A zero-one law for linear transformations of Levy noise

A L'evy noise on ${\mathbb{R}}^d$ assigns a random real “mass” $\mathrm{\Pi }(B)$ to each Borel subset $B$ of ${\mathbb{R}}^d$ with finite Lebesgue measure. The distribution of $\mathrm{\Pi }(B)$ only depends on the Lebesgue measure of $B$, and if $B_1,\mathrm{.}\mathrm{.}\mathrm{.},B_n$ is a finite collection of pairwise disjoint sets, then the random variables $\mathrm{\Pi }(B_1),\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\Pi }(B_n)$ are independent with $\Pi(B_1 \cup

… \cup B_n) = \Pi(B_1) + … + \Pi(B_n)$ almost surely. In particular, the distribution of $\mathrm{\Pi }\circ g$ is the same as that of $\mathrm{\Pi }$ when $g$ is a bijective transformation of ${\mathbb{R}}^d$ that preserves Lebesgue measure. It follows from the Hewitt–Savage zero–one law that any event which is almost surely invariant under the mappings $\mathrm{\Pi }\mapsto \mathrm{\Pi }\circ g$ for every Lebesgue measure preserving bijection $g$ of ${\mathbb{R}}^d$ must have probability 0 or 1. We investigate whether certain smaller groups of Lebesgue measure preserving bijections also possess this property. We show that if $d \ge 2$, the L'evy noise is not purely deterministic, and the group consists of linear transformations and is closed, then the invariant events all have probability 0 or 1 if and only if the group is not compact.