INF m. positive_integral (\<lambda>x. Real (?diff (m + n) x)))"
+ also have "\<dots> \<le> (SUP n. INF m. (\<integral>\<^isup>+ x. Real (?diff (m + n) x)))"
using u'_borel w u unfolding integrable_def
by (auto intro!: positive_integral_lim_INF)
- also have "\<dots> = positive_integral (\<lambda>x. Real (2 * w x)) -
- (INF n. SUP m. positive_integral (\<lambda>x. Real \<bar>u (m + n) x - u' x\<bar>))"
+ 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>))"
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. positive_integral (\<lambda>x. Real \<bar>u (m + n) x - u' x\<bar>) =
- Real (integral (\<lambda>x. \<bar>u (n + m) x - u' x\<bar>))"
+ 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 "(SUP n. INF m. positive_integral (\<lambda>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>)) \<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)
@@ -2168,15 +2168,15 @@
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 (\<lambda>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>) < r"
using diff by (auto simp: integral_positive)
show "\<exists>N. \<forall>n\<ge>N. norm (integral (u n) - integral u') < r"
proof (safe intro!: exI[of _ N])
fix n assume "N \<le> n"
- have "\<bar>integral (u n) - integral u'\<bar> = \<bar>integral (\<lambda>x. u n x - u' x)\<bar>"
+ 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 (\<lambda>x. \<bar>u n x - u' x\<bar>)" using u `integrable u'`
+ also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar>)" using u `integrable 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
@@ -2187,9 +2187,9 @@
lemma (in measure_space) integral_sums:
assumes borel: "\<And>i. integrable (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 (\<lambda>x. \<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 (\<lambda>x. (\<Sum>i. f i x))" (is ?integral)
+ and "(\<lambda>i. integral (f i)) sums (\<integral>x. (\<Sum>i. f i x))" (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
@@ -2198,7 +2198,7 @@
let "?w y" = "if y \<in> space M then w y else 0"
- obtain x where abs_sum: "(\<lambda>i. integral (\<lambda>x. \<bar>f i x\<bar>)) sums x"
+ obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar>)) sums x"
using sums unfolding summable_def ..
have 1: "\<And>n. integrable (\<lambda>x. \<Sum>i = 0..<n. f i x)"
@@ -2221,7 +2221,7 @@
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 (\<lambda>x. \<bar>f i x\<bar>))"
+ have *: "\<And>n. integral (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar>))"
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 .
@@ -2275,15 +2275,15 @@
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 (\<lambda>x. \<Sum>r. ?F r x)" "integrable f = integrable (\<lambda>x. \<Sum>r. ?F r x)"
+ have int_f: "integral f = (\<integral>x. \<Sum>r. ?F r x)" "integrable f = integrable (\<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 (\<lambda>x. \<bar>?F r x\<bar>) = integral (\<lambda>x. \<bar>enum r\<bar> * indicator (?A r) x)"
+ have "(\<integral>x. \<bar>?F r x\<bar>) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x)"
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 (\<lambda>x. \<bar>?F r x\<bar>) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
+ finally have "(\<integral>x. \<bar>?F r x\<bar>) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
by (simp add: abs_mult_pos real_pextreal_pos) }
note int_abs_F = this
@@ -2306,7 +2306,7 @@
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 (\<lambda>x. f x) =
+ and "(\<integral>x. f x) =
(\<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"
@@ -2336,7 +2336,7 @@
lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
"positive_integral f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
proof -
- have *: "positive_integral f = positive_integral (\<lambda>x. \<Sum>y\<in>space M. f y * indicator {y} x)"
+ have *: "positive_integral f = (\<integral>\<^isup>+ x. \<Sum>y\<in>space M. f y * indicator {y} x)"
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)
@@ -2347,8 +2347,8 @@
and "integral f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
proof -
have [simp]:
- "positive_integral (\<lambda>x. Real (f x)) = (\<Sum>x \<in> space M. Real (f x) * \<mu> {x})"
- "positive_integral (\<lambda>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)) = (\<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
by (simp add: setsum_\<omega> integrable_def)
--- a/src/HOL/Probability/Positive_Extended_Real.thy Fri Jan 14 14:21:26 2011 +0100
+++ b/src/HOL/Probability/Positive_Extended_Real.thy Fri Jan 14 14:21:48 2011 +0100
@@ -1603,6 +1603,136 @@
qed
end
+lemma minus_omega[simp]: "x - \<omega> = 0" by (cases x) auto
+
+lemma open_pextreal_alt: "open A \<longleftrightarrow>
