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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2008 • Question 4

Question

Math 6210 • Real Analysis Fall 2008 • Question 4

Let \(f:[0,1]\to \mathbb R_{\ge 0}\) be a measurable integrable function \(f\in L^1([0,1])\) (with respect to the Lebesgue measure). Show that \[ \int _{[0,1]} x f(x)\,dx =\int _{[0,1]}\left (\int _{[y,1]} f(x)\,dx\right )\,dy \]

Raw LaTeX

Let \(f:[0,1]\to\mathbb R_{\ge0}\) be a measurable integrable
function \(f\in L^1([0,1])\) (with respect to the Lebesgue measure). Show that
\[
\int_{[0,1]} x f(x)\,dx
=\int_{[0,1]}\left(\int_{[y,1]} f(x)\,dx\right)\,dy
\]

Topic Tags

Lebesgue Integration Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory