Facts About The Darboux Integral
A refinement of the partition
is a partition
such that for every i with
there is an integer r(i) such that
In other words, to make a refinement, cut the subintervals into smaller pieces and do not remove any existing cuts. If
is a refinement of
then
and
If P1, P2 are two partitions of the same interval (one need not be a refinement of the other), then
- .
It follows that
Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if
and
together make a tagged partition
(as in the definition of the Riemann integral), and if the Riemann sum of ƒ corresponding to P and T is R, then
From the previous fact, Riemann integrals are at least as strong as Darboux integrals: If the Darboux integral exists, then the upper and lower Darboux sums corresponding to a sufficiently fine partition will be close to the value of the integral, so any Riemann sum over the same partition will also be close to the value of the integral. It is not hard to see that there is a tagged partition that comes arbitrarily close to the value of the upper Darboux integral or lower Darboux integral, and consequently, if the Riemann integral exists, then the Darboux integral must exist as well.
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