Shannon-McMillan-Breiman theorem

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.