# Proving that if 2 functions are reimann integrable over a subset, then so is their product

All of these integrals are riemann not lebesgue integrals.

Problem statement:

Let $f$ and $g$ be real-valued functions on a subset $A$ of $E^n$. Show that if $\int_{A} f$ and $\int_{A} g$ exist then $\int_{A} fg$ exists.

I have already proven that if $f$ and $g$ are integrable on a closed interval $I$ of $E^n$ then so is $fg$ so maybe we can use this fact?

• And what is $E^n$? – Paul Sinclair Dec 6 '15 at 22:15
• $E^n$ is a metric space. – Chair Dec 7 '15 at 13:44
• How are you defining integration on an arbitrary metric space? What do you mean by an "Interval" in an arbitrary metric space? Or is there more to $E^n$ than just "metric space"? – Paul Sinclair Dec 7 '15 at 21:23