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No2(2004) - MATH 587 Assignment 2 Due(1 Let f and g be...

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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 ) if ω A c , where A is any set in F . Show that h is Borel measurable. (2) Let a n , and let B be a Borel subset of n . Show that a + B = { a + x | x B } and B = { cx | x B } are also Borel sets. Hint: define B = { B n | x + B ∈ B ( 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 ( 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 ) ( , B ( )) be measurable and non-negative. Define (a) h dµ = sup { f dµ : f is simple and 0 f h } , (b) h dµ = lim n →∞ f n , where { f n , n 1 } is a sequence of simple functions such that 0 f n f . Show that these two definitions are identical. (5) (a) Let (Ω , F , µ ) be a measure space and (Ω , F ) a measurable space. Let T : Ω Ω be measurable with respect to F and F . Define µ T ( A ) = µ ( T 1 (
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