Typicality and the Role of the Lebesgue Measure in Statistical Mechanics

Consider a …nite collection of marbles. The statement "half the marbles are white" is about counting, and not about the probability of drawing a white marble from the collection. The question is whether nonprobabilistic counting notions such as half, or vast majority can make sense, and preserve their meaning when extended to the realm of the continuum. In this paper we argue that the Lebesgue measure provides the proper non-probabilistic extension, which is as natural, and in a sense uniquely forced, as the extension of the concept of cardinal number to in…nite sets by Cantor. To accomplish this a di¤erent way of constructing the Lebesgue