--- a/src/HOL/Probability/Essential_Supremum.thy Fri Oct 21 11:45:35 2016 +0200
+++ b/src/HOL/Probability/Essential_Supremum.thy Fri Oct 21 11:02:36 2016 +0200
@@ -1,4 +1,5 @@
(* Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr
+ Author: Johannes Hölzl (TUM) -- ported to Limsup
License: BSD
*)
@@ -6,6 +7,9 @@
imports "../Analysis/Analysis"
begin
+lemma ae_filter_eq_bot_iff: "ae_filter M = bot \<longleftrightarrow> emeasure M (space M) = 0"
+ by (simp add: AE_iff_measurable trivial_limit_def)
+
section {*The essential supremum*}
text {*In this paragraph, we define the essential supremum and give its basic properties. The
@@ -14,7 +18,31 @@
it allows for neater statements in general. This is a prerequisiste to define the space $L^\infty$.*}
definition esssup::"'a measure \<Rightarrow> ('a \<Rightarrow> 'b::{second_countable_topology, dense_linorder, linorder_topology, complete_linorder}) \<Rightarrow> 'b"
- where "esssup M f = (if f \<in> borel_measurable M then Inf {z. emeasure M {x \<in> space M. f x > z} = 0} else top)"
+ where "esssup M f = (if f \<in> borel_measurable M then Limsup (ae_filter M) f else top)"
+
+lemma esssup_non_measurable: "f \<notin> M \<rightarrow>\<^sub>M borel \<Longrightarrow> esssup M f = top"
+ by (simp add: esssup_def)
+
+lemma esssup_eq_AE:
+ assumes f: "f \<in> M \<rightarrow>\<^sub>M borel" shows "esssup M f = Inf {z. AE x in M. f x \<le> z}"
+ unfolding esssup_def if_P[OF f] Limsup_def
+proof (intro antisym INF_greatest Inf_greatest; clarsimp)
+ fix y assume "AE x in M. f x \<le> y"
+ then have "(\<lambda>x. f x \<le> y) \<in> {P. AE x in M. P x}"
+ by simp
+ then show "(INF P:{P. AE x in M. P x}. SUP x:Collect P. f x) \<le> y"
+ by (rule INF_lower2) (auto intro: SUP_least)
+next
+ fix P assume P: "AE x in M. P x"
+ show "Inf {z. AE x in M. f x \<le> z} \<le> (SUP x:Collect P. f x)"
+ proof (rule Inf_lower; clarsimp)
+ show "AE x in M. f x \<le> (SUP x:Collect P. f x)"
+ using P by (auto elim: eventually_mono simp: SUP_upper)
+ qed
+qed
+
+lemma esssup_eq: "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> esssup M f = Inf {z. emeasure M {x \<in> space M. f x > z} = 0}"
+ by (auto simp add: esssup_eq_AE not_less[symmetric] AE_iff_measurable[OF _ refl] intro!: arg_cong[where f=Inf])
lemma esssup_zero_measure:
"emeasure M {x \<in> space M. f x > esssup M f} = 0"
@@ -23,13 +51,12 @@
then show ?thesis by auto
next
case False
- then have [measurable]: "f \<in> borel_measurable M" unfolding esssup_def by meson
+ then have f[measurable]: "f \<in> M \<rightarrow>\<^sub>M borel" unfolding esssup_def by meson
have "esssup M f < top" using False by (auto simp: less_top)
have *: "{x \<in> space M. f x > z} \<in> null_sets M" if "z > esssup M f" for z
proof -
have "\<exists>w. w < z \<and> emeasure M {x \<in> space M. f x > w} = 0"
- using `z > esssup M f` unfolding esssup_def apply auto
- by (metis (mono_tags, lifting) Inf_less_iff mem_Collect_eq)
+ using `z > esssup M f` f by (auto simp: esssup_eq Inf_less_iff)
then obtain w where "w < z" "emeasure M {x \<in> space M. f x > w} = 0" by auto
then have a: "{x \<in> space M. f x > w} \<in> null_sets M" by auto
have b: "{x \<in> space M. f x > z} \<subseteq> {x \<in> space M. f x > w}" using `w < z` by auto
@@ -46,127 +73,44 @@
finally show ?thesis by auto
qed
-lemma esssup_AE:
- "AE x in M. f x \<le> esssup M f"
-proof (cases "f \<in> borel_measurable M")
- case True
- show ?thesis
- apply (rule AE_I[OF _ esssup_zero_measure[of _ f]]) using True by auto
-next
- case False
- then have "esssup M f = top" unfolding esssup_def by auto
- then show ?thesis by auto
-qed
+lemma esssup_AE: "AE x in M. f x \<le> esssup M f"
+proof (cases "f \<in> M \<rightarrow>\<^sub>M borel")