+ (\<forall>x\<in>A. \<exists>e>0. {x-e <..< x+e} \<subseteq> A) \<and> (\<omega> \<in> A \<longrightarrow> (\<exists>x\<ge>0. {Real x <..} \<subseteq> A))"
+proof -
+ have *: "(\<exists>T. open T \<and> (Real ` (T\<inter>{0..}) = A - {\<omega>})) \<longleftrightarrow>
+ open (real ` (A - {\<omega>}) \<union> {..<0})"
+ proof safe
+ fix T assume "open T" and A: "Real ` (T \<inter> {0..}) = A - {\<omega>}"
+ have *: "(\<lambda>x. real (Real x)) ` (T \<inter> {0..}) = T \<inter> {0..}"
+ by auto
+ have **: "T \<inter> {0..} \<union> {..<0} = T \<union> {..<0}" by auto
+ show "open (real ` (A - {\<omega>}) \<union> {..<0})"
+ unfolding A[symmetric] image_image * ** using `open T` by auto
+ next
+ assume "open (real ` (A - {\<omega>}) \<union> {..<0})"
+ moreover have "Real ` ((real ` (A - {\<omega>}) \<union> {..<0}) \<inter> {0..}) = A - {\<omega>}"
+ apply auto
+ apply (case_tac xb)
+ apply auto
+ apply (case_tac x)
+ apply (auto simp: image_iff)
+ apply (erule_tac x="Real r" in ballE)
+ apply auto
+ done
+ ultimately show "\<exists>T. open T \<and> Real ` (T \<inter> {0..}) = A - {\<omega>}" by auto
+ qed
+ also have "\<dots> \<longleftrightarrow> (\<forall>x\<in>A. \<exists>e>0. {x-e <..< x+e} \<subseteq> A)"
+ proof (intro iffI ballI open_subopen[THEN iffD2])
+ fix x assume *: "\<forall>x\<in>A. \<exists>e>0. {x - e<..<x + e} \<subseteq> A" and x: "x \<in> real ` (A - {\<omega>}) \<union> {..<0}"
+ show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
+ proof (cases rule: linorder_cases)
+ assume "x < 0" then show ?thesis by (intro exI[of _ "{..<0}"]) auto
+ next
+ assume "x = 0" with x
+ have "0 \<in> A"
+ apply auto by (case_tac x) auto
+ with * obtain e where "e > 0" "{0 - e<..<0 + e} \<subseteq> A" by auto
+ then have "{..<e} \<subseteq> A" using `0 \<in> A`
+ apply auto
+ apply (case_tac "x = 0")
+ by (auto dest!: pextreal_zero_lessI)
+ then have *: "{..<e} \<subseteq> A - {\<omega>}" by auto
+ def e' \<equiv> "if e = \<omega> then 1 else real e"
+ then have "0 < e'" using `e > 0` by (cases e) auto
+ have "{0..<e'} \<subseteq> real ` (A - {\<omega>})"
+ proof (cases e)
+ case infinite
+ then have "{..<e} = UNIV - {\<omega>}" by auto
+ then have A: "A - {\<omega>} = UNIV - {\<omega>}" using * by auto
+ show ?thesis unfolding e'_def infinite A
+ apply safe
+ apply (rule_tac x="Real x" in image_eqI)
+ apply auto
+ done
+ next
+ case (preal r)
+ then show ?thesis unfolding e'_def using *
+ apply safe
+ apply (rule_tac x="Real x" in image_eqI)
+ by (auto simp: subset_eq)
+ qed
+ then have "{0..<e'} \<union> {..<0} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}" by auto
+ moreover have "{0..<e'} \<union> {..<0} = {..<e'}" using `0 < e'` by auto
+ ultimately have "{..<e'} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}" by simp
+ then show ?thesis using `0 < e'` `x = 0` by auto
+ next
+ assume "0 < x"
+ with x have "Real x \<in> A" apply auto by (case_tac x) auto
+ with * obtain e where "0 < e" and e: "{Real x - e<..<Real x + e} \<subseteq> A" by auto
+ show ?thesis
+ proof cases
+ assume "e < Real x"
+ with `0 < e` obtain r where r: "e = Real r" "0 < r" by (cases e) auto
+ then have "r < x" using `e < Real x` `0 < e` by (auto split: split_if_asm)
+ then have "{x - r <..< x + r} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
+ using e unfolding r
+ apply (auto simp: subset_eq)
+ apply (rule_tac x="Real xa" in image_eqI)
+ by auto
+ then show ?thesis using `0 < r` by (intro exI[of _ "{x - r <..< x + r}"]) auto
+ next
+ assume "\<not> e < Real x"
+ moreover then have "Real x - e = 0" by (cases e) auto
+ moreover have "\<And>z. 0 < z \<Longrightarrow> z * 2 < 3 * x \<Longrightarrow> Real z < Real x + e"
+ using `\<not> e < Real x` by (cases e) auto
+ ultimately have "{0 <..< x + x / 2} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
+ using e
+ apply (auto simp: subset_eq)
+ apply (erule_tac x="Real xa" in ballE)
+ apply (auto simp: not_less)
+ apply (rule_tac x="Real xa" in image_eqI)
+ apply auto
+ done
+ moreover have "x \<in> {0 <..< x + x / 2}" using `0 < x` by auto
+ ultimately show ?thesis by (intro exI[of _ "{0 <..< x + x / 2}"]) auto
+ qed
+ qed
+ next
+ fix x assume x: "x \<in> A" "open (real ` (A - {\<omega>}) \<union> {..<0})"
+ then show "\<exists>e>0. {x - e<..<x + e} \<subseteq> A"
+ proof (cases x)
+ case infinite then show ?thesis by (intro exI[of _ 2]) auto
+ next
+ case (preal r)
+ with `x \<in> A` have r: "r \<in> real ` (A - {\<omega>}) \<union> {..<0}" by force
+ from x(2)[unfolded open_real, THEN bspec, OF r]
+ obtain e where e: "0 < e" "\<And>x'. \<bar>x' - r\<bar> < e \<Longrightarrow> x' \<in> real ` (A - {\<omega>}) \<union> {..<0}"
+ by auto
