Riemann-Stieltjes integral |
In mathematics, the Riemann-Stieltjes integral is a
generalization of the Riemann integral. The Riemann-Stieltjes
integral of a real-valued function f of a real variable with respect
to a nondecreasing real function g is denoted by
-
and defined to be the limit as the mesh of the
partition of the interval [a, b]
approaches zero, of the sum
-
where ci is in the ith subinterval [xi,
xi+1]. In order that this Riemann-Stieltjes integral exist it is necessary that f and
g do not share any points of discontinuity in common. The two functions f and g are respectively
called the integrand and the integrator.
For another formulation of integration that is more general, see Lebesgue integration.
Properties and relation to the Riemann integral
If g should happen to be everywhere differentiable, then
the integral is no different from the Riemann integral
-
However, g may have jump discontinuities, or may have derivative zero almost everywhere while still being
continuous and nonconstant (for example, g could be the celebrated Cantor function), in either of which cases the Riemann-Stieltjes integral is not captured by any expression
involving derivatives of g.
The Riemann-Stieltjes integral admits integration by
parts in the form
-
What if g is not monotone?
Somewhat more generally, one may define a Riemann-Stieltjes integral with respect to any function g of bounded variation, since every such function can be written uniquely as a
difference between two nondecreasing functions; the integral is the corresponding difference between two Riemann-Stieltjes
integrals with respect to nondecreasing functions.
Application to probability theory
If g is the cumulative
probability distribution function of a random variable X
that has a probability density function
with respect to Lebesgue measure, and f is any function
for which the expected value E(|f(X)|) is finite,
then, as is well-known to students of probability theory, 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.e., all of the probability is accounted for by
point-masses), and even if the cumulative distribution function g is continuous, it does not work if g fails to
be absolutely continuous (again, the Cantor function may serve as an example of this failure). But the
identity
-
holds if g is any cumulative probability distribution function on the real line, no matter how
ill-behaved.
See also
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