Let $\mu$ be a measure on the $\sigma$ -algebra $\mathfrak B$ of Borel sets of a separable Hilbert space $X$ . An element $a\in X$ is called an admissible translation of $\mu$ if $\mu_a\ll\mu$ where $\mu_a$ is the measure obtained from $\mu$ under transformation of space $X\colon S_ax=x+a$ . In the paper, the set $M_\mu$ of admissible translations of $\mu$ and the form of the density $d\mu_a/d\mu$ are investigated. The class$\mathfrak M$of measures for which$M_\mu$contains the linear manifold dense in$X$is studied.$\mathfrak M$is shown to be a convex set. The set$\mathfrak K$of extreme points of$\mathfrak M$is found and it is proved that all the measures from$\mathfrak M$are mixtures of those from$\mathfrak K$.