  This study investigates the natural or intrinsic measure of a symbolic
dynamical system $\Sigma$. The measure $\mu([i_{1},i_{2},...,i_{n}])$ of a
pattern $[i_{1},i_{2},...,i_{n}]$ in $\Sigma$ is an asymptotic ratio of
$[i_{1},i_{2},...,i_{n}]$, which arises in all patterns of length $n$ within
very long patterns, such that in a typical long pattern, the pattern
$[i_{1},i_{2},...,i_{n}]$ appears with frequency
$\mu([i_{1},i_{2},...,i_{n}])$. When $\Sigma=\Sigma(A)$ is a shift of finite
type and $A$ is an irreducible $N\times N$ non-negative matrix, the measure
$\mu$ is the Parry measure. $\mu$ is ergodic with maximum entropy. The result
holds for sofic shift $\mathcal{G}=(G,\mathcal{L})$, which is irreducible. The
result can be extended to $\Sigma(A)$, where $A$ is a countably infinite matrix
that is irreducible, aperiodic and positive recurrent. By using the Krieger
cover, the natural measure of a general shift space is studied in the way of a
countably infinite state of sofic shift, including context free shift. The
Perron-Frobenius Theorem for non-negative matrices plays an essential role in
this study.
