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If we have one function $f$ which is Lebesgue integrable and one function $g$ which is both measurable and bounded, is the product of $f \cdot g$ Lebesgue integrable or not?

Thx in advance

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2 Answers 2

A measurable function $u$ is lebesgue integrable iff there exists a (positive) lebesgue integrable function $w$ such that $$\forall x \in \mathbb{R}: |u(x)| \leq w(x)$$

In your case: Since $g$ is bounded there exists a constant $c>0$ such that $|g(x)| \leq c$ for all $x \in \mathbb{R}$. Try to find an estimate for

$$|\underbrace{f(x) \cdot g(x)}_{=:u(x)}|$$

such that the upper bound is still a lebesgue integrable function. (Note that $\lambda \cdot f$ is integrable for all $\lambda \in \mathbb{R}$.)

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Sorry, I don't quite get your characterization of lebesgue integrable functions. Do you mean "a measurable function is lebesgue integrable, iff..."? otherwise the characteristic function of a vitali set seems to be a counterexample. –  Nils Matthes Feb 3 '13 at 15:12
@NilsMatthes Yes, that's what I meant... I'll correct it. –  saz Feb 3 '13 at 15:13

The answer is yes. This is really a very simple exercise though,you should try to prove it yourself.

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Give him some hints :-), don't just say simple exercise do it yourself, it doesnt help him. –  Ram Feb 3 '13 at 14:08

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