Miscellany5 - Analysis 2 Miscellaneous Problems 5 1. Let...

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Unformatted text preview: Analysis 2 Miscellaneous Problems 5 1. Let (Ω, F , µ) be an arbitrary measure space and let 1 p < q < ∞. Show that if p < t < q then L p (Ω, F , µ) ∩ Lq (Ω, F , µ) ⊆ Lt (Ω, F , µ). 2. Let Ω be an uncountable set and F be its σ-algebra of countable or cocountable subsets. On (Ω, F ), let µ be counting measure and let ν be the measure taking value 0 on countable sets and 1 on uncountable sets. Show that ν µ and yet there is no f ∈ L(Ω, F , µ) such that ν = µ f . 3. Let (Ω, F ) be a measurable space on which (µn : n ∈ N) is a sequence of finite measures. Prove that (Ω, F ) carries a finite measure µ with respect to which µn is absolutely continuous for each n ∈ N. 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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