author  paulson 
Mon, 09 Nov 2009 15:50:15 +0000  
changeset 33533  40b44cb20c8c 
child 33535  b233f48a4d3d 
permissions  rwrr 
33533
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
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1 
header {*Borel Sets*} 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

2 

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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
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3 
theory Borel 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

4 
imports Measure 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

5 
begin 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

6 

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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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7 
text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*} 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

8 

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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

9 
definition borel_space where 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

10 
"borel_space = sigma (UNIV::real set) (range (\<lambda>a::real. {x. x \<le> a}))" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

11 

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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

12 
definition borel_measurable where 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
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13 
"borel_measurable a = measurable a borel_space" 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

14 

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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

15 
definition indicator_fn where 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

16 
"indicator_fn s = (\<lambda>x. if x \<in> s then 1 else (0::real))" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

17 

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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

18 
definition mono_convergent where 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

19 
"mono_convergent u f s \<equiv> 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

20 
(\<forall>x m n. m \<le> n \<and> x \<in> s \<longrightarrow> u m x \<le> u n x) \<and> 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

21 
(\<forall>x \<in> s. (\<lambda>i. u i x) > f x)" 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

22 

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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
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23 
lemma in_borel_measurable: 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

24 
"f \<in> borel_measurable M \<longleftrightarrow> 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

25 
sigma_algebra M \<and> 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
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26 
(\<forall>s \<in> sets (sigma UNIV (range (\<lambda>a::real. {x. x \<le> a}))). 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
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27 
f ` s \<inter> space M \<in> sets M)" 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
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28 
apply (auto simp add: borel_measurable_def measurable_def borel_space_def) 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

29 
apply (metis PowD UNIV_I Un_commute sigma_algebra_sigma subset_Pow_Union subset_UNIV subset_Un_eq) 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

30 
done 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

31 

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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
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32 
lemma (in sigma_algebra) borel_measurable_const: 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
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33 
"(\<lambda>x. c) \<in> borel_measurable M" 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

34 
by (auto simp add: in_borel_measurable prems) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

35 

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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

36 
lemma (in sigma_algebra) borel_measurable_indicator: 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
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37 
assumes a: "a \<in> sets M" 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

38 
shows "indicator_fn a \<in> borel_measurable M" 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

39 
apply (auto simp add: in_borel_measurable indicator_fn_def prems) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

40 
apply (metis Diff_eq Int_commute a compl_sets) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

41 
done 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

42 

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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

43 
lemma Collect_eq: "{w \<in> X. f w \<le> a} = {w. f w \<le> a} \<inter> X" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

44 
by (metis Collect_conj_eq Collect_mem_eq Int_commute) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

45 

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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

46 
lemma (in measure_space) borel_measurable_le_iff: 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

47 
"f \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)" 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

48 
proof 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

49 
assume f: "f \<in> borel_measurable M" 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

50 
{ fix a 
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New theory Probability/Borel.thy, and some associated lemmas
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parents:
diff
changeset

51 
have "{w \<in> space M. f w \<le> a} \<in> sets M" using f 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

52 
apply (auto simp add: in_borel_measurable sigma_def Collect_eq) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

53 
apply (drule_tac x="{x. x \<le> a}" in bspec, auto) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

54 
apply (metis equalityE rangeI subsetD sigma_sets.Basic) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

55 
done 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

56 
} 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

57 
thus "\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M" by blast 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

58 
next 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

59 
assume "\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

60 
thus "f \<in> borel_measurable M" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

61 
apply (simp add: borel_measurable_def borel_space_def Collect_eq) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

62 
apply (rule measurable_sigma, auto) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

63 
done 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

64 
qed 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

65 

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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

66 
lemma Collect_less_le: 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

67 
"{w \<in> X. f w < g w} = (\<Union>n. {w \<in> X. f w \<le> g w  inverse(real(Suc n))})" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

68 
proof auto 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

69 
fix w 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

70 
assume w: "f w < g w" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

71 
hence nz: "g w  f w \<noteq> 0" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

