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How did Norbert Wiener use measure theory in the mathematical study of Brownian motion?

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In the mathematical study of Brownian motion, Norbert Wiener utilized measure theory to provide the first mathematically rigorous treatment of the process in 1923.1

By applying measure-theoretic concepts to the space of all continuous functions on an interval, Wiener demonstrated that the set of differentiable functions within that space has a measure of zero. In probabilistic terms, this implies that a function chosen at random is almost certainly nondifferentiable.2

This discovery was pivotal for understanding the physical phenomenon of Brownian motion, as it proved that a particle undergoing such motion follows a path that is continuous but nowhere differentiable. Consequently, the particle lacks a well-defined velocity or tangent at any point in time, a finding that clarified the fundamental ideas originally proposed by Albert Einstein regarding the irregular motion of particles buffeted by molecules.12