A Remark on Nonatomic Measures

Abstract

In this paper the following result is proved: THEOREM. If $\lambda$ is a measure on a product $\sigma$-algebra $\mathfrak{U} \times \mathscr{B}$ and $\lambda_1$ and $\lambda_2$ are the corresponding marginals on $\mathfrak{U}$ and $\mathscr{B}$ respectively, then $\lambda$ is nonatomic $\operatorname{iff}$ at least one of $\lambda_1$ and $\lambda_2$ is nonatomic.