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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2015 • Question 5

Question

Math 6210 • Real Analysis Spring 2015 • Question 5

(a) Let \(A\subset \mathbb R\). Prove that if \(\lambda (A)>0\), then there exists \(\delta _0>0\) so that for all \(0\le \delta \le \delta _0\) there exists \(a\) such that both \(a\) and \(a+\delta \) are in \(A\). Note, \(a\) is allowed to depend on \(\delta \).

(b) Show that if \(A\subset \mathbb R\) is closed under addition and subtraction, then \(\lambda (A)=0\) or \(A=\mathbb R\).

Raw LaTeX

(a) Let \(A\subset\mathbb R\). Prove that if \(\lambda(A)>0\), then
there exists \(\delta_0>0\) so that for all \(0\le\delta\le\delta_0\) there
exists \(a\) such that both \(a\) and \(a+\delta\) are in \(A\).
Note, \(a\) is allowed to depend on \(\delta\).

(b) Show that if \(A\subset\mathbb R\) is closed under addition and subtraction,
then \(\lambda(A)=0\) or \(A=\mathbb R\).

Topic Tags

Lebesgue Measure and Translation Invariance