A. P. Morse’s Blankets

Abstract

We now consider some of A. P. Morse’s blankets (I. 3.3). Throughout this chapter R denotes a metric space metrized by δ. At times, R and δ will be specialized. The terms bounded, open, Borel, etc., will be used relative to δ. We denote by δ (A) the δ-diameter of an arbitrary set A ⊂ R. The spreads of all Morse’s blankets are families of bounded Borel sets. He also introduces a Carathéodory measure function (outer measure) φ, finite on bounded sets [cf. 44, 43]. We may regard our measure μ, defined on the Borel sets and finite on bounded Borel sets, to have been induced by such a function φ, we let μ^* denote, as usual, the completion of μ. Since μ^* agrees with μ on the bounded Borel subsets of R, we never need to refer explicitly to φ.