- #1

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_{i})m(E

_{i}) where the E

_{i}are disjoint intervals whose union is the whole interval.

At least when I learned it, the Riemann integral was partitioned and tags were taken from with the interval [x

_{i}, x

_{i+1}] which does not form a disjoint collection of intervals.

Can you allow tags to come only from the open interval (x

_{i}, x

_{i+1}) for some [itex]i[/itex] and from the closed intervals for others, so that we do in fact get a disjoint collection whose union is [a,b]. (this obviously can't work generally for Darboux sums are the inf and sup might not be contained in the image of the interval).