- #1
sammycaps
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So the beginning of Rudin's Real and Complex Analysis states that the Riemann integral on an interval [itex][a,b][/itex] can be approximated by sums of the form [itex]\Sigma[/itex][itex]\stackrel{i=1}{n}[/itex]f(ti)m(Ei) where the Ei 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 [xi, xi+1] which does not form a disjoint collection of intervals.
Can you allow tags to come only from the open interval (xi, xi+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).
At least when I learned it, the Riemann integral was partitioned and tags were taken from with the interval [xi, xi+1] which does not form a disjoint collection of intervals.
Can you allow tags to come only from the open interval (xi, xi+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).