  Given a von Neumann algebra $M$ we introduce so called central extension
$mix(M)$ of $M$. We show that
  $mix(M)$ is a *-subalgebra in the algebra $LS(M)$ of all locally measurable
operators with respect to $M,$ and this algebra coincides with $LS(M)$ if and
only if $M $ does not admit type II direct summands. We prove that if $M$ is a
properly infinite von Neumann algebra then every additive derivation on the
algebra $mix(M)$ is inner. This implies that on the algebra $LS(M)$, where $M$
is a type I$_\infty$ or a type III von Neumann algebra, all additive
derivations are inner derivations.
