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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2018 • Question 2

Question

Math 6010 • Linear Models Spring 2018 • Question 2

Let \(\mu \) be a translation invariant \(\sigma \)-finite measure on \((\mathbb {R},\mathcal {B}(\mathbb {R}))\). Assume that there exist \(\alpha <\beta \) such that \(\mu ((\alpha ,\beta ])<\infty \). Prove that there exists a finite positive constant \(c\) such that \(c\mu \) is Lebesgue measure.

Raw LaTeX

Let \(\mu\) be a translation invariant \(\sigma\)-finite measure on
\((\mathbb{R},\mathcal{B}(\mathbb{R}))\). Assume that there exist \(\alpha<\beta\)
such that \(\mu((\alpha,\beta])<\infty\). Prove that there exists a finite
positive constant \(c\) such that \(c\mu\) is Lebesgue measure.

Topic Tags

Lebesgue Differentiation via Martingales Uniform Integrability and Truncation Methods