Math 6210 • Real Analysis

Let \(\mathcal M\) be a \(\sigma \)-algebra of subsets of \(X\) and \(\mu \) a measure on \(\mathcal M\). Let \(A_1\subseteq A_2\subseteq \cdots \) be a sequence of sets in \(\mathcal M\). Let \[ A=\bigcup _{i=1}^{\infty } A_i. \] Prove that \(\lim _{i\to \infty }\mu (A_i)=\mu (A)\).

Now assume that \(\mu (X)<\infty \), and let \(A_1\supseteq A_2\supseteq \cdots \) be a sequence of sets in \(\mathcal M\). Let \[ A=\bigcap _{i=1}^{\infty } A_i. \] Prove that \(\lim _{i\to \infty }\mu (A_i)=\mu (A)\). Give an example where this fails if \(\mu (X)=\infty \).