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$$\lim_{n \to \infty}\frac1n \sum_{i=0}^{n-1} \varphi \circ f^i(x) = \mathbb{E}_{\mu}(\varphi \mid \mathcal{I})(x) \quad\text{a.e.}$$

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A proof of Pointwise Birkhoff Ergodic Theorem in Lean

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