Riemann–Stieltjes integral

In the calculus the Stieltjesintegral denotes a significant generalization of the Riemann integral or a concretization of the concept of Lebesgue integral. It was named (1856-1894) after the Dutch mathematician Jean Thomas Stieltjes. The Stieltjesintegral, for the term of the integrator is fundamental, applies to many fields, especially in physics and stochastics.

The Riemann - Stieltjes integral for monotone integrators

It had a real interval and two functions. It is assumed that the integrand is bounded and, for the integrator (not necessarily strictly ) increases monotonically. The corresponding Riemann - Stieltjes integral of with respect to the interval as the Riemann integral over fine decompositions of the interval or upper and lower sums ( qv) defined. However, the formulas for the upper and lower sum for Stieltjes integrals noisy place

Now

Converging upper and lower sum for sufficiently fine partitions to the same value, ie with respect to Riemann - Stieltjes integrable and the common limit is called the value of the integral. The notation for this

The integrator regulates So how heavily weighted in different places. Instead integrator therefore the name Weight function is common. Obviously, the ordinary Riemann integral can now be regarded as a special case of the Riemann - Stieltjes integral with for all (identity).

The Riemann - Stieltjes integral exists, for example, the case of continuous function even with the Cantor function as an integrator (which is a monotonically increasing from 0 to 1 function that is constant almost everywhere, ie up to an uncountable set of measure zero ).

The Lebesgue - Stieltjes integral

The Lebesgue - Stieltjes integral is a special case of the Lebesgue integral. This is integrated over a Borel measure that is defined by the function in the case of monotonous Lebesgue Stieltjes integral and will be hereinafter referred to. The measure is defined by its values ​​on intervals:

Here, the left-side and right-hand limit of the function at the point. If the identity, it is the Lebesgue measure. Is respect to this measure Lebesgue - integrable, then one defines the associated Lebesgue - Stieltjes integral as

With the right side is to be interpreted as an ordinary Lebesgue integral.

Non - monotonic integrators

For a limited quantity of non monotone increasing integrators the Stieltjes integral can also be defined sense, namely to those with finite variation. Functions of finite variation can be represented as a difference of two monotone increasing functions namely always, so where are monotonically increasing. The corresponding Stieltjes integral (optionally in the Riemann or Lebesgue sense) is then defined as

It can be shown that this definition makes sense, ie well-defined ( ie independent of the particular choice of decomposition ) is.

Properties

• As the Riemann and Lebesgue integral is the Stieltjes integral linear in the integrand:

• Furthermore, the Stieltjes integral is also linear in the integrator, ie

• The integral is invariant under translations of the integrator, ie

• Step functions as integrators: Is steady and a step function that has jumps of height in points, so true

• Is continuously differentiable, the following applies

• Is absolutely continuous, then almost everywhere differentiable, the derivative is integrable and it applies here:

• For the Riemann - Stieltjes integral following rule applies for the partial integration: