author | hoelzl |
Fri, 02 Nov 2012 14:23:40 +0100 | |
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parent 50001 | 382bd3173584 |
child 50003 | 8c213922ed49 |
permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Borel_Space.thy |
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Author: Johannes Hölzl, TU München |
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Author: Armin Heller, TU München |
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*) |
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header {*Borel spaces*} |
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theory Borel_Space |
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imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis" |
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begin |
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|
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section "Generic Borel spaces" |
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definition borel :: "'a::topological_space measure" where |
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"borel = sigma UNIV {S. open S}" |
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abbreviation "borel_measurable M \<equiv> measurable M borel" |
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lemma in_borel_measurable: |
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"f \<in> borel_measurable M \<longleftrightarrow> |
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(\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)" |
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by (auto simp add: measurable_def borel_def) |
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|
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lemma in_borel_measurable_borel: |
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"f \<in> borel_measurable M \<longleftrightarrow> |
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(\<forall>S \<in> sets borel. |
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f -` S \<inter> space M \<in> sets M)" |
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by (auto simp add: measurable_def borel_def) |
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lemma space_borel[simp]: "space borel = UNIV" |
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unfolding borel_def by auto |
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lemma space_in_borel[measurable]: "UNIV \<in> sets borel" |
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unfolding borel_def by auto |
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lemma pred_Collect_borel[measurable (raw)]: "Sigma_Algebra.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel" |
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unfolding borel_def pred_def by auto |
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lemma borel_open[simp, measurable (raw generic)]: |
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assumes "open A" shows "A \<in> sets borel" |
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proof - |
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have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms . |
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thus ?thesis unfolding borel_def by auto |
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qed |
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lemma borel_closed[simp, measurable (raw generic)]: |
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assumes "closed A" shows "A \<in> sets borel" |
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proof - |
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have "space borel - (- A) \<in> sets borel" |
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using assms unfolding closed_def by (blast intro: borel_open) |
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thus ?thesis by simp |
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qed |
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lemma borel_insert[measurable]: |
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"A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t2_space measure)" |
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unfolding insert_def by (rule Un) auto |
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lemma borel_comp[intro, simp, measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel" |
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unfolding Compl_eq_Diff_UNIV by simp |
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lemma borel_measurable_vimage: |
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fixes f :: "'a \<Rightarrow> 'x::t2_space" |
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assumes borel[measurable]: "f \<in> borel_measurable M" |
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shows "f -` {x} \<inter> space M \<in> sets M" |
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by simp |
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|
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lemma borel_measurableI: |
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fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space" |
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assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M" |
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shows "f \<in> borel_measurable M" |
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unfolding borel_def |
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proof (rule measurable_measure_of, simp_all) |
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fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M" |
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using assms[of S] by simp |
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qed |
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lemma borel_singleton[simp, intro]: |
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fixes x :: "'a::t1_space" |
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shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel" |
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proof (rule insert_in_sets) |
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show "{x} \<in> sets borel" |
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using closed_singleton[of x] by (rule borel_closed) |
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qed simp |
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lemma borel_measurable_const[simp, intro, measurable (raw)]: |
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"(\<lambda>x. c) \<in> borel_measurable M" |
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by auto |
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lemma borel_measurable_indicator[simp, intro!]: |
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assumes A: "A \<in> sets M" |
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shows "indicator A \<in> borel_measurable M" |
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unfolding indicator_def [abs_def] using A |
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by (auto intro!: measurable_If_set) |
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lemma borel_measurable_indicator'[measurable]: |
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"{x\<in>space M. x \<in> A} \<in> sets M \<Longrightarrow> indicator A \<in> borel_measurable M" |
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unfolding indicator_def[abs_def] |
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by (auto intro!: measurable_If) |
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lemma borel_measurable_indicator_iff: |
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"(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M" |
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(is "?I \<in> borel_measurable M \<longleftrightarrow> _") |
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proof |
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assume "?I \<in> borel_measurable M" |
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then have "?I -` {1} \<inter> space M \<in> sets M" |
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unfolding measurable_def by auto |
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also have "?I -` {1} \<inter> space M = A \<inter> space M" |
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unfolding indicator_def [abs_def] by auto |
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finally show "A \<inter> space M \<in> sets M" . |
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next |
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assume "A \<inter> space M \<in> sets M" |
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moreover have "?I \<in> borel_measurable M \<longleftrightarrow> |
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(indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M" |
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by (intro measurable_cong) (auto simp: indicator_def) |
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ultimately show "?I \<in> borel_measurable M" by auto |
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qed |
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||
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lemma borel_measurable_subalgebra: |
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assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N" |
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shows "f \<in> borel_measurable M" |
