Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1992 • Question 2

Question

Math 6210 • Real Analysis 1992 • Question 2

Let \((X,\mathcal M,\mu )\) be a measure space. Suppose \(\mu \) is finite and positive, and it satisfies the following property:

(*) for each \(A\in \mathcal M\), such that \(\mu (A)>0\), there exists \(B\in \mathcal M\) such that \(B\subset A\) and \(0<\mu (B)<\mu (A)\).

Prove that for each \(r>0\) there exists \(C\in \mathcal M\) such that \[ 0<\mu (C)<r. \]

Raw LaTeX

Let \((X,\mathcal M,\mu)\) be a measure space. Suppose \(\mu\) is finite
and positive, and it satisfies the following property:

(*) for each \(A\in\mathcal M\), such that \(\mu(A)>0\), there exists
\(B\in\mathcal M\) such that \(B\subset A\) and \(0<\mu(B)<\mu(A)\).

Prove that for each \(r>0\) there exists \(C\in\mathcal M\) such that
\[
0<\mu(C)<r.
\]

\newpage

Topic Tags

Absolute Continuity and Singular Measures Measure Theory Outer Measures and Caratheodory Measurability Product Measures and Fubini-Tonelli Theorems