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Let $f$ be integrable over $(0,a)$ and $h(x)=\int_a^x\frac{f(t)}{t}dt$. Show that $h$ is integrable over $(0,a)$, and $\int_0^a h(x)dx=\int_0^a f(x)dx$. There is a hint to use Fubini-Tonelli Theorem, but I still have no idea how to deal with this question.

Can someone help me with the proof? Thanks!

Toad Jiang
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  • $f(t) = \frac{f(t)}{t} \cdot t$ so maybe you could use Fubini-Tonelli in the context of integration by parts? (see e.g. here: http://math.stackexchange.com/a/830581/327486) – Chill2Macht Nov 18 '16 at 20:13
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    If you insert the definition of $h$ into $\int_0^a h(x),dx$, can you see an option for Fubini-Tonelli? – Daniel Fischer Nov 18 '16 at 20:22

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