Assume that $E, E_1,E_2, … $ are Jordan measurable sets in $\mathbb{R}^n$, $E_1,…,E_n$ are disjoint, $E=\bigcup_{n=1}^\infty E_n$, a function $f:\mathbb{R}^n\to \mathbb{R}^n$ is Riemann integrable on $E, E_1, E_2, …$.
Is it possible to prove the following equality without the axiom of choice?
$$\int_E f=\sum_{n=1}^\infty \int _{E_n}f$$
I know that if we assume AC, there is the Lebesgue measure and we can prove the equality using it.