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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 1

Question

Math 6210 • Real Analysis Spring 2020 • Question 1

Let \((X,\mathcal M)\) be a measurable space and \(\mu ,\nu \) two measures on it. Suppose that for every \(E\in \mathcal M\), \[ \mu (E)=0\quad \Rightarrow \quad \nu (E)=0. \] (a) Assuming \(\nu (X)<\infty \), prove that for every \(\epsilon >0\) there is \(\delta >0\) such that \[ \mu (E)<\delta \quad \Rightarrow \quad \nu (E)<\epsilon . \] You are not allowed to use the Radon-Nikodym theorem for this problem.

(b) Find an example where the above statement is false when \(\nu (X)=\infty \).

Raw LaTeX

Let \((X,\mathcal M)\) be a measurable space and \(\mu,\nu\) two
measures on it. Suppose that for every \(E\in\mathcal M\),
\[
\mu(E)=0\quad\Rightarrow\quad\nu(E)=0.
\]
(a) Assuming \(\nu(X)<\infty\), prove that for every \(\epsilon>0\) there is
\(\delta>0\) such that
\[
\mu(E)<\delta\quad\Rightarrow\quad\nu(E)<\epsilon.
\]
You are not allowed to use the Radon-Nikodym theorem for this problem.

(b) Find an example where the above statement is false when \(\nu(X)=\infty\).

Topic Tags

Absolute Continuity and Singular Measures Measurable Functions Outer Measures and Caratheodory Measurability Product Measures and Fubini-Tonelli Theorems