For $A \subseteq \mathbb{R^n}$, $B \subseteq A$, and $f : A \rightarrow \mathbb{R^n}$, I know that it is not true in general that if $f$ is Riemann integrable on $A$ then $f$ is Riemann integrable on $B$.
My question is regarding the case where where $A$ and $B$ are both Jordan regions (that is, $vol(\partial A) = 0$ and $vol(\partial B) = 0$). In this case, is it true that $f$ being integrable on $A$ implies $f$ being integrable on $B$?
My intuition is telling me that this is indeed true since any counterexample to the general claim seems to involve a non-Jordan region (i.e. $[0,1] \cap \mathbb{Q}$). However, I am having trouble coming up with an argument that this is indeed true.
I would appreciate any input/suggestions on tackling this question!
