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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2006 • Question 2

Question

Math 6210 • Real Analysis 2006 • Question 2

We say that \(f_n\to f\) in measure if for all \(\epsilon >0\) there exists an \(N>0\) such that if \(n\ge N\) then \[ \mu \bigl (\{x\mid |f_n(x)-f(x)|>\epsilon \}\bigr )<\epsilon . \] Show that if \(f_n\to f\) pointwise and \(\mu (X)<\infty \) then \(f_n\to f\) in measure.

Raw LaTeX

We say that \(f_n\to f\) in measure if for all \(\epsilon>0\) there
exists an \(N>0\) such that if \(n\ge N\) then
\[
\mu\bigl(\{x\mid |f_n(x)-f(x)|>\epsilon\}\bigr)<\epsilon.
\]
Show that if \(f_n\to f\) pointwise and \(\mu(X)<\infty\) then \(f_n\to f\) in
measure.

Topic Tags

Absolute Continuity and Singular Measures Almost Everywhere and Convergence in Measure Measure Theory Outer Measures and Caratheodory Measurability