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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2021 • Question 3

Question

Math 6210 • Real Analysis Fall 2021 • Question 3

Show that there does not exist a Lebesgue measurable subset \(A \subset \mathbb {R}\) such that for every interval \((a, b) \subset \mathbb {R}\) we have \[ m(A \cap (a, b)) = \frac {b-a}{2} \] where \(m\) denotes Lebesgue measure.

Raw LaTeX

Show that there does not exist a Lebesgue measurable subset \(A \subset \mathbb{R}\) such
that for every interval \((a, b) \subset \mathbb{R}\) we have
\[
m(A \cap (a, b)) = \frac{b-a}{2}
\]
where \(m\) denotes Lebesgue measure.

Topic Tags

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