Now that I am dealing with Ergodic Theory in my spare time, I found a bit interesting application in an old book of Optics from the first half of the past century. Max Born wrote that there are two kind of ensembles in optics: stationary and ergodic. In fact, when we deal with interference and diffraction with partially coherent light, we have a relation for the signal disturbance as a function of the analytic signal given by
This sort of averages and the existence of this integral for all bounded operator is the main problem finding counterexamples of mean ergodic spaces.
Actually, I am reading this book trying to have more ideas about the mathematical context when the first problems in Ergodic Theory were posted. I already knew that Boltzmann‘s Statistical Mechanics had considerations of dynamical systems where the macroscopical equipment only allowed to get data in the form of the above integral. It’s really interesting to find ergodic spaces in Optics and other branches of Physics.