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author | hoelzl |

Tue, 18 Oct 2016 23:47:33 +0200 | |

changeset 64293 | 256298544491 |

parent 64292 | bad166cb5121 |

child 64309 | 1bde86d10013 |

add missing file Essential_Supremum.thy

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Probability/Essential_Supremum.thy Tue Oct 18 23:47:33 2016 +0200 @@ -0,0 +1,199 @@ +(* Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr + License: BSD +*) + +theory Essential_Supremum +imports "../Analysis/Analysis" +begin + +section {*The essential supremum*} + +text {*In this paragraph, we define the essential supremum and give its basic properties. The +essential supremum of a function is its maximum value if one is allowed to throw away a set +of measure $0$. It is convenient to define it to be infinity for non-measurable functions, as +it allows for neater statements in general. This is a prerequisiste to define the space $L^\infty$.*} + +definition esssup::"'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" + where "esssup M f = (if f \<in> borel_measurable M then Inf {z. emeasure M {x \<in> space M. f x > z} = 0} else \<infinity>)" + +lemma esssup_zero_measure: + "emeasure M {x \<in> space M. f x > esssup M f} = 0" +proof (cases "esssup M f = \<infinity>") + case True + then show ?thesis by auto +next + case False + then have [measurable]: "f \<in> borel_measurable M" unfolding esssup_def by meson + have "esssup M f < \<infinity>" using False by auto + 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) + 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 + show ?thesis using null_sets_subset[OF a _ b] by simp + qed + obtain u::"nat \<Rightarrow> ereal" where u: "\<And>n. u n > esssup M f" "u \<longlonglongrightarrow> esssup M f" + using approx_from_above_dense_linorder[OF `esssup M f < \<infinity>`] by auto + have "{x \<in> space M. f x > esssup M f} = (\<Union>n. {x \<in> space M. f x > u n})" + using u apply auto + apply (metis (mono_tags, lifting) order_tendsto_iff eventually_mono LIMSEQ_unique) + using less_imp_le less_le_trans by blast + also have "... \<in> null_sets M" + using *[OF u(1)] by auto + 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 = \<infinity>" unfolding esssup_def by auto + then show ?thesis by auto +qed + +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 + +lemma esssup_non_measurable: + assumes "f \<notin> borel_measurable M" + shows "esssup M f = \<infinity>" +using assms unfolding esssup_def by auto + +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: + 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_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 + +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_cmult: + assumes "c > (0::real)" + shows "esssup M (\<lambda>x. c * f x) = 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 = \<infinity>" using False unfolding esssup_def by auto + then have *: "c * esssup M f = \<infinity>" using assms by (simp add: ennreal_mult_eq_top_iff) + 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 + then show ?thesis using * by auto +qed + +lemma esssup_add: + "esssup M (\<lambda>x. f x + g x) \<le> esssup M f + esssup M g" +proof (cases "f \<in> borel_measurable M \<and> g \<in> borel_measurable M") + case True + then have [measurable]: "(\<lambda>x. f x + g x) \<in> borel_measurable M" by auto + have "f x + g x \<le> esssup M f + esssup M g" if "f x \<le> esssup M f" "g x \<le> esssup M g" for x + using that ereal_add_mono by auto + then have "AE x in M. f x + g x \<le> esssup M f + esssup M g" + using esssup_AE[of f M] esssup_AE[of g M] by auto + then show ?thesis using esssup_I by auto +next + case False + then have "esssup M f + esssup M g = \<infinity>" unfolding esssup_def by auto + then show ?thesis by auto +qed + +lemma esssup_zero_space: + assumes "emeasure M (space M) = 0" + "f \<in> borel_measurable M" + shows "esssup M f = - \<infinity>" +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 + +end +