Math 6210 • Real Analysis

\begin{enumerate}

\item
Let \((X, \mathcal{M})\) be a measurable space, \(\mu, \nu\) positive measures on \((X, \mathcal{M})\).
Suppose \(\nu (X) < \infty\). Prove that \(\nu\)
\(\ll \mu\) if and only if for every \(\epsilon > 0\)
there exists \(\delta > 0\) such that \(\mu(E) < \delta\) implies \(\nu(E) < \epsilon\).


\item
For \(\mu\) the Lebesgue measure on \(\mathbb{R}\) find a positive (but infinite) measure \(\nu\) such that \(\nu\)
\(\ll \mu\) but the \(\epsilon\)-\(\delta\) statement in (a) fails.

\end{enumerate}