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Let $f\geq 0$ be a measurable function which is finite almost everywhere.

For each $k\in\mathbb{Z}$, define, $E_k=\lbrace x|f(x)>2^k\rbrace, F_k=\lbrace x|2^k\leq f(x)<2^{k+1}\rbrace$.

Is it true that $\sum_{k=-\infty}^{\infty}2^km(E_k)<\infty$ if and only if $\sum_{k=-\infty}^{\infty}2^km(F_k)<\infty$?

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    You say this is for homework, how can we help? What do you get about the question, what are you stuck with?2011-03-29
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    hint: $\sum_{l \geq k+1}m(F_l) \leq m(E_k) \leq \sum_{l \geq k}m(F_l)$.2011-03-29
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    I'm stuck with how can I control $\sum_{k=-\infty}^\infty \sum_{l\geq k}2^k m(F_k)$. Becasue there are too many repeated terms.2011-03-29
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    We have $2^km(E_k)\leq \sum_{l\geq k}2^km(F_l)\leq \sum_{l\geq k} 2^lm(F_l)$. If we denote $s_k=\sum_{l\geq k} 2^lm(F_l)$, then we know $s_k$ converges to $0$ but we do not know how fast $s_k$ converges to 0 which I need to conclude that $\sum_{k=-\infty}^\infty s_k<\infty$.2011-03-29
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    change the order of summation, and add with respect to $k$ first.2011-03-29

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