  For a measurable space ($X,\mathcal{A}$), let $\mathcal{M}(X,\mathcal{A})$ be
the corresponding ring of all real valued measurable functions and let $\mu$ be
a measure on ($X,\mathcal{A}$). In this paper, we generalize the so-called
$m_{\mu}$ and $U_{\mu}$ topologies on $\mathcal{M}(X,\mathcal{A})$ via an ideal
$I$ in the ring $\mathcal{M}(X,\mathcal{A})$. The generalized versions will be
referred to as the $m_{\mu_{I}}$ and $U_{\mu_{I}}$ topology, respectively,
throughout the paper. $L_{I}^{\infty} \left(\mu\right)$ stands for the subring
of $\mathcal{M}(X,\mathcal{A})$ consisting of all functions that are
essentially $I$-bounded (over the measure space ($X,\mathcal{A}, \mu$)). Also
let $I_{\mu} (X,\mathcal{A}) = \big \{ f \in \mathcal{M}(X,\mathcal{A}) : \,
\text{for every} \, g \in \mathcal{M}(X,\mathcal{A}), fg \, \, \text{is
essentially} \, I$-$\text{bounded} \big \}$. Then $I_{\mu} (X,\mathcal{A})$ is
an ideal in $\mathcal{M}(X,\mathcal{A})$ containing $I$ and contained in
$L_{I}^{\infty} \left(\mu\right)$. It is also shown that $I_{\mu}
(X,\mathcal{A})$ and $L_{I}^{\infty} \left(\mu\right)$ are the components of
$0$ in the spaces $m_{\mu_{I}}$ and $U_{\mu_{I}}$, respectively. Additionally,
we obtain a chain of necessary and sufficient conditions as to when these two
topologies coincide.

Functional Analysis