Excerpt from Measure-Theoretic Foundations of Statistical Mechanics Section 1, the mathematical theory, may be viewed as a small addition to measure theory and ergodic theory; in fact, most of the results may be considered corollaries of Birkhoff's ergodic theorem. In the beginning of that section, we use the theory of differentiation of set functions to arrive at a formula which is the natural generalization of the 'microcanon ical distribution'. This approach, which is very simple, is made possible by the introduction of the notion of a 'complete set of invariant functions' In order to treat a system in interaction with its environment we introduce a precise definition of 'weak interaction'; and in deriving the generalized form of the 'canonical distribution' we use Grad's definition of a 'large component' rather than Khinchin's approach based on the central limit theorem. In our development, the exponential law of the canonical distri bution appears as a natural consequence of an attempt to extend our results by the use of the equivalent measures. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.