+ case True then show ?thesis
+ by (intro AE_I[OF _ esssup_zero_measure[of _ f]]) auto
+qed (simp add: esssup_non_measurable)
lemma esssup_pos_measure:
- assumes "f \<in> borel_measurable M" "z < esssup M f"
- shows "emeasure M {x \<in> space M. f x > z} > 0"
-using assms Inf_less_iff mem_Collect_eq not_gr_zero unfolding esssup_def by force
+ "f \<in> borel_measurable M \<Longrightarrow> z < esssup M f \<Longrightarrow> emeasure M {x \<in> space M. f x > z} > 0"
+ using Inf_less_iff mem_Collect_eq not_gr_zero by (force simp: esssup_eq)
-lemma esssup_non_measurable:
- assumes "f \<notin> borel_measurable M"
- shows "esssup M f = top"
-using assms unfolding esssup_def by auto
+lemma esssup_I [intro]: "f \<in> borel_measurable M \<Longrightarrow> AE x in M. f x \<le> c \<Longrightarrow> esssup M f \<le> c"
+ unfolding esssup_def by (simp add: Limsup_bounded)
-lemma esssup_I [intro]:
- assumes "f \<in> borel_measurable M" "AE x in M. f x \<le> c"
- shows "esssup M f \<le> c"
-proof -
- have "emeasure M {x \<in> space M. \<not> f x \<le> c} = 0"
- apply (rule AE_E2[OF assms(2)]) using assms(1) by simp
- then have *: "emeasure M {x \<in> space M. f x > c} = 0"
- by (metis (mono_tags, lifting) Collect_cong not_less)
- show ?thesis unfolding esssup_def using assms apply simp by (rule Inf_lower, simp add: *)
-qed
+lemma esssup_AE_mono: "f \<in> borel_measurable M \<Longrightarrow> AE x in M. f x \<le> g x \<Longrightarrow> esssup M f \<le> esssup M g"
+ by (auto simp: esssup_def Limsup_mono)
-lemma esssup_AE_mono:
- assumes "f \<in> borel_measurable M" "AE x in M. f x \<le> g x"
- shows "esssup M f \<le> esssup M g"
-proof (cases "g \<in> borel_measurable M")
- case False
- then show ?thesis unfolding esssup_def by auto
-next
- case True
- have "AE x in M. f x \<le> esssup M g"
- using assms(2) esssup_AE[of g M] by auto
- then show ?thesis using esssup_I assms(1) by auto
-qed
-
-lemma esssup_mono:
- assumes "f \<in> borel_measurable M" "\<And>x. f x \<le> g x"
- shows "esssup M f \<le> esssup M g"
-apply (rule esssup_AE_mono) using assms by auto
+lemma esssup_mono: "f \<in> borel_measurable M \<Longrightarrow> (\<And>x. f x \<le> g x) \<Longrightarrow> esssup M f \<le> esssup M g"
+ by (rule esssup_AE_mono) auto
lemma esssup_AE_cong:
- assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
- and "AE x in M. f x = g x"
- shows "esssup M f = esssup M g"
-proof -
- have "esssup M f \<le> esssup M g"
- using esssup_AE_mono[OF assms(1), of g] assms(3) by (simp add: eq_iff)
- moreover have "esssup M g \<le> esssup M f"
- using esssup_AE_mono[OF assms(2), of f] assms(3) by (simp add: eq_iff)
- ultimately show ?thesis by simp
-qed
+ "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow> esssup M f = esssup M g"
+ by (auto simp: esssup_def intro!: Limsup_eq)
-lemma esssup_const:
- assumes "emeasure M (space M) \<noteq> 0"
- shows "esssup M (\<lambda>x. c) = c"
-proof -
- have "emeasure M {x \<in> space M. (\<lambda>x. c) x > z} = (if c > z then emeasure M (space M) else 0)" for z
- by auto
- then have "{z. emeasure M {x \<in> space M. (\<lambda>x. c) x > z} = 0} = {c..}" using assms by auto
- then have "esssup M (\<lambda>x. c) = Inf {c..}" unfolding esssup_def by auto
- then show ?thesis by auto
-qed
+lemma esssup_const: "emeasure M (space M) \<noteq> 0 \<Longrightarrow> esssup M (\<lambda>x. c) = c"
+ by (simp add: esssup_def Limsup_const ae_filter_eq_bot_iff)
-lemma esssup_cmult:
- assumes "c > (0::real)"
- shows "esssup M (\<lambda>x. c * f x::ereal) = c * esssup M f"
-proof (cases "f \<in> borel_measurable M")
- case True
- then have a [measurable]: "f \<in> borel_measurable M" by simp
- then have b [measurable]: "(\<lambda>x. c * f x) \<in> borel_measurable M" by simp
