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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2024 • Question 1

Question

Math 6210 • Real Analysis Fall 2024 • Question 1

Let

\[ A=\left \{\sum _{n=1}^{\infty }\frac {x_n}{3^n}\mid x_n=0\text { or }1\right \}\subset [0,1]. \]

Show that \(A\) has Lebesgue measure 0 but

\[ A+A:=\{x+y\mid x,y\in A\}=[0,1]. \]

Raw LaTeX

Let

\[
A=\left\{\sum_{n=1}^{\infty}\frac{x_n}{3^n}\mid x_n=0\text{ or }1\right\}\subset[0,1].
\]

Show that \(A\) has Lebesgue measure 0 but

\[
A+A:=\{x+y\mid x,y\in A\}=[0,1].
\]

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Lebesgue Measure and Translation Invariance Measure Theory