Exponential valuations on lattice polygons

We classify translatively exponential and $\mathrm {GL}(2,\mathbb Z)$ covariant valuations on lattice polygons with values in the space of real (complex) measurable 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 $\mathrm {SL}(2,\mathbb Z)$ on $\mathbb R^2$ , and some elementary properties of the Fibonacci numbers.