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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2010 • Question 5

Question

Math 6210 • Real Analysis Spring 2010 • Question 5

Let \((X,\mathcal M,\mu )\) be a measure space such that \(\mu (X)<\infty \). Let \(f:X\to X\) be measure preserving in the sense that \(\mu (E)=\mu (f(E))\) for all \(E\in \mathcal M\). Prove that, for any \(E\in \mathcal M\), the set of points \(x\) of \(E\) such that the \(n\)th iterate \(f^n(x)\) is not in \(E\) for all \(n>0\) has zero measure.

Hint: Suppose not and obtain a contradiction with the fact that \(\mu (X)<\infty \).

Raw LaTeX

Let \((X,\mathcal M,\mu)\) be a measure space such that
\(\mu(X)<\infty\). Let \(f:X\to X\) be measure preserving in the sense that
\(\mu(E)=\mu(f(E))\) for all \(E\in\mathcal M\). Prove that, for any
\(E\in\mathcal M\), the set of points \(x\) of \(E\) such that the \(n\)th iterate
\(f^n(x)\) is not in \(E\) for all \(n>0\) has zero measure.

Hint: Suppose not and obtain a contradiction with the fact that
\(\mu(X)<\infty\).

Topic Tags

Absolute Continuity and Singular Measures Measure Theory Outer Measures and Caratheodory Measurability Zeros of Holomorphic Functions and the Identity Theorem