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The theory of probability has made much progress recently in the direction of completely mathematical formulations of its methods and results.J The purpose of this paper is to make a further contribution in this direction.In order to analyze the results of repeated trials of an experiment, a certain space of infinitely many dimensions is the proper tool.This space is discussed in the first section of the paper.In the second section, the results of the first are applied to obtain for the first time a complete proof of the validity of the method of maximum likelihood of R. A. Fisher, which is used in statistics to estimate the true probability distribution when the results of a repeated (1934), pp.376-379.
J. L. Doob (Mon,) studied this question.