Measure and Probability Theory
ثبت نشده
چکیده
1 Describing measurable sets 8 1.1 Families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Restricted algebras . . . . . . . . . . . . . . . . . . . . 12 1.1.3 Sigma Algebras . . . . . . . . . . . . . . . . . . . . . . 12 1.1.4 Binary Unions . . . . . . . . . . . . . . . . . . . . . . 14 1.1.5 Initial Sigma Algebra . . . . . . . . . . . . . . . . . . 15 1.1.6 Disjoint families . . . . . . . . . . . . . . . . . . . . . 17 1.1.7 Ring generated by a semiring . . . . . . . . . . . . . . 19 1.1.8 A Two-Element Series . . . . . . . . . . . . . . . . . . 20 1.1.9 Closed CDI . . . . . . . . . . . . . . . . . . . . . . . . 21 1.1.10 Dynkin systems . . . . . . . . . . . . . . . . . . . . . . 23 1.1.11 Intersection sets systems . . . . . . . . . . . . . . . . . 23 1.1.12 Smallest Dynkin systems . . . . . . . . . . . . . . . . 24 1.1.13 Induction rule for intersection-stable generators . . . . 25 1.2 Measure type . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2.1 Constructing simple ′a measure . . . . . . . . . . . . . 28 1.2.2 Measurable functions . . . . . . . . . . . . . . . . . . . 28 1.2.3 Counting space . . . . . . . . . . . . . . . . . . . . . . 30 1.2.4 Extend measure . . . . . . . . . . . . . . . . . . . . . 32 1.2.5 Supremum of a set of σ-algebras . . . . . . . . . . . . 32 1.3 The smallest σ-algebra regarding a function . . . . . . . . . . 33 1.3.1 Restricted Space Sigma Algebra . . . . . . . . . . . . 35 1.4 Measurability prover . . . . . . . . . . . . . . . . . . . . . . . 38 1.5 Measurability for (co)inductive predicates . . . . . . . . . . . 43
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