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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2009 • Question 1

Question

Math 6210 • Real Analysis Fall 2009 • Question 1

Let \(\{f_n\}\), \(f\) be measurable functions on a measure space \((X,\mu )\) such that \[ f_1\ge f_2\ge \cdots \ge f\ge 0 \] Suppose that \(\int f_n\,d\mu \to \int f\,d\mu \) and that \(f_1\) is integrable. Prove that \(f_n\to f\) almost everywhere. Show by example, that this conclusion may be false if \(f_1\) is not integrable.

Raw LaTeX

Let \(\{f_n\}\), \(f\) be measurable functions on a measure space
\((X,\mu)\) such that
\[
f_1\ge f_2\ge\cdots\ge f\ge0
\]
Suppose that \(\int f_n\,d\mu\to\int f\,d\mu\) and that \(f_1\) is integrable.
Prove that \(f_n\to f\) almost everywhere. Show by example, that this conclusion
may be false if \(f_1\) is not integrable.

Topic Tags

Absolute Continuity and Singular Measures Almost Everywhere and Convergence in Measure Measurable Functions Measure Theory