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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1999 • Question 2

Question

Math 6210 • Real Analysis 1999 • Question 2

Let \(\mu \) be a measure on a measure space \(X\) and let \(f\ge 0\) be a measurable function on \(X\). Prove that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty \phi (y)\,dy \] where \(\phi (y)=\mu (\{x:f(x)\ge y\})\).

Raw LaTeX

Let \(\mu\) be a measure on a measure space \(X\) and let \(f\ge 0\) be a
measurable function on \(X\). Prove that
\[
\int_X f(x)\,d\mu(x)=\int_0^\infty \phi(y)\,dy
\]
where \(\phi(y)=\mu(\{x:f(x)\ge y\})\).

Topic Tags

Absolute Continuity and Singular Measures Complex Integration and Real Integral Evaluation Lebesgue Integration Measure Theory