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### Ergodic_Theory: tuned some names

parent 24b6785e3e55
 (* File: Trivial_Measure.thy File: ME_Library_Complement.thy Author: Manuel Eberl, TU München *) theory Trivial_Measure theory ME_Library_Complement imports "HOL-Analysis.Analysis" begin ... ... @@ -33,13 +33,29 @@ lemma measurable_trivial_measure_iff: unfolding measurable_def by (auto simp: sets_trivial_measure) definition (in sigma_algebra) induced_algebra :: "('b \ 'a) \ 'b set \ 'b set set" where "induced_algebra f \' = sigma_sets \' {f - A \ \' |A. A \ M}" subsection \Pullback algebras\ lemma (in sigma_algebra) sigma_algebra_induced: "sigma_algebra \' (induced_algebra f \')" unfolding induced_algebra_def by (rule sigma_algebra_sigma_sets) auto lemma (in sigma_algebra) in_induced_algebra: "A \ M \ f - A \ \' \ induced_algebra f \'" unfolding induced_algebra_def by (rule sigma_sets.Basic) auto text \ The pullback algebra $f^{-1}(\Sigma)$ of a \\\-algebra $(\Omega, \Sigma)$ is the smallest \\\-algebra such that $f$ is $f^{-1}(\Sigma)--\Sigma$-measurable. \ definition (in sigma_algebra) pullback_algebra :: "('b \ 'a) \ 'b set \ 'b set set" where "pullback_algebra f \' = sigma_sets \' {f - A \ \' |A. A \ M}" lemma pullback_algebra_minimal: assumes "f \ M \\<^sub>M N" shows "sets.pullback_algebra N f (space M) \ sets M" proof fix X assume "X \ sets.pullback_algebra N f (space M)" thus "X \ sets M" unfolding sets.pullback_algebra_def by induction (use assms in \auto simp: measurable_def\) qed lemma (in sigma_algebra) sigma_algebra_pullback: "sigma_algebra \' (pullback_algebra f \')" unfolding pullback_algebra_def by (rule sigma_algebra_sigma_sets) auto lemma (in sigma_algebra) in_pullback_algebra: "A \ M \ f - A \ \' \ pullback_algebra f \'" unfolding pullback_algebra_def by (rule sigma_sets.Basic) auto end \ No newline at end of file
 ... ... @@ -4,7 +4,7 @@ session "Ergodic_Theory" (AFP) = "HOL-Probability" + options [timeout = 900] theories SG_Library_Complement Trivial_Measure ME_Library_Complement Fekete Asymptotic_Density Measure_Preserving_Transformations ... ...
 ... ... @@ -4,7 +4,7 @@ *) section \The shift operator on an infinite product measure\ theory Shift_Operator imports Ergodicity Trivial_Measure imports Ergodicity ME_Library_Complement begin text \ ... ... @@ -33,9 +33,9 @@ lemma measurable_T [measurable]: "T \ P \\<^sub>M P" text \ The \n\-th tail algebra $\mathcal{T}_n$ is, in some sense, the algebra in which we forget all information about all $x_i$ with \i < n\. We simply change the product algebra of \P\ by replacing the algebra for each \i < n\ with the trivial algebra that contains only the empty set and the The \n\-th tail algebra $\mathcal{T}_n$ is, in some sense, the algebra in which we forget all information about all $x_i$ with \i < n\. We simply change the product algebra of \P\ by replacing the algebra for each \i < n\ with the trivial algebra that contains only the empty set and the entire space. \ definition tail_algebra :: "nat \ (nat \ 'a) measure" ... ... @@ -100,12 +100,12 @@ text \ Related to the tail algebra, we define the algebra induced by the \i\-th variable (i.e. the algebra that contains only information about the \i\-th variable): \ sublocale X: sigma_algebra "space P" "sets.induced_algebra M (\f. f i) (space P)" by (rule sets.sigma_algebra_induced) sublocale X: sigma_algebra "space P" "sets.pullback_algebra M (\f. f i) (space P)" by (rule sets.sigma_algebra_pullback) lemma indep_sets_induced_algebra: "P.indep_sets (\i. sets.induced_algebra M (\f. f i) (space P)) UNIV" using indep_vars unfolding P.indep_vars_def sets.induced_algebra_def by blast lemma indep_sets_pullback_algebra: "P.indep_sets (\i. sets.pullback_algebra M (\f. f i) (space P)) UNIV" using indep_vars unfolding P.indep_vars_def sets.pullback_algebra_def by blast text \ ... ... @@ -114,13 +114,13 @@ text \ \ lemma tail_algebra_subset: "sets (tail_algebra n) \ sigma_sets (space P) (\i\{n..