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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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up vote 1 down vote accepted

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.

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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

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