72 
by arith 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

73 
with w have "real(Suc(natceiling(inverse(g w  f w)))) > inverse(g w  f w)" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

74 
by (metis lessI order_le_less_trans real_natceiling_ge real_of_nat_less_iff) hence "inverse(real(Suc(natceiling(inverse(g w  f w))))) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

75 
< inverse(inverse(g w  f w))" 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

76 
by (metis less_iff_diff_less_0 less_imp_inverse_less linorder_neqE_ordered_idom nz positive_imp_inverse_positive real_le_anti_sym real_less_def w) 
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

77 
hence "inverse(real(Suc(natceiling(inverse(g w  f w))))) < g w  f w" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

78 
by (metis inverse_inverse_eq order_less_le_trans real_le_refl) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

79 
thus "\<exists>n. f w \<le> g w  inverse(real(Suc n))" using w 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

80 
by (rule_tac x="natceiling(inverse(g w  f w))" in exI, auto) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

81 
next 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

82 
fix w n 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

83 
assume le: "f w \<le> g w  inverse(real(Suc n))" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

84 
hence "0 < inverse(real(Suc n))" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

85 
by (metis inverse_real_of_nat_gt_zero) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

86 
thus "f w < g w" using le 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

87 
by arith 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

88 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

89 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

90 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

91 
lemma (in sigma_algebra) sigma_le_less: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

92 
assumes M: "!!a::real. {w \<in> space M. f w \<le> a} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

93 
shows "{w \<in> space M. f w < a} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

94 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

95 
show ?thesis using Collect_less_le [of "space M" f "\<lambda>x. a"] 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

96 
by (auto simp add: countable_UN M) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

97 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

98 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

99 
lemma (in sigma_algebra) sigma_less_ge: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

100 
assumes M: "!!a::real. {w \<in> space M. f w < a} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

101 
shows "{w \<in> space M. a \<le> f w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

102 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

103 
have "{w \<in> space M. a \<le> f w} = space M  {w \<in> space M. f w < a}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

104 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

105 
thus ?thesis using M 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

106 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

107 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

108 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

109 
lemma (in sigma_algebra) sigma_ge_gr: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

110 
assumes M: "!!a::real. {w \<in> space M. a \<le> f w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

111 
shows "{w \<in> space M. a < f w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

112 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

113 
show ?thesis using Collect_less_le [of "space M" "\<lambda>x. a" f] 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

114 
by (auto simp add: countable_UN le_diff_eq M) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

115 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

116 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

117 
lemma (in sigma_algebra) sigma_gr_le: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

118 
assumes M: "!!a::real. {w \<in> space M. a < f w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

119 
shows "{w \<in> space M. f w \<le> a} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

120 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

121 
have "{w \<in> space M. f w \<le> a} = space M  {w \<in> space M. a < f w}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

122 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

123 
thus ?thesis 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

124 
by (simp add: M compl_sets) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

125 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

126 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

127 
lemma (in measure_space) borel_measurable_gr_iff: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

128 
"f \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

129 
proof (auto simp add: borel_measurable_le_iff sigma_gr_le) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

130 
fix u 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

131 
assume M: "\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

132 
have "{w \<in> space M. u < f w} = space M  {w \<in> space M. f w \<le> u}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

133 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

134 
thus "{w \<in> space M. u < f w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

135 
by (force simp add: compl_sets countable_UN M) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

136 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

137 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

138 
lemma (in measure_space) borel_measurable_less_iff: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

139 
"f \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

140 
proof (auto simp add: borel_measurable_le_iff sigma_le_less) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

141 
fix u 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

142 
assume M: "\<forall>a. {w \<in> space M. f w < a} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

143 
have "{w \<in> space M. f w \<le> u} = space M  {w \<in> space M. u < f w}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

144 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

145 
thus "{w \<in> space M. f w \<le> u} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

146 
using Collect_less_le [of "space M" "\<lambda>x. u" f] 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

147 
by (force simp add: compl_sets countable_UN le_diff_eq sigma_less_ge M) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

148 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

149 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

150 
lemma (in measure_space) borel_measurable_ge_iff: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

