Let's $A$ be such that $\lambda (A) = 0$ (Lebesgue measure). I want to prove that for every measurable function $f$,
$\int_A f(x) \lambda(dx) = 0$
I did the following : $|\int_A f(x) \lambda(dx)| <= \lambda(A) \times \sup_{x \in \mathbb{R}} |f(x)| = 0 \times \sup_{x \in \mathbb{R}} |f(x)| = 0$
For that to be true I have to assume that $0 \times \infty = 0$.
Is it an assumption we always do in measure theory, and why ? Or is there an other way to prove what I intended to prove ?