In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes both Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework.
They find common application in probability and stochastic processes, and in certain branches of analysis including potential theory.
To start, assume that f is non-negative and g is monotone non-decreasing and right-continuous.
By Carathéodory's extension theorem, there is a unique Borel measure μg on [a, b] which agrees with w on every interval I.
This functional can then be extended to the class of all non-negative functions by setting For Borel measurable functions, one has and either side of the identity then defines the Lebesgue–Stieltjes integral of h. The outer measure μg is defined via where χA is the indicator function of A. Integrators of bounded variation are handled as above by decomposing into positive and negative variations.
Suppose that γ : [a, b] → R2 is a rectifiable curve in the plane and ρ : R2 → [0, ∞) is Borel measurable.
Then we may define the length of γ with respect to the Euclidean metric weighted by ρ to be where
This notion is quite useful for various applications: for example, in muddy terrain the speed in which a person can move may depend on how deep the mud is.
The concept of extremal length uses this notion of the ρ-length of curves and is useful in the study of conformal mappings.
An alternative result, of significant importance in the theory of stochastic calculus is the following.
This result can be seen as a precursor to Itô's lemma, and is of use in the general theory of stochastic integration.
The final term is ΔU(t)ΔV(t) = d[U, V],which arises from the quadratic covariation of U and V. (The earlier result can then be seen as a result pertaining to the Stratonovich integral.)
This is particularly common in probability theory when v is the cumulative distribution function of a real-valued random variable X, in which case (See the article on Riemann–Stieltjes integration for more detail on dealing with such cases.)