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The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.## References

Suppose that $(X,\; mathcal\{B\},\; mu)$ is a probability space, that $T\; :\; X\; to\; X$ is a (possibly noninvertible) measure-preserving transformation, and that $f\; in\; L^1(mu)$. Define $f^*$ by

- $f^*\; =\; sup\_\{N=1ldotsinfty\}\; frac1N\; sum\_\{i=0\}^\{N-1\}\; f\; circ\; T^i.$

- $int\_\{f^*\; >\; lambda\}\; f\; ,dmu\; ge\; lambda\; cdot\; mu\{\; f^*\; >\; lambda\}$

This theorem is used to prove the point-wise ergodic theorem.

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Last updated on Monday May 26, 2008 at 15:47:48 PDT (GMT -0700)

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Last updated on Monday May 26, 2008 at 15:47:48 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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