On one problem concerned with the arithmetic of probability measures on spheres

Let $\mathscr P_\nu$ be a topological semigroup of sequences $t=\{t_n\}$ of the form (I) under pointwise multiplication and the topology of pointwise convergence. For $\nu=(n-2)/2$, $n=3,4,\dots$ the semigroup $\mathscr P_\nu$ is isomorphic to the convolution semigroup of probability measures on $\mathrm{SO}(n)$ bi-invariant under the action of $\mathrm{SO}(n-1$). Some sufficient conditions for an element $t\in\mathscr P_\nu$ be indecomposable are given. It is showed that the set of indecomposable elements of $\mathscr P_\nu$ is dense in $\mathscr P_\nu$. It is proved that the set of elements of $\mathscr P_\nu$ without indecomposable factors consists of the elements $v=\{P_n^\nu(0)\}$ and $W(c)=\{W_n\}$, $W_{2k}=1$, $W_{2k+1}=c$, $c\in[-1,1]$ ($k=0,1,2,\dots$). This is the solution of one problem posed by J. Lamperti in 1968.