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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1996 • Question 1

Question

Math 6210 • Real Analysis Fall 1996 • Question 1

Let \((X,\mathfrak M,\mu )\) be a measure space, and let \(A_1\supset A_2\supset \cdots \supset A_n\supset \cdots \) be a sequence of sets in \(\mathfrak M\).

Is it always true that \(\mu (\bigcap _{n=1}^{\infty }A_n)=\lim _{n\to \infty }\mu (A_n)\)? If so, give a proof. Otherwise give a counterexample.

Raw LaTeX

Let \((X,\mathfrak M,\mu)\) be a measure space, and let
\(A_1\supset A_2\supset\cdots\supset A_n\supset\cdots\) be a sequence of sets in
\(\mathfrak M\).

Is it always true that
\(\mu(\bigcap_{n=1}^{\infty}A_n)=\lim_{n\to\infty}\mu(A_n)\)? If so, give a
proof. Otherwise give a counterexample.

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measure Theory Outer Measures and Caratheodory Measurability