Definition
There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. In research papers and advanced books signed measures are usually only allowed to take finite values, while undergraduate textbooks often allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures".
Given a measurable space (X, Σ), that is, a set X with a sigma algebra Σ on it, an extended signed measure is a function
such that and is sigma additive, that is, it satisfies the equality
for any sequence A1, A2, ..., An, ... of disjoint sets in Σ. One consequence is that any extended signed measure can take +∞ as value, or it can take −∞ as value, but both are not available. The expression ∞ − ∞ is undefined and must be avoided.
A finite signed measure is defined in the same way, except that it is only allowed to take real values. That is, it cannot take +∞ or −∞.
Finite signed measures form a vector space, while extended signed measures are not even closed under addition, which makes them rather hard to work with. On the other hand, measures are extended signed measures, but are not in general finite signed measures.
Read more about this topic: Signed Measure
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