Generalized Distribution Functions: The $\sigma$-Lower Finite Case

Abstract

A mass $m(x) \geqq 0$ is assigned to each point $x$ of a partially ordered countable set $X$. It is further assumed that $M(x) = \sum_{y\leqq x} m(y) < \infty$ for each $x \in X. M$ is called a distribution function. For certain sets $X$, it is shown that $M$ determines $m$. For others, $M$ need not determine $m$ uniquely. A theory is presented for $\sigma$-lower finite spaces (sets), which are defined in the paper. Such spaces are locally finite. That is, each interval $\lbrack x, y \rbrack = \{z \in X: x \leqq z \leqq y\}$ has a finite number of points. Mobius functions, which have been defined for locally finite spaces, are used throughout. Distribution functions on a particular $\sigma$-lower finite space arise naturally from boundary crossing problems analyzed by Doob and Anderson. The theory is applied to this example and to another.