Entropy is the only finitely observable invariant

Our main purpose is to present a surprising new characterization of the Shannon entropy of stationary ergodic processes. We will use two basic concepts: isomorphism of stationary processes and a notion of finite observability, and we will see how one is led, inevitably, to Shannon's entropy. A function $J$ with values in some metric space, defined on all finite-valued, stationary, ergodic processes is said to be finitely observable (FO) if there is a sequence of functions $S_{n}(x_{1},x_{2},...,x_{n})$ that for all processes $\mathcal{X}$ converges to $J(\mathcal{X})$ for almost every realization $x_{1}^{\infty}$ of $\mathcal{X}$. It is called an invariant if it returns the same value for isomorphic processes. We show that any finitely observable invariant is necessarily a continuous function of the entropy. Several extensions of this result will also be given.