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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1998 • Question 2

Question

Math 6210 • Real Analysis 1998 • Question 2

Let \(\mu \) be a measure on \(X\). Let \(f_1\ge f_2\ge \cdots f_n\ge \cdots \ge 0\) be a sequence of measurable functions. Assume \(\lim _{n\to \infty }\int _X f_n\,d\mu =0\). Show that \(f_n(x)\to 0\) a.e.

Raw LaTeX

Let \(\mu\) be a measure on \(X\). Let
\(f_1\ge f_2\ge \cdots f_n\ge \cdots \ge 0\) be a sequence of measurable
functions. Assume \(\lim_{n\to\infty}\int_X f_n\,d\mu=0\). Show that
\(f_n(x)\to 0\) a.e.

Topic Tags

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