Equivalent definition of Riemann integral. We can define Riemann integral using Darboux sums. In particular, we partition a segment in this way: $a = x_0 < x_1 < ... < x_{n-1} < x_n = b$ (1).
But can we take any partition of $[a; b]$ such that $[a; b] = \sqcup_{k=1}^nE_k$, $E_k$ is Jordan measurable,  and replace $\Delta x_i$ with $\mu_J(E_k)$ (2).
In our lecture we prove that $(2) \implies (1)$, and our teacher omitted prove $(1) \implies (2)$ because it's "a tricky exercise". I'm confused a bit...
We can replace $[x_i; x_{i+1}]$ with $[x_i; x_{i + 1})$ to get disjoint union and with a couple of equations we're done. Or it's not so simple?
The question is: does (1) implies (2) and is it difficult to prove?
 A: Suppose $f$ is Riemann integrable under definition (1).  Given any Jordan measureable set $E \subseteq [a,b]$, we define the integral of $f$ over $E$ as
$$\tag{*}\int_E f = \int_a^b f(x) \chi_{E}(x) \, dx,$$
The integral on the RHS of (*) exists since the boundary of $E$ has zero content and if $f$ is Riemann integrable on $[a,b]$, then $f \chi_{E}$ is discontinuous on a set of at most measure zero.  We also have the usual property that
$$\inf_{x \in E} f(x) \cdot\mu(E) \leqslant \int_E f \leqslant\sup_{x \in E} f(x) \cdot \mu(E)$$
Let $\mathcal{S} = \{P=\{E_k\}_{k=1}^n |\, n \in \mathbb{N}, \,\cup_{k=1}^nE_k = [a,b], \, E_k  \text{ Jordan measureable}   \}$, the collection of Jordan measureable partitions of $[a,b]$ and define upper and lower sums as
$$U_{\mathcal{S}}(P,f) = \sum_{k=1}^n \sup_{x \in E_k}f(x) \cdot \mu(E_k), \,\,\, L_{\mathcal{S}}(P,f) = \sum_{k=1}^n \sup_{x \in E_k}f(x) \cdot \mu(E_k)$$
Definition (2) of Riemann integrability is
$$\tag{**}\int_a^b f(x) \, dx = \sup_{P \in \mathcal{S}}\, L_{\mathcal{S}}(P,f) = \inf_{P \in \mathcal{S}}\, U_{\mathcal{S}}(P,f)$$
To prove that (1) implies (2), note that for any partition $P = \{E_k\}_{k=1}^n$
$$\int_a^b f(x) \, dx = \sum_{k=1}^n\int_{E_k}f \geqslant \sum_{k=1}^n \inf_{x \in E_k}f(x) \cdot \mu(E_k) = L_{\mathcal{S}}(P,f),$$
and, thus, $\int_a^b f(x) \, dx) \geqslant \sup_{P \in \mathcal{S}} L_{\mathcal{S}}(P,f)$.  To prove that the first equality in (**) holds, we must show that for any $\epsilon > 0$ there exists a partition $P \in \mathcal{S}$ such that
$$\int_a^b f(x) \, dx - \epsilon \leqslant L_{\mathcal{S}}(P,f)$$
As you suggested this is obvious by taking $P = \{[a,x_1), [x_1,x_2), \ldots, [x_{n-1},b]\}$and using Riemnann integrability of $f$ under definition (1).
In a similar way, the second equality of (**) can be proved.
