Abstract
A set function is a real function defined on a collection of subsets of an underlying measurable space. In this chapter we consider set functions that have some of the properties ascribed to area. The main property is additivity. The area of two regions that do not overlap is the sum of their areas. A charge is any nonnegative set function that is additive in this sense. A measure is a charge that is countably additive. That is, the area of a sequence of disjoint regions is the infinite series of their areas. A probability measure is a measure that assigns measure one to the entire set. Charges and measures are intimately entwined with integration, which we take up in Chapter 9. But here we study them in their own right.
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References
The Ultrafilter Theorem 2.16 implies that for any infinite set there is a probability charge that assigns mass zero to each point. Every free ultrafilter defines a charge assuming only the values zero and one; see Lemma 13.37.
An interesting question is whether there are any measurable cardinals at all. The question is still open, but it is known that if there are any measurable cardinals, they must be so large that you will never encounter them; see T. Jech [124, Chapter 5]. Surprisingly there are some straightforward questions regarding probability measures on metrics spaces whose answers depend on the existence of measurable cardinals. For instance, the question of whether the support of a Borel probability measure is separable is such a question; see P. Billingsley [33].
Even restricting attention to charges does not enable us to measure all the subsets of ℝ3 in a way that is invariant under both translation and rotation. This observation is due to F. Hausdorff [109]. The famous Banach-Tarski Paradox [25] (see page 12) is a refinement of his work.
The σ in these definitions stands for (infinite) sequences.
It may seem more natural to call any signed charge a measure and then specialize to say countably additive measures or positive measures. In fact, many authors refer to charges as finitely additive measures. The terminology we use has the virtue of brevity. Not every author follows these conventions, so beware.
Some authors use the term totally finite to indicate a finite measure defined on an algebra, rather than just on a semiring of sets.
This is at odds with the standard Cartesian product notation. However, this is the notation used by most authors and we retain it. You should not have any problem understanding its meaning from the context of the discussion.
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© 1994 Springer-Verlag Berlin Heidelberg
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Aliprantis, C.D., Border, K.C. (1994). Charges and measures. In: Infinite Dimensional Analysis. Studies in Economic Theory, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03004-2_8
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DOI: https://doi.org/10.1007/978-3-662-03004-2_8
Publisher Name: Springer, Berlin, Heidelberg
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