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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2010 • Question 2

Question

Math 6210 • Real Analysis Fall 2010 • Question 2

Suppose \((X,\mathcal M,\mu )\) is a measure space and fix \(p\) and \(q\) finite such that \[ \frac {1}{p}+\frac {1}{q}=1. \] Let \(f_1,f_2,\ldots \) be a sequence of functions in \(L^p(X)\) converging (in \(L^p\)) to \(f\), and let \(g_1,g_2,\ldots \) be a sequence of functions in \(L^q\) converging (in \(L^q\)) to \(g\). Prove that the sequence \(f_1g_1,f_2g_2,\ldots \) converges to \(fg\) in \(L^1\). Does the same conclusion hold if \(p=1\) and \(q=\infty \)?

Raw LaTeX

Suppose \((X,\mathcal M,\mu)\) is a measure space and fix \(p\) and
\(q\) finite such that
\[
\frac{1}{p}+\frac{1}{q}=1.
\]
Let \(f_1,f_2,\ldots\) be a sequence of functions in \(L^p(X)\) converging (in
\(L^p\)) to \(f\), and let \(g_1,g_2,\ldots\) be a sequence of functions in \(L^q\)
converging (in \(L^q\)) to \(g\). Prove that the sequence
\(f_1g_1,f_2g_2,\ldots\) converges to \(fg\) in \(L^1\). Does the same conclusion
hold if \(p=1\) and \(q=\infty\)?

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measure Theory Outer Measures and Caratheodory Measurability