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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 2

Question

Math 6210 • Real Analysis Spring 2020 • Question 2

Let \((X,\mathcal M,\mu )\) be a measure space and let \(f:X\to [0,\infty )\) be a measurable function. Define a new measure \(\nu \) by \[ \nu (E)=\int _E f\,d\mu \] for \(E\in \mathcal M\) (you don’t have to prove that \(\nu \) is a measure). Prove that for every positive measurable function \(g\), \[ \int _X g\,d\nu =\int _X f g\,d\mu . \]

Raw LaTeX

Let \((X,\mathcal M,\mu)\) be a measure space and let
\(f:X\to[0,\infty)\) be a measurable function. Define a new measure \(\nu\) by
\[
\nu(E)=\int_E f\,d\mu
\]
for \(E\in\mathcal M\) (you don't have to prove that \(\nu\) is a measure).
Prove that for every positive measurable function \(g\),
\[
\int_X g\,d\nu=\int_X f g\,d\mu.
\]

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Measurable Functions Measure Theory