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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2016 • Question 3

Question

Math 6210 • Real Analysis Spring 2016 • Question 3

Let \(f:\mathbb R^2\to \mathbb C\) be a Borel measurable function. Let \(g:\mathbb R^2\to \mathbb R^2\) be the map given by \(g(x,y)=(2x+y^2,y)\) for \((x,y)\in \mathbb R^2\). If \(f\in L^1(\lambda ^2)\), what is \[ \frac {\int _{\mathbb R^2}(f\circ g)\,d\lambda ^2} {\int _{\mathbb R^2}f\,d\lambda ^2}? \]

Raw LaTeX

Let \(f:\mathbb R^2\to\mathbb C\) be a Borel measurable function.
Let \(g:\mathbb R^2\to\mathbb R^2\) be the map given by
\(g(x,y)=(2x+y^2,y)\) for \((x,y)\in\mathbb R^2\). If \(f\in L^1(\lambda^2)\),
what is
\[
\frac{\int_{\mathbb R^2}(f\circ g)\,d\lambda^2}
     {\int_{\mathbb R^2}f\,d\lambda^2}?
\]

Topic Tags

Lebesgue Integration Lp Spaces and Sequence Spaces Measurable Functions