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Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #7 (due on Friday, December 10). 50 points are divided between 5 problems, 10 points each. #1. Let f ∈ L 1 ( R 1 ). Show that S n ( x ) = 1 n n 1 X j =0 f x + j n → S ( x ) = Z x +1 x f ( t ) dt in L 1 ( R 1 ) as n → ∞ , i.e. Z  S n ( x ) S ( x )  dx → as n → ∞ . Hint. Using Theorem 2.26, show that ω ( h ) := sup  t ≤ h Z R 1  f ( x + t ) f ( x )  dx → as h → + . Moreover, writing S n S in the form S n ( x ) S ( x ) = n 1 X j =0 1 /n Z f x + j n f x + j n + t dt, one can get Z R 1  S n ( x ) S ( x )  dx ≤ ω (1 /n ) . #2. Let f,f 1 ,f 2 ,...,f n ,... be measurable functions on a measure space ( X, M ,μ ) with μ ( X ) = 1, such that f n ( x ) → f ( x ) as n → ∞ for all x ∈ X . Suppose that Z  f n  1+ α dμ ≤ C for all n, with some constants α > and C > . Show that Z  f n f  dμ → as n → ∞ ....
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 Fall '08
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 Math, ej, measure, Lebesgue integration, lim sup Ej, nonnegative measurable function

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