Let $f:[a_1,b_1]\times \cdots \times[a_n,b_n] \rightarrow \mathbb{R}$ be Riemann integrable. Prove that is $f$ Lebesgue integrable.
Proof: $$Q:= [a_1,b_1]\times \cdots \times [a_n,b_n].$$ For simple function $g = \sum_{j=1}^{N} c_j \chi_{Q_j}$ ,where are $Q_j$ parallelepipeds contained in $Q$, Riemann and Lebesgue integrals coincide. Call such function as step function. Let choose sequence $P_k$ partition of cuboid $Q$ that is $P_{k+1}$ subdivision of partition $P_k$ and diameter of $P_k \to 0$ when $k \to \infty$. Let $$Q_{j}^{k}, \quad 1 \leq j \leq M_k,$$ be a partition cuboid of $P_k$ and let $$m_{j}^{k} = \inf_{Q_{j}^{k}}f, \quad M_{j}^{k} = \sup_{Q_{j}^{k}}f.$$
Define functions $g_k, G_k$ on $Q$ in this way: $$g_k :=m_{j}^{k}, \quad G_k :=M_{j}^{k},$$ for $x \in Q_{j}^{k}$. Now we have lower Darboux sum of function $f$ with regard to $P_k$: $$s_k = \int_{Q} g_k \, dx = \int_{Q}g_k \, dm_n,$$ Riemann and Lebesgue integral step function $g_k$ coincide. Same conclusion for upper Darboux sum $$\int_{Q}G_k \, dx = \int_{Q}G_k \, dm_n.$$
We have $g_k$ increasing and $G_k$ decreasing, so $g_k \rightarrow g$, $G_k \rightarrow G$ where $g(x),G(x)$ are measurable. $f$ is Riemann integrabile implies $$\int_{Q} g_k \, dm_n \to I = \int_{Q} f(x) \, dx \text{ and } \int_{Q} G_k \, dm_n \to I = \int_{Q} f(x) \,dx.$$
From theorem dominated convergence
$$I = \lim_{k \to \infty} \int_{Q}G_k \, dm_n = \int_{Q}G \, dm_n$$
$$I = \lim_{k \to \infty}\int_{Q}g_k \, dm_n = \int_{Q}g \, dm_n,$$
so that
$$\int_{Q} (G-g) dm_n = 0 \text{ and } (G-g) \geq 0.$$
So $G(x)=g(x)$ a.e. and $g\leq f \leq G$.
$g(x),G(x)$ are Lebesgue integrable. $\Box$
$\textbf{Question: Prove that is function $f$ continuous at every point x where g(x)= G(x)? } $