Radon Nikodym Theorem
Overview
Absolute Continuity
Given two measures {% \mu %} and {% \nu %} on a measure space ({% S %}, {% \sum %}), then the
measure {% \nu %} is absolutely continuous with respect to {% \mu %}, written {% \nu < < \mu %}
if {% \mu(A) = 0 %} implies that {% \nu(A) = 0 %}
Statement
Given tow {% \sigma - finite %} measures on the same measure space ({% S %}, {% \sum %}) with {% \nu << \mu %},
then there exists a non-negative measurable function {% f : S \rightarrow [0,\infty] %} such that
{% \nu(A) = \int_A f d \mu %}
Relationship to Riemann Stieltjes
The Radon Nikodym theorem can be seen when constructing a
Riemann Stieltjes
measure.
{% \int dF(x) = \int f(x) dx %}
