Riemann-Stieltjes integrals

The integral

$$I = \int_{0}^{\infty} e^{-\Lambda(v)} d \Lambda^{-1}(v)$$

is a Riemann-Stieltjes integral(or just Stieltjes integral) which is defined as:

$$I = \int_{a}^{b} f(x) dg(x)$$

Here, $f(x)$ and $g(x)$ are functions, and the integral is taken over the interval $[a, b]$. In this case, $dg(x)$ represents the measure of integration as determined by the function $g(x)$. In your example, $f(v) = e^{-\Lambda(v)}$, and $g(v) = \Lambda^{-1}(v)$.

If the function $g(v) = \Lambda^{-1}(v)$ has a continuous derivative, denoted as $\frac{d\Lambda^{-1}(v)}{dv}$, you can convert the Riemann-Stieltjes integral to a Riemann integral using the chain rule:

$$I = \int_{0}^{\infty} e^{-\Lambda(v)} d\Lambda^{-1}(v) = \int_{0}^{\infty} e^{-\Lambda(v)} \frac{d\Lambda^{-1}(v)}{dv} dv$$

Now, the integral is a standard Riemann integral with respect to the variable $v$. You can evaluate this integral using standard techniques or numerical methods if an analytical solution is not possible.

So, when an integral is with respect to a function rather than a variable, it means that the measure of integration is based on the function itself, and it is represented as a Riemann-Stieltjes integral. You can evaluate this integral by converting it to a standard Riemann integral if the function has a continuous derivative, and then proceed to use standard techniques or numerical methods for evaluation.

Application to functional analysis

The Riemann-Stieltjes integral plays an important role in various areas of mathematics including but not limited to:

1. F. Riesz's theorem: The Riemann-Stieltjes integral appears in the original formulation of F. Riesz's theorem, which characterizes the dual space of the Banach space $C[a,b]$ of continuous functions on the interval $[a,b]$. In this context, the dual space consists of Riemann-Stieltjes integrals against functions of bounded variation. Later, as you pointed out, the theorem was reformulated in terms of measures, giving rise to the concept of Radon measures.

2. Spectral theorem for self-adjoint operators: The Riemann-Stieltjes integral also appears in the formulation of the spectral theorem for non-compact, self-adjoint (or more generally, normal) operators in a Hilbert space. The spectral theorem expresses a self-adjoint operator as an integral of its spectral projections with respect to a spectral measure. The spectral measure, in this case, is a mapping from the Borel subsets of the spectrum to the orthogonal projections in the Hilbert space, and the Riemann-Stieltjes integral is used to define the integral with respect to the spectral family of projections.

Both of these instances demonstrate the importance and versatility of the Riemann-Stieltjes integral in various branches of mathematics, including functional analysis and operator theory.