Miscellany3

# Miscellany3 - Analysis 2 Miscellaneous Problems 3 1. Let...

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Unformatted text preview: Analysis 2 Miscellaneous Problems 3 1. Let (Ω, F , µ) be a measure space. Deﬁne F to comprise all those A ⊆ Ω for which there exist L, U ∈ F such that L ⊆ A ⊆ U and µ(U L) = 0 and then deﬁne µ(A) to be the common value µ(L) = µ(U ). Prove that (Ω, F , µ) is a measure space that is complete in the sense that each subset of a null set is null. 2. Let (An )∞ 1 be a sequence in F . For each N let BN be the set comprising all n= ω ∈ Ω such that ω ∈ An for at least N values of n. Prove that ∞ N µ( BN ) µ(An ). n =1 Recover the fact that if ∞ n=1 µ(An ) < ∞ then µ(lim supn An ) = 0. 3. Decide whether the measurable space (R, P(R)) carries a probability measure µ that satisﬁes: (i) if t ∈ R then µ({t}) = 0; (ii) if A ⊆ R then µ(A) = 0 or µ(A) = 1. 1 ...
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## This note was uploaded on 07/08/2011 for the course MHF 3202 taught by Professor Larson during the Spring '09 term at University of Florida.

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