= Shannon-McMillan-Breiman theorem
{c}
For a probability <measure-preserving system> and a countable <measurable partition> $\xi$ of finite entropy, $-n^{-1}\log\mu(\xi_0^{n-1}(x))$ converges <almost everywhere> and in $L^1$ to an invariant function whose integral is $h_\mu(T,\xi)$. For an <ergodic transformation>, the limit is the constant $h_\mu(T,\xi)$. In general it is $\mathbb E[I_\mu(\xi\mid\bigvee_{j=1}^\infty T^{-j}\xi)\mid\mathcal I]$. The <Martingale convergence theorem>, <maximal inequality for conditional information functions>, and <triangular ergodic averaging lemma> prove this directly, without invertibility.
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