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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2005 • Question 4

Question

Math 6210 • Real Analysis 2005 • Question 4

Construct a sequence of continuous real-valued functions \(f_n\) on \([0,1]\) such that \(0\le f_n\le 1\) and \[ \lim _{n\to \infty }\int _0^1 f_n(x)\,dx=0, \] but such that the sequence \(\{f_n(x)\}\) converges for no \(x\in [0,1]\).

Raw LaTeX

Construct a sequence of continuous real-valued functions \(f_n\) on
\([0,1]\) such that \(0\le f_n\le 1\) and
\[
\lim_{n\to\infty}\int_0^1 f_n(x)\,dx=0,
\]
but such that the sequence \(\{f_n(x)\}\) converges for no \(x\in[0,1]\).

Topic Tags

Almost Everywhere and Convergence in Measure Lebesgue Integration Lp Spaces and Sequence Spaces