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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 2

Question

Math 6210 • Real Analysis 2004 • Question 2

Let \(\{f_n\}\) be a sequence of complex valued measurable functions defined on a finite interval \((a,b)\subset \mathbb R\) (”measurable” means with respect to the Lebesgue measure). Suppose that there exists a constant \(M<\infty \) such that \(\int _a^b |f_n(x)|^2\,dx<M\) for all \(n\), and \(f_n(x)\to 0\) a. e. Prove that \[ \lim _{n\to \infty }\int _a^b f_n(x)\,dx=0. \] Show also that this assertion fails in general if one replaces the interval \((a,b)\) by \(\mathbb R\) (give a counterexample).

Raw LaTeX

Let \(\{f_n\}\) be a sequence of complex valued measurable functions
defined on a finite interval \((a,b)\subset \mathbb R\) ("measurable" means with
respect to the Lebesgue measure). Suppose that there exists a constant
\(M<\infty\) such that \(\int_a^b |f_n(x)|^2\,dx<M\) for all \(n\), and
\(f_n(x)\to 0\) a. e. Prove that
\[
\lim_{n\to\infty}\int_a^b f_n(x)\,dx=0.
\]
Show also that this assertion fails in general if one replaces the interval
\((a,b)\) by \(\mathbb R\) (give a counterexample).

Topic Tags

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