+ show ?thesis using `0 < e` preal
+ proof (auto intro!: exI[of _ "Real e"] simp: greaterThanLessThan_iff not_less
+ split: split_if_asm)
+ fix z assume *: "Real (r - e) < z" "z < Real (r + e)"
+ then obtain q where [simp]: "z = Real q" "0 \<le> q" by (cases z) auto
+ with * have "r - e < q" "q < r + e" by (auto split: split_if_asm)
+ with e(2)[of q] have "q \<in> real ` (A - {\<omega>}) \<union> {..<0}" by auto
+ then show "z \<in> A" using `0 \<le> q` apply auto apply (case_tac x) apply auto done
+ next
+ fix z assume *: "0 < z" "z < Real (r + e)" "r \<le> e"
+ then obtain q where [simp]: "z = Real q" and "0 < q" by (cases z) auto
+ with * have "q < r + e" by (auto split: split_if_asm)
+ moreover have "r - e < q" using `r \<le> e` `0 < q` by auto
+ ultimately have "q \<in> real ` (A - {\<omega>}) \<union> {..<0}" using e(2)[of q] by auto
+ then show "z \<in> A" using `0 < q` apply auto apply (case_tac x) apply auto done
+ qed
+ qed
+ qed
+ finally show ?thesis unfolding open_pextreal_def by simp
+qed
+
lemma open_pextreal_lessThan[simp]:
"open {..< a :: pextreal}"
proof (cases a)
--- a/src/HOL/Probability/Probability_Space.thy Fri Jan 14 14:21:26 2011 +0100
+++ b/src/HOL/Probability/Probability_Space.thy Fri Jan 14 14:21:48 2011 +0100
@@ -957,7 +957,7 @@
assumes borel: "X \<in> borel_measurable M"
and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr> sets := N \<rparr>)"
shows "\<exists>Y\<in>borel_measurable (M\<lparr> sets := N \<rparr>). \<forall>C\<in>N.
- positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)"
+ (\<integral>\<^isup>+x. Y x * indicator C x) = (\<integral>\<^isup>+x. X x * indicator C x)"
proof -
interpret P: prob_space "M\<lparr> sets := N \<rparr>" \<mu>
using prob_space_subalgebra[OF N_subalgebra] .
@@ -994,7 +994,7 @@
definition (in prob_space)
"conditional_expectation N X = (SOME Y. Y\<in>borel_measurable (M\<lparr>sets:=N\<rparr>)
- \<and> (\<forall>C\<in>N. positive_integral (\<lambda>x. Y x * indicator C x) = positive_integral (\<lambda>x. X x * indicator C x)))"
+ \<and> (\<forall>C\<in>N. (\<integral>\<^isup>+x. Y x * indicator C x) = (\<integral>\<^isup>+x. X x * indicator C x)))"
abbreviation (in prob_space)
"conditional_prob N A \<equiv> conditional_expectation N (indicator A)"
@@ -1006,8 +1006,8 @@
shows borel_measurable_conditional_expectation:
"conditional_expectation N X \<in> borel_measurable (M\<lparr> sets := N \<rparr>)"
and conditional_expectation: "\<And>C. C \<in> N \<Longrightarrow>
- positive_integral (\<lambda>x. conditional_expectation N X x * indicator C x) =
- positive_integral (\<lambda>x. X x * indicator C x)"
+ (\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x) =
+ (\<integral>\<^isup>+x. X x * indicator C x)"
(is "\<And>C. C \<in> N \<Longrightarrow> ?eq C")
proof -
note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
--- a/src/HOL/Probability/Product_Measure.thy Fri Jan 14 14:21:26 2011 +0100
+++ b/src/HOL/Probability/Product_Measure.thy Fri Jan 14 14:21:48 2011 +0100
@@ -784,7 +784,7 @@
note SUPR_C = this
ultimately show "?C f \<in> borel_measurable M1"
by (simp cong: measurable_cong)
- have "positive_integral (\<lambda>x. SUP i. F i x) = (SUP i. positive_integral (F i))"
+ have "positive_integral (\<lambda>x. (SUP i. F i x)) = (SUP i. positive_integral (F i))"
using F_borel F_mono
by (auto intro!: positive_integral_monotone_convergence_SUP[symmetric])
also have "(SUP i. positive_integral (F i)) =
@@ -1935,8 +1935,7 @@
proof (unfold integrable_def, intro conjI)
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"
using borel by auto
- have "positive_integral (\<lambda>x. Real (abs (?f x))) =
- positive_integral (\<lambda>x. \<Prod>i\<in>I. Real (abs (f i (x i))))"
+ have "positive_integral (\<lambda>x. Real (abs (?f x))) = positive_integral (\<lambda>x. \<Prod>i\<in>I. Real (abs (f i (x i))))"
by (simp add: Real_setprod abs_setprod)
also have "\<dots> = (\<Prod>i\<in>I. M.positive_integral i (\<lambda>x. Real (abs (f i x))))"
using f by (subst product_positive_integral_setprod) auto
--- a/src/HOL/Probability/Radon_Nikodym.thy Fri Jan 14 14:21:26 2011 +0100
+++ b/src/HOL/Probability/Radon_Nikodym.thy Fri Jan 14 14:21:48 2011 +0100
@@ -114,7 +114,7 @@
qed
lemma (in measure_space) density_is_absolutely_continuous:
- assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
shows "absolutely_continuous \<nu>"
using assms unfolding absolutely_continuous_def
by (simp add: positive_integral_null_set)
@@ -289,10 +289,10 @@
lemma (in finite_measure) Radon_Nikodym_finite_measure:
assumes "finite_measure M \<nu>"
assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
proof -
interpret M': finite_measure M \<nu> using assms(1) .
- def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A}"
+ def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A}"
have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
hence "G \<noteq> {}" by auto
{ fix f g assume f: "f \<in> G" and g: "g \<in> G"
@@ -308,16 +308,16 @@
have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
by (auto simp: indicator_def max_def)
- hence "positive_integral (\<lambda>x. max (g x) (f x) * indicator A x) =
- positive_integral (\<lambda>x. g x * indicator (?A \<inter> A) x) +
- positive_integral (\<lambda>x. f x * indicator ((space M - ?A) \<inter> A) x)"
+ hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x) =
+ (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x) +
+ (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x)"
using f g sets unfolding G_def
by (auto cong: positive_integral_cong intro!: positive_integral_add borel_measurable_indicator)
also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)"
using f g sets unfolding G_def by (auto intro!: add_mono)
also have "\<dots> = \<nu> A"
using M'.measure_additive[OF sets] union by auto
- finally show "positive_integral (\<lambda>x. max (g x) (f x) * indicator A x) \<le> \<nu> A" .
+ finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x) \<le> \<nu> A" .
qed }
note max_in_G = this
{ fix f g assume "f \<up> g" and f: "\<And>i. f i \<in> G"
@@ -331,17 +331,17 @@
hence "\<And>i. (\<lambda>x. f i x * indicator A x) \<in> borel_measurable M"
using f_borel by (auto intro!: borel_measurable_indicator)
from positive_integral_isoton[OF isoton_indicator[OF `f \<up> g`] this]
- have SUP: "positive_integral (\<lambda>x. g x * indicator A x) =
- (SUP i. positive_integral (\<lambda>x. f i x * indicator A x))"
+ have SUP: "(\<integral>\<^isup>+x. g x * indicator A x) =
+ (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x))"
unfolding isoton_def by simp
- show "positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A" unfolding SUP
+ show "(\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A" unfolding SUP
using f `A \<in> sets M` unfolding G_def by (auto intro!: SUP_leI)
qed }
note SUP_in_G = this
let ?y = "SUP g : G. positive_integral g"
have "?y \<le> \<nu> (space M)" unfolding G_def
proof (safe intro!: SUP_leI)
- fix g assume "\<forall>A\<in>sets M. positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A"
+ fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A"
from this[THEN bspec, OF top] show "positive_integral g \<le> \<nu> (space M)"
by (simp cong: positive_integral_cong)
qed
@@ -384,7 +384,7 @@
by (auto intro!: SUP_mono positive_integral_mono Max_ge)
qed
finally have int_f_eq_y: "positive_integral f = ?y" .
- let "?t A" = "\<nu> A - positive_integral (\<lambda>x. f x * indicator A x)"
+ let "?t A" = "\<nu> A - (\<integral>\<^isup>+x. f x * indicator A x)"
have "finite_measure M ?t"
proof
show "?t {} = 0" by simp
@@ -392,26 +392,26 @@
show "countably_additive M ?t" unfolding countably_additive_def
proof safe
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A"
- have "(\<Sum>\<^isub>\<infinity> n. positive_integral (\<lambda>x. f x * indicator (A n) x))
- = positive_integral (\<lambda>x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x))"
+ have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x))
+ = (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x))"
using `range A \<subseteq> sets M`
by (rule_tac positive_integral_psuminf[symmetric]) (auto intro!: borel_measurable_indicator)
- also have "\<dots> = positive_integral (\<lambda>x. f x * indicator (\<Union>n. A n) x)"
+ also have "\<dots> = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x)"
apply (rule positive_integral_cong)
apply (subst psuminf_cmult_right)
unfolding psuminf_indicator[OF `disjoint_family A`] ..
- finally have "(\<Sum>\<^isub>\<infinity> n. positive_integral (\<lambda>x. f x * indicator (A n) x))
- = positive_integral (\<lambda>x. f x * indicator (\<Union>n. A n) x)" .
+ finally have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x))
+ = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x)" .
moreover have "(\<Sum>\<^isub>\<infinity>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
using M'.measure_countably_additive A by (simp add: comp_def)
- moreover have "\<And>i. positive_integral (\<lambda>x. f x * indicator (A i) x) \<le> \<nu> (A i)"
+ moreover have "\<And>i. (\<integral>\<^isup>+x. f x * indicator (A i) x) \<le> \<nu> (A i)"
using A `f \<in> G` unfolding G_def by auto
moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<omega>" using M'.finite_measure A by (simp add: countable_UN)
moreover {
- have "positive_integral (\<lambda>x. f x * indicator (\<Union>i. A i) x) \<le> \<nu> (\<Union>i. A i)"
+ have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x) \<le> \<nu> (\<Union>i. A i)"
using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pextreal_less_\<omega>)
- finally have "positive_integral (\<lambda>x. f x * indicator (\<Union>i. A i) x) \<noteq> \<omega>"
+ finally have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x) \<noteq> \<omega>"
by (simp add: pextreal_less_\<omega>) }
ultimately
show "(\<Sum>\<^isub>\<infinity> n. ?t (A n)) = ?t (\<Union>i. A i)"
@@ -433,9 +433,9 @@
hence pos_M: "0 < \<mu> (space M)"
using ac top unfolding absolutely_continuous_def by auto
moreover
- have "positive_integral (\<lambda>x. f x * indicator (space M) x) \<le> \<nu> (space M)"
+ have "(\<integral>\<^isup>+x. f x * indicator (space M) x) \<le> \<nu> (space M)"
using `f \<in> G` unfolding G_def by auto
- hence "positive_integral (\<lambda>x. f x * indicator (space M) x) \<noteq> \<omega>"
+ hence "(\<integral>\<^isup>+x. f x * indicator (space M) x) \<noteq> \<omega>"
using M'.finite_measure_of_space by auto
moreover
def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
@@ -462,30 +462,30 @@
let "?f0 x" = "f x + b * indicator A0 x"
{ fix A assume A: "A \<in> sets M"
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
- have "positive_integral (\<lambda>x. ?f0 x * indicator A x) =
- positive_integral (\<lambda>x. f x * indicator A x + b * indicator (A \<inter> A0) x)"
+ have "(\<integral>\<^isup>+x. ?f0 x * indicator A x) =
+ (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x)"
by (auto intro!: positive_integral_cong simp: field_simps indicator_inter_arith)
- hence "positive_integral (\<lambda>x. ?f0 x * indicator A x) =
- positive_integral (\<lambda>x. f x * indicator A x) + b * \<mu> (A \<inter> A0)"
+ hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x) =
+ (\<integral>\<^isup>+x. f x * indicator A x) + b * \<mu> (A \<inter> A0)"
using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A
by (simp add: borel_measurable_indicator positive_integral_add positive_integral_cmult_indicator) }
note f0_eq = this
{ fix A assume A: "A \<in> sets M"
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
- have f_le_v: "positive_integral (\<lambda>x. f x * indicator A x) \<le> \<nu> A"
+ have f_le_v: "(\<integral>\<^isup>+x. f x * indicator A x) \<le> \<nu> A"
using `f \<in> G` A unfolding G_def by auto
note f0_eq[OF A]
- also have "positive_integral (\<lambda>x. f x * indicator A x) + b * \<mu> (A \<inter> A0) \<le>
- positive_integral (\<lambda>x. f x * indicator A x) + ?t (A \<inter> A0)"
+ also have "(\<integral>\<^isup>+x. f x * indicator A x) + b * \<mu> (A \<inter> A0) \<le>
+ (\<integral>\<^isup>+x. f x * indicator A x) + ?t (A \<inter> A0)"
using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
by (auto intro!: add_left_mono)
- also have "\<dots> \<le> positive_integral (\<lambda>x. f x * indicator A x) + ?t A"
+ also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x) + ?t A"
using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
by (auto intro!: add_left_mono)
also have "\<dots> \<le> \<nu> A"
using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`]
- by (cases "positive_integral (\<lambda>x. f x * indicator A x)", cases "\<nu> A", auto)
- finally have "positive_integral (\<lambda>x. ?f0 x * indicator A x) \<le> \<nu> A" . }
+ by (cases "(\<integral>\<^isup>+x. f x * indicator A x)", cases "\<nu> A", auto)
+ finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x) \<le> \<nu> A" . }
hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def
by (auto intro!: borel_measurable_indicator borel_measurable_pextreal_add borel_measurable_pextreal_times)
have real: "?t (space M) \<noteq> \<omega>" "?t A0 \<noteq> \<omega>"
@@ -525,11 +525,11 @@
show ?thesis
proof (safe intro!: bexI[of _ f])
fix A assume "A\<in>sets M"
- show "\<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
proof (rule antisym)
- show "positive_integral (\<lambda>x. f x * indicator A x) \<le> \<nu> A"
+ show "(\<integral>\<^isup>+x. f x * indicator A x) \<le> \<nu> A"
using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
- show "\<nu> A \<le> positive_integral (\<lambda>x. f x * indicator A x)"
+ show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x)"
using upper_bound[THEN bspec, OF `A \<in> sets M`]
by (simp add: pextreal_zero_le_diff)
qed
@@ -669,7 +669,7 @@
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
assumes "measure_space M \<nu>"
assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
proof -
interpret v: measure_space M \<nu> by fact
from split_space_into_finite_sets_and_rest[OF assms]
@@ -680,7 +680,7 @@
and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<omega>" by force
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
- \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f x * indicator (Q i \<inter> A) x))"
+ \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x))"
proof
fix i
have indicator_eq: "\<And>f x A. (f x :: pextreal) * indicator (Q i \<inter> A) x * indicator (Q i) x
@@ -702,17 +702,17 @@
by (auto simp: R.absolutely_continuous_def absolutely_continuous_def)
from finite_measure.Radon_Nikodym_finite_measure[OF fm fmv this]
obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M"
- and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. (f x * indicator (Q i) x) * indicator A x)"
+ and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. (f x * indicator (Q i) x) * indicator A x)"
unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
- \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f x * indicator (Q i \<inter> A) x))"
+ \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x))"
by (fastsimp intro!: exI[of _ "\<lambda>x. f x * indicator (Q i) x"] positive_integral_cong
simp: indicator_def)
qed
from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M"
and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
- \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f i x * indicator (Q i \<inter> A) x)"
+ \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x)"
by auto
let "?f x" =
"(\<Sum>\<^isub>\<infinity> i. f i x * indicator (Q i) x) + \<omega> * indicator Q0 x"
@@ -728,7 +728,7 @@
f i x * indicator (Q i \<inter> A) x"
"\<And>x i. (indicator A x * indicator Q0 x :: pextreal) =
indicator (Q0 \<inter> A) x" by (auto simp: indicator_def)
- have "positive_integral (\<lambda>x. ?f x * indicator A x) =
+ have "(\<integral>\<^isup>+x. ?f x * indicator A x) =
(\<Sum>\<^isub>\<infinity> i. \<nu> (Q i \<inter> A)) + \<omega> * \<mu> (Q0 \<inter> A)"
unfolding f[OF `A \<in> sets M`]
apply (simp del: pextreal_times(2) add: field_simps *)
@@ -755,7 +755,7 @@
using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN)
moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
using `A \<in> sets M` sets_into_space Q0 by auto
- ultimately show "\<nu> A = positive_integral (\<lambda>x. ?f x * indicator A x)"
+ ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x)"
using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
by simp
qed
@@ -764,14 +764,14 @@
lemma (in sigma_finite_measure) Radon_Nikodym:
assumes "measure_space M \<nu>"
assumes "absolutely_continuous \<nu>"
- shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
proof -
from Ex_finite_integrable_function
obtain h where finite: "positive_integral h \<noteq> \<omega>" and
borel: "h \<in> borel_measurable M" and
pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
"\<And>x. x \<in> space M \<Longrightarrow> h x < \<omega>" by auto
- let "?T A" = "positive_integral (\<lambda>x. h x * indicator A x)"
+ let "?T A" = "(\<integral>\<^isup>+x. h x * indicator A x)"
from measure_space_density[OF borel] finite
interpret T: finite_measure M ?T
unfolding finite_measure_def finite_measure_axioms_def
@@ -792,7 +792,7 @@
then have "(\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
using f_borel by (auto intro: borel_measurable_pextreal_times borel_measurable_indicator)
from positive_integral_translated_density[OF borel this]
- show "\<nu> A = positive_integral (\<lambda>x. h x * f x * indicator A x)"
+ show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x)"
unfolding fT[OF `A \<in> sets M`] by (simp add: field_simps)
qed
qed
@@ -802,7 +802,7 @@
lemma (in measure_space) finite_density_unique:
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
and fin: "positive_integral f < \<omega>"
- shows "(\<forall>A\<in>sets M. positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. g x * indicator A x))
+ shows "(\<forall>A\<in>sets M. (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. g x * indicator A x))
\<longleftrightarrow> (AE x. f x = g x)"
(is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
proof (intro iffI ballI)
@@ -817,7 +817,7 @@
and g_fin: "positive_integral g < \<omega>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
let ?N = "{x\<in>space M. g x < f x}"
have N: "?N \<in> sets M" using borel by simp
- have "?P (\<lambda>x. (f x - g x)) ?N = positive_integral (\<lambda>x. f x * indicator ?N x - g x * indicator ?N x)"
+ have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x)"
by (auto intro!: positive_integral_cong simp: indicator_def)
also have "\<dots> = ?P f ?N - ?P g ?N"
proof (rule positive_integral_diff)
@@ -848,7 +848,7 @@
lemma (in finite_measure) density_unique_finite_measure:
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
- assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. f' x * indicator A x)"
+ assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. f' x * indicator A x)"
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
shows "AE x. f x = f' x"
proof -
@@ -876,13 +876,13 @@
have 2: "AE x. ?f Q0 x = ?f' Q0 x"
proof (rule AE_I')
{ fix f :: "'a \<Rightarrow> pextreal" assume borel: "f \<in> borel_measurable M"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
let "?A i" = "Q0 \<inter> {x \<in> space M. f x < of_nat i}"
have "(\<Union>i. ?A i) \<in> null_sets"
proof (rule null_sets_UN)
fix i have "?A i \<in> sets M"
using borel Q0(1) by auto
- have "?\<nu> (?A i) \<le> positive_integral (\<lambda>x. of_nat i * indicator (?A i) x)"
+ have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. of_nat i * indicator (?A i) x)"
unfolding eq[OF `?A i \<in> sets M`]
by (auto intro!: positive_integral_mono simp: indicator_def)
also have "\<dots> = of_nat i * \<mu> (?A i)"
@@ -912,38 +912,38 @@
lemma (in sigma_finite_measure) density_unique:
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
- assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. f' x * indicator A x)"
+ assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. f' x * indicator A x)"
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
shows "AE x. f x = f' x"
proof -
obtain h where h_borel: "h \<in> borel_measurable M"
and fin: "positive_integral h \<noteq> \<omega>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>"
using Ex_finite_integrable_function by auto
- interpret h: measure_space M "\<lambda>A. positive_integral (\<lambda>x. h x * indicator A x)"
+ interpret h: measure_space M "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x)"
using h_borel by (rule measure_space_density)
- interpret h: finite_measure M "\<lambda>A. positive_integral (\<lambda>x. h x * indicator A x)"
+ interpret h: finite_measure M "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x)"
by default (simp cong: positive_integral_cong add: fin)
- interpret f: measure_space M "\<lambda>A. positive_integral (\<lambda>x. f x * indicator A x)"
+ interpret f: measure_space M "\<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x)"
using borel(1) by (rule measure_space_density)
- interpret f': measure_space M "\<lambda>A. positive_integral (\<lambda>x. f' x * indicator A x)"
+ interpret f': measure_space M "\<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x)"
using borel(2) by (rule measure_space_density)
{ fix A assume "A \<in> sets M"
then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pextreal)} = A"
using pos sets_into_space by (force simp: indicator_def)
- then have "positive_integral (\<lambda>xa. h xa * indicator A xa) = 0 \<longleftrightarrow> A \<in> null_sets"
+ then have "(\<integral>\<^isup>+x. h x * indicator A x) = 0 \<longleftrightarrow> A \<in> null_sets"
using h_borel `A \<in> sets M` by (simp add: positive_integral_0_iff) }
note h_null_sets = this
{ fix A assume "A \<in> sets M"
- have "positive_integral (\<lambda>x. h x * (f x * indicator A x)) =
+ have "(\<integral>\<^isup>+x. h x * (f x * indicator A x)) =
f.positive_integral (\<lambda>x. h x * indicator A x)"
using `A \<in> sets M` h_borel borel
by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
also have "\<dots> = f'.positive_integral (\<lambda>x. h x * indicator A x)"
by (rule f'.positive_integral_cong_measure) (rule f)
- also have "\<dots> = positive_integral (\<lambda>x. h x * (f' x * indicator A x))"
+ also have "\<dots> = (\<integral>\<^isup>+x. h x * (f' x * indicator A x))"
using `A \<in> sets M` h_borel borel
by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
- finally have "positive_integral (\<lambda>x. h x * (f x * indicator A x)) = positive_integral (\<lambda>x. h x * (f' x * indicator A x))" . }
+ finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x)) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x))" . }
then have "h.almost_everywhere (\<lambda>x. f x = f' x)"
using h_borel borel
by (intro h.density_unique_finite_measure[OF borel])
@@ -955,7 +955,7 @@
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
assumes \<nu>: "measure_space M \<nu>" and f: "f \<in> borel_measurable M"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
shows "sigma_finite_measure M \<nu> \<longleftrightarrow> (AE x. f x \<noteq> \<omega>)"
proof
assume "sigma_finite_measure M \<nu>"
@@ -965,10 +965,10 @@
and fin: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>" by auto
have "AE x. f x * h x \<noteq> \<omega>"
proof (rule AE_I')
- have "\<nu>.positive_integral h = positive_integral (\<lambda>x. f x * h x)"
+ have "\<nu>.positive_integral h = (\<integral>\<^isup>+x. f x * h x)"
by (simp add: \<nu>.positive_integral_cong_measure[symmetric, OF eq[symmetric]])
(intro positive_integral_translated_density f h)
- then have "positive_integral (\<lambda>x. f x * h x) \<noteq> \<omega>"
+ then have "(\<integral>\<^isup>+x. f x * h x) \<noteq> \<omega>"
using h(2) by simp
then show "(\<lambda>x. f x * h x) -` {\<omega>} \<inter> space M \<in> null_sets"
using f h(1) by (auto intro!: positive_integral_omega borel_measurable_vimage)
@@ -1018,12 +1018,12 @@
next
fix n obtain i j where
[simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
- have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) \<noteq> \<omega>"
+ have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) \<noteq> \<omega>"
proof (cases i)
case 0
have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
using AE by (rule AE_mp) (auto intro!: AE_cong simp: A_def `i = 0`)
- then have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) = 0"
+ then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) = 0"
using A_in_sets f
apply (subst positive_integral_0_iff)
apply fast
@@ -1034,8 +1034,8 @@
then show ?thesis by simp
next
case (Suc n)
- then have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) \<le>
- positive_integral (\<lambda>x. of_nat (Suc n) * indicator (Q j) x)"
+ then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) \<le>
+ (\<integral>\<^isup>+x. of_nat (Suc n) * indicator (Q j) x)"
by (auto intro!: positive_integral_mono simp: indicator_def A_def)
also have "\<dots> = of_nat (Suc n) * \<mu> (Q j)"
using Q by (auto intro!: positive_integral_cmult_indicator)
@@ -1052,7 +1052,7 @@
definition (in sigma_finite_measure)
"RN_deriv \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and>
- (\<forall>A \<in> sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x))"
+ (\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x))"
lemma (in sigma_finite_measure) RN_deriv_cong:
assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> \<mu>' A = \<mu> A" "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
@@ -1069,7 +1069,7 @@
assumes "measure_space M \<nu>"
assumes "absolutely_continuous \<nu>"
shows "RN_deriv \<nu> \<in> borel_measurable M" (is ?borel)
- and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)"
+ and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)"
(is "\<And>A. _ \<Longrightarrow> ?int A")
proof -
note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
@@ -1082,13 +1082,13 @@
lemma (in sigma_finite_measure) RN_deriv_positive_integral:
assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
and f: "f \<in> borel_measurable M"
- shows "measure_space.positive_integral M \<nu> f = positive_integral (\<lambda>x. RN_deriv \<nu> x * f x)"
+ shows "measure_space.positive_integral M \<nu> f = (\<integral>\<^isup>+x. RN_deriv \<nu> x * f x)"
proof -
interpret \<nu>: measure_space M \<nu> by fact
have "\<nu>.positive_integral f =
- measure_space.positive_integral M (\<lambda>A. positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)) f"
+ measure_space.positive_integral M (\<lambda>A. (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)) f"
by (intro \<nu>.positive_integral_cong_measure[symmetric] RN_deriv(2)[OF \<nu>, symmetric])
- also have "\<dots> = positive_integral (\<lambda>x. RN_deriv \<nu> x * f x)"
+ also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv \<nu> x * f x)"
by (intro positive_integral_translated_density RN_deriv[OF \<nu>] f)
finally show ?thesis .
qed
@@ -1096,11 +1096,11 @@
lemma (in sigma_finite_measure) RN_deriv_unique:
assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
and f: "f \<in> borel_measurable M"
- and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
+ and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)"
shows "AE x. f x = RN_deriv \<nu> x"
proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
fix A assume A: "A \<in> sets M"
- show "positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)"
+ show "(\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)"
unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
qed
@@ -1130,10 +1130,10 @@
using f by (auto simp: bij_inv_def indicator_def)
have "\<nu> (f ` (f -` A \<inter> S)) = sf.positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) x * indicator (f -` A \<inter> S) x)"
using `A \<in> sets M` by (force intro!: sf.RN_deriv(2)[OF \<nu>' ac])
- also have "\<dots> = positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)"
+ also have "\<dots> = (\<integral>\<^isup>+x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)"
unfolding positive_integral_vimage_inv[OF f]
by (simp add: * cong: positive_integral_cong)
- finally show "\<nu> A = positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)"
+ finally show "\<nu> A = (\<integral>\<^isup>+x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)"
unfolding A_f[OF `A \<in> sets M`] .
qed
@@ -1150,7 +1150,7 @@
lemma (in sigma_finite_measure)
assumes \<nu>: "sigma_finite_measure M \<nu>" "absolutely_continuous \<nu>"
and f: "f \<in> borel_measurable M"
- shows RN_deriv_integral: "measure_space.integral M \<nu> f = integral (\<lambda>x. real (RN_deriv \<nu> x) * f x)" (is ?integral)
+ shows RN_deriv_integral: "measure_space.integral M \<nu> f = (\<integral>x. real (RN_deriv \<nu> x) * f x)" (is ?integral)
and RN_deriv_integrable: "measure_space.integrable M \<nu> f \<longleftrightarrow> integrable (\<lambda>x. real (RN_deriv \<nu> x) * f x)" (is ?integrable)
proof -
interpret \<nu>: sigma_finite_measure M \<nu> by fact
@@ -1164,7 +1164,7 @@
then have "RN_deriv \<nu> x * Real (f x) = Real (real (RN_deriv \<nu> x) * f x)"
using * by (simp add: Real_real) }
note * = this
- have "positive_integral (\<lambda>x. RN_deriv \<nu> x * Real (f x)) = positive_integral (\<lambda>x. Real (real (RN_deriv \<nu> x) * f x))"
+ have "(\<integral>\<^isup>+x. RN_deriv \<nu> x * Real (f x)) = (\<integral>\<^isup>+x. Real (real (RN_deriv \<nu> x) * f x))"
apply (rule positive_integral_cong_AE)
apply (rule AE_mp[OF RN_deriv_finite[OF \<nu>]])
by (auto intro!: AE_cong simp: *) }
@@ -1182,7 +1182,7 @@
proof -
note deriv = RN_deriv[OF assms(1, 2)]
from deriv(2)[OF `{x} \<in> sets M`]
- have "\<nu> {x} = positive_integral (\<lambda>w. RN_deriv \<nu> x * indicator {x} w)"
+ have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv \<nu> x * indicator {x} w)"
by (auto simp: indicator_def intro!: positive_integral_cong)
thus ?thesis using positive_integral_cmult_indicator[OF `{x} \<in> sets M`]
by auto