  We prove that if $\mathcal M$ is a properly infinite von Neumann algebra and
$LS(\mathcal M)$ is the local measurable operator algebra affiliated with
$\mathcal M$, then every Jordan derivation from $LS(\mathcal M)$ into itself is
continuous with respect to the local measure topology $t(\mathcal M)$. We
construct an extension of a Jordan derivation from $\mathcal M$ into
$LS(\mathcal M)$ up to a Jordan derivation from $LS(\mathcal M)$ into itself.
Moreover, we prove that if $\mathcal M$ is a properly von Neumann algebra and
$\mathcal A$ is a subalgebra of $LS(\mathcal M)$ such that $\mathcal
M\subset\mathcal A$, then every Jordan derivation from $\mathcal A$ into
$LS(\mathcal M)$ is continuous with respect to the local measure topology
$t(\mathcal M)$.
