Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 2

Question

Math 6210 • Real Analysis 2004 • Question 2

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\) such that \(\mu (X)=1\). Let \(f\) be a bounded measurable function on \(X\). Show that \(f\in L^p(X)\) for \(1\le p<\infty \) and \(\|f\|_p\le \|f\|_q\) for \(1<p<q<\infty \).

Raw LaTeX

Let \(\mu\) be a positive measure on a measure space \((X,\Sigma)\) such
that \(\mu(X)=1\). Let \(f\) be a bounded measurable function on \(X\). Show that
\(f\in L^p(X)\) for \(1\le p<\infty\) and \(\|f\|_p\le \|f\|_q\) for
\(1<p<q<\infty\).

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory