--- a/src/HOL/Probability/Radon_Nikodym.thy Mon Dec 07 16:48:10 2015 +0000
+++ b/src/HOL/Probability/Radon_Nikodym.thy Mon Dec 07 20:19:59 2015 +0100
@@ -2,7 +2,7 @@
Author: Johannes Hölzl, TU München
*)
-section {*Radon-Nikod{\'y}m derivative*}
+section \<open>Radon-Nikod{\'y}m derivative\<close>
theory Radon_Nikodym
imports Bochner_Integration
@@ -80,7 +80,7 @@
by (cases rule: ereal2_cases[of "n N" "emeasure M (A N)"])
(simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide)
finally show "n N * emeasure M (A N) \<le> (1 / 2) ^ Suc N" .
- show "0 \<le> n N * emeasure M (A N)" using n[of N] `A N \<in> sets M` by (simp add: emeasure_nonneg)
+ show "0 \<le> n N * emeasure M (A N)" using n[of N] \<open>A N \<in> sets M\<close> by (simp add: emeasure_nonneg)
qed
finally show "integral\<^sup>N M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto
next
@@ -121,12 +121,12 @@
and "absolutely_continuous M M'" "AE x in M. P x"
shows "AE x in M'. P x"
proof -
- from `AE x in M. P x` obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
+ from \<open>AE x in M. P x\<close> obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
unfolding eventually_ae_filter by auto
show "AE x in M'. P x"
proof (rule AE_I')
show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp
- from `absolutely_continuous M M'` show "N \<in> null_sets M'"
+ from \<open>absolutely_continuous M M'\<close> show "N \<in> null_sets M'"
using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto
qed
qed
@@ -167,14 +167,14 @@
fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
hence "?d (A n \<union> B) = ?d (A n) + ?d B"
- using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq)
+ using \<open>A n \<in> sets M\<close> finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq)
also have "\<dots> \<le> ?d (A n) - e" using dB by simp
finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
qed }
note dA_epsilon = this
{ fix n have "?d (A (Suc n)) \<le> ?d (A n)"
proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
- case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
+ case True from dA_epsilon[OF this] show ?thesis using \<open>0 < e\<close> by simp
next
case False
hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le)
@@ -214,13 +214,13 @@
fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
qed
have A: "incseq A" by (auto intro!: incseq_SucI)
- from finite_Lim_measure_incseq[OF _ A] `range A \<subseteq> sets M`
+ from finite_Lim_measure_incseq[OF _ A] \<open>range A \<subseteq> sets M\<close>
M'.finite_Lim_measure_incseq[OF _ A]
have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
by (auto intro!: tendsto_diff simp: sets_eq)
obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
- have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
+ have "real n \<le> - ?d (\<Union>i. A i) / e" using \<open>0<e\<close> by (simp add: field_simps)
ultimately show ?thesis by auto
qed
qed
@@ -258,7 +258,7 @@
by (auto simp add: mono_iff_le_Suc)
show ?thesis
proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
- show "(\<Inter>i. A i) \<in> sets M" using `\<And>n. A n \<in> sets M` by auto
+ show "(\<Inter>i. A i) \<in> sets M" using \<open>\<And>n. A n \<in> sets M\<close> by auto
have "decseq A" using A by (auto intro!: decseq_SucI)
from A(1) finite_Lim_measure_decseq[OF _ this] N.finite_Lim_measure_decseq[OF _ this]
have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq)
@@ -299,10 +299,10 @@
let ?A = "{x \<in> space M. f x \<le> g x}"
have "?A \<in> sets M" using f g unfolding G_def by auto
fix A assume "A \<in> sets M"
- hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
+ hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using \<open>?A \<in> sets M\<close> by auto
hence sets': "?A \<inter> A \<in> sets N" "(space M - ?A) \<inter> A \<in> sets N" by (auto simp: sets_eq)
have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
- using sets.sets_into_space[OF `A \<in> sets M`] by auto
+ using sets.sets_into_space[OF \<open>A \<in> sets M\<close>] by auto
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)
@@ -333,11 +333,11 @@
(\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
by (intro nn_integral_cong) (simp split: split_indicator)
also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))"
- using `incseq f` f `A \<in> sets M`
+ using \<open>incseq f\<close> f \<open>A \<in> sets M\<close>
by (intro nn_integral_monotone_convergence_SUP)
(auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A"
- using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def)
+ using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_def)
qed }
note SUP_in_G = this
let ?y = "SUP g : G. integral\<^sup>N M g"
@@ -347,7 +347,7 @@
from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)"
by (simp cong: nn_integral_cong)
qed
- from SUP_countable_SUP [OF `G \<noteq> {}`, of "integral\<^sup>N M"] guess ys .. note ys = this
+ from SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this
then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n"
proof safe
fix n assume "range ys \<subseteq> integral\<^sup>N M ` G"
@@ -365,7 +365,7 @@
case 0 thus ?case by simp fact
next
case (Suc i)
- with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case
+ with Suc gs_not_empty \<open>gs (Suc i) \<in> G\<close> show ?case
by (auto simp add: atMost_Suc intro!: max_in_G)
qed }
note g_in_G = this
@@ -374,7 +374,7 @@
from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def .
then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def
- using g_in_G `incseq ?g`
+ using g_in_G \<open>incseq ?g\<close>
by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def)
also have "\<dots> = ?y"
proof (rule antisym)
@@ -385,12 +385,12 @@
qed
finally have int_f_eq_y: "integral\<^sup>N M f = ?y" .
have "\<And>x. 0 \<le> f x"
- unfolding f_def using `\<And>i. gs i \<in> G`
+ unfolding f_def using \<open>\<And>i. gs i \<in> G\<close>
by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def)
let ?t = "\<lambda>A. N A - (\<integral>\<^sup>+x. ?F A x \<partial>M)"
let ?M = "diff_measure N (density M f)"
have f_le_N: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. ?F A x \<partial>M) \<le> N A"
- using `f \<in> G` unfolding G_def by auto
+ using \<open>f \<in> G\<close> unfolding G_def by auto
have emeasure_M: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure ?M A = ?t A"
proof (subst emeasure_diff_measure)
from f_le_N[of "space M"] show "finite_measure N" "finite_measure (density M f)"
@@ -406,9 +406,9 @@
have ac: "absolutely_continuous M ?M" unfolding absolutely_continuous_def
proof
fix A assume A_M: "A \<in> null_sets M"
- with `absolutely_continuous M N` have A_N: "A \<in> null_sets N"
+ with \<open>absolutely_continuous M N\<close> have A_N: "A \<in> null_sets N"
unfolding absolutely_continuous_def by auto
- moreover from A_M A_N have "(\<integral>\<^sup>+ x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` by (auto simp: G_def)
+ moreover from A_M A_N have "(\<integral>\<^sup>+ x. ?F A x \<partial>M) \<le> N A" using \<open>f \<in> G\<close> by (auto simp: G_def)
ultimately have "N A - (\<integral>\<^sup>+ x. ?F A x \<partial>M) = 0"
using nn_integral_nonneg[of M] by (auto intro!: antisym)
then show "A \<in> null_sets ?M"
@@ -430,7 +430,7 @@
using emeasure_nonneg[of M "space M"] by (simp add: le_less)
moreover
have "(\<integral>\<^sup>+x. f x * indicator (space M) x \<partial>M) \<le> N (space M)"
- using `f \<in> G` unfolding G_def by auto
+ using \<open>f \<in> G\<close> unfolding G_def by auto
hence "(\<integral>\<^sup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>"
using M'.finite_emeasure_space by auto
moreover
@@ -452,31 +452,31 @@
note bM_le_t = this
let ?f0 = "\<lambda>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
+ hence "A \<inter> A0 \<in> sets M" using \<open>A0 \<in> sets M\<close> by auto
have "(\<integral>\<^sup>+x. ?f0 x * indicator A x \<partial>M) =
(\<integral>\<^sup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
by (auto intro!: nn_integral_cong split: split_indicator)
hence "(\<integral>\<^sup>+x. ?f0 x * indicator A x \<partial>M) =
(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) + b * emeasure M (A \<inter> A0)"
- using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G`
+ using \<open>A0 \<in> sets M\<close> \<open>A \<inter> A0 \<in> sets M\<close> A b \<open>f \<in> G\<close>
by (simp add: nn_integral_add nn_integral_cmult_indicator G_def) }
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: "(\<integral>\<^sup>+x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` A unfolding G_def by auto
+ hence "A \<inter> A0 \<in> sets M" using \<open>A0 \<in> sets M\<close> by auto
+ have f_le_v: "(\<integral>\<^sup>+x. ?F A x \<partial>M) \<le> N A" using \<open>f \<in> G\<close> A unfolding G_def by auto
note f0_eq[OF A]
also have "(\<integral>\<^sup>+x. ?F A x \<partial>M) + b * emeasure M (A \<inter> A0) \<le> (\<integral>\<^sup>+x. ?F A x \<partial>M) + ?M (A \<inter> A0)"
- using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
+ using bM_le_t[OF \<open>A \<inter> A0 \<in> sets M\<close>] \<open>A \<in> sets M\<close> \<open>A0 \<in> sets M\<close>
by (auto intro!: add_left_mono)
also have "\<dots> \<le> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) + ?M A"
- using emeasure_mono[of "A \<inter> A0" A ?M] `A \<in> sets M` `A0 \<in> sets M`
+ using emeasure_mono[of "A \<inter> A0" A ?M] \<open>A \<in> sets M\<close> \<open>A0 \<in> sets M\<close>
by (auto intro!: add_left_mono simp: sets_eq)
also have "\<dots> \<le> N A"
- unfolding emeasure_M[OF `A \<in> sets M`]
+ unfolding emeasure_M[OF \<open>A \<in> sets M\<close>]
using f_le_v N.emeasure_eq_measure[of A] nn_integral_nonneg[of M "?F A"]
by (cases "\<integral>\<^sup>+x. ?F A x \<partial>M", cases "N A") auto
finally have "(\<integral>\<^sup>+x. ?f0 x * indicator A x \<partial>M) \<le> N A" . }
- hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` by (auto simp: G_def)
+ hence "?f0 \<in> G" using \<open>A0 \<in> sets M\<close> b \<open>f \<in> G\<close> by (auto simp: G_def)
have int_f_finite: "integral\<^sup>N M f \<noteq> \<infinity>"
by (metis N.emeasure_finite ereal_infty_less_eq2(1) int_f_eq_y y_le)
have "0 < ?M (space M) - emeasure ?Mb (space M)"
@@ -484,25 +484,25 @@
by (simp add: b emeasure_density_const)
(simp add: M'.emeasure_eq_measure emeasure_eq_measure pos_M b_def)
also have "\<dots> \<le> ?M A0 - b * emeasure M A0"
- using space_less_A0 `A0 \<in> sets M` b
+ using space_less_A0 \<open>A0 \<in> sets M\<close> b
by (cases b) (auto simp add: b emeasure_density_const sets_eq M'.emeasure_eq_measure emeasure_eq_measure)
finally have 1: "b * emeasure M A0 < ?M A0"
- by (metis M'.emeasure_real `A0 \<in> sets M` bM_le_t diff_self ereal_less(1) ereal_minus(1)
+ by (metis M'.emeasure_real \<open>A0 \<in> sets M\<close> bM_le_t diff_self ereal_less(1) ereal_minus(1)
less_eq_ereal_def mult_zero_left not_square_less_zero subset_refl zero_ereal_def)
with b have "0 < ?M A0"
by (metis M'.emeasure_real MInfty_neq_PInfty(1) emeasure_real ereal_less_eq(5) ereal_zero_times
ereal_mult_eq_MInfty ereal_mult_eq_PInfty ereal_zero_less_0_iff less_eq_ereal_def)
- then have "emeasure M A0 \<noteq> 0" using ac `A0 \<in> sets M`
+ then have "emeasure M A0 \<noteq> 0" using ac \<open>A0 \<in> sets M\<close>
by (auto simp: absolutely_continuous_def null_sets_def)
then have "0 < emeasure M A0" using emeasure_nonneg[of M A0] by auto
hence "0 < b * emeasure M A0" using b by (auto simp: ereal_zero_less_0_iff)
with int_f_finite have "?y + 0 < integral\<^sup>N M f + b * emeasure M A0" unfolding int_f_eq_y
- using `f \<in> G`
+ using \<open>f \<in> G\<close>
by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 nn_integral_nonneg)
- also have "\<dots> = integral\<^sup>N M ?f0" using f0_eq[OF sets.top] `A0 \<in> sets M` sets.sets_into_space
+ also have "\<dots> = integral\<^sup>N M ?f0" using f0_eq[OF sets.top] \<open>A0 \<in> sets M\<close> sets.sets_into_space
by (simp cong: nn_integral_cong)
finally have "?y < integral\<^sup>N M ?f0" by simp
- moreover from `?f0 \<in> G` have "integral\<^sup>N M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
+ moreover from \<open>?f0 \<in> G\<close> have "integral\<^sup>N M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
ultimately show False by auto
qed
let ?f = "\<lambda>x. max 0 (f x)"
@@ -512,7 +512,7 @@
by (simp add: sets_eq)
fix A assume A: "A\<in>sets (density M ?f)"
then show "emeasure (density M ?f) A = emeasure N A"
- using `f \<in> G` A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A]
+ using \<open>f \<in> G\<close> A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A]
by (cases "integral\<^sup>N M (?F A)")
(auto intro!: antisym simp add: emeasure_density G_def emeasure_M density_max_0[symmetric])
qed auto
@@ -599,7 +599,7 @@
also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))"
proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
- using `N A \<noteq> \<infinity>` O_sets A by auto
+ using \<open>N A \<noteq> \<infinity>\<close> O_sets A by auto
qed (fastforce intro!: incseq_SucI)
also have "\<dots> \<le> ?a"
proof (safe intro!: SUP_least)
@@ -609,13 +609,13 @@
from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A"
using emeasure_subadditive[of "?O i" N A] A O_sets by (auto simp: sets_eq)
with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>"
- using `N A \<noteq> \<infinity>` by auto
+ using \<open>N A \<noteq> \<infinity>\<close> by auto
qed
then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
qed
finally have "emeasure M A = 0"
unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure)
- with `emeasure M A \<noteq> 0` show ?thesis by auto
+ with \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto
qed
qed }
{ fix i show "N (Q i) \<noteq> \<infinity>"
@@ -624,7 +624,7 @@
unfolding Q_def using Q'[of 0] by simp
next
case (Suc n)
- with `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
+ with \<open>?O n \<in> ?Q\<close> \<open>?O (Suc n) \<in> ?Q\<close>
emeasure_Diff[OF _ _ _ O_mono, of N n] emeasure_nonneg[of N "(\<Union>x\<le>n. Q' x)"]
show ?thesis
by (auto simp: sets_eq ereal_minus_eq_PInfty_iff Q_def)
@@ -671,7 +671,7 @@
show "sets (?N i) = sets (?M i)" by (simp add: sets_eq)
have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq)
show "absolutely_continuous (?M i) (?N i)"
- using `absolutely_continuous M N` `Q i \<in> sets M`
+ using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close>
by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq
intro!: absolutely_continuous_AE[OF sets_eq])
qed
@@ -700,31 +700,31 @@
have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
"\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x"
- using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times)
+ using borel Qi Q0(1) \<open>A \<in> sets M\<close> by (auto intro!: borel_measurable_ereal_times)
have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)"
using borel by (intro nn_integral_cong) (auto simp: indicator_def)
also have "\<dots> = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M (Q0 \<inter> A)"
- using borel Qi Q0(1) `A \<in> sets M`
+ using borel Qi Q0(1) \<open>A \<in> sets M\<close>
by (subst nn_integral_add) (auto simp del: ereal_infty_mult
simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le)
also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)"
- by (subst integral_eq[OF `A \<in> sets M`], subst nn_integral_suminf) auto
+ by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto
finally have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" .
moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)"
- using Q Q_sets `A \<in> sets M`
+ using Q Q_sets \<open>A \<in> sets M\<close>
by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
moreover have "\<infinity> * emeasure M (Q0 \<inter> A) = N (Q0 \<inter> A)"
proof -
- have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
+ have "Q0 \<inter> A \<in> sets M" using Q0(1) \<open>A \<in> sets M\<close> by blast
from in_Q0[OF this] show ?thesis by auto
qed
moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
- using Q_sets `A \<in> sets M` Q0(1) by auto
+ using Q_sets \<open>A \<in> sets M\<close> Q0(1) by auto
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.sets_into_space Q0 by auto
+ using \<open>A \<in> sets M\<close> sets.sets_into_space Q0 by auto
ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)"
using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "Q0 \<inter> A"] by (simp add: sets_eq)
- with `A \<in> sets M` borel Q Q0(1) show "emeasure (density M ?f) A = N A"
+ with \<open>A \<in> sets M\<close> borel Q Q0(1) show "emeasure (density M ?f) A = N A"
by (auto simp: subset_eq emeasure_density)
qed (simp add: sets_eq)
qed
@@ -752,7 +752,7 @@
with pos sets.sets_into_space have "AE x in M. x \<notin> A"
by (elim eventually_elim1) (auto simp: not_le[symmetric])
then have "A \<in> null_sets M"
- using `A \<in> sets M` by (simp add: AE_iff_null_sets)
+ using \<open>A \<in> sets M\<close> by (simp add: AE_iff_null_sets)
with ac show "A \<in> null_sets N"
by (auto simp: absolutely_continuous_def)
qed (auto simp add: sets_eq)
@@ -762,7 +762,7 @@
by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq)
qed
-subsection {* Uniqueness of densities *}
+subsection \<open>Uniqueness of densities\<close>
lemma finite_density_unique:
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
@@ -848,13 +848,13 @@
fix i ::nat have "?A i \<in> sets M"
using borel Q0(1) by auto
have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
- unfolding eq[OF `?A i \<in> sets M`]
+ unfolding eq[OF \<open>?A i \<in> sets M\<close>]
by (auto intro!: nn_integral_mono simp: indicator_def)
also have "\<dots> = i * emeasure M (?A i)"
- using `?A i \<in> sets M` by (auto intro!: nn_integral_cmult_indicator)
+ using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator)
also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by simp
finally have "?N (?A i) \<noteq> \<infinity>" by simp
- then show "?A i \<in> null_sets M" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
+ then show "?A i \<in> null_sets M" using in_Q0[OF \<open>?A i \<in> sets M\<close>] \<open>?A i \<in> sets M\<close> by auto
qed
also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
by (auto simp: less_PInf_Ex_of_nat)
@@ -894,21 +894,21 @@
then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
using pos(1) sets.sets_into_space by (force simp: indicator_def)
then have "(\<integral>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
- using h_borel `A \<in> sets M` h_nn by (subst nn_integral_0_iff) auto }
+ using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto }
note h_null_sets = this
{ fix A assume "A \<in> sets M"
have "(\<integral>\<^sup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?fM)"
- using `A \<in> sets M` h_borel h_nn f f'
+ using \<open>A \<in> sets M\<close> h_borel h_nn f f'
by (intro nn_integral_density[symmetric]) auto
also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)"
by (simp_all add: density_eq)
also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)"
- using `A \<in> sets M` h_borel h_nn f f'
+ using \<open>A \<in> sets M\<close> h_borel h_nn f f'
by (intro nn_integral_density) auto
finally have "(\<integral>\<^sup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * (f' x * indicator A x) \<partial>M)"
by (simp add: ac_simps)
then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)"
- using `A \<in> sets M` h_borel h_nn f f'
+ using \<open>A \<in> sets M\<close> h_borel h_nn f f'
by (subst (asm) (1 2) nn_integral_density[symmetric]) auto }
then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M])
@@ -1025,7 +1025,7 @@
proof (cases i)
case 0
have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0"
- using AE by (auto simp: A_def `i = 0`)
+ using AE by (auto simp: A_def \<open>i = 0\<close>)
from nn_integral_cong_AE[OF this] show ?thesis by simp
next
case (Suc n)
@@ -1050,7 +1050,7 @@
apply (auto simp: max_def intro!: measurable_If)
done
-subsection {* Radon-Nikodym derivative *}
+subsection \<open>Radon-Nikodym derivative\<close>
definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ereal" where
"RN_deriv M N =
@@ -1164,11 +1164,11 @@
next
fix X assume "X \<in> (\<lambda>A. T' -` A \<inter> space ?M')`F"
then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto
- have "X \<in> sets M'" using F T' `A\<in>F` by auto
+ have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto
moreover
- have Fi: "A \<in> sets M" using F `A\<in>F` by auto
+ have Fi: "A \<in> sets M" using F \<open>A\<in>F\<close> by auto
ultimately show "emeasure ?M' X \<noteq> \<infinity>"
- using F T T' `A\<in>F` by (simp add: emeasure_distr)
+ using F T T' \<open>A\<in>F\<close> by (simp add: emeasure_distr)
qed (insert F, auto)
qed
have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M"
@@ -1291,7 +1291,7 @@
and x: "{x} \<in> sets M"
shows "N {x} = RN_deriv M N x * emeasure M {x}"
proof -
- from `{x} \<in> sets M`
+ from \<open>{x} \<in> sets M\<close>
have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong)
with x density_RN_deriv[OF ac] RN_deriv_nonneg[of M N] show ?thesis