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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2023 • Question 4

Question

Math 6210 • Real Analysis Spring 2023 • Question 4

Let \((X,\mathcal {M},\mu )\) be a measure space. Give an example of \(f_1,\ldots :X\to \mathbb {C}\), measurable, which converge in measure to the zero function but do not converge \(\mu \)-a.e. Show that there exists a subsequence \(n_1,n_2,\ldots \) so that \(f_{n_1},\ldots \) converges almost everywhere to the zero function.

Raw LaTeX

Let \((X,\mathcal{M},\mu)\) be a measure space. Give an example of \(f_1,\ldots:X\to\mathbb{C}\),
measurable, which converge in measure to the zero function but do not
converge \(\mu\)-a.e. Show that there exists a subsequence \(n_1,n_2,\ldots\) so that
\(f_{n_1},\ldots\) converges almost everywhere to the zero function.

Topic Tags

Absolute Continuity and Singular Measures Almost Everywhere and Convergence in Measure Measurable Functions Measure Theory