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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2009 • Question 1

Question

Math 6210 • Real Analysis Spring 2009 • Question 1

Decide whether the following statement is true or false. If true, prove it, if false, provide a counterexample.

Let \(\{f_n\}\) be a sequence of measurable functions (with respect to Lebesgue measure) with \(0\le f_n\le 1\) such that \(\int _0^1 f_n(x)\,dx\to 0\). Then \(f_n\to 0\) almost everywhere.

Raw LaTeX

Decide whether the following statement is true or false. If true,
prove it, if false, provide a counterexample.

Let \(\{f_n\}\) be a sequence of measurable functions (with respect to Lebesgue
measure) with \(0\le f_n\le1\) such that
\(\int_0^1 f_n(x)\,dx\to0\). Then \(f_n\to0\) almost everywhere.

Topic Tags

Almost Everywhere and Convergence in Measure Lebesgue Integration Lebesgue Measure and Translation Invariance Measurable Functions