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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2019 • Question 5

Question

Math 6210 • Real Analysis Fall 2019 • Question 5

Let \(X=[0,1]\) with the usual measure. For \(n=1,2,\ldots \), let \(f_n=n^{1/p}\cdot \chi _{[0,1/n]}\). Prove that the sequence \(f_n\) does not converge weakly to \(0\) in \(L^1([0,1])\).

Raw LaTeX

Let \(X=[0,1]\) with the usual measure. For \(n=1,2,\ldots\), let
\(f_n=n^{1/p}\cdot\chi_{[0,1/n]}\). Prove that the sequence \(f_n\) does not
converge weakly to \(0\) in \(L^1([0,1])\).

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measure Theory Weak Topologies and Weak Convergence