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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Ask a homework question - tutors are online