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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2016 • Question 4

Question

Math 6210 • Real Analysis Fall 2016 • Question 4

Let \((X,\mathcal M,\mu )\) be a measure space, and suppose that \(A_n\in \mathcal M\), \(n=1,\ldots \), with \[ \sum _{n=1}^{\infty }\mu (A_n)<\infty . \] Show that the set of \(x\) belonging to infinitely many \(A_n\) has measure zero.

Raw LaTeX

Let \((X,\mathcal M,\mu)\) be a measure space, and suppose that
\(A_n\in\mathcal M\), \(n=1,\ldots\), with
\[
\sum_{n=1}^{\infty}\mu(A_n)<\infty.
\]
Show that the set of \(x\) belonging to infinitely many \(A_n\) has measure zero.

Topic Tags

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