151 
"f \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

152 
proof (auto simp add: borel_measurable_less_iff sigma_le_less sigma_ge_gr sigma_gr_le) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

153 
fix u 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

154 
assume M: "\<forall>a. {w \<in> space M. f w < a} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

155 
have "{w \<in> space M. u \<le> f w} = space M  {w \<in> space M. f w < u}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

156 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

157 
thus "{w \<in> space M. u \<le> f w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

158 
by (force simp add: compl_sets countable_UN M) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

159 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

160 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

161 
lemma (in measure_space) affine_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

162 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

163 
shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

164 
proof (cases rule: linorder_cases [of b 0]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

165 
case equal thus ?thesis 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

166 
by (simp add: borel_measurable_const) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

167 
next 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

168 
case less 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

169 
{ 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

170 
fix w c 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

171 
have "a + g w * b \<le> c \<longleftrightarrow> g w * b \<le> c  a" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

172 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

173 
also have "... \<longleftrightarrow> (ca)/b \<le> g w" using less 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

174 
by (metis divide_le_eq less less_asym) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

175 
finally have "a + g w * b \<le> c \<longleftrightarrow> (ca)/b \<le> g w" . 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

176 
} 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

177 
hence "\<And>w c. a + g w * b \<le> c \<longleftrightarrow> (ca)/b \<le> g w" . 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

178 
thus ?thesis using less g 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

179 
by (simp add: borel_measurable_ge_iff [of g]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

180 
(simp add: borel_measurable_le_iff) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

181 
next 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

182 
case greater 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

183 
hence 0: "\<And>x c. (g x * b \<le> c  a) \<longleftrightarrow> (g x \<le> (c  a) / b)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

184 
by (metis mult_imp_le_div_pos le_divide_eq) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

185 
have 1: "\<And>x c. (a + g x * b \<le> c) \<longleftrightarrow> (g x * b \<le> c  a)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

186 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

187 
from greater g 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

188 
show ?thesis 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

189 
by (simp add: borel_measurable_le_iff 0 1) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

190 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

191 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

192 
definition 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

193 
nat_to_rat_surj :: "nat \<Rightarrow> rat" where 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

194 
"nat_to_rat_surj n = (let (i,j) = nat_to_nat2 n 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

195 
in Fract (nat_to_int_bij i) (nat_to_int_bij j))" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

196 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

197 
lemma nat_to_rat_surj: "surj nat_to_rat_surj" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

198 
proof (auto simp add: surj_def nat_to_rat_surj_def) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

199 
fix y 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

200 
show "\<exists>x. y = (\<lambda>(i, j). Fract (nat_to_int_bij i) (nat_to_int_bij j)) (nat_to_nat2 x)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

201 
proof (cases y) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

202 
case (Fract a b) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

203 
obtain i where i: "nat_to_int_bij i = a" using surj_nat_to_int_bij 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

204 
by (metis surj_def) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

205 
obtain j where j: "nat_to_int_bij j = b" using surj_nat_to_int_bij 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

206 
by (metis surj_def) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

207 
obtain n where n: "nat_to_nat2 n = (i,j)" using nat_to_nat2_surj 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

208 
by (metis surj_def) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

209 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

210 
from Fract i j n show ?thesis 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

211 
by (metis prod.cases prod_case_split) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

212 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

213 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

214 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

215 
lemma rats_enumeration: "\<rat> = range (of_rat o nat_to_rat_surj)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

216 
using nat_to_rat_surj 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

217 
by (auto simp add: image_def surj_def) (metis Rats_cases) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

218 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

219 
lemma (in measure_space) borel_measurable_less_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

220 
assumes f: "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

221 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

222 
shows "{w \<in> space M. f w < g w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

223 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

224 
have "{w \<in> space M. f w < g w} = 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

225 
(\<Union>r\<in>\<rat>. {w \<in> space M. f w < r} \<inter> {w \<in> space M. r < g w })" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

226 
by (auto simp add: Rats_dense_in_real) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

227 
thus ?thesis using f g 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

228 
by (simp add: borel_measurable_less_iff [of f] 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