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using assms unfolding measurable_def by auto |
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lemma borel_measurable_continuous_on1: |
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fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" |
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assumes "continuous_on UNIV f" |
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shows "f \<in> borel_measurable borel" |
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apply(rule borel_measurableI) |
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using continuous_open_preimage[OF assms] unfolding vimage_def by auto |
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section "Borel spaces on euclidean spaces" |
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||
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lemma borel_measurable_euclidean_component'[measurable]: |
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"(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel" |
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by (intro continuous_on_euclidean_component continuous_on_id borel_measurable_continuous_on1) |
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lemma borel_measurable_euclidean_component: |
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"(f :: 'a \<Rightarrow> 'b::euclidean_space) \<in> borel_measurable M \<Longrightarrow>(\<lambda>x. f x $$ i) \<in> borel_measurable M" |
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by simp |
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lemma [simp, intro, measurable]: |
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fixes a b :: "'a\<Colon>ordered_euclidean_space" |
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shows lessThan_borel: "{..< a} \<in> sets borel" |
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and greaterThan_borel: "{a <..} \<in> sets borel" |
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and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel" |
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and atMost_borel: "{..a} \<in> sets borel" |
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and atLeast_borel: "{a..} \<in> sets borel" |
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and atLeastAtMost_borel: "{a..b} \<in> sets borel" |
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and greaterThanAtMost_borel: "{a<..b} \<in> sets borel" |
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and atLeastLessThan_borel: "{a..<b} \<in> sets borel" |
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unfolding greaterThanAtMost_def atLeastLessThan_def |
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by (blast intro: borel_open borel_closed)+ |
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lemma |
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shows hafspace_less_borel[simp, intro]: "{x::'a::euclidean_space. a < x $$ i} \<in> sets borel" |
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and hafspace_greater_borel[simp, intro]: "{x::'a::euclidean_space. x $$ i < a} \<in> sets borel" |
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and hafspace_less_eq_borel[simp, intro]: "{x::'a::euclidean_space. a \<le> x $$ i} \<in> sets borel" |
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and hafspace_greater_eq_borel[simp, intro]: "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel" |
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by simp_all |
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lemma borel_measurable_less[simp, intro, measurable]: |
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fixes f :: "'a \<Rightarrow> real" |
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assumes f: "f \<in> borel_measurable M" |
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assumes g: "g \<in> borel_measurable M" |
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shows "{w \<in> space M. f w < g w} \<in> sets M" |
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proof - |
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have "{w \<in> space M. f w < g w} = {x \<in> space M. \<exists>r. f x < of_rat r \<and> of_rat r < g x}" |
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using Rats_dense_in_real by (auto simp add: Rats_def) |
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with f g show ?thesis |
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169 |
by simp |
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170 |
qed |
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lemma [simp, intro]: |
38656 | 173 |
fixes f :: "'a \<Rightarrow> real" |
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174 |
assumes f[measurable]: "f \<in> borel_measurable M" |
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175 |
assumes g[measurable]: "g \<in> borel_measurable M" |
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176 |
shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M" |
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177 |
and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M" |
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178 |
and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M" |
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179 |
unfolding eq_iff not_less[symmetric] by measurable+ |
38656 | 180 |
|
181 |
subsection "Borel space equals sigma algebras over intervals" |
|
182 |
||
183 |
lemma rational_boxes: |
|
184 |
fixes x :: "'a\<Colon>ordered_euclidean_space" |
|
185 |
assumes "0 < e" |
|
186 |
shows "\<exists>a b. (\<forall>i. a $$ i \<in> \<rat>) \<and> (\<forall>i. b $$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e" |
|
187 |
proof - |
|
188 |
def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))" |
|
189 |
then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos) |
|
190 |
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i") |
|
191 |
proof |
|
192 |
fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e |
|
193 |
show "?th i" by auto |
|
194 |
qed |
|
195 |
from choice[OF this] guess a .. note a = this |
|
196 |
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i") |
|
197 |
proof |
|
198 |
fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e |
|
199 |
show "?th i" by auto |
|
200 |
qed |
|
201 |
from choice[OF this] guess b .. note b = this |
|
202 |
{ fix y :: 'a assume *: "Chi a < y" "y < Chi b" |
|
203 |
have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)" |
|
204 |
unfolding setL2_def[symmetric] by (rule euclidean_dist_l2) |
|
205 |
also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))" |
|
206 |
proof (rule real_sqrt_less_mono, rule setsum_strict_mono) |
|
207 |
fix i assume i: "i \<in> {..<DIM('a)}" |
|
208 |
have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto |
|
209 |
moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto |
|
210 |
moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto |
|
211 |
ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto |
|
212 |
then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))" |
|
213 |
unfolding e'_def by (auto simp: dist_real_def) |
|
214 |
then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>" |
|
215 |
by (rule power_strict_mono) auto |
|
216 |
then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)" |
|
217 |
by (simp add: power_divide) |
|
218 |
qed auto |
|
219 |
also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive) |
|
220 |
finally have "dist x y < e" . } |
|
221 |
with a b show ?thesis |
|
222 |
apply (rule_tac exI[of _ "Chi a"]) |
|
223 |
apply (rule_tac exI[of _ "Chi b"]) |
|
224 |
using eucl_less[where 'a='a] by auto |
|
225 |
qed |
|
226 |
||
227 |
lemma ex_rat_list: |
|
228 |
fixes x :: "'a\<Colon>ordered_euclidean_space" |
|
229 |
assumes "\<And> i. x $$ i \<in> \<rat>" |
|
230 |
shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)" |
|
231 |
proof - |
|
232 |
have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast |
|
233 |
from choice[OF this] guess r .. |
|
234 |
then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"]) |
|
235 |
qed |
|
236 |
||
237 |
lemma open_UNION: |
|
238 |
fixes M :: "'a\<Colon>ordered_euclidean_space set" |
|
239 |
assumes "open M" |
|
240 |
shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M} |
|
241 |
(\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})" |
|
242 |
(is "M = UNION ?idx ?box") |
|
243 |
proof safe |
|
244 |
fix x assume "x \<in> M" |
|
245 |
obtain e where e: "e > 0" "ball x e \<subseteq> M" |
|
246 |
using openE[OF assms `x \<in> M`] by auto |
|
247 |
then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a $$ i \<in> \<rat>" "\<And>i. b $$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e" |
|
248 |
using rational_boxes[OF e(1)] by blast |
|
249 |
then obtain p q where pq: "length p = DIM ('a)" |
|
250 |
"length q = DIM ('a)" |
|
251 |
"\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i" |
|
252 |
using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast |
|
253 |
hence p: "Chi (of_rat \<circ> op ! p) = a" |
|
254 |
using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a] |
|
255 |
unfolding o_def by auto |
|
256 |
from pq have q: "Chi (of_rat \<circ> op ! q) = b" |
|
257 |
using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b] |
|
258 |
unfolding o_def by auto |
|
259 |
have "x \<in> ?box (p, q)" |
|
260 |
using p q ab by auto |
|
261 |
thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto |
|
262 |
qed auto |
|
263 |
||
47694 | 264 |
lemma borel_sigma_sets_subset: |
265 |
"A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel" |
