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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.