I thougt about a modification of the construction of the Riemann-Darboux-integral in the following way: For a non-negative function $f\colon [a,b] \to \mathbb{R}$ take the upper and lower Darboux-integrals and if they coincide call the function integrable with the corresponding integral (as usual). But now the difference: For the partions of the interval $[a,b]$ needed for constructing the upper and lower Darboux sums use instead of disjoint intervals
(i) a finite collection of Jordan-measurable sets OR
(ii) an countable sequence of Jordan-measuarble sets
For (i) I found a post here on MSE that this is equivalent to the standard Darboux construction using an partion of $[a,b]$ consisting of intervals (Equivalent definition of Riemann integral.). For the case (ii) I am not really sure if it is still equivalent...I came up with this question because the Jordan-content is a premeasure and hence $\sigma$-additive (but the Jordan-contented sets doesn't form a $\sigma$-algebra). And I considered a non-negative function for $f$ to avoid trouble with the infinite partion. Do you know something about the equivalence? (I discussed something similar for Lebesgue-integrals and simple function constructed by a countable partion here: Equality of an integral of a measurable function) Thank you!