|
266 |
using sigma_sets_subset[of A borel] by simp |
|
267 |
||
268 |
lemma borel_eq_sigmaI1: |
|
269 |
fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set" |
|
270 |
assumes borel_eq: "borel = sigma UNIV X" |
|
271 |
assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range F))" |
|
272 |
assumes F: "\<And>i. F i \<in> sets borel" |
|
273 |
shows "borel = sigma UNIV (range F)" |
|
274 |
unfolding borel_def |
|
275 |
proof (intro sigma_eqI antisym) |
|
276 |
have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel" |
|
277 |
unfolding borel_def by simp |
|
278 |
also have "\<dots> = sigma_sets UNIV X" |
|
279 |
unfolding borel_eq by simp |
|
280 |
also have "\<dots> \<subseteq> sigma_sets UNIV (range F)" |
|
281 |
using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto |
|
282 |
finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (range F)" . |
|
283 |
show "sigma_sets UNIV (range F) \<subseteq> sigma_sets UNIV {S. open S}" |
|
284 |
unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto |
|
285 |
qed auto |
|
38656 | 286 |
|
47694 | 287 |
lemma borel_eq_sigmaI2: |
288 |
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" |
|
289 |
and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set" |
|
290 |
assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))" |
|
291 |
assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))" |
|
292 |
assumes F: "\<And>i j. F i j \<in> sets borel" |
|
293 |
shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))" |
|
294 |
using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F="(\<lambda>(i, j). F i j)"]) auto |
|
295 |
||
296 |
lemma borel_eq_sigmaI3: |
|
297 |
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set" |
|
298 |
assumes borel_eq: "borel = sigma UNIV X" |
|
299 |
assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))" |
|
300 |
assumes F: "\<And>i j. F i j \<in> sets borel" |
|
301 |
shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))" |
|
302 |
using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto |
|
303 |
||
304 |
lemma borel_eq_sigmaI4: |
|
305 |
fixes F :: "'i \<Rightarrow> 'a::topological_space set" |
|
306 |
and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set" |
|
307 |
assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))" |
|
308 |
assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range F))" |
|
309 |
assumes F: "\<And>i. F i \<in> sets borel" |
|
310 |
shows "borel = sigma UNIV (range F)" |
|
311 |
using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F=F]) auto |
|
312 |
||
313 |
lemma borel_eq_sigmaI5: |
|
314 |
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set" |
|
315 |
assumes borel_eq: "borel = sigma UNIV (range G)" |
|
316 |
assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))" |
|
317 |
assumes F: "\<And>i j. F i j \<in> sets borel" |
|
318 |
shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))" |
|
319 |
using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto |
|
38656 | 320 |
|
321 |
lemma halfspace_gt_in_halfspace: |
|
47694 | 322 |
"{x\<Colon>'a. a < x $$ i} \<in> sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))" |
323 |
(is "?set \<in> ?SIGMA") |
|
38656 | 324 |
proof - |
47694 | 325 |
interpret sigma_algebra UNIV ?SIGMA |
326 |
by (intro sigma_algebra_sigma_sets) simp_all |
|
327 |
have *: "?set = (\<Union>n. UNIV - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})" |
|
38656 | 328 |
proof (safe, simp_all add: not_less) |
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329 |
fix x :: 'a assume "a < x $$ i" |
38656 | 330 |
with reals_Archimedean[of "x $$ i - a"] |
331 |
obtain n where "a + 1 / real (Suc n) < x $$ i" |
|
332 |
by (auto simp: inverse_eq_divide field_simps) |
|
333 |
then show "\<exists>n. a + 1 / real (Suc n) \<le> x $$ i" |
|
334 |
by (blast intro: less_imp_le) |
|
335 |
next |
|
336 |
fix x n |
|
337 |
have "a < a + 1 / real (Suc n)" by auto |
|
338 |
also assume "\<dots> \<le> x" |
|
339 |
finally show "a < x" . |
|
340 |
qed |
|
47694 | 341 |
show "?set \<in> ?SIGMA" unfolding * |
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|
342 |
by (auto del: Diff intro!: Diff) |
40859 | 343 |
qed |
38656 | 344 |
|
47694 | 345 |
lemma borel_eq_halfspace_less: |
346 |
"borel = sigma UNIV (range (\<lambda>(a, i). {x::'a::ordered_euclidean_space. x $$ i < a}))" |
|
347 |
(is "_ = ?SIGMA") |
|
348 |
proof (rule borel_eq_sigmaI3[OF borel_def]) |
|
349 |
fix S :: "'a set" assume "S \<in> {S. open S}" |
|
350 |
then have "open S" by simp |
|
351 |
from open_UNION[OF this] |
|
352 |
obtain I where *: "S = |
|
353 |
(\<Union>(a, b)\<in>I. |
|
354 |
(\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter> |
|
355 |
(\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))" |
|
356 |
unfolding greaterThanLessThan_def |
|
357 |
unfolding eucl_greaterThan_eq_halfspaces[where 'a='a] |
|
358 |
unfolding eucl_lessThan_eq_halfspaces[where 'a='a] |
|
359 |
by blast |
|
360 |
show "S \<in> ?SIGMA" |
|
361 |
unfolding * |
|
362 |
by (safe intro!: countable_UN Int countable_INT) (auto intro!: halfspace_gt_in_halfspace) |
|
363 |
qed auto |
|
38656 | 364 |
|
47694 | 365 |
lemma borel_eq_halfspace_le: |
366 |
"borel = sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i \<le> a}))" |
|
367 |
(is "_ = ?SIGMA") |
|
368 |
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less]) |
|
369 |
fix a i |
|
370 |
have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})" |
|
371 |
proof (safe, simp_all) |
|
372 |
fix x::'a assume *: "x$$i < a" |
|
373 |
with reals_Archimedean[of "a - x$$i"] |
|
374 |
obtain n where "x $$ i < a - 1 / (real (Suc n))" |
|
375 |
by (auto simp: field_simps inverse_eq_divide) |
|
376 |
then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))" |
|
377 |
by (blast intro: less_imp_le) |
|
378 |
next |
|
379 |
fix x::'a and n |
|
380 |
assume "x$$i \<le> a - 1 / real (Suc n)" |
|
381 |
also have "\<dots> < a" by auto |
|
382 |
finally show "x$$i < a" . |
|
383 |
qed |
|
384 |
show "{x. x$$i < a} \<in> ?SIGMA" unfolding * |
|
385 |
by (safe intro!: countable_UN) auto |
|
386 |
qed auto |
|
38656 | 387 |
|
47694 | 388 |
lemma borel_eq_halfspace_ge: |
389 |
"borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i}))" |
|
390 |
(is "_ = ?SIGMA") |
|
391 |
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less]) |
|
392 |
fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto |
|
393 |
show "{x. x$$i < a} \<in> ?SIGMA" unfolding * |
|
394 |
by (safe intro!: compl_sets) auto |
|
395 |
qed auto |
|
38656 | 396 |
|
47694 | 397 |
lemma borel_eq_halfspace_greater: |
398 |
"borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a < x $$ i}))" |
|
399 |
(is "_ = ?SIGMA") |
|
400 |
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le]) |
|
401 |
fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto |
|
402 |
show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding * |
|
403 |
by (safe intro!: compl_sets) auto |
|
404 |
qed auto |
|
405 |
||
406 |
lemma borel_eq_atMost: |
|
407 |
"borel = sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))" |
|
408 |
(is "_ = ?SIGMA") |
|
409 |
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le]) |
|
410 |
fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA" |
|
38656 | 411 |
proof cases |
47694 | 412 |
assume "i < DIM('a)" |
38656 | 413 |
then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})" |
414 |
proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm) |
|
415 |
fix x |
|
416 |
from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat .. |
|
417 |
then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k" |
|
418 |
by (subst (asm) Max_le_iff) auto |
|
419 |
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k" |
|
420 |
by (auto intro!: exI[of _ k]) |
|
421 |
qed |
|
47694 | 422 |
show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding * |
423 |
by (safe intro!: countable_UN) auto |
|
424 |
qed (auto intro: sigma_sets_top sigma_sets.Empty) |
|
425 |
qed auto |
|
38656 | 426 |
|
47694 | 427 |
lemma borel_eq_greaterThan: |
428 |
"borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {a<..}))" |
|
429 |
(is "_ = ?SIGMA") |
|
430 |
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le]) |
|
431 |
fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA" |
|
38656 | 432 |
proof cases |
47694 | 433 |
assume "i < DIM('a)" |
434 |
have "{x::'a. x$$i \<le> a} = UNIV - {x::'a. a < x$$i}" by auto |
|
38656 | 435 |
also have *: "{x::'a. a < x$$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)` |
436 |
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm) |
|
437 |
fix x |
|
44666 | 438 |
from reals_Archimedean2[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"] |
38656 | 439 |
guess k::nat .. note k = this |
440 |
{ fix i assume "i < DIM('a)" |
|
441 |
then have "-x$$i < real k" |
|
442 |
using k by (subst (asm) Max_less_iff) auto |
|
443 |
then have "- real k < x$$i" by simp } |
|
444 |
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia" |
|
445 |
by (auto intro!: exI[of _ k]) |
|
446 |
qed |
|
47694 | 447 |
finally show "{x. x$$i \<le> a} \<in> ?SIGMA" |
38656 | 448 |
apply (simp only:) |
449 |
apply (safe intro!: countable_UN Diff) |
|
47694 | 450 |
apply (auto intro: sigma_sets_top) |
46731 | 451 |
done |
47694 | 452 |
qed (auto intro: sigma_sets_top sigma_sets.Empty) |
453 |
qed auto |
|
40859 | 454 |
|
47694 | 455 |
lemma borel_eq_lessThan: |
456 |
"borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {..<a}))" |
|
457 |
(is "_ = ?SIGMA") |
|
458 |
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge]) |
|
459 |
fix a i show "{x. a \<le> x$$i} \<in> ?SIGMA" |
|
