Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1999 • Question 1

Question

Math 6210 • Real Analysis 1999 • Question 1

Let \(f\) be a non-negative measurable function and \(\mu \) a finite positive measure on a measure space \((X,\Sigma )\). Prove that \[ \lim _{n\to \infty }\int \frac {f^n(x)}{1+f^n(x)}\,d\mu (x) =\mu (E)+\frac {1}{2}\mu (F), \] where \(E=\{x\in X; f(x)>1\}\) and \(F=\{x\in X; f(x)=1\}\).

Raw LaTeX

Let \(f\) be a non-negative measurable function and \(\mu\) a finite
positive measure on a measure space \((X,\Sigma)\). Prove that
\[
\lim_{n\to\infty}\int \frac{f^n(x)}{1+f^n(x)}\,d\mu(x)
=\mu(E)+\frac{1}{2}\mu(F),
\]
where \(E=\{x\in X; f(x)>1\}\) and \(F=\{x\in X; f(x)=1\}\).

Topic Tags

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