Product of two Lebesgue measurable functions that can take on $\pm \infty$

Question

Let $f$ and $g$ be 2 Lebesgue measurable functions on $\mathbb{R}^{n}$ that we allow to take the infinite values $+\infty$ and $-\infty$. If $f$ and $g$ are both finite valued, that is, $-\infty < f(x) < \infty$ (and similarly for $g$) we know that $f(x)g(x)$ is measurable. Is this true if we have $-\infty \leq f(x) \leq \infty$ (similarly for $g$)?

Answer 1

Yes, it is still measurable. The values $\pm\infty$ don't have any special status regarding measurability of functions. We think of them as just two more points in the target space.