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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2024 • Question 1

Question

Math 6210 • Real Analysis Spring 2024 • Question 1

Let \(X\) be a set and \(\mu ^*\) be an outer measure on \(X\). A set \(A\subset X\) is called \(\mu ^*\)-measurable if for every \(S\subset X\) we have

\[ \mu ^*(S)=\mu ^*(S\cap A)+\mu ^*(S\setminus A). \]

Show that \(\mu ^*\) is countably additive on the \(\mu ^*\) measurable sets.

Raw LaTeX

Let \(X\) be a set and \(\mu^*\) be an outer measure on \(X\). A set \(A\subset X\) is called
\(\mu^*\)-measurable if for every \(S\subset X\) we have

\[
\mu^*(S)=\mu^*(S\cap A)+\mu^*(S\setminus A).
\]


Show that \(\mu^*\) is countably additive on the \(\mu^*\) measurable sets.

Topic Tags

Absolute Continuity and Singular Measures Measurable Functions Measure Theory Outer Measures and Caratheodory Measurability