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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2005 • Question 2

Question

Math 6210 • Real Analysis 2005 • Question 2

Let \((\mathcal M,\mu ,X)\) be a positive measure space. Let \(\mathcal M'\) denote the collection of subsets (in \(\mathcal M\)) whose measures are finite. Prove that \[ L^2(X)\subset L^1(X) \] if and only if \[ \sup _{E\in \mathcal M'}\mu (E)<\infty . \]

Raw LaTeX

Let \((\mathcal M,\mu,X)\) be a positive measure space. Let
\(\mathcal M'\) denote the collection of subsets (in \(\mathcal M\)) whose measures
are finite. Prove that
\[
L^2(X)\subset L^1(X)
\]
if and only if
\[
\sup_{E\in\mathcal M'}\mu(E)<\infty.
\]

Topic Tags

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