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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Summer 2017 • Question 1

Question

Math 6210 • Real Analysis Summer 2017 • Question 1

Recall that (a version of) Lusin’s theorem states that if \(f:[0,1]\to \mathbb C\) is measurable then for all \(\epsilon >0\) there exists \(g:[0,1]\to \mathbb C\) continuous so that \[ \lambda (\{x\in [0,1]: g(x)\ne f(x)\})<\epsilon . \] Prove that Lusin’s Theorem implies that if \(f:\mathbb R\to \mathbb C\) is measurable then there exists a continuous function \(g:\mathbb R\to \mathbb C\) so that \[ \lambda (\{x:f(x)\ne g(x)\})\le 1. \]

Raw LaTeX

Recall that (a version of) Lusin's theorem states that if
\(f:[0,1]\to\mathbb C\) is measurable then for all \(\epsilon>0\) there exists
\(g:[0,1]\to\mathbb C\) continuous so that
\[
\lambda(\{x\in[0,1]: g(x)\ne f(x)\})<\epsilon.
\]
Prove that Lusin's Theorem implies that if \(f:\mathbb R\to\mathbb C\) is
measurable then there exists a continuous function \(g:\mathbb R\to\mathbb C\)
so that
\[
\lambda(\{x:f(x)\ne g(x)\})\le1.
\]

Topic Tags

Measurable Functions