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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2019 • Question 1

Question

Math 6210 • Real Analysis Fall 2019 • Question 1

Let \(f\ge 0\) be an integrable function on a measure space \(X\). Prove that, for every \(\epsilon >0\), there exists a set \(E\subset X\) of finite measure such that \[ \int _E f>\left (\int _X f\right )-\epsilon . \]

Raw LaTeX

Let \(f\ge0\) be an integrable function on a measure space \(X\).
Prove that, for every \(\epsilon>0\), there exists a set \(E\subset X\) of finite
measure such that
\[
\int_E f>\left(\int_X f\right)-\epsilon.
\]

Topic Tags

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