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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2025 • Question 1

Question

Math 6210 • Real Analysis Spring 2025 • Question 1

Prove the following variant of Egorov’s theorem that applies to sets of infinite measure. Let \((X,\mathcal {M},\mu )\) be a measure space and let \(g\in L^1(\mu )\). Suppose \(f_n:X\to \mathbb {R}\) are measurable with \(|f_n|\le g\) and \(f_n\to f\) pointwise. Then for every \(\epsilon >0\) there is a set \(E\subset X\) of measure \(<\epsilon \) so that \(f_n\to f\) uniformly on \(X\setminus E\).

Raw LaTeX

Prove the following variant of Egorov's theorem that applies to sets of
infinite measure. Let \((X,\mathcal{M},\mu)\) be a measure space and let \(g\in L^1(\mu)\).
Suppose \(f_n:X\to\mathbb{R}\) are measurable with \(|f_n|\le g\) and \(f_n\to f\) pointwise.
Then for every \(\epsilon>0\) there is a set \(E\subset X\) of measure \(<\epsilon\) so that \(f_n\to f\)
uniformly on \(X\setminus E\).

Topic Tags

Absolute Continuity and Singular Measures Almost Everywhere and Convergence in Measure Lp Spaces and Sequence Spaces Measure Theory