The product of two Riemann integrable functions is integrable.

Asked May 03 '16 at 17:19

Active Jun 02 '19 at 21:31

Viewed 1,133 times

This question may have been asked here.

But I would like to know if I can prove it in the following way. If not, why?

Proof: Let functions $f,g: S\to\Bbb R$ be integrable, where $S\subset\Bbb R^n$. Let $A$ be the set of discontinuities of $f$, and let $B$ be the set of discontinuities of $g$. Then the product function $fg$ can only be discontinuous on $A\cup B$, since the product of continuous functions is still continuous on $\Bbb R^n\backslash (A\cup B)$.

By integrability, we know that $A,B$ are of measure 0. Hence $A\cup B$ is of measure 0. Therefore the set of discontinuities of $fg$ is of measure 0. Hence $fg$ is integrable over $S$.$\blacksquare$

edited Jun 02 '19 at 21:31

asked May 03 '16 at 17:19

Sekots Reivan

Did you mean to say $f,g:S\to \Bbb R$? – BigbearZzz May 03 '16 at 17:22
That works, for bounded functions. But you need to know (the proof of) Lebesgue's criterion for Riemann integrability, while the other proof doesn't use that. – Silvia Ghinassi May 03 '16 at 17:24
@BigbearZzz I am trying to prove the more general case. – Sekots Reivan May 03 '16 at 17:26
So what is $fg$ then? – BigbearZzz May 03 '16 at 17:27
@BigbearZzz you are right! – Sekots Reivan May 03 '16 at 17:28
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@SilviaGhinassi The definition of the Riemann integral only applies to bounded functions. (There are extensions to this concept to unbounded functions, but for functions to which these apply the original definition can no longer be used). – Thomas May 03 '16 at 17:29
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@Thomas Good point. My bad. The main part of my previous comment still holds. – Silvia Ghinassi May 03 '16 at 17:30
@Thomas Thanks for replying. Does Riemann integral apply to bounded functions over unbounded domains? – Sekots Reivan May 03 '16 at 17:31
@LTW The original definition does only apply to bounded functions on bounded domains. Also with regard to the domain extensions exist, but they are a ****. – Thomas May 03 '16 at 17:36

The product of two Riemann integrable functions is integrable.

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