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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2023 • Question 2

Question

Math 6210 • Real Analysis Fall 2023 • Question 2

Let \(f:\mathbb {R}\to \mathbb {C}\) be Lebesgue measurable. Assume that \(f\) is invariant under translations by rational numbers. Show that \(f\) is constant Lebesgue almost everywhere.

Raw LaTeX

Let \(f:\mathbb{R}\to\mathbb{C}\) be Lebesgue measurable. Assume that \(f\) is invariant
under translations by rational numbers. Show that \(f\) is constant Lebesgue
almost everywhere.

Topic Tags

Almost Everywhere and Convergence in Measure Lebesgue Integration Lebesgue Measure and Translation Invariance Measurable Functions