Darboux Integral - Facts About The Darboux Integral

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.