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

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 $g$)?

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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.
Well, they have special status in that it is not quite clear how $(-\infty)\cdot(+\infty)$ should be defined... However, it doesn't really matter for this specific issue. – t.b. Mar 7 '12 at 10:19