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let $f_n(x)$, for $n \ge 1$, be a sequence of continuous nonnegative functions on [0,1] such that $\lim_{n\to\infty} \int_{o}^{1} f_n(x)=0$.

Which of the following statements is always correct?

A. $f_n \to 0 $ uniformly on [0,1]

B. $f_n$ may not converge uniformly but converges to $0$ pointwise

C. $f_n$ will converge pointwise and the limit may be nonzero

D. $f_n$ is not guaranteed to have a pointwise limit

i know for sure that A is not true.

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  • $\begingroup$ Nice... May be you should ask 4th and 21st :P $\endgroup$
    – user87543
    Dec 10, 2013 at 11:32

1 Answer 1

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Consider $f_n$ to be given such that $f_n(1/2) = n$, and $f_n$ is supported on $(1/2-1/2n^2, 1/2+1/2n^2)$, and $f_n(x) \leq n$ for all $x$. (Think of an isosceles triangle whose apex is at $1/2$). Hence, $$ \int_0^1 f_n \leq n\frac{1}{n^2} \to 0 $$ but $f_n(1/2)$ does not converge to any real number.

So $D$ is true.

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