On admissible translations of measures in Hilbert space

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 $\mathfrak K$.