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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 1

Question

Math 6010 • Linear Models Fall 2018 • Question 1

Let \(\mu \) be a translation invariant \(\sigma \)-additive measure on \((\mathbb {R},\mathcal {B}(\mathbb {R}))\) such that \(\mu ((0,1])<\infty \). Prove that there exists a \(c\in (0,\infty )\) such that \(c\mu \) is Lebesgue measure.

Raw LaTeX

Let \(\mu\) be a translation invariant \(\sigma\)-additive measure on
\((\mathbb{R},\mathcal{B}(\mathbb{R}))\) such that \(\mu((0,1])<\infty\). Prove
that there exists a \(c\in(0,\infty)\) such that \(c\mu\) is Lebesgue measure.

Topic Tags

Lebesgue Differentiation via Martingales Uniform Integrability and Truncation Methods