Pointwise Convergence Implies Uniform?

Question

Let $f_{n} \to 0$ pointwise on the interval $[-A,A]$ where $f_{n}$ are continuous and uniformly bounded. Can we show the convergence is uniform?

Thanks, any help appreciated.

Indeed the 'best' similar result in some sense is the Arzela-Ascoli Theorem, which you might be interested to learn. — Edward Hughes, Apr 10 '12 at 23:50

David Mitra · Accepted Answer · 2012-04-11T00:18:11.443

2

No. Choose two sequences $(a_n)$ and $(b_n)$ of real numbers so that $0<a_n<b_n<1$ for each $n$ and $\lim\limits_{n\rightarrow \infty} a_n=1$. For each $n$, let $f_n$ be the function on $[0,1]$ whose graph consists of the straight line segments from $(0,0)$ to $(a_n,0)$, from $(a_n,0)$ to $(b_n,1)$, and from $(b_n,1)$ to $(1,0)$.

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edited Apr 11 '12 at 00:18

answered Apr 10 '12 at 23:49

David Mitra

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score 2 · Answer 2 · answered Sep 09 '12 at 09:32

Pointwise convergence does not imply uniform convergence. We have, however, Egorov's theorem, which for the given example states:

Suppose that $f_n$ are measurable and converges almost everywhere to some function $f$ on $[-A,A]$. For every $\epsilon>0$ there exists a measurable set $B_\epsilon\subset [-A,A]$ with measure (total length) $\mu(B_\epsilon) < \epsilon$, such that the convergence is uniform on $[-A,A] \setminus B_\epsilon$.

Pointwise Convergence Implies Uniform?

2 Answers2