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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2006 • Question 1

Question

Math 6210 • Real Analysis 2006 • Question 1

Assume that \(f\) is non-negative and \[ \int _X f\,d\mu <\infty . \] Show that for every \(\epsilon >0\) there exists a \(\delta >0\) such that if \(A\) is a measurable subset of \(X\) such that \(\mu (A)\le \delta \) then \[ \int _A f\,d\mu \le \epsilon . \]

Raw LaTeX

Assume that \(f\) is non-negative and
\[
\int_X f\,d\mu<\infty.
\]
Show that for every \(\epsilon>0\) there exists a \(\delta>0\) such that if \(A\) is a
measurable subset of \(X\) such that \(\mu(A)\le \delta\) then
\[
\int_A f\,d\mu\le \epsilon.
\]

Topic Tags

Lebesgue Integration Measurable Functions