equal
deleted
inserted
replaced
1110 shows "AE x in M. Q x = P x" |
1110 shows "AE x in M. Q x = P x" |
1111 using assms by auto |
1111 using assms by auto |
1112 |
1112 |
1113 lemma AE_impI: |
1113 lemma AE_impI: |
1114 "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x" |
1114 "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x" |
1115 by (cases P) auto |
1115 by fastforce |
1116 |
1116 |
1117 lemma AE_measure: |
1117 lemma AE_measure: |
1118 assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M") |
1118 assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M") |
1119 shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)" |
1119 shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)" |
1120 proof - |
1120 proof - |
1559 by (cases "A \<in> M") (auto simp: measure_notin_sets emeasure_notin_sets) |
1559 by (cases "A \<in> M") (auto simp: measure_notin_sets emeasure_notin_sets) |
1560 |
1560 |
1561 lemma enn2real_plus:"a < top \<Longrightarrow> b < top \<Longrightarrow> enn2real (a + b) = enn2real a + enn2real b" |
1561 lemma enn2real_plus:"a < top \<Longrightarrow> b < top \<Longrightarrow> enn2real (a + b) = enn2real a + enn2real b" |
1562 by (simp add: enn2real_def plus_ennreal.rep_eq real_of_ereal_add less_top |
1562 by (simp add: enn2real_def plus_ennreal.rep_eq real_of_ereal_add less_top |
1563 del: real_of_ereal_enn2ereal) |
1563 del: real_of_ereal_enn2ereal) |
|
1564 |
|
1565 lemma enn2real_sum:"(\<And>i. i \<in> I \<Longrightarrow> f i < top) \<Longrightarrow> enn2real (sum f I) = sum (enn2real \<circ> f) I" |
|
1566 by (induction I rule: infinite_finite_induct) (auto simp: enn2real_plus) |
1564 |
1567 |
1565 lemma measure_eq_AE: |
1568 lemma measure_eq_AE: |
1566 assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" |
1569 assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" |
1567 assumes A: "A \<in> sets M" and B: "B \<in> sets M" |
1570 assumes A: "A \<in> sets M" and B: "B \<in> sets M" |
1568 shows "measure M A = measure M B" |
1571 shows "measure M A = measure M B" |