This paper investigates aspects of measure and randomness in the context of locale theory . We prove that every measure 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 . Furthermore, when μ is a finite measure with μ=M, the σ-locale X has a smallest σ-sublocale of measure M . 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 in the lattice of sublocales, assigns a weight to every subset of Rn. 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 are satisfied. (shrink)