Measure, randomness and sublocales

Abstract

This paper investigates aspects of measure and randomness in the context of locale theory (point-free topology). We prove that every measure (σ-continuous valuation) μ, on the σ-frame of opens of a fitted σ-locale X, extends to a measure on the lattice of all σ-sublocales of X (Theorem 1). Furthermore, when μ is a finite measure with μ(X)=M, the σ-locale X has a smallest σ-sublocale of measure M (Theorem 2). In particular, when μ is a probability measure, X has a smallest σ-sublocale of measure 1. All σ prefixes can be dropped from these statements whenever X is a strongly Lindelöf locale, as is the case in the following applications. When μ is the Lebesgue measure on the Euclidean space Rn, Theorem 1 produces an isometry-invariant measure that, via the inclusion of the powerset P(Rn) in the lattice of sublocales, assigns a weight to every subset of Rn. (Contradiction is avoided because disjoint subsets need not be disjoint as sublocales.) When μ is the uniform probability measure on Cantor space {0,1}ω, the smallest measure-1 sublocale, given by Theorem 2, provides a canonical locale of random sequences, where randomness means that all probabilistic laws (measure-1 properties) are satisfied.