LebesgueStieltjesintegrál

The Lebesgue-Stieltjes integral is a generalization of the Riemann integral and the Lebesgue integral. It was developed by Henri Lebesgue and Thomas Joannes Stieltjes. This type of integral allows for integration with respect to a more general function, known as a integrator function, rather than just the variable of integration. The notation for the Lebesgue-Stieltjes integral of a function f with respect to a function g over an interval [a, b] is often written as $\int_a^b f(x) dg(x)$.

The key idea is that the "width" of the intervals in the Riemann sum is replaced by the "change in the integrator function" over those intervals. For a simple partition $a = x_0 < x_1 < \dots < x_n = b$, a Riemann-Stieltjes sum would look like $\sum_{i=1}^n f(c_i) (g(x_i) - g(x_{i-1}))$, where $c_i$ is a point in $[x_{i-1}, x_i]$. The Lebesgue-Stieltjes integral is the limit of such sums as the partition becomes finer, under certain conditions on f and g.

If the integrator function g is differentiable, and its derivative $g'$ exists, then the Lebesgue-Stieltjes integral can be related to a standard Lebesgue integral: $\int_a^b f(x) dg(x) = \int_a^b f(x) g'(x) dx$, provided $g'$ is integrable. This connection highlights its role as a unifying concept in integration theory. The Lebesgue-Stieltjes integral has applications in probability theory, where it is used to define expected values, and in functional analysis.