By Leadbetter R., Cambanis S., Pipiras V.

ISBN-10: 1107020409

ISBN-13: 9781107020405

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Extra resources for A Basic Course in Measure and Probability: Theory for Applications

Sample text

A set function μ deﬁned on a class E of sets is called additive if μ(E ∪ F) = μ(E) + μ(F) whenever E ∈ E, F ∈ E, E ∪ F ∈ E, E ∩ F = ∅. μ deﬁned on E is called ﬁnitely additive (countably additive) if μ(∪n1 Ei ) = n ∞ ∞ 1 μ(Ei )) whenever Ei are disjoint sets of E for i = 1 μ(Ei ) (μ(∪1 Ei ) = 1, 2, . . , n (i = 1, 2, . ), whose union ∪n1 Ei (∪∞ 1 Ei ) is also in E. μ is called a ﬁnite set function on E if |μ(E)| < ∞ for each E ∈ E. μ is called σ-ﬁnite on E if, for each E ∈ E there is a sequence {En } of sets of E with E ⊂ ∪∞ n=1 En and |μ(En )| < ∞; that is, if E can be “covered” by a sequence of sets En ∈ E with |μ(En )| < ∞.

N (i = 1, 2, . ), whose union ∪n1 Ei (∪∞ 1 Ei ) is also in E. μ is called a ﬁnite set function on E if |μ(E)| < ∞ for each E ∈ E. μ is called σ-ﬁnite on E if, for each E ∈ E there is a sequence {En } of sets of E with E ⊂ ∪∞ n=1 En and |μ(En )| < ∞; that is, if E can be “covered” by a sequence of sets En ∈ E with |μ(En )| < ∞. It will also be useful to talk about extensions and restrictions of a set function μ on a class E since one often needs either to “extend” the definition of μ to a class larger than E, or restrict attention to some subclass of E.

If also F–E ∈ P and μ(E) is ﬁnite, then F = E∪(F–E) and μ(F) = μ(E)+ μ(F – E) so that μ(F) – μ(E) = μ(F – E), showing that μ is subtractive. 2 If μ is a measure on a ring R, if E ∈ R, and {Ei } is any ∞ sequence of sets of R such that E ⊂ ∪∞ 1 μ(Ei ). 3). Thus μ(E) = ∞ 1 μ(Gi ) ≤ ∞ 1 μ(Ei ) since μ is monotone and Gi ⊂ E ∩ Ei ⊂ Ei . The next result establishes a reverse inequality for disjoint sequences. 3 If μ is a measure on a ring R, if E ∈ R, and if {Ei } is a ∞ disjoint sequence of sets in R such that ∪∞ 1 μ(Ei ) ≤ μ(E).