}. sets.induced_algebra M (\f. f i) (space P))" sigma_sets (space P) (\i\{n..}. sets.pullback_algebra M (\f. f i) (space P))" proof - have "sets (tail_algebra n) = sigma_sets (space P) (prod_algebra UNIV (\i. if i < n then trivial_measure (space M) else M))" by (simp add: tail_algebra_def sets_PiM PiE_def Pi_def P_def space_PiM) also have "\ \ sigma_sets (space P) (\i\{n..}. sets.induced_algebra M (\f. f i) (space P))" also have "\ \ sigma_sets (space P) (\i\{n..}. sets.pullback_algebra M (\f. f i) (space P))" proof (intro sigma_sets_mono subsetI) fix C assume "C \ prod_algebra UNIV (\i. if i < n then trivial_measure (space M) else M)" then obtain C' ... ... @@ -143,7 +143,7 @@ proof - hence "C \ space P" using sets.sets_into_space by blast show "C \ sigma_sets (space P) (\i\{n..}. sets.induced_algebra M (\f. f i) (space P))" show "C \ sigma_sets (space P) (\i\{n..}. sets.pullback_algebra M (\f. f i) (space P))" proof (cases "C = {}") case False have "C = (\i\{n..}. (\f. f i) - C' i) \ space P" ... ... @@ -174,7 +174,7 @@ proof - (\i\{n..}. (\f. f i) - C' i \ space P)" by blast also have "\ \ sigma_sets (space P) (\i\{n..}. sets.induced_algebra M (\f. f i) (space P))" also have "\ \ sigma_sets (space P) (\i\{n..}. sets.pullback_algebra M (\f. f i) (space P))" (is "_ \ ?rhs") proof (intro sigma_sets_INTER, goal_cases) fix i show "(\f. f i) - C' i \ space P \ ?rhs" ... ... @@ -194,17 +194,17 @@ proof - qed (auto intro: sigma_sets.Empty) next case i: True have "(\f. f i) - C' i \ space P \ sets.induced_algebra M (\f. f i) (space P)" using C'_2[OF i] by (intro sets.in_induced_algebra) auto have "(\f. f i) - C' i \ space P \ sets.pullback_algebra M (\f. f i) (space P)" using C'_2[OF i] by (intro sets.in_pullback_algebra) auto thus ?thesis using i by blast qed next have "C \ space P" if "C \ sets.induced_algebra M (\f. f i) (space P)" for i C have "C \ space P" if "C \ sets.pullback_algebra M (\f. f i) (space P)" for i C proof - show ?thesis by (rule sigma_sets_into_sp) (use that X.space_closed[of i] in auto) qed thus "(\i\{n..}. sets.induced_algebra M (\f. f i) (space P))\ Pow (space P)" thus "(\i\{n..}. sets.pullback_algebra M (\f. f i) (space P))\ Pow (space P)" by auto qed auto ... ... @@ -220,12 +220,12 @@ text \ by the variables: \ lemma Invariants_subset_tail_algebra: "sets Invariants \ P.tail_events (\i. sets.induced_algebra M (\f. f i) (space P))" "sets Invariants \ P.tail_events (\i. sets.pullback_algebra M (\f. f i) (space P))" proof fix A assume A: "A \ sets Invariants" have A': "A \ P.events" using A unfolding Invariants_sets by simp_all show "A \ P.tail_events (\i. sets.induced_algebra M (\f. f i) (space P))" show "A \ P.tail_events (\i. sets.pullback_algebra M (\f. f i) (space P))" unfolding P.tail_events_def proof safe fix n :: nat ... ... @@ -242,9 +242,9 @@ proof also have "(T ^^ n) - A \ space (tail_algebra n) \ sets (tail_algebra n)" by (rule measurable_sets[OF measurable_funpow_T' A']) also have "sets (tail_algebra n) \ sigma_sets (space P) (\i\{n..}. sets.induced_algebra M (\f. f i) (space P))" sigma_sets (space P) (\i\{n..}. sets.pullback_algebra M (\f. f i) (space P))" by (rule tail_algebra_subset) finally show "A \ sigma_sets (space P) (\i\{n..}. sets.induced_algebra M (\f. f i) (space P))" . finally show "A \ sigma_sets (space P) (\i\{n..}. sets.pullback_algebra M (\f. f i) (space P))" . qed qed ... ... @@ -257,7 +257,7 @@ proof have A': "A \ P.events" using A unfolding Invariants_sets by simp_all have "P.prob A = 0 \ P.prob A = 1" using X.sigma_algebra_axioms indep_sets_induced_algebra using X.sigma_algebra_axioms indep_sets_pullback_algebra by (rule P.kolmogorov_0_1_law) (use A Invariants_subset_tail_algebra in blast) thus "A \ null_sets P \ space P - A \ null_sets P" by (rule disj_forward) (use A'(1) P.prob_compl[of A] in \auto simp: P.emeasure_eq_measure\) ... ...
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