No2(2004) - MATH 587 Assignment 2 Due October 26, 2004 (1)...

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Unformatted text preview: MATH 587 Assignment 2 Due October 26, 2004 (1) Let f and g be extended real-valued Borel measurable functions on (Ω , F ), and define h ( ω ) = ( = f ( ω ) if ω ∈ A , g ( ω ) i f ω ∈ A c , where A is any set in F . Show that h is Borel measurable. (2) Let a ∈ R n , and let B be a Borel subset of R n . Show that a + B = { a + x | x ∈ B } and B = { cx | x ∈ B } are also Borel sets. Hint: define B = { B ⊂ R n | x + B ∈ B ( R n ) } and show that B is a σ-algebra containing all open intervals. This is sometimes called the good sets principle . (If you want, do this problem for n = 1.) (3) Let µ be Lebesgue measure on B ( R n ). With reference to the preceding problem, show that µ ( a + B ) = µ ( B ) and µ ( − B ) = µ ( B ) for all Borel sets B . (Again, do for n = 1 only if you want.) (4) Let h : (Ω , F , µ ) → ( R , B ( R )) be measurable and non-negative. Define (a) R h dµ = sup { R f dµ : f is simple and 0 ≤ f ≤ h } , (b) R h dµ = lim n →∞ R f...
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