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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2010 • Question 1

Question

Math 6210 • Real Analysis Spring 2010 • Question 1

Let \([0,1]\) be the unit interval equipped with Lebesgue measure and fix \(1\le p<\infty \). Suppose \(f_1,f_2,\ldots \) is a sequence in \(L^p([0,1])\) which is Cauchy with respect to the \(L^p\) norm.

(a) Prove that there is a subsequence \(g_1,g_2,\ldots \) of \(f_1,f_2,\ldots \) such that \(\lim _{i\to \infty }g_i(x)\) exists for almost all \(x\).

(b) Show (by producing an example of a Cauchy sequence \(f_1,f_2,\ldots \) in \(L^p([0,1])\)) that \(\lim _{i\to \infty }f_i(x)\) can fail to exist for almost all \(x\).

Raw LaTeX

Let \([0,1]\) be the unit interval equipped with Lebesgue measure
and fix \(1\le p<\infty\). Suppose \(f_1,f_2,\ldots\) is a sequence in
\(L^p([0,1])\) which is Cauchy with respect to the \(L^p\) norm.

(a) Prove that there is a subsequence \(g_1,g_2,\ldots\) of
\(f_1,f_2,\ldots\) such that \(\lim_{i\to\infty}g_i(x)\) exists for almost all
\(x\).

(b) Show (by producing an example of a Cauchy sequence \(f_1,f_2,\ldots\) in
\(L^p([0,1])\)) that \(\lim_{i\to\infty}f_i(x)\) can fail to exist for almost all
\(x\).

Topic Tags

Almost Everywhere and Convergence in Measure Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory