Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2023 • Question 6

Question

Math 6210 • Real Analysis Fall 2023 • Question 6

Let \((X,\mathcal {B},\mu )\) be a finite measure space and \(f:X\to \mathbb {R}\) be measurable.

  1. Show that if \(g_n(x)=(f(x))^n\) and \(\int g_n\,d\mu \) is uniformly bounded for all \(n\) then \(|f(x)|\le 1\) (\(\mu \)-a.e.).
  2. Show that \(\int _X g_n\,d\mu \) is independent of \(n\) iff there is \(A\in \mathcal {B}\) so that \(f=\chi _A\) (\(\mu \) a.e.).

Raw LaTeX

Let \((X,\mathcal{B},\mu)\) be a finite measure space and \(f:X\to\mathbb{R}\) be measurable.
\begin{enumerate}

\item
Show that if \(g_n(x)=(f(x))^n\) and \(\int g_n\,d\mu\) is uniformly bounded for
all \(n\) then \(|f(x)|\le 1\) (\(\mu\)-a.e.).



\item
Show that \(\int_X g_n\,d\mu\) is independent of \(n\) iff there is \(A\in\mathcal{B}\) so that
\(f=\chi_A\) (\(\mu\) a.e.).


\end{enumerate}

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Measurable Functions Measure Theory