Darboux integral

In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function.

Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.

[1] The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral.

Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.

[2] Moreover, the definition is readily extended to defining Riemann–Stieltjes integration.

The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function

The upper and lower integrals are in turn the infimum and supremum, respectively, of upper and lower (Darboux) sums which over- and underestimate, respectively, the "area under the curve."

In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of f in each subinterval of the partition.

These ideas are made precise below: A partition of an interval

The upper Darboux sum of

is The lower Darboux sum of

is The lower and upper Darboux sums are often called the lower and upper sums.

The upper Darboux integral of f is The lower Darboux integral of f is In some literature, an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively: and like Darboux sums they are sometimes simply called the lower and upper integrals.

If Uf = Lf, then we call the common value the Darboux integral.

[4] We also say that f is Darboux-integrable or simply integrable and set An equivalent and sometimes useful criterion for the integrability of f is to show that for every ε > 0 there exists a partition Pε of [a, b] such that[5] Suppose we want to show that the function

equally sized subintervals each of length

, the infimum on any particular subinterval is given by its starting point.

Likewise the supremum on any particular subinterval is given by its end point.

Thus the lower Darboux sum on a partition

is given by similarly, the upper Darboux sum is given by Since Thus for given any

To find the value of the integral note that Suppose we have the Dirichlet function

defined as Since the rational and irrational numbers are both dense subsets of

takes on the value of 0 and 1 on every subinterval of any partition.

we have from which we can see that the lower and upper Darboux integrals are unequal.

such that for all i = 0, …, n 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.

then and If P1, P2 are two partitions of the same interval (one need not be a refinement of the other), then and it follows that Riemann sums always lie between the corresponding lower and upper Darboux sums.

together make a tagged partition (as in the definition of the Riemann integral), and if the Riemann sum of

corresponding to P and T, 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.

There is (see below) 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.

, we have Taking limits of both sides, Similarly, (with a different sequences of tags) Thus, we have which means that the Darboux integral exists and equals