This short note gives an introduction to the Riemann-Stieltjes integral on R and Rn. Some natural and important applications in probability. Definitions. Riemann Stieltjes Integration. Existence and Integrability Criterion. References. Riemann Stieltjes Integration – Definition and. Existence of Integral. Note. In this section we define the Riemann-Stieltjes integral of function f with respect to function g. When g(x) = x, this reduces to the Riemann.

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In mathematicsthe Riemann—Stieltjes integral is a generalization of the Riemann integralnamed after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in by Stieltjes.

## Riemann–Stieltjes integral

Hildebrandt calls it the Pollard—Moore—Stieltjes integral. The Riemann—Stieltjes integral can be efficiently handled using an appropriate generalization of Darboux sums. Furthermore, f is Riemann—Stieltjes integrable with respect to g in the classical sense if.

Then the Integralle can be evaluated as. The Riemann—Stieltjes integral admits integration by parts in the form. The best simple existence theorem states that if f is continuous and g is of bounded variation on [ ab ], then the integral exists.

### Stieltjes Integral — from Wolfram MathWorld

If g is not of bounded variation, then there will be continuous functions which cannot be integrated with respect to g. In general, the integral is not well-defined if f and g share any points of discontinuitybut this sufficient condition is not necessary. If g is the cumulative probability distribution function of a random variable X that has a probability density function with respect to Lebesgue measureand f is any function for which the expected value E f X is finite, then the probability density function of X is the derivative of g and we have.

But this formula does not work if X does not have a probability density function with respect to Lebesgue measure. In particular, it does not work if the distribution of X is discrete i.

In particular, no matter how ill-behaved re cumulative distribution function g of a random variable Xif the moment E X n exists, then it is equal to. The Riemann—Stieltjes integral appears in the original formulation of F.

## Stieltjes Integral

Riesz’s theorem which represents the dual space of the Ibtegrale space C [ ab ] of continuous functions in an interval [ ab ] as Riemann—Stieltjes integrals against functions of bounded variation. Later, that theorem was reformulated in terms of measures.

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. In this theorem, the integral is considered with respect to a spectral family of projections. Nagy for details.

An important generalization is the Lebesgue—Stieltjes integral which generalizes the Riemann—Stieltjes integral in a way analogous to how the Wtieltjes integral generalizes the Riemann integral. If improper Riemann—Stieltjes integrals are allowed, the Lebesgue integral is not strictly more general than the Riemann—Stieltjes integral. This generalization plays a role in the study of semigroupsvia the Laplace—Stieltjes transform. From Wikipedia, the free encyclopedia.

Rudinpages — Integration by parts Integration by substitution Inverse function integration Order of integration calculus trigonometric substitution Integration by partial fractions Integration by reduction formulae Integration using parametric derivatives Integration using Euler’s formula Differentiation under the integral sign Contour integration.

Improper integral Gaussian integral.

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