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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2017 • Question 1

Question

Math 6210 • Real Analysis Fall 2017 • Question 1

Prove or disprove: If \((X,\mathcal M,\mu )\) is a measure space, \(A_1,\ldots \) are measurable, and \[ \sum _{i=1}^{\infty }\mu (A_i)^2<\infty , \] then \[ \mu \left (\bigcap _{m=1}^{\infty }\bigcup _{i=m}^{\infty } A_i\right )=0. \]

Raw LaTeX

Prove or disprove: If \((X,\mathcal M,\mu)\) is a measure space,
\(A_1,\ldots\) are measurable, and
\[
\sum_{i=1}^{\infty}\mu(A_i)^2<\infty,
\]
then
\[
\mu\left(\bigcap_{m=1}^{\infty}\bigcup_{i=m}^{\infty} A_i\right)=0.
\]

Topic Tags

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