Kolmogorov's Extension Theorem for Infinite Measures

Abstract

We consider the extension problem of a self-consistent family of infinite measures to a completely additive measure. For probability measures, Kolmogorov's extension theorem assures that the extension is uniquely possible. Our results are as follows:
(a) For σ-finite measures, we can reduce the problem to the case of probability measures, so that the extension is uniquely possible. As an application, on an infinite dimensional vector space we can construct such a measure that is invariant both under rotations and homotheties with respect to the origin. It is obtained as the limit of n-dimensional measure:
\frac{1}{|x|ⁿ}dx₁dx₂...dx_n
Also we shall discuss about the Lorentz invariant measure on an infinite dimensional space.
(b) If measures are not σ-finite, under the additional condition (EC) in §6, the extension is possible but not unique. We shall mention about the largest and the smallest extension. As an application, we can consider the symbolic representation of a flow {T_t} defined on an infinite measure space X, namely constructing an appropriate product space W^R and an appropriate measure on W^R, T_t on X is represented by a shift S_t on W^R: w(·)→ w(·+t).
§1  Kolmogorov's extension theorem
§2  Reduction to finite measures
§3  Rotationally invariant measure
§4  (0, ∞)-type measures
§5  Lorentz invariant measure
§6  Non σ-finite case
§7  σ-finite plus essentially infinite case
§8  Symbolic representation of flows