Exponential valuations on lattice polygons

We classify translatively exponential and GL(2,Z) covariant valuations on lattice polygons valued at measurable real functions. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The argument uses the ergodicity of the linear action of SL(2,Z) on R2, and some elementary properties of the Fibonacci numbers.