Is the product of a measurable and bounded function with a Lebesgue integrable function Lebesgue integrable? 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
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 $c&gt;0$ 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 $\lambda \in \mathbb{R}$.)
The answer is yes. This is really a very simple exercise though,you should try to prove it yourself.