Zero-one law for symmetric convolution semigroups of measures on groups (Q1266768)

The authors suggest a new method of proving the zero-one laws for a convolution semigroup of probability measures on a nonabelian metric group. The main result is: Let \(G\) be a complete separable metric group, \((\mu_t)_{t>0}\) be a symmetric convolution semigroup of measures on \(G\) with the Lévy measure \(\nu\), \(H\) be a Borel subgroup of \(G\). If \(\nu (H^c) = \infty\), then \(\mu_t(H)=0\) for all \(t>0\). If \(\nu (H^c)=0\), then either \(\mu_t(H)=0\) for all \(t>0\), or \(\mu_t(H) = 1\) for all \(t>0\). If in addition \((\mu_t)_{t>0}\) is Gaussian, then either \(\mu_t(H) = 0 \) for all \(t>0\), or \(\mu_t(H) = 1 \) for all \(t>0\).