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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2022 • Question 4

Question

Math 6210 • Real Analysis Spring 2022 • Question 4

Let \((X, \mathcal {M}, \mu )\) be a measure space with \(\mu (X) < \infty \). Let \(f : X \to X\) be a measure preserving bijection, in the sense that \(\mu (A) = \mu (f(A))\) for every \(A \in \mathcal {M}\). Prove that for every \(E \in \mathcal {M}\)

\[ \{x \in E \mid f^{n}(x) \notin E \text { for all } n > 0\} \]

has measure 0.

Raw LaTeX

Let \((X, \mathcal{M}, \mu)\) be a measure space with \(\mu(X) < \infty\). Let \(f : X \to X\) be a
measure preserving bijection, in the sense that \(\mu(A) = \mu(f(A))\) for every
\(A \in \mathcal{M}\). Prove that for every \(E \in \mathcal{M}\)

\[
\{x \in E \mid f^{n}(x) \notin E \text{ for all } n > 0\}
\]


has measure 0.

Topic Tags

Absolute Continuity and Singular Measures Measure Theory Outer Measures and Caratheodory Measurability Product Measures and Fubini-Tonelli Theorems