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A L\'evy noise on $\mathbb{R}^d$ assigns a random real "mass" $\Pi(B)$ toeach Borel subset $B$ of $\mathbb{R}^d$ with finite Lebesgue measure. Thedistribution of $\Pi(B)$ only depends on the Lebesgue measure of $B$, and if$B_1, ..., B_n$ is a finite collection of pairwise disjoint sets, then therandom variables $\Pi(B_1), ..., \Pi(B_n)$ are independent with $\Pi(B_1 \cup>... \cup B_n) = \Pi(B_1) + ... + \Pi(B_n)$ almost surely. In particular, thedistribution of $\Pi \circ g$ is the same as that of $\Pi$ when $g$ is abijective transformation of $\mathbb{R}^d$ that preserves Lebesgue measure. Itfollows from the Hewitt--Savage zero--one law that any event which is almostsurely invariant under the mappings $\Pi \mapsto \Pi \circ g$ for everyLebesgue measure preserving bijection $g$ of $\mathbb{R}^d$ must haveprobability 0 or 1. We investigate whether certain smaller groups of Lebesguemeasure 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 oflinear transformations and is closed, then the invariant events all haveprobability 0 or 1 if and only if the group is not compact.

A zero-one law for linear transformations of Levy noise