ABSTRACT
Theorem 2.14 in Chapter 8 shows that there exists no σ-additive translation invariant measure μ on the σ-algebra of all subsets of R such that μ([a, b]) = b—a for every interval [a, b) in R. This raises the question whether there exists any σ-additive measure on P{R), or for that matter on P(S) for any infinite set S. Of course, the counting measure, which allows the value μ(S) = ∞, is a trivial example, so we formulate the problem differently. We allow a measure to have only finite values, and we further require (without loss of generality) that μ(S) = 1.
