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All of these integrals are riemann not lebesgue integrals.

Problem statement:

Part 1

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.

Part 2

Let $f$ be a real-valued function on a subset $A$ of $E^n$ and let $B \subset A$ Show that if $\int_{A} f$ exists and B has volume then $\int_{B} f$ 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?

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  • $\begingroup$ Ask just one question and smaller title may help you. $\endgroup$ – Hoseyn Heydari Dec 4 '15 at 17:28
  • $\begingroup$ @hosenheydari Ok thank you ! $\endgroup$ – Chair Dec 6 '15 at 15:09

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