# Integrability on $A$ implies integrability on $B\subset A$?

Let $f:A\subset\mathbb{R}^n\to\mathbb{R}$ be Riemann integrable on $A$. If $B\subset A$ does the Riemann integral$$\int_B f(x_1,\ldots,x_n)dx_1\ldots dx_n$$exist finite and, if it does, how can it be proved?

I think a hint may come from a particular case that I find here where it is said that, at least in the particular case where $A=[a,b]\subset\mathbb{R}$ and $B=[a',b']\subset [a,b]$, since $\int_A f(x_1,\ldots,x_n)dx_1\ldots dx_n$ and $\int_A \chi_B(x_1,\ldots,x_n)dx_1\ldots dx_n$ exist, then $\int_A \chi_B(x_1,\ldots,x_n)f(x_1,\ldots,x_n)dx_1\ldots dx_n$ also does and, by definition, is identical to $\int_B f(x_1,\ldots,x_n)dx_1\ldots dx_n$. Nevertheless, I am not sure whether these considerations are true for $B\subset A\subset\mathbb{R}^n$ with $n\ge 2$ and, if they are, I would have to know how to prove that the product of two Riemann integrable functions is Riemann integrable. As a side note, I would not be able to understand a proof based on Lebesgue integration because I have no knowledge of how Riemann and Lebesgue integrals are related if $n>1$. I heartily thank any answerer!

Take $A=[0,1]$, $f(x)=1$ and $B=\mathbb{Q}\cap [0,1]$.
The Riemann integral on $A$ is equal to one while the Riemann integral on $B$ is undefined.
• If we add opportune constraints on $B$, would the integrability orf $f$ on $B$ follow? I think that it is more correct to ask a separate question for that: math.stackexchange.com/questions/1535549/… – Self-teaching worker Nov 18 '15 at 17:52