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

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... ) that for all processes X converges to J (X ) for almost every realization x ∞ 1 of 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.