- have a: "{x \<in> space M. c * z < c * f x} = {x \<in> space M. z < f x}" for z::ereal
- by (meson assms ereal_less(2) ereal_mult_left_mono ereal_mult_strict_left_mono less_ereal.simps(4) less_imp_le not_less)
- have *: "{z::ereal. emeasure M {x \<in> space M. ereal c * f x > z} = 0} = {c * z| z::ereal. emeasure M {x \<in> space M. f x > z} = 0}"
- proof (auto)
- fix y assume *: "emeasure M {x \<in> space M. y < c * f x} = 0"
- define z where "z = y / c"
- have **: "y = c * z" unfolding z_def using assms by (simp add: ereal_mult_divide)
- then have "y = c * z \<and> emeasure M {x \<in> space M. z < f x} = 0"
- using * unfolding ** unfolding a by auto
- then show "\<exists>z. y = ereal c * z \<and> emeasure M {x \<in> space M. z < f x} = 0"
- by auto
- next
- fix z assume *: "emeasure M {x \<in> space M. z < f x} = 0"
- then show "emeasure M {x \<in> space M. c * z < c * f x} = 0"
- using a by auto
- qed
- have "esssup M (\<lambda>x. c * f x) = Inf {z::ereal. emeasure M {x \<in> space M. c * f x > z} = 0}"
- unfolding esssup_def using b by auto
- also have "... = Inf {c * z| z::ereal. emeasure M {x \<in> space M. f x > z} = 0}"
- using * by auto
- also have "... = ereal c * Inf {z. emeasure M {x \<in> space M. f x > z} = 0}"
- apply (rule ereal_Inf_cmult) using assms by auto
- also have "... = c * esssup M f"
- unfolding esssup_def by auto
- finally show ?thesis by simp
-next
- case False
- have "esssup M f = top" using False unfolding esssup_def by auto
- then have *: "c * esssup M f = \<infinity>" using assms by (simp add: ennreal_mult_eq_top_iff top_ereal_def)
- have "(\<lambda>x. c * f x) \<notin> borel_measurable M"
- proof (rule ccontr)
- assume "\<not> (\<lambda>x. c * f x) \<notin> borel_measurable M"
- then have [measurable]: "(\<lambda>x. c * f x) \<in> borel_measurable M" by simp
- then have "(\<lambda>x. (1/c) * (c * f x)) \<in> borel_measurable M" by measurable
- moreover have "(1/c) * (c * f x) = f x" for x
- by (metis "*" PInfty_neq_ereal(1) divide_inverse divide_self_if ereal_zero_mult mult.assoc mult.commute mult.left_neutral one_ereal_def times_ereal.simps(1) zero_ereal_def)
- ultimately show False using False by auto
- qed
- then have "esssup M (\<lambda>x. c * f x) = \<infinity>" unfolding esssup_def by (simp add: top_ereal_def)
- then show ?thesis using * by auto
+lemma esssup_cmult: assumes "c > (0::real)" shows "esssup M (\<lambda>x. c * f x::ereal) = c * esssup M f"
+proof -
+ have "(\<lambda>x. ereal c * f x) \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> f \<in> M \<rightarrow>\<^sub>M borel"
+ proof (subst measurable_cong)
+ fix \<omega> show "f \<omega> = ereal (1/c) * (ereal c * f \<omega>)"
+ using \<open>0 < c\<close> by (cases "f \<omega>") auto
+ qed auto
+ then have "(\<lambda>x. ereal c * f x) \<in> M \<rightarrow>\<^sub>M borel \<longleftrightarrow> f \<in> M \<rightarrow>\<^sub>M borel"
+ by(safe intro!: borel_measurable_ereal_times borel_measurable_const)
+ with \<open>0<c\<close> show ?thesis
+ by (cases "ae_filter M = bot")
+ (auto simp: esssup_def bot_ereal_def top_ereal_def Limsup_ereal_mult_left)
qed
lemma esssup_add:
@@ -186,14 +130,8 @@
qed
lemma esssup_zero_space:
- assumes "emeasure M (space M) = 0"
- "f \<in> borel_measurable M"
- shows "esssup M f = (- \<infinity>::ereal)"
-proof -
- have "emeasure M {x \<in> space M. f x > - \<infinity>} = 0"
- using assms(1) emeasure_mono emeasure_eq_0 by fastforce
- then show ?thesis unfolding esssup_def using assms(2) Inf_eq_MInfty by auto
-qed
+ "emeasure M (space M) = 0 \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> esssup M f = (- \<infinity>::ereal)"
+ by (simp add: esssup_def ae_filter_eq_bot_iff[symmetric] bot_ereal_def)
end