40859 | 460 |
proof cases |
461 |
fix a i assume "i < DIM('a)" |
|
47694 | 462 |
have "{x::'a. a \<le> x$$i} = UNIV - {x::'a. x$$i < a}" by auto |
40859 | 463 |
also have *: "{x::'a. x$$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)` |
464 |
proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm) |
|
465 |
fix x |
|
44666 | 466 |
from reals_Archimedean2[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] |
40859 | 467 |
guess k::nat .. note k = this |
468 |
{ fix i assume "i < DIM('a)" |
|
469 |
then have "x$$i < real k" |
|
470 |
using k by (subst (asm) Max_less_iff) auto |
|
471 |
then have "x$$i < real k" by simp } |
|
472 |
then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia < real k" |
|
473 |
by (auto intro!: exI[of _ k]) |
|
474 |
qed |
|
47694 | 475 |
finally show "{x. a \<le> x$$i} \<in> ?SIGMA" |
40859 | 476 |
apply (simp only:) |
477 |
apply (safe intro!: countable_UN Diff) |
|
47694 | 478 |
apply (auto intro: sigma_sets_top) |
46731 | 479 |
done |
47694 | 480 |
qed (auto intro: sigma_sets_top sigma_sets.Empty) |
40859 | 481 |
qed auto |
482 |
||
483 |
lemma borel_eq_atLeastAtMost: |
|
47694 | 484 |
"borel = sigma UNIV (range (\<lambda>(a,b). {a..b} \<Colon>'a\<Colon>ordered_euclidean_space set))" |
485 |
(is "_ = ?SIGMA") |
|
486 |
proof (rule borel_eq_sigmaI5[OF borel_eq_atMost]) |
|
487 |
fix a::'a |
|
488 |
have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})" |
|
489 |
proof (safe, simp_all add: eucl_le[where 'a='a]) |
|
490 |
fix x |
|
491 |
from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"] |
|
492 |
guess k::nat .. note k = this |
|
493 |
{ fix i assume "i < DIM('a)" |
|
494 |
with k have "- x$$i \<le> real k" |
|
495 |
by (subst (asm) Max_le_iff) (auto simp: field_simps) |
|
496 |
then have "- real k \<le> x$$i" by simp } |
|
497 |
then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i" |
|
498 |
by (auto intro!: exI[of _ k]) |
|
499 |
qed |
|
500 |
show "{..a} \<in> ?SIGMA" unfolding * |
|
501 |
by (safe intro!: countable_UN) |
|
502 |
(auto intro!: sigma_sets_top) |
|
40859 | 503 |
qed auto |
504 |
||
505 |
lemma borel_eq_greaterThanLessThan: |
|
47694 | 506 |
"borel = sigma UNIV (range (\<lambda> (a, b). {a <..< b} :: 'a \<Colon> ordered_euclidean_space set))" |
40859 | 507 |
(is "_ = ?SIGMA") |
47694 | 508 |
proof (rule borel_eq_sigmaI1[OF borel_def]) |
509 |
fix M :: "'a set" assume "M \<in> {S. open S}" |
|
510 |
then have "open M" by simp |
|
511 |
show "M \<in> ?SIGMA" |
|
512 |
apply (subst open_UNION[OF `open M`]) |
|
513 |
apply (safe intro!: countable_UN) |
|
514 |
apply auto |
|
515 |
done |
|
38656 | 516 |
qed auto |
517 |
||
42862 | 518 |
lemma borel_eq_atLeastLessThan: |
47694 | 519 |
"borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA") |
520 |
proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan]) |
|
521 |
have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto |
|
522 |
fix x :: real |
|
523 |
have "{..<x} = (\<Union>i::nat. {-real i ..< x})" |
|
524 |
by (auto simp: move_uminus real_arch_simple) |
|
525 |
then show "{..< x} \<in> ?SIGMA" |
|
526 |
by (auto intro: sigma_sets.intros) |
|
40859 | 527 |
qed auto |
528 |
||
47694 | 529 |
lemma borel_measurable_halfspacesI: |
38656 | 530 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
47694 | 531 |
assumes F: "borel = sigma UNIV (range F)" |
532 |
and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M" |
|
533 |
and S: "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M" |
|
38656 | 534 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)" |
535 |
proof safe |
|
536 |
fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M" |
|
537 |
then show "S a i \<in> sets M" unfolding assms |
|
47694 | 538 |
by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1)) |
38656 | 539 |
next |
540 |
assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M" |
|
541 |
{ fix a i have "S a i \<in> sets M" |
|
542 |
proof cases |
|
543 |
assume "i < DIM('c)" |
|
544 |
with a show ?thesis unfolding assms(2) by simp |
|
545 |
next |
|
546 |
assume "\<not> i < DIM('c)" |
|
47694 | 547 |
from S[OF this] show ?thesis . |
38656 | 548 |
qed } |
47694 | 549 |
then show "f \<in> borel_measurable M" |
550 |
by (auto intro!: measurable_measure_of simp: S_eq F) |
|
38656 | 551 |
qed |
552 |
||
47694 | 553 |
lemma borel_measurable_iff_halfspace_le: |
38656 | 554 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
555 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)" |
|
40859 | 556 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto |
38656 | 557 |
|
47694 | 558 |
lemma borel_measurable_iff_halfspace_less: |
38656 | 559 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
560 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)" |
|
40859 | 561 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto |
38656 | 562 |
|
47694 | 563 |
lemma borel_measurable_iff_halfspace_ge: |
38656 | 564 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
565 |
shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)" |
|
40859 | 566 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto |
38656 | 567 |
|
47694 | 568 |
lemma borel_measurable_iff_halfspace_greater: |
38656 | 569 |
fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space" |
570 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)" |
|
47694 | 571 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto |
38656 | 572 |
|
47694 | 573 |
lemma borel_measurable_iff_le: |
38656 | 574 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)" |
575 |
using borel_measurable_iff_halfspace_le[where 'c=real] by simp |
|
576 |
||
47694 | 577 |
lemma borel_measurable_iff_less: |
38656 | 578 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)" |
579 |
using borel_measurable_iff_halfspace_less[where 'c=real] by simp |
|
580 |
||
47694 | 581 |
lemma borel_measurable_iff_ge: |
38656 | 582 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
583 |
using borel_measurable_iff_halfspace_ge[where 'c=real] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
584 |
by simp |
38656 | 585 |
|
47694 | 586 |
lemma borel_measurable_iff_greater: |
38656 | 587 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)" |
588 |
using borel_measurable_iff_halfspace_greater[where 'c=real] by simp |
|
589 |
||
47694 | 590 |
lemma borel_measurable_euclidean_space: |
39087
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
591 |
fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
592 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
593 |
proof safe |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
594 |
fix i assume "f \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
595 |
then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M" |
41025 | 596 |
by (auto intro: borel_measurable_euclidean_component) |
39087
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
597 |
next |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
598 |
assume f: "\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
599 |
then show "f \<in> borel_measurable M" |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
600 |
unfolding borel_measurable_iff_halfspace_le by auto |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
601 |
qed |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
602 |
|
38656 | 603 |
subsection "Borel measurable operators" |
604 |
||
49774 | 605 |
lemma borel_measurable_continuous_on: |
606 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" |
|
607 |
assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M" |
|
608 |
shows "(\<lambda>x. f (g x)) \<in> borel_measurable M" |
|
609 |
using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def) |
|
610 |
||
611 |
lemma borel_measurable_continuous_on_open': |
|
612 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space" |
|
613 |
assumes cont: "continuous_on A f" "open A" |
|
614 |
shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _") |
|
615 |
proof (rule borel_measurableI) |
|
616 |
fix S :: "'b set" assume "open S" |
|
617 |
then have "open {x\<in>A. f x \<in> S}" |
|
618 |
by (intro continuous_open_preimage[OF cont]) auto |
|
619 |
then have *: "{x\<in>A. f x \<in> S} \<in> sets borel" by auto |
|
620 |
have "?f -` S \<inter> space borel = |
|
621 |
{x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})" |
|
622 |
by (auto split: split_if_asm) |
|
623 |
also have "\<dots> \<in> sets borel" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
624 |
using * `open A` by auto |
49774 | 625 |
finally show "?f -` S \<inter> space borel \<in> sets borel" . |
626 |
qed |
|
627 |
||
628 |
lemma borel_measurable_continuous_on_open: |
|
629 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space" |
|
630 |
assumes cont: "continuous_on A f" "open A" |
|
631 |
assumes g: "g \<in> borel_measurable M" |
|
632 |
shows "(\<lambda>x. if g x \<in> A then f (g x) else c) \<in> borel_measurable M" |
|
633 |
using measurable_comp[OF g borel_measurable_continuous_on_open'[OF cont], of c] |
|
634 |
by (simp add: comp_def) |
|
635 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
636 |
lemma borel_measurable_uminus[simp, intro, measurable (raw)]: |
49774 | 637 |
fixes g :: "'a \<Rightarrow> real" |
638 |
assumes g: "g \<in> borel_measurable M" |
|
639 |
shows "(\<lambda>x. - g x) \<in> borel_measurable M" |
|
640 |
by (rule borel_measurable_continuous_on[OF _ g]) (auto intro: continuous_on_minus continuous_on_id) |
|
641 |
||
642 |
lemma euclidean_component_prod: |
|
643 |
fixes x :: "'a :: euclidean_space \<times> 'b :: euclidean_space" |
|
644 |
shows "x $$ i = (if i < DIM('a) then fst x $$ i else snd x $$ (i - DIM('a)))" |
|
645 |
