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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2024 • Question 6

Question

Math 6210 • Real Analysis Spring 2024 • Question 6

Let \((X,\mathcal {M})\) be a measurable space and \(\mu \) and \(\nu \) be two finite measures on \((X,\mathcal {M})\). Show that the following are equivalent

  • \(\nu \ll \mu \)
  • Whenever \(A_1,\ldots \in \mathcal {M}\) satisfy \(\lim _{n\to \infty }\mu (A_n)=0\) we have \(\lim _{n\to \infty }\nu (A_n)=0\).

Raw LaTeX

Let \((X,\mathcal{M})\) be a measurable space and \(\mu\) and \(\nu\) be two finite measures
on \((X,\mathcal{M})\). Show that the following are equivalent
\begin{itemize}
\item \(\nu\ll\mu\)
\item Whenever \(A_1,\ldots\in\mathcal{M}\) satisfy \(\lim_{n\to\infty}\mu(A_n)=0\) we have \(\lim_{n\to\infty}\nu(A_n)=0\).
\end{itemize}

Topic Tags

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