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Measure, randomness and sublocales
Simpson, Alex
This paper investigates aspects of measure and randomness in the context of locale theory (point-free topology). We prove that every measure (▫$\sigma $▫-continuous valuation) ▫$\mu$▫ on the ...▫$\sigma$▫-frame of opens of a fitted ▫$\sigma$▫-locale ▫$X$▫ extends to a measure on the lattice of all ▫$\sigma$▫-sublocales of ▫$X$▫ (Theorem 1). Furthermore, when ▫$\mu$▫ is a finite measure with ▫$\mu (X) = M$▫, the ▫$\sigma$▫-locale ▫$X$▫ has a smallest ▫$\sigma$▫-sublocale of measure ▫$M$▫ (Theorem 2). In particular, when ▫$\mu$▫ is a probability measure, ▫$X$▫ has a smallest ▫$\sigma$▫-sublocale of measure 1. All ▫$\sigma$▫ prefixes can be dropped from these statements whenever ▫$X$▫ is a strongly Lindelöf locale, as is the case in the following applications. When ▫$\mu$▫ is the Lebesgue measure on the Euclidean space ▫$\mathbb{R}^{n}$▫, Theorem 1 produces an isometry-invariant measure that, via the inclusion of the powerset ▫$\mathcal{P}(\mathbb{R}^{n})$▫ in the lattice of sublocales, assigns a weight to every subset of ▫$\mathbb{R}^{n}$▫. (Contradiction is avoided because disjoint subsets need not be disjoint as sublocales.) When▫ $\mu$▫ is the uniform probability measure on Cantor space ▫$\{0,1\}^{\omega }$▫, 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.
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