  Let $\mathcal{M}(X,\mathcal{A})$ be the ring of all real valued measurable
functions defined over the measurable space $(X,\mathcal{A})$. Given an ideal
$I$ in $\mathcal{M}(X,\mathcal{A})$ and a measure
$\mu:\mathcal{A}\to[0,\infty]$, we introduce the $U_\mu^I$-topology and the
$m_\mu^I$-topology on $\mathcal{M}(X,\mathcal{A})$ as generalized versions of
the topology of uniform convergence or the $U$-topology and the $m$-topology on
$\mathcal{M}(X,\mathcal{A})$ respectively. With $I=\mathcal{M}(X,\mathcal{A})$,
these two topologies reduce to the $U_\mu$-topology and the $m_\mu$-topology on
$\mathcal{M}(X,\mathcal{A})$ respectively, already considered before. If $I$ is
a countably generated ideal in $\mathcal{M}(X,\mathcal{A})$, then the
$U_\mu^I$-topology and the $m_\mu^I$-topology coincide if and only if
$X\setminus \bigcap Z[I]$ is a $\mu$-bounded subset of $X$. The components of
$0$ in $\mathcal{M}(X,\mathcal{A})$ in the $U_\mu^I$-topology and the
$m_\mu^I$-topology are realized as $I\cap L^\infty(X,\mathcal{A},\mu)$ and
$I\cap L_\psi(X,\mathcal{A},\mu)$ respectively. Here
$L^\infty(X,\mathcal{A},\mu)$ is the set of all functions in
$\mathcal{M}(X,\mathcal{A})$ which are essentially $\mu$-bounded over $X$ and
$L_\psi(X,\mathcal{A},\mu)=\{f\in \mathcal{M}(X,\mathcal{A}): ~\forall
g\in\mathcal{M}(X,\mathcal{A}), f.g\in L^\infty(X,\mathcal{A},\mu)\}$. It is
established that an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ is dense in the
$U_\mu$-topology if and only if it is dense in the $m_\mu$-topology and this
happens when and only when there exists $Z\in Z[I]$ such that $\mu(Z)=0$.
Furthermore, it is proved that $I$ is closed in $\mathcal{M}(X,\mathcal{A})$ in
the $m_\mu$-topology if and only if it is a $Z_\mu$-ideal in the sense that if
$f\equiv g$ almost everywhere on $X$ with $f\in I$ and
$g\in\mathcal{M}(X,\mathcal{A})$, then $g\in I$.
