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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2016 • Question 2

Question

Math 6210 • Real Analysis Spring 2016 • Question 2

A family of complex-valued functions \(\mathcal S\) on \([0,1]\) is uniformly integrable if for all \(\epsilon >0\) there exists \(\delta >0\) so that \(\lambda (A)<\delta \) implies \(\left |\int _A f\,d\lambda \right |<\epsilon \) for all \(f\in \mathcal S\).

Let \((f_i;i\in \mathbb N)\) be a uniformly integrable sequence. Assume that \[ \lim _{i\to \infty }f_i(x)=g(x)\quad \text {for all }x\in [0,1]. \] Prove that \[ \lim _{i\to \infty }\int _0^1 f_i\,d\lambda =\int _0^1 g\,d\lambda . \]

Raw LaTeX

A family of complex-valued functions \(\mathcal S\) on \([0,1]\) is
\textit{uniformly integrable} if for all \(\epsilon>0\) there exists
\(\delta>0\) so that \(\lambda(A)<\delta\) implies
\(\left|\int_A f\,d\lambda\right|<\epsilon\) for all \(f\in\mathcal S\).

Let \((f_i;i\in\mathbb N)\) be a uniformly integrable sequence. Assume that
\[
\lim_{i\to\infty}f_i(x)=g(x)\quad\text{for all }x\in[0,1].
\]
Prove that
\[
\lim_{i\to\infty}\int_0^1 f_i\,d\lambda=\int_0^1 g\,d\lambda.
\]

Topic Tags

Lebesgue Integration Lp Spaces and Sequence Spaces Normal Families and Montel's Theorem