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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2008 • Question 1

Question

Math 6210 • Real Analysis Spring 2008 • Question 1

Let \((X,\mu )\) be a measure space and \(f\ge 0\) an integrable function on \(X\). Prove that for \(\epsilon >0\), there exists \(\delta >0\) such that whenever \(\mu (A)<\delta \), then \(\int _A f<\epsilon \).

Raw LaTeX

Let \((X,\mu)\) be a measure space and \(f\ge0\) an integrable
function on \(X\). Prove that for \(\epsilon>0\), there exists \(\delta>0\) such
that whenever \(\mu(A)<\delta\), then \(\int_A f<\epsilon\).

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Measure Theory Outer Measures and Caratheodory Measurability