  A ring $S(X,\mathcal{A})$ of real valued $\mathcal{A}$-measurable functions
defined over a measurable space $(X,\mathcal{A})$ is called a $\chi$-ring if
for each $E\in \mathcal{A} $, the characteristic function $\chi_{E}\in
S(X,\mathcal{A})$. The set $\mathcal{U}_X$ of all $\mathcal{A}$-ultrafilters on
$X$ with the Stone topology $\tau$ is seen to be homeomorphic to an appropriate
quotient space of the set $\mathcal{M}_X$ of all maximal ideals in
$S(X,\mathcal{A})$ equipped with the hull-kernel topology $\tau_S$. It is
realized that $(\mathcal{U}_X,\tau)$ is homeomorphic to
$(\mathcal{M}_S,\tau_S)$ if and only if $S(X,\mathcal{A})$ is a Gelfand ring.
It is further observed that $S(X,\mathcal{A})$ is a Von-Neumann regular ring if
and only if each ideal in this ring is a $\mathcal{Z}_S$-ideal and
$S(X,\mathcal{A})$ is Gelfand when and only when every maximal ideal in it is a
$\mathcal{Z}_S$-ideal. A pair of topologies $u_\mu$-topology and
$m_\mu$-topology, are introduced on the set $S(X,\mathcal{A})$ and a few
properties are studied.
