Set Functions, Finite Additivity and Joint Distribution Function Representations

Abstract

Suppose that $F$ is a field of subsets of a set $U$, $N$ is a positive integer, ${\left\{{\alpha }_k\right\}}_{k=1}^N$ is a sequence of functions from $F$ into $\exp (\mathbf{R})$ and $\mu$ is a real, nonnegative - valued finitely additive function on $F$. Suppose that ${\mathrm{\Xi }}^N$ is the set of all $N$ - dimensional subintervals of ${\mathbf{R}}^N$. It is shown that there is a nonnegative - valued function A from ${\mathrm{\Xi }}^N\times F$ into $\mathbf{R}$ such that for each $V$ in $F$, $A(\cdot ,V)$ is finitely additive on ${\mathrm{\Xi }}^N$, such that if for $k=1,\dots ,N,{\alpha }_k$ is $\mu$ - summable (see “Fields of Sets, Set Functions, Set Function Integrals, and Finite Additivity”, Internat. J. Math. & Math. Sci., Vol. 7 No. 2 (1984) pp 209 - 233) and $g$ is a real - valued function on ${\mathbf{R}}^N$ satisfying certain continuity and boundedness conditions, then

$${\displaystyle \int \dots }{\displaystyle {\int }_R[g{\displaystyle {\int }_{U}A(\cdot ,\cdot )]\to {\sigma }_{\mu }(g({\alpha }_1,\dots ,{\alpha }_N))(U),}}$$

$$R=[H_1;K_1]\times \cdots \times [H_N;K_N],\min \{-H_1,\dots ,-H_N,K_1,\dots ,K_N\}\to \infty ,$$

where ${\sigma }_{\mu }$ is the $\mu$-summability operator and all integrals are refinement - wise limits of the appropriate sums.