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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.
- Spring '09