Elliott-Morse measures and Kakutani's dichotomy theorem
We first extend Lebesgue's decomposition theorem for measures: If \(\mu\) is a \(\sigma\)-finite measure and \(\nu\) is an arbitrary measure on a measurable space \((X,{\mathcal A})\), then there exists a unique decomposition of \(\mu\) w.r.t. \(\nu\) into the sum of a singular part, a part with a density function and a diffuse part in a sense especially defined. We then study this decomposition in the case of two product measures \(\mu=\otimes_{k\in\mathbb{N}}\mu_ k\) and \(\eta=\otimes_{k\in\mathbb{N}}\eta_ k\), where \(\mu_ k\) is a probability measure and \(\eta_ k\) is an arbitrary measure \(\geq 0\) on a measurable space \((X_ k,{\mathcal A}_ k)\) such that each \(\mu_ k\) has a density function with respect to \(\eta_ k\). Here the infinite product \(\otimes_{k\in\mathbb{N}}\eta_ k\) is defined in the sense of \textit{E. O. Elliott} and \textit{A. P. Morse} [Trans. Am. Math. Soc. 110, 245-283 (1964; Zbl 0125.303)]. It turns out that, similarly as in Kakutani's classical theorem [\textit{S. Kakutani}, Ann. Math., II. Ser. 49, 214-224 (1948; Zbl 0030.02303)], \(\mu\) is of pure type with respect to \(\eta\), i.e., \(\mu\) and \(\eta\) are either mutually singular, or \(\mu\) has a density function with respect to \(\eta\), or else \(\mu\) is diffuse w.r.t. \(\eta\). We finally use these results to classify the infinite products \(\otimes_{k\in\mathbb{N}}\eta_ k\) according to their finiteness, \(\sigma\)- finiteness, and non-\(\sigma\)-finiteness.
scientific article; zbMATH DE number 108922
scientific article