MI_Problems

# MI_Problems - Measure and Integration Problems 1 Problem...

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Measure and Integration Problems 1 Problem 1.1 Let C be a collection of subsets of the set Ω . For each A σ ( C ) show that there is a countable subcollection C A of C such that A σ ( C A ) . Problem 1.2 Let , F ) and 0 , F 0 ) be measurable spaces and f : Ω Ω 0 a function. (a) Prove that f - 1 ( F 0 ) is a σ -algebra on Ω . (b) Show that f ( F ) need not be a σ -algebra even if f is surjective. 2 Problem 2.1 Find a measurable space , F ) carrying ﬁnite measures μ and ν such that μ (Ω) = ν (Ω) but such that { A ∈ F : μ ( A ) = ν ( A ) } is not a σ -algebra. Problem 2.2 Let , F ) be a measurable space on which ( μ n : n N ) is an increasing sequence of measures. Show that a measure μ on , F ) is deﬁned by the rule A ∈ F ⇒ μ ( A ) = lim μ n ( A ) . 3 Problem 3.1 Let φ : R 2 R be continuous and , F ) a measurable space. Show that if f, g m , F ) then φ ( f, g ) m , F ) . [In particular, f + g and fg are measurable.] Problem 3.2

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MI_Problems - Measure and Integration Problems 1 Problem...

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