Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $g$)?

share|improve this question

1 Answer 1

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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.