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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Summer 2017 • Question 4

Question

Math 6210 • Real Analysis Summer 2017 • Question 4

Show that if \(f:[0,\infty )\to \mathbb C\) is continuous and compactly supported, then \[ \lim _{\rho \to 1}\int |f(x^\rho )-f(x)|\,d\lambda =0. \] Use this to show that if \(f\in L^\infty ([1,2],\lambda )\), we have that \[ \lim _{\rho \to 1^+}\int |f(x^\rho )-f(x)|\,d\lambda =0. \]

Raw LaTeX

Show that if \(f:[0,\infty)\to\mathbb C\) is continuous and
compactly supported, then
\[
\lim_{\rho\to1}\int |f(x^\rho)-f(x)|\,d\lambda=0.
\]
Use this to show that if \(f\in L^\infty([1,2],\lambda)\), we have that
\[
\lim_{\rho\to1^+}\int |f(x^\rho)-f(x)|\,d\lambda=0.
\]

Topic Tags

Compactness in Metric and Normed Spaces Lebesgue Integration Lp Spaces and Sequence Spaces