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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2024 • Question 4

Question

Math 6210 • Real Analysis Fall 2024 • Question 4

Let \(\mu \) and \(\nu \) be two finite positive measures on a measurable space \((X,\mathcal {M})\). Suppose for every \(\epsilon >0\), there exists \(E\in \mathcal {M}\) such that \(\mu (E)<\epsilon \) and \(\nu (X\setminus E)<\epsilon \). Show that \(\mu \perp \nu \) (\(\mu \) and \(\nu \) are mutually singular).

Raw LaTeX

Let \(\mu\) and \(\nu\) be two finite positive measures on a measurable space \((X,\mathcal{M})\).
Suppose for every \(\epsilon>0\), there exists \(E\in\mathcal{M}\) such that \(\mu(E)<\epsilon\) and
\(\nu(X\setminus E)<\epsilon\). Show that \(\mu\perp\nu\) (\(\mu\) and \(\nu\) are mutually singular).

Topic Tags

Absolute Continuity and Singular Measures Hilbert Spaces and Orthogonality Lebesgue Measure and Translation Invariance Measurable Functions