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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2013 • Question 2

Question

Math 6210 • Real Analysis Spring 2013 • Question 2

Let \((X,\mathcal M,\mu )\) be a measure space, and let \(f\ge 0\) be an integrable function on \(X\). Suppose that \(f_n\ge 0\) is a measurable function, \(n=1,2,\ldots \), and that \[ f_1\le f_2\le \cdots \le f. \] If \(\int f_n\,d\mu \to \int f\,d\mu \), prove that \(f_n\to f\) a.e.

Raw LaTeX

Let \((X,\mathcal M,\mu)\) be a measure space, and let \(f\ge0\) be
an integrable function on \(X\). Suppose that \(f_n\ge0\) is a measurable
function, \(n=1,2,\ldots\), and that
\[
f_1\le f_2\le\cdots\le f.
\]
If \(\int f_n\,d\mu\to\int f\,d\mu\), prove that \(f_n\to f\) a.e.

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Measurable Functions Measure Theory