Unformatted text preview: [ α ∈ A B α = [ α ∈ A B α . #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 ² x + y 2 ³ ≥ 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 inﬁnite measure ν and ﬁnite measure μ on R 1 , such that ν ± μ , and for each δ > 0 there is an interval I ⊂ R 1 satisfying μ ( I ) < δ and ν ( I ) ≥ 1 . Hint. Try dμ = f dx, dν = g dx with diﬀerent positive f, g . Remark. By Theorem 3.5, such an example is impossible if ν is ﬁnite....
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 Fall '08
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 Math, 2 hours, 120 min, countable subset, g. Remark

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