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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2010 • Question 2

Question

Math 6210 • Real Analysis Spring 2010 • Question 2

Suppose \((X,\mathcal M,\mu )\) is a measure space and that \(L^p(X)\subset L^q(X)\) for some \(1\le p<q<\infty \).

(a) Show that the inclusion \(L^p(X)\to L^q(X)\) is bounded.

(b) Let \(\mathcal M'\) denote the subsets in \(\mathcal M\) of nonzero measure. Use (a) to prove that \[ \inf _{E'\in \mathcal M'}\mu (E')>0. \]

Raw LaTeX

Suppose \((X,\mathcal M,\mu)\) is a measure space and that
\(L^p(X)\subset L^q(X)\) for some \(1\le p<q<\infty\).

(a) Show that the inclusion \(L^p(X)\to L^q(X)\) is bounded.

(b) Let \(\mathcal M'\) denote the subsets in \(\mathcal M\) of nonzero measure.
Use (a) to prove that
\[
\inf_{E'\in\mathcal M'}\mu(E')>0.
\]

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measure Theory Zeros of Holomorphic Functions and the Identity Theorem