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Unformatted text preview: [ A B = [ A B . #2. Show that for arbitrary I := [ a, b ] (0 , ), ln x(ln x ) I I 1 , where f I := 1 ba b Z a f ( x ) dx. Hint. It is convenient to write ln xln y = x Z y dt t . #3. Let f ( x ) be a function in L 1 ([0 , 1]), such that f (0) , f (1) , and f x + y 2 f ( x ) + f ( y ) 2 for all x, y [0 , 1] . Show that f ( x ) 0 for all x [0 , 1]. #4. Give an example of innite measure and nite measure on R 1 , such that , and for each > 0 there is an interval I R 1 satisfying ( I ) < and ( I ) 1 . Hint. Try d = f dx, d = g dx with dierent positive f, g . Remark. By Theorem 3.5, such an example is impossible if is nite....
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 Fall '08
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 Math

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