  Let $\mathcal{A}$ be a field of subsets of a set $X$ and
$\EuScript{M}(X,\mathcal{A})$ be the ring of all real valued
$\mathcal{A}$-measurable functions on $X$. It is shown that
$\EuScript{M}(X,\mathcal{A})$ is self-injective if and only if $\mathcal{A}$ is
a complete and $\mathfrak{c}^+$- additive field of sets. This answers a
question raised in [H. Azadi, M. Henriksen and E. Momtahan, \textit{Some
properties of algebras of real valued measurable functions}, Acta Math. Hungar,
124, (2009), 15--23]. Also, it is observed that if $\mathcal{A}$ is a
$\sigma$-field, $\EuScript{M}(X,\mathcal{A})$ modulo its socle is
self-injective if and only if $\mathcal{A}$ is a complete and $\mathfrak{c}^+$-
additive field of sets with a finite number of atoms.
