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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2018 • Question 1

Question

Math 6210 • Real Analysis Spring 2018 • Question 1

Let \(\mu \) be the outer measure on \(\mathbb R\) defined by \[ \mu (A)=\inf \left \{ \sum _{i=1}^{\infty }(b_i-a_i)\ \middle |\ A\subset \bigcup _{i=1}^{\infty }(a_i,b_i],\ a_i\le b_i \right \} \] for every \(A\subset \mathbb R\). Prove that \(\mu ((a,b])=b-a\). You may use that the length of a segment covered by finitely many segments is less than the sum of lengths of segments in the covering.

Raw LaTeX

Let \(\mu\) be the outer measure on \(\mathbb R\) defined by
\[
\mu(A)=\inf\left\{
\sum_{i=1}^{\infty}(b_i-a_i)\ \middle|\ 
A\subset\bigcup_{i=1}^{\infty}(a_i,b_i],\ a_i\le b_i
\right\}
\]
for every \(A\subset\mathbb R\). Prove that \(\mu((a,b])=b-a\). You may use that
the length of a segment covered by finitely many segments is less than the
sum of lengths of segments in the covering.

Topic Tags

Absolute Continuity and Singular Measures Measure Theory Outer Measures and Caratheodory Measurability Product Measures and Fubini-Tonelli Theorems