ABSTRACT
Given any sigma-algebra A of subsets of K we shall define ,4-
measurability of functions in the usual way. In our applications A will be either Λ4σ or Λ4 which were introduced in §4.1.
For / a function on K let ( / > c) = / _1(c, oo) for each c in E , ( / > c) = / _ 1[c,oo),(/ < c) = / _1( -o o ,c ) ,( / < c) = / - 1(—oo, c], and ( / = c) = / - 1(c). / is ,4-m easu rab le if ( / > c) belongs to A for every c in E. For an equivalent definition ( / > c) may be replaced by any of the sets ( / > c), ( / < c), or ( / < c). If / is ^-measurable then f ~ 1(D) belongs to A for D any interval, open set, or closed set in E.