unfolding euclidean_component_def basis_prod_def inner_prod_def by auto |
|
646 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
647 |
lemma borel_measurable_Pair[simp, intro, measurable (raw)]: |
49774 | 648 |
fixes f :: "'a \<Rightarrow> 'b::ordered_euclidean_space" and g :: "'a \<Rightarrow> 'c::ordered_euclidean_space" |
649 |
assumes f: "f \<in> borel_measurable M" |
|
650 |
assumes g: "g \<in> borel_measurable M" |
|
651 |
shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M" |
|
652 |
proof (intro borel_measurable_iff_halfspace_le[THEN iffD2] allI impI) |
|
653 |
fix i and a :: real assume i: "i < DIM('b \<times> 'c)" |
|
654 |
have [simp]: "\<And>P A B C. {w. (P \<longrightarrow> A w \<and> B w) \<and> (\<not> P \<longrightarrow> A w \<and> C w)} = |
|
655 |
{w. A w \<and> (P \<longrightarrow> B w) \<and> (\<not> P \<longrightarrow> C w)}" by auto |
|
656 |
from i f g show "{w \<in> space M. (f w, g w) $$ i \<le> a} \<in> sets M" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
657 |
by (auto simp: euclidean_component_prod) |
49774 | 658 |
qed |
659 |
||
660 |
lemma continuous_on_fst: "continuous_on UNIV fst" |
|
661 |
proof - |
|
662 |
have [simp]: "range fst = UNIV" by (auto simp: image_iff) |
|
663 |
show ?thesis |
|
664 |
using closed_vimage_fst |
|
665 |
by (auto simp: continuous_on_closed closed_closedin vimage_def) |
|
666 |
qed |
|
667 |
||
668 |
lemma continuous_on_snd: "continuous_on UNIV snd" |
|
669 |
proof - |
|
670 |
have [simp]: "range snd = UNIV" by (auto simp: image_iff) |
|
671 |
show ?thesis |
|
672 |
using closed_vimage_snd |
|
673 |
by (auto simp: continuous_on_closed closed_closedin vimage_def) |
|
674 |
qed |
|
675 |
||
676 |
lemma borel_measurable_continuous_Pair: |
|
677 |
fixes f :: "'a \<Rightarrow> 'b::ordered_euclidean_space" and g :: "'a \<Rightarrow> 'c::ordered_euclidean_space" |
|
678 |
assumes [simp]: "f \<in> borel_measurable M" |
|
679 |
assumes [simp]: "g \<in> borel_measurable M" |
|
680 |
assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))" |
|
681 |
shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M" |
|
682 |
proof - |
|
683 |
have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto |
|
684 |
show ?thesis |
|
685 |
unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto |
|
686 |
qed |
|
687 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
688 |
lemma borel_measurable_add[simp, intro, measurable (raw)]: |
49774 | 689 |
fixes f g :: "'a \<Rightarrow> 'c::ordered_euclidean_space" |
690 |
assumes f: "f \<in> borel_measurable M" |
|
691 |
assumes g: "g \<in> borel_measurable M" |
|
692 |
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M" |
|
693 |
using f g |
|
694 |
by (rule borel_measurable_continuous_Pair) |
|
695 |
(auto intro: continuous_on_fst continuous_on_snd continuous_on_add) |
|
696 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
697 |
lemma borel_measurable_setsum[simp, intro, measurable (raw)]: |
49774 | 698 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real" |
699 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
|
700 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M" |
|
701 |
proof cases |
|
702 |
assume "finite S" |
|
703 |
thus ?thesis using assms by induct auto |
|
704 |
qed simp |
|
705 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
706 |
lemma borel_measurable_diff[simp, intro, measurable (raw)]: |
49774 | 707 |
fixes f :: "'a \<Rightarrow> real" |
708 |
assumes f: "f \<in> borel_measurable M" |
|
709 |
assumes g: "g \<in> borel_measurable M" |
|
710 |
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M" |
|
711 |
unfolding diff_minus using assms by fast |
|
712 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
713 |
lemma borel_measurable_times[simp, intro, measurable (raw)]: |
49774 | 714 |
fixes f :: "'a \<Rightarrow> real" |
715 |
assumes f: "f \<in> borel_measurable M" |
|
716 |
assumes g: "g \<in> borel_measurable M" |
|
717 |
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M" |
|
718 |
using f g |
|
719 |
by (rule borel_measurable_continuous_Pair) |
|
720 |
(auto intro: continuous_on_fst continuous_on_snd continuous_on_mult) |
|
721 |
||
722 |
lemma continuous_on_dist: |
|
723 |
fixes f :: "'a :: t2_space \<Rightarrow> 'b :: metric_space" |
|
724 |
shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. dist (f x) (g x))" |
|
725 |
unfolding continuous_on_eq_continuous_within by (auto simp: continuous_dist) |
|
726 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
727 |
lemma borel_measurable_dist[simp, intro, measurable (raw)]: |
49774 | 728 |
fixes g f :: "'a \<Rightarrow> 'b::ordered_euclidean_space" |
729 |
assumes f: "f \<in> borel_measurable M" |
|
730 |
assumes g: "g \<in> borel_measurable M" |
|
731 |
shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M" |
|
732 |
using f g |
|
733 |
by (rule borel_measurable_continuous_Pair) |
|
734 |
(intro continuous_on_dist continuous_on_fst continuous_on_snd) |
|
735 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
736 |
lemma borel_measurable_scaleR[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
737 |
fixes g :: "'a \<Rightarrow> 'b::ordered_euclidean_space" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
738 |
assumes f: "f \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
739 |
assumes g: "g \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
740 |
shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
741 |
by (rule borel_measurable_continuous_Pair[OF f g]) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
742 |
(auto intro!: continuous_on_scaleR continuous_on_fst continuous_on_snd) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
743 |
|
47694 | 744 |
lemma affine_borel_measurable_vector: |
38656 | 745 |
fixes f :: "'a \<Rightarrow> 'x::real_normed_vector" |
746 |
assumes "f \<in> borel_measurable M" |
|
747 |
shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M" |
|
748 |
proof (rule borel_measurableI) |
|
749 |
fix S :: "'x set" assume "open S" |
|
750 |
show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M" |
|
751 |
proof cases |
|
752 |
assume "b \<noteq> 0" |
|
44537
c10485a6a7af
make HOL-Probability respect set/pred distinction
huffman
parents:
44282
diff
changeset
|
753 |
with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S") |
c10485a6a7af
make HOL-Probability respect set/pred distinction
huffman
parents:
44282
diff
changeset
|
754 |
by (auto intro!: open_affinity simp: scaleR_add_right) |
47694 | 755 |
hence "?S \<in> sets borel" by auto |
38656 | 756 |
moreover |
757 |
from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S" |
|
758 |
apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all) |
|
40859 | 759 |
ultimately show ?thesis using assms unfolding in_borel_measurable_borel |
38656 | 760 |
by auto |
761 |
qed simp |
|
762 |
qed |
|
763 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
764 |
lemma borel_measurable_const_scaleR[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
765 |
"f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
766 |
using affine_borel_measurable_vector[of f M 0 b] by simp |
38656 | 767 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
768 |
lemma borel_measurable_const_add[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
769 |
"f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
770 |
using affine_borel_measurable_vector[of f M a 1] by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
771 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
772 |
lemma borel_measurable_setprod[simp, intro, measurable (raw)]: |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
773 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
774 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
775 |
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
776 |
proof cases |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
777 |
assume "finite S" |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
778 |
thus ?thesis using assms by induct auto |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
779 |
qed simp |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
780 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
781 |
lemma borel_measurable_inverse[simp, intro, measurable (raw)]: |
38656 | 782 |
fixes f :: "'a \<Rightarrow> real" |
49774 | 783 |
assumes f: "f \<in> borel_measurable M" |
35692 | 784 |
shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M" |
49774 | 785 |
proof - |
786 |
have *: "\<And>x::real. inverse x = (if x \<in> UNIV - {0} then inverse x else 0)" by auto |
|
787 |
show ?thesis |
|
788 |
apply (subst *) |
|
789 |
apply (rule borel_measurable_continuous_on_open) |
|
790 |
apply (auto intro!: f continuous_on_inverse continuous_on_id) |
|
791 |
done |
|
35692 | 792 |
qed |
793 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
794 |
lemma borel_measurable_divide[simp, intro, measurable (raw)]: |
38656 | 795 |
fixes f :: "'a \<Rightarrow> real" |
35692 | 796 |
assumes "f \<in> borel_measurable M" |
797 |
and "g \<in> borel_measurable M" |
|
798 |
shows "(\<lambda>x. f x / g x) \<in> borel_measurable M" |
|
799 |
unfolding field_divide_inverse |
|
38656 | 800 |
by (rule borel_measurable_inverse borel_measurable_times assms)+ |
801 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
802 |
lemma borel_measurable_max[intro, simp, measurable (raw)]: |
38656 | 803 |
fixes f g :: "'a \<Rightarrow> real" |
804 |
assumes "f \<in> borel_measurable M" |
|
805 |
assumes "g \<in> borel_measurable M" |
|
806 |
shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M" |
|
49774 | 807 |
unfolding max_def by (auto intro!: assms measurable_If) |
38656 | 808 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
