fence with the greater projection and thereby carry the most weight in the integral. When is a step function the fence has a rectangular "gate" of width 1 and height equal to . Thus the gate, and its projection, have area equal to , the value of the Riemann-Stieljes integral. Generalization An important generalization is the Lebesgue–Stieltjes integral, which generalizes the Riemann–Stieltjes integral in a way analogous to how the Lebesgue integral generalizes the Riemann integral. If improper Riemann–Stieltjes integrals are allowed, then the Lebesgue integral is not strictly more general than the Riemann–Stieltjes integral. The Riemann–Stieltjes integral also generalizes to the case when either the integrand ƒ or the integrator g take values in a Banach space. If takes values in the Banach space X, then it is natural to assume that it is of strongly bounded variation, meaning that the supremum being taken over all finite partitions of the interval [a,b]. This generalization plays a role in the study of semigroups, via the Laplace–Stieltjes transform. The Itô integral extends the Riemann–Stietjes integral to encompass integrands and integrators which are stochastic processes rather than simple functions; see also stochastic calculus. Generalized Riemann–Stieltjes integral A slight generalization is to consider in the above definition partitions P that refine another partition Pε, meaning that P arises from Pε by the addition of points, rather than from partitions with a finer mesh. Specifically, the generalized Riemann–Stieltjes integral of f with respect to g is a number A such that for every ε > 0 there exists a partition Pε such that for every partition P that refines Pε, for every choice of points ci in [xi, xi+1]. This generalization exhibits the Riemann–Stieltjes integral as the Moore–Smith limit on the directed set of partitions of [a, b] . A consequence is that with this definition, the integral can still be defined in cases where f and g have a point of discontinuity in common. Darboux sums The Riemann–Stieltjes integral can be efficiently handled using an appropriate generalization of Darboux sums. For a partition P and a nondecreasing function g on [a, b] define the upper Darboux sum of f with respect to g by and the lower sum by Then the generalized Riemann–Stieltjes of f with respect to g exists if and only if, for every ε > 0, there exists a partition P such that Furthermore, f is Riemann–Stieltjes integrable with respect