Motivated by the fact that σ-locales generalize measurable spaces and seeking to overcome some restrictions of measure theory, Alex Simpson [1] proposed an approach to measure theory in the framework of point-free topology. This talk will be a general survey of that approach while focusing on its motivations and advantages. Firstly, we will revise some basic concepts and general results of σ-locales, and present an equivalence between the category of sober measurable spaces (that contains most of the measurable spaces of relevance in measure theory) and the category of spatial Boolean σ-locales. Then, we will generalize the standard definition of measure to an arbitrary lattice with countable suprema; and given a measure μ on a σ-locale X, we will extend it to the co-frame of σ-sublocales of X, S(X), showing that under the condition of X being a fit σ-sublocale, μ is indeed a measure on S(X). In particular, we can extend the n-dimensional Lebesgue measure to a measure that not only assigns a value to each subset of ℝⁿ but also is invariant under the Euclidean group of isometries. Finally, if time permits, we will also approach the question "What are random sequences?", defining the space of random sequences and proposing a suitable candidate for a model of the phenomenon of randomness.

References: [1] A. Simpson, Measure, randomness and sublocales, Ann. Pure Appl. Logic 163 (2012), 1642-1659.