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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2018 • Question 2

Question

Math 6210 • Real Analysis Fall 2018 • Question 2

Let \((X,\mu )\) be a (not necessarily finite) measure space and \(f\ge 0\) be a \(\mu \) integrable function. Prove that, for every \(\epsilon >0\), there exists a set \(E\subset X\) of finite measure such that \[ \int _E f\,d\mu >\int _X f\,d\mu -\epsilon . \]

Raw LaTeX

Let \((X,\mu)\) be a (not necessarily finite) measure space and
\(f\ge0\) be a \(\mu\) integrable function. Prove that, for every
\(\epsilon>0\), there exists a set \(E\subset X\) of finite measure such that
\[
\int_E f\,d\mu>\int_X f\,d\mu-\epsilon.
\]

Topic Tags

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