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

Suppose that {\displaystyle } is a probability space, that T : X → X {\displaystyle T:X\to X} is a measure-preserving transformation, and that f ∈ L 1 {\displaystyle f\in L^{1}}. Define f ∗ {\displaystyle f^{*}} by

Then the maximal ergodic theorem states that

for any λ ∈ R.

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