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I have some problems proving that some functions are integrable. For example, if $f$ is a measurable function on $[0,\infty)$, let

$F(s) = \int_o^{\infty}\frac{f(x)}{(1+sx)^2} dx$.

How can I show that if $f(x)/x$ is integrable, then $F(s)$ is finite for almost every $s$ and that $F$ is integrable on $[0,\infty)$?


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up vote 1 down vote accepted

By the definition of $F(s)$ and Tonelli's theorem, $$\int_0^\infty|F(s)|ds\le \int_0^\infty\left(\int_0^\infty\frac{1}{(1+sx)^2}ds\right)|f(x)|dx=\int_0^\infty\frac{|f(x)|}{x}dx<\infty.$$

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