Soft outer measure and soft premeasure

Abstract

Molodtsov introduced the concept of soft set as a new mathematical tool for dealing with problems treating uncertainties, and, later, Riaz at al.
developed various concepts of soft sets. Here, we study the concept of soft semiring and soft premeasure and explore their basic properties. We define a $[0,\infty]$--valued function on some soft set $\mathcal{E}$, that is not a soft $\sigma$--algebra, and then we extend this function to a measure on the soft $\sigma$--algebra generated by $\mathcal{E}$. We also introduce various set functions like soft content and soft premeasures, and give their basic properties.