Nonmetric compact spaces and nonmeasurable processes

Abstract

Measure theory in compact Hausdorff spaces has some pathological aspects. Some natural functions are nonmeasurable. Others are measurable but have nonmeasurable ranges. Joint measurability is not preserved by the Kakutani-Nelson canonical representation of functions as elements of product spaces, even "almost", for the indicator function of the triangle below the diagonal of the unit square (empirical distribution function for sample size n = 1 n = 1 ).