Riemann Integral - Integrability

Integrability

A function on a compact interval is Riemann integrable if and only if it is bounded and continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). This is known as the Lebesgue integrability condition or Lebesgue's criterion for Riemann integrability or the Riemann—Lebesgue theorem. Note that this should not be confused with the notion of the Lebesgue integral of a function existing; the result is due to Lebesgue, and uses the notion of measure zero, but does not refer to or use Lebesgue measure more generally, or the Lebesgue integral.

The integrability condition can be proven in various ways, one of which is sketched below.

Proof

The proof is easiest using the Darboux integral definition of integrability (formally, the Riemann condition for integrability) – a function is Riemann integrable if and only if the upper and lower sums can be made arbitrarily close by choosing an appropriate partition. One direction is very brief by using the oscillation definition of continuity: if f is discontinuous on a set of positive measure, then for some ε, f has oscillation at least ε on a set Xε of positive measure so the upper and lower integrals of f differ by at least this is where oscillation is used. The converse direction is straightforward but longer. Conversely, if f is continuous almost everywhere, then for any partition of the interval first divide the partition into two sets of intervals, C and D, with D containing all discontinuous points and C containing the rest. Intuitively, the width of D can be made arbitrarily small, while the height of C can be made arbitrarily small. Formally, for any ε, one can choose a subpartition D′ such that discontinuities are contained in intervals of total length at most ε; then the lower sum and upper sum on D′ differ by at most where m and M are the infimum and supremum of f; this is where boundedness is used, and implicitly the equivalence of Jordan content zero and Lebesgue measure zero on a compact set (hence a finite partition can be used). On the rest (C′), the function is continuous on a compact interval, hence uniformly continuous, so a subpartition can be chosen such that on each subinterval, the function varies by at most ε, so the lower and upper sums differ by at most (this is where compactness is used). The total difference is thus bounded by which is a constant times ε, and hence can be made arbitrarily small, thus the function is Riemann integrable.

In particular, a countable set has measure zero, and thus a bounded function (on a compact interval) with only finitely many or countably infinitely many discontinuities is Riemann integrable.

An indicator function of a bounded set is Riemann-integrable if and only if the set is Jordan measurable.

If a real-valued function is monotone on the interval it is Riemann-integrable, since its set of discontinuities is denumerable, and therefore of Lebesgue measure zero.

If a real-valued function on is Riemann-integrable, it is Lebesgue-integrable. That is, Riemann-integrability is a stronger (meaning more difficult to satisfy) condition than Lebesgue-integrability.

If is a uniformly convergent sequence on with limit, then Riemann integrability of all implies Riemann integrability of, and

However, the Lebesgue monotone convergence theorem (on a monotone pointwise limit) does not hold.