Unformatted text preview: X ⊥ Y then X + Y is closed. 6. 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 r L ) = 0 and then deﬁne μ ( A ) to be the common value μ ( L ) = μ ( U ). Show that (Ω , F , μ ) is a measure space that is complete in the sense that each subset of a null set is null. 7. Let (Ω , F ,μ ) be an arbitrary measure space; let p,q,r > 1 satisfy 1 p + 1 q = 1 r . Prove that if f ∈ L p ( μ ) and g ∈ L q ( μ ) then fg ∈ L r ( μ ) with k fg k r 6 k f k p k g k q . 8. Let 1 6 p < q < ∞ . Show that L p ( μ ) * L q ( μ ) iﬀ (Ω , F ,μ ) contains measurable sets of arbitrarily small positive measure. 1...
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 Summer '11
 Robinson
 Hilbert space, Topological space, measure space, PhD Examination, null scalar sequences

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