Stieltjes integral

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

The Riemann-Stieltjes integral for monotonic integrators

Let with and two functions . It is assumed that , the integrand, is bounded and , the integrator, grows monotonically (not necessarily strictly) . The associated Riemann Stieltjes integral of relative to the interval as the Riemann integral via fine decompositions of the interval or the upper and lower sums (see there) defined. However, the formulas for the upper and lower sums for Stieltjes integrals are instead ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle a<b}$ ${\displaystyle f,h\colon [a,b]\to \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle h}$ ${\displaystyle f}$ ${\displaystyle h}$ ${\displaystyle [a,b]}$

${\displaystyle {\overline {S_{N}}}=\sum _{i=1}^{N}\sup {\Big \{}f(t):t\in [t_{i-1},t_{i}]{\Big \}}\cdot (t_{i}-t_{i-1})}$ (Upper sum) and

${\displaystyle {\underline {S_{N}}}=\sum _{i=1}^{N}\inf {\Big \{}f(t):t\in [t_{i-1},t_{i}]{\Big \}}\cdot (t_{i}-t_{i-1})}$ (Sub-total)

now

${\displaystyle {\overline {S_{N}}}=\sum _{i=1}^{N}\sup {\Big \{}f(t):t\in [t_{i-1},t_{i}]{\Big \}}\cdot (h(t_{i})-h(t_{i-1}))}$ (Stieltjes upper sum) and

${\displaystyle {\underline {S_{N}}}=\sum _{i=1}^{N}\inf {\Big \{}f(t):t\in [t_{i-1},t_{i}]{\Big \}}\cdot (h(t_{i})-h(t_{i-1}))}$ (Stieltjes sub-sum).

If the upper and lower sums converge towards the same value for sufficiently fine decompositions, then with respect to Riemann-Stieltjes is called integrable and the common limit value is called the value of the integral. The notation for this is ${\displaystyle f\!\,}$ ${\displaystyle h\!\,}$ ${\displaystyle [a,b]\!\,}$

${\displaystyle \int _{a}^{b}f\,\mathrm {d} h\quad {\text{oder}}\quad \int _{a}^{b}f(t)\,\mathrm {d} h(t).}$

The integrator regulates how heavily weighted is given in different places . Instead of integrator, the term weight function is therefore also common. Obviously, the ordinary Riemann integral can be understood as a special case of the Riemann-Stieltjes integral for all ( identity ). In contrast to the Riemann integral, the standard assumption is that the integrand function is continuous, but the integrator function can be more complicated: ${\displaystyle h}$ ${\displaystyle f}$ ${\displaystyle h(x)=x}$ ${\displaystyle x}$ ${\displaystyle f(t)}$ ${\displaystyle h(t)}$

Cantor function (10 iterations, continuous and monotonic, but nowhere differentiable with a positive derivative)

Heaviside function (discontinuous)

The Riemann-Stieltjes integral exists e.g. B. in the case of a continuous function even with the Cantor function as integrator (this is a monotonically increasing continuous function from 0 to 1, the derivative of which is 0 almost everywhere, namely up to an uncountable zero set). It even exists with a discontinuous but monotonous step function, for example for for all but for ( Heaviside function ). ${\displaystyle f}$ ${\displaystyle h}$ ${\displaystyle h,}$ ${\displaystyle h(t)=0}$ ${\displaystyle t<0}$ ${\displaystyle h(t)=1}$ ${\displaystyle t\geq 0}$

The Lebesgue-Stieltjes integral

The Lebesgue-Stieltjes integral is a special case of the Lebesgue integral . Here, integration is carried out using a Borel measure , which in the case of the Lebesgue-Stieltjes integral is defined by the monotonic function and is referred to below as . The dimension is determined by its values ​​at intervals: ${\displaystyle \mu }$ ${\displaystyle h}$ ${\displaystyle \mu _{h}}$ ${\displaystyle \mu _{h}}$

${\displaystyle {\begin{aligned}\mu _{h}([x,y[)&=h(y-)-h(x-),\\\mu _{h}([x,y])&=h(y+)-h(x-),\\\mu _{h}(]x,y])&=h(y+)-h(x+)\end{aligned}}}$

Here denotes the left-hand and right-hand limit values ​​of the function at the point . If the identity is, it is the Lebesgue measure . If Lebesgue can be integrated with regard to this measure , the associated Lebesgue-Stieltjes integral is defined as ${\displaystyle h(y-)}$ ${\displaystyle h(y+)}$ ${\displaystyle h}$ ${\displaystyle y}$ ${\displaystyle h}$ ${\displaystyle f}$ ${\displaystyle \mu _{h}}$

${\displaystyle \int _{a}^{b}f\,\mathrm {d} h=\int _{a}^{b}f\,\mathrm {d} \mu _{h},}$

where the right hand side is to be understood as an ordinary Lebesgue integral .

