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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.
 Spring '09
 LARSON

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