12 1. Measure theory 1.1.3. Connection with the Riemann integral. To conclude this sec- tion, we briefly discuss the relationship between Jordan measure and the Riemann integral (or the equivalent Darboux integral). For simplicity we will only discuss the classical one-dimensional Riemann integral on an in- terval [a, b], though one can extend the Riemann theory without much dif- ficulty to higher-dimensional integrals on Jordan measurable sets. (In later sections, this Riemann integral will be superceded by the Lebesgue integral.) Definition 1.1.5 (Riemann integrability). Let [a, b] be an interval of pos- itive length, and let f : [a, b] R be a function. A tagged partition P = ((x0,x1,...,xn), (x1,...,xn)) of [a, b] is a finite sequence of real numbers a = x0 x1 . . . xn = b, together with additional numbers xi−1 xi xi for each i = 1,...,n. We abbreviate xi xi−1 as δxi. The quantity Δ(P) := sup1≤i≤n δxi will be called the norm of the tagged partition. The Riemann sum R(f, P) of f with respect to the tagged partition P is defined as R(f, P) := n i=1 f(xi ∗)δxi. We say that f is Riemann integrable on [a, b] if there exists a real number, denoted b a f(x) dx and referred to as the Riemann integral of f on [a, b], for which we have b a f(x) dx = lim Δ(P)→0 R(f, P) by which we mean that for every ε 0 there exists δ 0 such that |R(f, P)− b a f(x) dx| ε for every tagged partition P with Δ(P) δ. If [a, b] is an interval of zero length, we adopt the convention that every function f : [a, b] R is Riemann integrable, with a Riemann integral of zero. Note that unbounded functions cannot be Riemann integrable (why?). The above definition, while geometrically natural, can be awkward to use in practice. A more convenient formulation of the Riemann integral can be formulated using some additional machinery. Exercise 1.1.20 (Piecewise constant functions). Let [a, b] be an interval. A piecewise constant function f : [a, b] R is a function for which there exists a partition of [a, b] into finitely many intervals I1,...,In, such that f is equal to a constant ci on each of the intervals Ii. If f is piecewise constant, show that the expression n i=1 ci|Ii|
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