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I'm given with the following data: $\int_0^{\infty} f(x) dx $ is finite , g(x) is a bounded and continous function at $(0,\infty ) $ .

Prove that $ \int_0 ^{\infty} f(x)g(x)dx $ converges...

Ok. So I know that $ mf(x) \leq f(x)g(x) \leq Mf(x) $ for some constants $m,M$ . So if the integral of f(x)g(x) exists, its value must be between the integral of both sides...

But why does $f(x)g(x) $ must be integrable?

Thanks !

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1 Answer 1

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You basically have it already. Let $m$ be the minimum of $g$ on $(0, \infty)$ and $M$ the maximum. Then, since $m f(x) \le g(x) f(x) \le M f(x)$, as you said the integral in the middle is bounded by the outer integrals if $f(x) g(x)$ is integrable.

But what do you need to be integrable? Continuity almost everywhere and boundedness. Since $f$ is integrable, it's continuous almost everywhere and bounded; we know that $g$ is continuous and bounded. So their product must be continuous almost everywhere and bounded, hence integrable.

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Thanks, but what if I haven't learn about Riemann Measure? How can I show the product is integrable ? Thanks !!! –  theMissingIngredient Feb 10 '13 at 19:28
    
You could manually break up the integral at each point of discontinuity in $f$ (of which there can be only countably many). Each section of $f$ is then continuous, and you can use the fact that continuous and bounded functions are integrable to show that each section of $f g$ is going to be integrable, then just add them together. –  Dougal Feb 10 '13 at 19:43

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