Let $f,g:\mathbb{R}\to \mathbb{R}$ Lebesgue integrable. Prove: $B(x,y)=f(x)\cdot g(y)$ lebesgue integrable on $\mathbb{R}^2$
$f,g$ Lebesgue integrable $\iff$ $f,g$ are measurable and $\int |f|<\infty$ ,$\int |g|<\infty$
$$\int \int |fg|dxdy=\int|f|dx\int |g|dy=\|f\|_1\|g\|_1<\infty$$
Using Tonelli's theorem