An uncertainty principle for Möbius inversion on posets

We give conditions for a locally finite poset $P$ to have the property that for any functions $f:P\to \mathbf{C}$ and $g:P\to \mathbf{C}$ not identically zero and linked by the Möbius inversion formula, the support of at least one of $f$ and $g$ is infinite. This generalises and gives an entirely poset-theoretic proof of a result of Pollack. Various examples and non-examples are discussed.