229 
borel_measurable_gr_iff [of g]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

230 
(blast intro: gen_countable_UN [OF rats_enumeration]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

231 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

232 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

233 
lemma (in measure_space) borel_measurable_leq_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

234 
assumes f: "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

235 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

236 
shows "{w \<in> space M. f w \<le> g w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

237 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

238 
have "{w \<in> space M. f w \<le> g w} = space M  {w \<in> space M. g w < f w}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

239 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

240 
thus ?thesis using f g 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

241 
by (simp add: borel_measurable_less_borel_measurable compl_sets) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

242 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

243 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

244 
lemma (in measure_space) borel_measurable_eq_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

245 
assumes f: "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

246 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

247 
shows "{w \<in> space M. f w = g w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

248 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

249 
have "{w \<in> space M. f w = g w} = 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

250 
{w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

251 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

252 
thus ?thesis using f g 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

253 
by (simp add: borel_measurable_leq_borel_measurable Int) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

254 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

255 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

256 
lemma (in measure_space) borel_measurable_neq_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

257 
assumes f: "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

258 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

259 
shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

260 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

261 
have "{w \<in> space M. f w \<noteq> g w} = space M  {w \<in> space M. f w = g w}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

262 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

263 
thus ?thesis using f g 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

264 
by (simp add: borel_measurable_eq_borel_measurable compl_sets) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

265 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

266 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

267 
lemma (in measure_space) borel_measurable_plus_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

268 
assumes f: "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

269 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

270 
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

271 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

272 
have 1:"!!a. {w \<in> space M. a \<le> f w + g w} = {w \<in> space M. a + (g w) * 1 \<le> f w}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

273 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

274 
have "!!a. (\<lambda>w. a + (g w) * 1) \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

275 
by (rule affine_borel_measurable [OF g]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

276 
hence "!!a. {w \<in> space M. (\<lambda>w. a + (g w) * 1)(w) \<le> f w} \<in> sets M" using f 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

277 
by (rule borel_measurable_leq_borel_measurable) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

278 
hence "!!a. {w \<in> space M. a \<le> f w + g w} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

279 
by (simp add: 1) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

280 
thus ?thesis 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

281 
by (simp add: borel_measurable_ge_iff) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

282 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

283 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

284 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

285 
lemma (in measure_space) borel_measurable_square: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

286 
assumes f: "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

287 
shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

288 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

289 
{ 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

290 
fix a 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

291 
have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

292 
proof (cases rule: linorder_cases [of a 0]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

293 
case less 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

294 
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

295 
by auto (metis less order_le_less_trans power2_less_0) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

296 
also have "... \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

297 
by (rule empty_sets) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

298 
finally show ?thesis . 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

299 
next 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

300 
case equal 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

301 
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

302 
{w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

303 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

304 
also have "... \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

305 
apply (insert f) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

306 
apply (rule Int) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

307 
apply (simp add: borel_measurable_le_iff) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

308 
apply (simp add: borel_measurable_ge_iff) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

309 
done 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

310 
finally show ?thesis . 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

311 
next 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

312 
case greater 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

313 
have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = ( sqrt a \<le> f x & f x \<le> sqrt a)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

314 
by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

315 
real_sqrt_le_iff real_sqrt_power) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

316 
hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

317 
{w \<in> space M. (sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

318 
using greater by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

319 
also have "... \<in> sets M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

320 
apply (insert f) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

321 
apply (rule Int) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

322 
apply (simp add: borel_measurable_ge_iff) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

323 
apply (simp add: borel_measurable_le_iff) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

324 
done 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

325 
finally show ?thesis . 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

326 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

327 
} 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

328 
thus ?thesis by (auto simp add: borel_measurable_le_iff) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

329 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

330 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

331 
lemma times_eq_sum_squares: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

332 
fixes x::real 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

333 
shows"x*y = ((x+y)^2)/4  ((xy)^ 2)/4" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

334 
by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

335 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

336 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

337 
lemma (in measure_space) borel_measurable_uminus_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

338 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

339 
shows "(\<lambda>x.  g x) \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

