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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2010 • Question 1

Question

Math 6210 • Real Analysis Fall 2010 • Question 1

Suppose \((X,\mathcal M,\mu )\) is a measure space and \(f:X\to \mathbb R\) is a real-valued function on \(X\). Suppose further that \(E_r:=\{x\mid f(x)>r\}\) is measureable for each rational number \(r\). Either prove the following assertion or find a counter-example: \(f\) is measureable.

Raw LaTeX

Suppose \((X,\mathcal M,\mu)\) is a measure space and
\(f:X\to\mathbb R\) is a real-valued function on \(X\). Suppose further that
\(E_r:=\{x\mid f(x)>r\}\) is measureable for each rational number \(r\). Either
prove the following assertion or find a counter-example: \(f\) is measureable.

Topic Tags

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