809 |
lemma borel_measurable_min[intro, simp, measurable (raw)]: |
38656 | 810 |
fixes f g :: "'a \<Rightarrow> real" |
811 |
assumes "f \<in> borel_measurable M" |
|
812 |
assumes "g \<in> borel_measurable M" |
|
813 |
shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M" |
|
49774 | 814 |
unfolding min_def by (auto intro!: assms measurable_If) |
38656 | 815 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
816 |
lemma borel_measurable_abs[simp, intro, measurable (raw)]: |
38656 | 817 |
assumes "f \<in> borel_measurable M" |
818 |
shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M" |
|
819 |
proof - |
|
820 |
have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def) |
|
821 |
show ?thesis unfolding * using assms by auto |
|
822 |
qed |
|
823 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
824 |
lemma borel_measurable_nth[simp, intro, measurable (raw)]: |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
825 |
"(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel" |
49774 | 826 |
using borel_measurable_euclidean_component' |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
827 |
unfolding nth_conv_component by auto |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
828 |
|
47694 | 829 |
lemma convex_measurable: |
42990
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
830 |
fixes a b :: real |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
831 |
assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
832 |
assumes q: "convex_on { a <..< b} q" |
49774 | 833 |
shows "(\<lambda>x. q (X x)) \<in> borel_measurable M" |
42990
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
834 |
proof - |
49774 | 835 |
have "(\<lambda>x. if X x \<in> {a <..< b} then q (X x) else 0) \<in> borel_measurable M" (is "?qX") |
836 |
proof (rule borel_measurable_continuous_on_open[OF _ _ X(1)]) |
|
42990
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
837 |
show "open {a<..<b}" by auto |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
838 |
from this q show "continuous_on {a<..<b} q" |
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
839 |
by (rule convex_on_continuous) |
41830 | 840 |
qed |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
841 |
also have "?qX \<longleftrightarrow> (\<lambda>x. q (X x)) \<in> borel_measurable M" |
42990
3706951a6421
composition of convex and measurable function is measurable
hoelzl
parents:
42950
diff
changeset
|
842 |
using X by (intro measurable_cong) auto |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
843 |
finally show ?thesis . |
41830 | 844 |
qed |
845 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
846 |
lemma borel_measurable_ln[simp, intro, measurable (raw)]: |
49774 | 847 |
assumes f: "f \<in> borel_measurable M" |
848 |
shows "(\<lambda>x. ln (f x)) \<in> borel_measurable M" |
|
41830 | 849 |
proof - |
850 |
{ fix x :: real assume x: "x \<le> 0" |
|
851 |
{ fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto } |
|
49774 | 852 |
from this[of x] x this[of 0] have "ln 0 = ln x" |
853 |
by (auto simp: ln_def) } |
|
854 |
note ln_imp = this |
|
855 |
have "(\<lambda>x. if f x \<in> {0<..} then ln (f x) else ln 0) \<in> borel_measurable M" |
|
856 |
proof (rule borel_measurable_continuous_on_open[OF _ _ f]) |
|
857 |
show "continuous_on {0<..} ln" |
|
858 |
by (auto intro!: continuous_at_imp_continuous_on DERIV_ln DERIV_isCont |
|
41830 | 859 |
simp: continuous_isCont[symmetric]) |
860 |
show "open ({0<..}::real set)" by auto |
|
861 |
qed |
|
49774 | 862 |
also have "(\<lambda>x. if x \<in> {0<..} then ln x else ln 0) = ln" |
863 |
by (simp add: fun_eq_iff not_less ln_imp) |
|
41830 | 864 |
finally show ?thesis . |
865 |
qed |
|
866 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
867 |
lemma borel_measurable_log[simp, intro, measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
868 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M" |
49774 | 869 |
unfolding log_def by auto |
41830 | 870 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
871 |
lemma measurable_count_space_eq2_countable: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
872 |
fixes f :: "'a => 'c::countable" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
873 |
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
874 |
proof - |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
875 |
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
876 |
then have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
877 |
by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
878 |
moreover assume "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
879 |
ultimately have "f -` X \<inter> space M \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
880 |
using `X \<subseteq> A` by (simp add: subset_eq del: UN_simps) } |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
881 |
then show ?thesis |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
882 |
unfolding measurable_def by auto |
47761 | 883 |
qed |
884 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
885 |
lemma measurable_real_floor[measurable]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
886 |
"(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)" |
47761 | 887 |
proof - |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
888 |
have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real a \<le> x \<and> x < real (a + 1))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
889 |
by (auto intro: floor_eq2) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
890 |
then show ?thesis |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
891 |
by (auto simp: vimage_def measurable_count_space_eq2_countable) |
47761 | 892 |
qed |
893 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
894 |
lemma measurable_real_natfloor[measurable]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
895 |
"(natfloor :: real \<Rightarrow> nat) \<in> measurable borel (count_space UNIV)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
896 |
by (simp add: natfloor_def[abs_def]) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
897 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
898 |
lemma measurable_real_ceiling[measurable]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
899 |
"(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
900 |
unfolding ceiling_def[abs_def] by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
901 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
902 |
lemma borel_measurable_real_floor: "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
903 |
by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
904 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
905 |
lemma borel_measurable_real_natfloor[intro, simp]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
906 |
"f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
907 |
by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
908 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
909 |
subsection "Borel space on the extended reals" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
910 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
911 |
lemma borel_measurable_ereal[simp, intro, measurable (raw)]: |
43920 | 912 |
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M" |
49774 | 913 |
using continuous_on_ereal f by (rule borel_measurable_continuous_on) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
914 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
915 |
lemma borel_measurable_real_of_ereal[simp, intro, measurable (raw)]: |
49774 | 916 |
fixes f :: "'a \<Rightarrow> ereal" |
917 |
assumes f: "f \<in> borel_measurable M" |
|
918 |
shows "(\<lambda>x. real (f x)) \<in> borel_measurable M" |
|
919 |
proof - |
|
920 |
have "(\<lambda>x. if f x \<in> UNIV - { \<infinity>, - \<infinity> } then real (f x) else 0) \<in> borel_measurable M" |
|
921 |
using continuous_on_real |
|
922 |
by (rule borel_measurable_continuous_on_open[OF _ _ f]) auto |
|
923 |
also have "(\<lambda>x. if f x \<in> UNIV - { \<infinity>, - \<infinity> } then real (f x) else 0) = (\<lambda>x. real (f x))" |
|
924 |
by auto |
|
925 |
finally show ?thesis . |
|
926 |
qed |
|
927 |
||
928 |
lemma borel_measurable_ereal_cases: |
|
929 |
fixes f :: "'a \<Rightarrow> ereal" |
|
930 |
assumes f: "f \<in> borel_measurable M" |
|
931 |
assumes H: "(\<lambda>x. H (ereal (real (f x)))) \<in> borel_measurable M" |
|
932 |
shows "(\<lambda>x. H (f x)) \<in> borel_measurable M" |
|
933 |
proof - |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
934 |
let ?F = "\<lambda>x. if f x = \<infinity> then H \<infinity> else if f x = - \<infinity> then H (-\<infinity>) else H (ereal (real (f x)))" |
49774 | 935 |
{ fix x have "H (f x) = ?F x" by (cases "f x") auto } |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
936 |
with f H show ?thesis by simp |
47694 | 937 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
938 |
|
49774 | 939 |
lemma |
940 |
fixes f :: "'a \<Rightarrow> ereal" assumes f[simp]: "f \<in> borel_measurable M" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
941 |
shows borel_measurable_ereal_abs[intro, simp, measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
942 |
and borel_measurable_ereal_inverse[simp, intro, measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
943 |
and borel_measurable_uminus_ereal[intro, measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M" |
49774 | 944 |
by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If) |
945 |
||
946 |
lemma borel_measurable_uminus_eq_ereal[simp]: |
|
947 |
"(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r") |
|
948 |
proof |
|
949 |
assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp |
|
950 |
qed auto |
|
951 |
||
952 |
lemma set_Collect_ereal2: |
|
953 |
fixes f g :: "'a \<Rightarrow> ereal" |
|
954 |
assumes f: "f \<in> borel_measurable M" |
|
955 |
assumes g: "g \<in> borel_measurable M" |
|
956 |
assumes H: "{x \<in> space M. H (ereal (real (f x))) (ereal (real (g x)))} \<in> sets M" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
957 |
"{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
958 |
"{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
959 |
"{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
960 |
"{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel" |
49774 | 961 |
shows "{x \<in> space M. H (f x) (g x)} \<in> sets M" |
962 |
proof - |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
963 |
let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = -\<infinity> then H y (-\<infinity>) else H y (ereal (real (g x)))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
964 |
let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = -\<infinity> then ?G (-\<infinity>) x else ?G (ereal (real (f x))) x" |
49774 | 965 |
{ fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto } |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
966 |
note * = this |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
967 |
from assms show ?thesis |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
968 |
by (subst *) (simp del: space_borel split del: split_if) |
49774 | 969 |
qed |
970 |
||
971 |
lemma |
|
972 |
fixes f g :: "'a \<Rightarrow> ereal" |
|
973 |
assumes f: "f \<in> borel_measurable M" |
|
974 |
assumes g: "g \<in> borel_measurable M" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
975 |
shows borel_measurable_ereal_le[intro,simp,measurable(raw)]: "{x \<in> space M. f x \<le> g x} \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
976 |
and borel_measurable_ereal_less[intro,simp,measurable(raw)]: "{x \<in> space M. f x < g x} \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
977 |
and borel_measurable_ereal_eq[intro,simp,measurable(raw)]: "{w \<in> space M. f w = g w} \<in> sets M" |
49774 | 978 |
and borel_measurable_ereal_neq[intro,simp]: "{w \<in> space M. f w \<noteq> g w} \<in> sets M" |
979 |
using f g by (auto simp: f g set_Collect_ereal2[OF f g] intro!: sets_Collect_neg) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
980 |
|
47694 | 981 |
lemma borel_measurable_ereal_iff: |
43920 | 982 |
shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
983 |
proof |
43920 | 984 |
assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M" |
985 |
from borel_measurable_real_of_ereal[OF this] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
986 |
show "f \<in> borel_measurable M" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
987 |
qed auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
988 |
|
47694 | 989 |
lemma borel_measurable_ereal_iff_real: |
43923 | 990 |
fixes f :: "'a \<Rightarrow> ereal" |
991 |
shows "f \<in> borel_measurable M \<longleftrightarrow> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
992 |
((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
993 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
994 |
assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
995 |
have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
996 |
with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all |
46731 | 997 |
let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
998 |
have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto |
43920 | 999 |
also have "?f = f" by (auto simp: fun_eq_iff ereal_real) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1000 |
finally show "f \<in> borel_measurable M" . |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1001 |
qed simp_all |
41830 | 1002 |
|
47694 | 1003 |
lemma borel_measurable_eq_atMost_ereal: |
43923 | 1004 |
fixes f :: "'a \<Rightarrow> ereal" |
1005 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1006 |
proof (intro iffI allI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1007 |
assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1008 |
show "f \<in> borel_measurable M" |
43920 | 1009 |
unfolding borel_measurable_ereal_iff_real borel_measurable_iff_le |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1010 |
proof (intro conjI allI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1011 |
fix a :: real |
43920 | 1012 |
{ fix x :: ereal assume *: "\<forall>i::nat. real i < x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1013 |
have "x = \<infinity>" |
43920 | 1014 |
proof (rule ereal_top) |
44666 | 1015 |
fix B from reals_Archimedean2[of B] guess n .. |
43920 | 1016 |
then have "ereal B < real n" by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1017 |
with * show "B \<le> x" by (metis less_trans less_imp_le) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1018 |
qed } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1019 |
then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1020 |
by (auto simp: not_le) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1021 |
then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1022 |
by (auto simp del: UN_simps) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1023 |
moreover |
43923 | 1024 |
have "{-\<infinity>::ereal} = {..-\<infinity>}" by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1025 |
then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto |
43920 | 1026 |
moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M" |
1027 |
using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1028 |
moreover have "{w \<in> space M. real (f w) \<le> a} = |
43920 | 1029 |
(if a < 0 then {w \<in> space M. f w \<le> ereal a} - f -` {-\<infinity>} \<inter> space M |
1030 |
else {w \<in> space M. f w \<le> ereal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r") |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1031 |
proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1032 |
ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto |
35582 | 1033 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1034 |
qed (simp add: measurable_sets) |
35582 | 1035 |
|
47694 | 1036 |
lemma borel_measurable_eq_atLeast_ereal: |
43920 | 1037 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1038 |
proof |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1039 |
assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1040 |
moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}" |
43920 | 1041 |
by (auto simp: ereal_uminus_le_reorder) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1042 |
ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M" |
43920 | 1043 |
unfolding borel_measurable_eq_atMost_ereal by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1044 |
then show "f \<in> borel_measurable M" by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1045 |
qed (simp add: measurable_sets) |
35582 | 1046 |
|
49774 | 1047 |
lemma greater_eq_le_measurable: |
1048 |
fixes f :: "'a \<Rightarrow> 'c::linorder" |
|
1049 |
shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M" |
|
1050 |
proof |
|
1051 |
assume "f -` {a ..} \<inter> space M \<in> sets M" |
|
1052 |
moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto |
|
1053 |
ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto |
|
1054 |
next |
|
1055 |
assume "f -` {..< a} \<inter> space M \<in> sets M" |
|
1056 |
moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto |
|
1057 |
ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto |
|
1058 |
qed |
|
1059 |
||
47694 | 1060 |
lemma borel_measurable_ereal_iff_less: |
43920 | 1061 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)" |
1062 |
unfolding borel_measurable_eq_atLeast_ereal greater_eq_le_measurable .. |
|
38656 | 1063 |
|
49774 | 1064 |
lemma less_eq_ge_measurable: |
1065 |
fixes f :: "'a \<Rightarrow> 'c::linorder" |
|
1066 |
shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M" |
|
1067 |
proof |
|
1068 |
assume "f -` {a <..} \<inter> space M \<in> sets M" |
|
1069 |
moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto |
|
1070 |
ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto |
|
1071 |
next |
|
1072 |
assume "f -` {..a} \<inter> space M \<in> sets M" |
|
1073 |
moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto |
|
1074 |
ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto |
|
1075 |
qed |
|
1076 |
||
47694 | 1077 |
lemma borel_measurable_ereal_iff_ge: |
43920 | 1078 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)" |
1079 |
unfolding borel_measurable_eq_atMost_ereal less_eq_ge_measurable .. |
|
38656 | 1080 |
|
49774 | 1081 |
lemma borel_measurable_ereal2: |
1082 |
fixes f g :: "'a \<Rightarrow> ereal" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1083 |
assumes f: "f \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1084 |
assumes g: "g \<in> borel_measurable M" |
49774 | 1085 |
assumes H: "(\<lambda>x. H (ereal (real (f x))) (ereal (real (g x)))) \<in> borel_measurable M" |
1086 |
"(\<lambda>x. H (-\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M" |
|
1087 |
"(\<lambda>x. H (\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M" |
|
1088 |
"(\<lambda>x. H (ereal (real (f x))) (-\<infinity>)) \<in> borel_measurable M" |
|
1089 |
"(\<lambda>x. H (ereal (real (f x))) (\<infinity>)) \<in> borel_measurable M" |
|
1090 |
shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1091 |
proof - |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1092 |
let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = - \<infinity> then H y (-\<infinity>) else H y (ereal (real (g x)))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1093 |
