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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2002 • Question 4

Question

Math 6210 • Real Analysis 2002 • Question 4

Let \(X\) be a measure space with \(\mu (X)=1\). Let \(f\) be a measurable function in \(L^\infty (X)\). Note that \(\|f\|_p\le \|f\|_\infty \) for all \(p\ge 1\). In particular, \(f\) can be considered an element of \(L^p(X)\) for all \(p\ge 1\). Show that \[ \lim _{p\to \infty }\|f\|_p=\|f\|_\infty . \]

Raw LaTeX

Let \(X\) be a measure space with \(\mu(X)=1\). Let \(f\) be a measurable
function in \(L^\infty(X)\). Note that \(\|f\|_p\le \|f\|_\infty\) for all
\(p\ge 1\). In particular, \(f\) can be considered an element of \(L^p(X)\) for all
\(p\ge 1\). Show that
\[
\lim_{p\to\infty}\|f\|_p=\|f\|_\infty.
\]

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measurable Functions Measure Theory