340 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

341 
have "(\<lambda>x.  g x) = (\<lambda>x. 0 + (g x) * 1)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

342 
by simp 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

343 
also have "... \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

344 
by (fast intro: affine_borel_measurable g) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

345 
finally show ?thesis . 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

346 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

347 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

348 
lemma (in measure_space) borel_measurable_times_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

349 
assumes f: "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

350 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

351 
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

352 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

353 
have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

354 
by (fast intro: affine_borel_measurable borel_measurable_square 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

355 
borel_measurable_plus_borel_measurable f g) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

356 
have "(\<lambda>x. ((f x + g x) ^ 2 * inverse 4)) = 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

357 
(\<lambda>x. 0 + ((f x + g x) ^ 2 * inverse 4))" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

358 
by (simp add: Ring_and_Field.minus_divide_right) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

359 
also have "... \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

360 
by (fast intro: affine_borel_measurable borel_measurable_square 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

361 
borel_measurable_plus_borel_measurable 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

362 
borel_measurable_uminus_borel_measurable f g) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

363 
finally have 2: "(\<lambda>x. ((f x + g x) ^ 2 * inverse 4)) \<in> borel_measurable M" . 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

364 
show ?thesis 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

365 
apply (simp add: times_eq_sum_squares real_diff_def) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

366 
using 1 2 apply (simp add: borel_measurable_plus_borel_measurable) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

367 
done 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

368 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

369 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

370 
lemma (in measure_space) borel_measurable_diff_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

371 
assumes f: "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

372 
assumes g: "g \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

373 
shows "(\<lambda>x. f x  g x) \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

374 
unfolding real_diff_def 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

375 
by (fast intro: borel_measurable_plus_borel_measurable 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

376 
borel_measurable_uminus_borel_measurable f g) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

377 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

378 
lemma (in measure_space) mono_convergent_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

379 
assumes u: "!!n. u n \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

380 
assumes mc: "mono_convergent u f (space M)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

381 
shows "f \<in> borel_measurable M" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

382 
proof  
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

383 
{ 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

384 
fix a 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

385 
have 1: "!!w. w \<in> space M & f w <= a \<longleftrightarrow> w \<in> space M & (\<forall>i. u i w <= a)" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

386 
proof safe 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

387 
fix w i 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

388 
assume w: "w \<in> space M" and f: "f w \<le> a" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

389 
hence "u i w \<le> f w" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

390 
by (auto intro: SEQ.incseq_le 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

391 
simp add: incseq_def mc [unfolded mono_convergent_def]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

392 
thus "u i w \<le> a" using f 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

393 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

394 
next 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

395 
fix w 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

396 
assume w: "w \<in> space M" and u: "\<forall>i. u i w \<le> a" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

397 
thus "f w \<le> a" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

398 
by (metis LIMSEQ_le_const2 mc [unfolded mono_convergent_def]) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

399 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

400 
have "{w \<in> space M. f w \<le> a} = {w \<in> space M. (\<forall>i. u i w <= a)}" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

401 
by (simp add: 1) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

402 
also have "... = (\<Inter>i. {w \<in> space M. u i w \<le> a})" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

403 
by auto 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

404 
also have "... \<in> sets M" using u 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

405 
by (auto simp add: borel_measurable_le_iff intro: countable_INT) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

406 
finally have "{w \<in> space M. f w \<le> a} \<in> sets M" . 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

407 
} 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

408 
thus ?thesis 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

409 
by (auto simp add: borel_measurable_le_iff) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

410 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

411 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

412 
lemma (in measure_space) borel_measurable_SIGMA_borel_measurable: 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

413 
assumes s: "finite s" 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

414 
shows "(!!i. i \<in> s ==> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) s) \<in> borel_measurable M" using s 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

415 
proof (induct s) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

416 
case empty 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

417 
thus ?case 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

418 
by (simp add: borel_measurable_const) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

419 
next 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

420 
case (insert x s) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

421 
thus ?case 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

422 
by (auto simp add: borel_measurable_plus_borel_measurable) 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

423 
qed 
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

424 

40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset

425 
end 