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Sheet7 - m n → ∞ Show that there exists a real-valued...

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Statistics 330/600 Due: Thursday 3 March 2005: Sheet 7 *(7.1) (Integration by parts) UGMTP Problem 4.16, frst two parts. IF you attempt the third part you should assume that µ has density f and ν has density g , both with respect to Lebesgue measure. You should also interpret dF / dt to mean f ( t ) and dG / dt to mean g ( t ) . (7.2) ( ² , δ version oF absolute continuity on felds; don’t need E countable) UGMTP Problem 3.6. Hints: Read Example 2.5. Also, how is this Problem related to Example 3.3? (7.3) (Absolutely continuous but no density) UGMTP Problem 3.2. *(7.4) (Compare with UGMTP Problem 2.19.) ±or a measure space ( X , A ,µ) , suppose { f n : n N } is a Cauchy sequence in L 2 (µ) , that is, k f n f m k 2 0asm
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Unformatted text preview: ( m , n ) → ∞ . Show that there exists a real-valued Function f in L 2 (µ) For which k f n − f k 2 → 0 as n → ∞ , by the Following steps. (i) Show that sup n ∈ N k f n k 2 < ∞ . (ii) Show that there is no loss oF generality in supposing f n ≥ 0 For all n . Hint: Consider f ± n . (iii) Show that { f 2 n : n ∈ N } is a Cauchy sequence in L 1 (µ) . (iv) Show that there exists a Function g in L 1 (µ) with g ≥ 0 and µ | f 2 n − g | → 0. (v) Show that √ g ∈ L 2 (µ) and k f n − √ g k 2 → 0. Hint: | f n − √ g | ≤ f n + √ g . (vi) Anything more to do?...
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