Suppose Ωis a finite set equipped with the algebra of all subsets. Let μand νbe two measures on

step 1 Suppose Omega is a finite set, then we know the cardinality of Omega, which is denoted as the ab

Suppose Ωis a finite set equipped with the algebra of all...

Suppose $\Omega$ is a finite set equipped with the algebra of all subsets. Let $\mu$ and $\nu$ be two measures on $\Omega$ such that $\mu(\{\omega\}) \neq 0, \nu(\{\omega\}) \neq 0$, for all $\omega \in \Omega$. Decide under which conditions $\mu$ dominates $\nu$ and find $\frac{d \nu}{d \mu}$.

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Key Concepts

Measures on Finite Sets

On a finite set with all subsets measurable, measures can be fully described by their values at the individual points. If every singleton has a nonzero measure, then any subset's measure is just the sum over its elements. This greatly simplifies concepts such as absolute continuity and the Radon-Nikodym derivative because the measures are determined by their values on singletons.

Absolute Continuity

Absolute continuity of one measure with respect to another means that whenever the dominating measure assigns a value of zero to a measurable set, the other measure also assigns zero to that set. In finite spaces, this is equivalent to saying that the support (the set of points which have nonzero measure) of the second measure is contained within the support of the first measure.

Radon-Nikodym Theorem

The Radon-Nikodym theorem provides conditions under which we can express one measure as an integral with respect to another measure. Specifically, if one measure is absolutely continuous with respect to another, there exists a function, called the Radon-Nikodym derivative, that serves as the density of the first measure with respect to the second. In a finite space, this derivative can be computed explicitly for each point.

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