Atoms and a Saks type decomposition in effect algebras

Mona Khare, Akhilesh Kumar Singh

The present paper deals with the study of the notion of an atom of a function $m$ defined on an effect algebra $L$ with values in $[0,\infty]$; a few examples of atoms for null-additive as well as for non-null-additive functions are also given. We have proved a Saks type decomposition theorem for an element a with $m(a)>0$ (for a suitable $m$), which does not contain any atom of $m$, in a $\sigma$-complete effect algebra $L$. A characterization for a measure $\mu$, to be non-atomic ($\mu$ is defined on a $\sigma$-complete effect algebra with values in $[0,\infty]$) is established and a result for a non-atomic measure $\mu$ is proved, which has resemblance with the Intermediate Value Theorem for continuous functions.