Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

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 $\lambda \in \mathbb{R}$.)

share|improve this answer
    
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.

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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.