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