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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2022 • Question 1

Question

Math 6210 • Real Analysis Spring 2022 • Question 1

Prove the Borel-Cantelli lemma: Let \((X, \mathcal {M}, \mu )\) be a measure space and let \(A_{i}\) be measurable sets such that \(\sum _{i=1}^{\infty }\mu (A_{i}) < \infty \). Then the set

\[ \Omega = \{x \in X \mid x \in A_{i}\text { for infinitely many } i\} \]

of points that belong to infinitely many \(A_{i}\) has measure 0.

Raw LaTeX

Prove the Borel-Cantelli lemma: Let \((X, \mathcal{M}, \mu)\) be a measure space and
let \(A_{i}\) be measurable sets such that \(\sum_{i=1}^{\infty}\mu(A_{i}) < \infty\). Then the set

\[
\Omega = \{x \in X \mid x \in A_{i}\text{ for infinitely many } i\}
\]


of points that belong to infinitely many \(A_{i}\) has measure 0.

Topic Tags

Absolute Continuity and Singular Measures Measurable Functions Measure Theory Outer Measures and Caratheodory Measurability