We consider the locally measure topology $t(\mathcal{M})$ on the *-algebra
$LS(\mathcal{M})$ of all locally measurable operators affiliated with a von
Neumann algebra $\mathcal{M}$. We prove that $t(\mathcal{M})$ coincides with
the $(o)$-topology on $LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^*=T\}$ if
and only if the algebra $\mathcal{M}$ is $\sigma$-finite and a finite algebra.
We study relationships between the topology $t(\mathcal{M})$ and various
topologies generated by faithful normal semifinite traces on $\mathcal{M}$.