Signed Measures

Why

Can we view the set of measures as a vector space?

Not quite: the difference of two measures may take negative values on some set. This functional will be countably additive, however, and so behaves similar to a measure.

Definition

Suppose $\mathcal{F} $ is a sigma algebra. Then $f: \mathcal{F} \to \Rbar$ is countably additive if the result of the function applied to the union of a disjoint countable family of distinguished sets is the limit of the partial sums of the results of the function applied to each of the sets individually. The limit of the partial sums must exist irregardless of the summand order.

A signed measure is an extended-real-valued function on a sigma algebra that is (1) zero on the empty set and (2) countably additive. We call the result of the function applied to a set in the sigma algebra the signed measure (or when no ambiguity arises, the measure) of the set.

When speaking of a measure, which is non-negative, in contrast to a signed measure, we will call the former a nonnegative measure (or positive measure).

Basic properties

A signed measure never takes both positive infinity and negative infinity.

Let $(X, \mathcal{A} )$ be a measurable space. Let $\mu : \mathcal{A} \to \eri$ be a signed measure. First, suppose $\mu (X)$ is finite, Then by Proposition~\ref{prop:finitesignedmeasures} $\mu $ is finite for each $A \in \mathcal{A} $.

Suppose $\mu (X) = \infty$. Let $A \in \mathcal{A} $. As before, $\mu (X) = \mu (A) + \mu (X - A)$. Since $\mu (X) = +\infty$, then both of $\mu (A)$ and $\mu (X-A)$ must be either finite or $+\infty$. Argue similarly for $\mu (X) = -\infty$.