A Solution of Ulam's Conjecture on the Existence of Invariant Measures and its Applications

Abstract

This chapter presents a solution of Ulam's conjecture on the existence of invariant measures and its applications. It describes the Kryloff–Bogoliuboff theorem. For transformations τ: X → X continuous on a topological space X, the problem of the existence of an invariant measure was solved in 1937 by Kryloff and Bogoliuboff. According to their theorem, such a measure defined on a σ-algebra of Borel sets exists when the topological space X is compact. The Kryloff–Bogoliuboff theorem becomes banal if transformation τ has a fixed point. The invariant measure can be concentrated only on this point. Therefore, there arises a natural problem of seeking some more interesting measures. It is advantageous to find a measure absolutely continuous with respect to the Lebesgue measure. The support of such a measure has a positive Lebesgue measure. An absolutely continuous measure also has the additional advantage that every bounded measurable function is integrable with respect to this measure; therefore, it is possible to apply the ergodic theorem to a rather broad class of functions. The problem of constructing the invariant measure is, in fact, the problem of seeking approximate solutions. An example connected with the technology of foundry casting is also presented.