RiemannStieltjesintegrál

The Riemann–Stieltjes integral is a generalization of the Riemann integral. It was introduced by Thomas Joannes Stieltjes. The integral is defined as the limit of Riemann sums, but instead of using the width of the subintervals, it uses the variation of another function. Specifically, the Riemann–Stieltjes integral of a function f with respect to a function g, denoted by $\int_a^b f(x) \, dg(x)$, is defined as the limit of sums of the form $\sum_{i=1}^n f(c_i) [g(x_i) - g(x_{i-1})]$ as the partition of the interval $[a, b]$ becomes arbitrarily fine, where $c_i$ is a point in the $i$-th subinterval $[x_{i-1}, x_i]$.

The function g is often called the integrator. If g is a continuously differentiable function, then the Riemann–Stieltjes integral coincides with the ordinary Riemann integral: $\int_a^b f(x) \, dg(x) = \int_a^b f(x) g'(x) \, dx$. This integral is particularly useful when the integrator function g has jumps or is not differentiable, which is common in probability theory and functional analysis. For instance, when g represents a cumulative distribution function, the Riemann–Stieltjes integral can be used to compute expected values.

The existence of the Riemann–Stieltjes integral depends on the properties of both f and g. If f is continuous and g is of bounded variation, then the integral exists. If f is Riemann integrable and g is continuous, the integral also exists. This generalization allows for a broader range of applications than the standard Riemann integral.