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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2003 • Question 2

Question

Math 6210 • Real Analysis 2003 • Question 2

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\) and let \(f\) be a complex valued function on \(X\) which is integrable with respect to \(\mu \). For each \(n\), set \[ E_n=\{x\in X:1/n\le |f(x)|\le n\}. \] Then each \(E_n\) is a measurable set on which \(|f|\) is bounded. Prove that

(a) \(\mu (E_n)\) is finite for each \(n\); and

(b) \(\displaystyle \lim _{n\to \infty }\int _{E_n} f(x)\,d\mu (x) =\int _X f(x)\,d\mu (x)\).

Raw LaTeX

Let \(\mu\) be a positive measure on a measure space \((X,\Sigma)\) and
let \(f\) be a complex valued function on \(X\) which is integrable with respect to
\(\mu\). For each \(n\), set
\[
E_n=\{x\in X:1/n\le |f(x)|\le n\}.
\]
Then each \(E_n\) is a measurable set on which \(|f|\) is bounded. Prove that

(a) \(\mu(E_n)\) is finite for each \(n\); and

(b) \(\displaystyle\lim_{n\to\infty}\int_{E_n} f(x)\,d\mu(x)
=\int_X f(x)\,d\mu(x)\).

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Lebesgue Measure and Translation Invariance Measure Theory