let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = - \<infinity> then ?G (-\<infinity>) x else ?G (ereal (real (f x))) x" |
49774 | 1094 |
{ fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto } |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1095 |
note * = this |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1096 |
from assms show ?thesis unfolding * by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1097 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1098 |
|
49774 | 1099 |
lemma |
1100 |
fixes f :: "'a \<Rightarrow> ereal" assumes f: "f \<in> borel_measurable M" |
|
1101 |
shows borel_measurable_ereal_eq_const: "{x\<in>space M. f x = c} \<in> sets M" |
|
1102 |
and borel_measurable_ereal_neq_const: "{x\<in>space M. f x \<noteq> c} \<in> sets M" |
|
1103 |
using f by auto |
|
38656 | 1104 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1105 |
lemma [intro, simp, measurable(raw)]: |
43920 | 1106 |
fixes f :: "'a \<Rightarrow> ereal" |
49774 | 1107 |
assumes [simp]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1108 |
shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1109 |
and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1110 |
and borel_measurable_ereal_min: "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1111 |
and borel_measurable_ereal_max: "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M" |
49774 | 1112 |
by (auto simp add: borel_measurable_ereal2 measurable_If min_def max_def) |
1113 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1114 |
lemma [simp, intro, measurable(raw)]: |
49774 | 1115 |
fixes f g :: "'a \<Rightarrow> ereal" |
1116 |
assumes "f \<in> borel_measurable M" |
|
1117 |
assumes "g \<in> borel_measurable M" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1118 |
shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1119 |
and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M" |
49774 | 1120 |
unfolding minus_ereal_def divide_ereal_def using assms by auto |
38656 | 1121 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1122 |
lemma borel_measurable_ereal_setsum[simp, intro,measurable (raw)]: |
43920 | 1123 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal" |
41096 | 1124 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
1125 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M" |
|
1126 |
proof cases |
|
1127 |
assume "finite S" |
|
1128 |
thus ?thesis using assms |
|
1129 |
by induct auto |
|
49774 | 1130 |
qed simp |
38656 | 1131 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1132 |
lemma borel_measurable_ereal_setprod[simp, intro,measurable (raw)]: |
43920 | 1133 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal" |
38656 | 1134 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
41096 | 1135 |
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M" |
38656 | 1136 |
proof cases |
1137 |
assume "finite S" |
|
41096 | 1138 |
thus ?thesis using assms by induct auto |
1139 |
qed simp |
|
38656 | 1140 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1141 |
lemma borel_measurable_SUP[simp, intro,measurable (raw)]: |
43920 | 1142 |
fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> ereal" |
38656 | 1143 |
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M" |
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41096
diff
changeset
|
1144 |
shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M") |
43920 | 1145 |
unfolding borel_measurable_ereal_iff_ge |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1146 |
proof |
38656 | 1147 |
fix a |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1148 |
have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})" |
46884 | 1149 |
by (auto simp: less_SUP_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1150 |
then show "?sup -` {a<..} \<inter> space M \<in> sets M" |
38656 | 1151 |
using assms by auto |
1152 |
qed |
|
1153 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1154 |
lemma borel_measurable_INF[simp, intro,measurable (raw)]: |
43920 | 1155 |
fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> ereal" |
38656 | 1156 |
assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M" |
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41096
diff
changeset
|
1157 |
shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M") |
43920 | 1158 |
unfolding borel_measurable_ereal_iff_less |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1159 |
proof |
38656 | 1160 |
fix a |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1161 |
have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})" |
46884 | 1162 |
by (auto simp: INF_less_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1163 |
then show "?inf -` {..<a} \<inter> space M \<in> sets M" |
38656 | 1164 |
using assms by auto |
1165 |
qed |
|
1166 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1167 |
lemma [simp, intro, measurable (raw)]: |
43920 | 1168 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1169 |
assumes "\<And>i. f i \<in> borel_measurable M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1170 |
shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1171 |
and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M" |
49774 | 1172 |
unfolding liminf_SUPR_INFI limsup_INFI_SUPR using assms by auto |
35692 | 1173 |
|
49774 | 1174 |
lemma borel_measurable_ereal_LIMSEQ: |
1175 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
|
1176 |
assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x" |
|
1177 |
and u: "\<And>i. u i \<in> borel_measurable M" |
|
1178 |
shows "u' \<in> borel_measurable M" |
|
47694 | 1179 |
proof - |
49774 | 1180 |
have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)" |
1181 |
using u' by (simp add: lim_imp_Liminf[symmetric]) |
|
1182 |
then show ?thesis by (simp add: u cong: measurable_cong) |
|
47694 | 1183 |
qed |
1184 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1185 |
lemma borel_measurable_psuminf[simp, intro, measurable (raw)]: |
43920 | 1186 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1187 |
assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1188 |
shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1189 |
apply (subst measurable_cong) |
43920 | 1190 |
apply (subst suminf_ereal_eq_SUPR) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1191 |
apply (rule pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1192 |
using assms by auto |
39092 | 1193 |
|
1194 |
section "LIMSEQ is borel measurable" |
|
1195 |
||
47694 | 1196 |
lemma borel_measurable_LIMSEQ: |
39092 | 1197 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" |
1198 |
assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x" |
|
1199 |
and u: "\<And>i. u i \<in> borel_measurable M" |
|
1200 |
shows "u' \<in> borel_measurable M" |
|
1201 |
proof - |
|
43920 | 1202 |
have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)" |
46731 | 1203 |
using u' by (simp add: lim_imp_Liminf) |
43920 | 1204 |
moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M" |
39092 | 1205 |
by auto |
43920 | 1206 |
ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff) |
39092 | 1207 |
qed |
1208 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1209 |
lemma sets_Collect_Cauchy[measurable]: |
49774 | 1210 |
fixes f :: "nat \<Rightarrow> 'a => real" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1211 |
assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M" |
49774 | 1212 |
shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1213 |
unfolding Cauchy_iff2 using f by auto |
49774 | 1214 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1215 |
lemma borel_measurable_lim[measurable (raw)]: |
49774 | 1216 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1217 |
assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M" |
49774 | 1218 |
shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M" |
1219 |
proof - |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1220 |
def u' \<equiv> "\<lambda>x. lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1221 |
then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))" |
49774 | 1222 |
by (auto simp: lim_def convergent_eq_cauchy[symmetric]) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1223 |
have "u' \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1224 |
proof (rule borel_measurable_LIMSEQ) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1225 |
fix x |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1226 |
have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" |
49774 | 1227 |
by (cases "Cauchy (\<lambda>i. f i x)") |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1228 |
(auto simp add: convergent_eq_cauchy[symmetric] convergent_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1229 |
then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) ----> u' x" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1230 |
unfolding u'_def |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1231 |
by (rule convergent_LIMSEQ_iff[THEN iffD1]) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1232 |
qed measurable |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1233 |
then show ?thesis |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1234 |
unfolding * by measurable |
49774 | 1235 |
qed |
1236 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1237 |
lemma borel_measurable_suminf[measurable (raw)]: |
49774 | 1238 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1239 |
assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M" |
49774 | 1240 |
shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1241 |
unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp |
49774 | 1242 |
|
1243 |
end |