Overview
The Riemann Stieltjes integral is an extension of the Riemann integral.
Definition
The Riemann Stieltjes integral is an extension of the Riemann integral which gives meaning to the following integral
{% \int g(x) dF(x) %}
where {% F(x) %} is a
monotonically
increasing or decreasing function.
Similar to the procedure for computing a Riemann integral, the domain is subdivided into intervals, and then the following approximation is constructed.
{% \sum g(c_i) \times (F(x_{i+1}) - F(x_i)) %}
where {% c_i \in [x_i, x_{i+1}] %}
The integral is defined as the value of the limit of the above expression as {% \Delta x \rightarrow 0 %} . Under suitable conditions, the integral is equal to the following Riemann integral:
{% \int g(x) \frac{dF(x)}{dx} dx %}
Conditions
The following conditions are required for the Riemann Stieltjes integral to exist
- The Quadratic Variation of {% F %} is zero.
Topics
- Relationship to Riemann Integral: includes a discussion of the Dirac Delta Function