Unformatted text preview: = k A n . #2. Let f ( x,y ) be a function deﬁned on the unit square { ≤ x ≤ 1 , ≤ y ≤ 1 } , which is continuous in each variable separately. Show that f is a Borel measurable function of ( x,y ). #3. Let f ( x ) be a right continuous function on R 1 , i.e. lim x → x ,x>x f ( x ) = f ( x ) for all x ∈ R 1 . (a). Show that f is measurable. (b). Show that f is continuous a.e. in R 1 . #4 (Problem 21 on p.59). Let ( X, M ,μ ) be a measure space, and let f,f 1 ,f 2 ,... be nonnegative functions functions in L 1 ( μ ), such that f n → f a.e. as n → ∞ . Show that if Z f n dμ → Z f dμ, then Z  f nf  dμ → . #5 (compare with Problem 31(d) on p.60). Show that for a > 0, ∞ Z eax x1 sin xdx = arctan ( a1 ) . Hint. Apply Theorem 2.27(b)....
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 Fall '08
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 Math, lim, lim sup

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