Anderson inequality is strict for Gaussian and stable measures

Abstract:

Let $\mu$ be a symmetric Gaussian measure on a separable Banach space $(E,\left \| \bullet \right \|)$. Denote $U = \{ x:\left \| x \right \| < 1\}$. Then for every $x \in {\text {supp}}\;\mu ,x \ne 0$, the function $t \to \mu (U + tx)$ is strictly decreasing for $t \in (0,\infty )$. The same property holds for symmetric p-stable measures on E. Using this property we answer a question of W. Linde : if $\smallint _{U + z} {xd\mu (x) = 0}$, then $z = 0$.