Nonaccessible filters in measure algebras and functionals on ${\displaystyle L^{\infty }(\lambda )\cdot }$

In a nonatomic measure algebra, we construct a nonprincipal filter with the inaccessibility property considered by Kunen [7]. Using that filter we define two "pathological" functionals on ${\displaystyle L^{\infty }{\left(\lambda \right)}^{\ast }}$. It follows that the Banach space ${\displaystyle L^{\infty }\left(\lambda \right)}$ is not realcompact whenever the measure ${\displaystyle \lambda }$ is not separable.