Riemann Integral - Generalizations

Generalizations

It is easy to extend the Riemann integral to functions with values in the Euclidean vector space Rn for any n. The integral is defined by linearity; in other words, if ƒ = (ƒ₁,…,ƒ_n) then

In particular, since the complex numbers are a real vector space, this allows the integration of complex valued functions.

The Riemann integral is only defined on bounded intervals, and it does not extend well to unbounded intervals. The simplest possible extension is to define such an integral as a limit, in other words, as an improper integral. We could set:

Unfortunately, this does not work well. Translation invariance, the fact that the Riemann integral of the function should not change if we move the function left or right, is lost. For example, let ƒ(x) = −1 for all x < 0, ƒ(0) = 0 and ƒ(x) = 1 for all x > 0 then

for all x. But if we shift ƒ(x) to the right by one unit to get ƒ(x−1), we get

for all x > 1. Since this is unacceptable, we could try the definition:

Then if we attempt to integrate the function ƒ above, we get +∞, because we take the limit b → ∞ first. If we reverse the order of the limits, then we get −∞.

This is also unacceptable, so we could require that the integral exists and gives the same value regardless of the order. Even this does not give us what we want, because the Riemann integral no longer commutes with uniform limits. For example, let ƒ_n(x) = n−1 on (0,n) and 0 everywhere else. For all n we have

But ƒ_n converges uniformly to zero, so the integral of lim(ƒ_n) is zero. Consequently

Even though this is the correct value, it shows that the most important criterion for exchanging limits and (proper) integrals is false for improper integrals. This makes the Riemann integral unworkable in applications.

A better route is to abandon the Riemann integral for the Lebesgue integral. The definition of the Lebesgue integral is not obviously a generalization of the Riemann integral, but it is not hard to prove that every Riemann-integrable function is Lebesgue-integrable and that the values of the two integrals agree whenever they are both defined. Moreover, a function ƒ defined on a bounded interval is Riemann-integrable if and only if it is bounded and the set of points where ƒ is discontinuous has Lebesgue measure zero.

An integral which is in fact a direct generalization of the Riemann integral is the Henstock–Kurzweil integral.

Another way of generalizing the Riemann integral is to replace the factors x_k+1 − x_k in the definition of a Riemann sum by something else; roughly speaking, this gives the interval of integration a different notion of length. This is the approach taken by the Riemann–Stieltjes integral.