Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 1

Question

Math 6210 • Real Analysis 2004 • Question 1

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\), and let \(f\ge 0\) be a function which is integrable with respect to \(\mu \). Let \(f_1\le f_2\le \cdots \le f_n\le \cdots \) be a sequence of real valued, measurable functions on \(X\), such that \(f_n\le f\) for each \(n\) and \(\int _X f_n\,d\mu \to \int _X f\,d\mu \). Show that \(f_n\to f\) almost everywhere.

Raw LaTeX

Let \(\mu\) be a positive measure on a measure space \((X,\Sigma)\), and
let \(f\ge 0\) be a function which is integrable with respect to \(\mu\). Let
\(f_1\le f_2\le\cdots\le f_n\le\cdots\) be a sequence of real valued, measurable
functions on \(X\), such that \(f_n\le f\) for each \(n\) and
\(\int_X f_n\,d\mu\to\int_X f\,d\mu\). Show that \(f_n\to f\) almost everywhere.

Topic Tags

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