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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 4

Question

Math 6210 • Real Analysis 2004 • Question 4

Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\ge 0\) be a measurable function on \(X\). Define \(g(t)=\mu (\{x:f(x)\ge t\})\). Show that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty g(t)\,dt. \]

Raw LaTeX

Let \((X,\mathfrak M,\mu)\) be a positive measure space, and let
\(f\ge 0\) be a measurable function on \(X\). Define
\(g(t)=\mu(\{x:f(x)\ge t\})\). Show that
\[
\int_X f(x)\,d\mu(x)=\int_0^\infty g(t)\,dt.
\]

Topic Tags

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