Non-monotonic integrators

For a limited amount of non-monotonically increasing integrators the Stieltjes integral can be also defined sense, namely for those with finite variation on . Functions of finite variation can always be represented as the difference between two monotonically increasing functions, that is, where are monotonically increasing. The associated Stieltjes integral (optionally in the Riemann or Lebesgue sense) is then defined as ${\displaystyle [a,b]}$ ${\displaystyle h=h_{1}-h_{2}}$ ${\displaystyle h_{1},h_{2}\colon [a,b]\to \mathbb {R} }$

${\displaystyle \int _{a}^{b}f\,\mathrm {d} h:=\int _{a}^{b}f\,\mathrm {d} h_{1}-\int _{a}^{b}f\,\mathrm {d} h_{2}.}$

It can be shown that this definition makes sense; H. is well-defined (i.e. independent of the particular choice of decomposition).

properties

• Like the Riemann and Lebesgue integrals, the Stieltjes integral is also linear in the integrand:

  ${\displaystyle \int _{a}^{b}(\alpha f+\beta g)\,\mathrm {d} h=\alpha \int _{a}^{b}f\,\mathrm {d} h+\beta \int _{a}^{b}g\,\mathrm {d} h}$

for constants if the considered integrals exist. ${\displaystyle \alpha ,\beta \in \mathbb {R} }$

• Furthermore, the Stieltjes integral is also linear in the integrator, i.e.

  ${\displaystyle \int _{a}^{b}f\,\mathrm {d} (\alpha g+\beta h)=\alpha \int _{a}^{b}f\,\mathrm {d} g+\beta \int _{a}^{b}f\,\mathrm {d} h}$

for constants and functions of finite variation. ${\displaystyle \alpha ,\beta \in \mathbb {R} }$ ${\displaystyle g,h}$

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

  ${\displaystyle \int _{a}^{b}f\,\mathrm {d} (h+c)=\int _{a}^{b}f\,\mathrm {d} h}$

for constants . ${\displaystyle c}$

• Step functions as integrators: If continuous and a step function that has jumps in height at the points , then applies ${\displaystyle f}$ ${\displaystyle h}$ ${\displaystyle t_{1},\ldots ,t_{n}\in \,]a,b[}$ ${\displaystyle \Delta h_{1},\ldots ,\Delta h_{n}\in \mathbb {R} }$

  ${\displaystyle \int _{a}^{b}f\,\mathrm {d} h=\sum _{i=1}^{n}f(t_{i})\Delta h_{i}.}$

• If it is continuously differentiable , then we have ${\displaystyle h}$

  ${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} h(x)=\int _{a}^{b}f(x)h'(x)\,\mathrm {d} x.}$

(In Lebesgue's sense: is the density of .) ${\displaystyle h\!\,'}$ ${\displaystyle \mu _{h}}$

• Is absolutely continuous , it is differentiable almost everywhere, the derivation can be integrated and it also applies here: ${\displaystyle h}$ ${\displaystyle h}$ ${\displaystyle h'}$

  ${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} h(x)=\int _{a}^{b}f(x)h'(x)\,\mathrm {d} x.}$

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

  ${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} h(x)=f(b)h(b)-f(a)h(a)-\int _{a}^{b}h(x)\,\mathrm {d} f(x).}$

literature

• Isidor P. Natanson: Theory of the functions of a real variable. Unchanged reprint of the 4th edition. Harri Deutsch, Thun et al. 1981, ISBN 3-87-144-217-8 .

Individual evidence

1. ^ Wolfgang Walter : Analysis. Volume 2. 5th, expanded edition. Springer, Berlin et al. 2002, ISBN 3-540-42953-0 , p. 193 f.