example, sequence of measurable

**xianghu21** · December 13th 2009, 01:09 PM

Give an example in $[0, 1]$ of a sequence of measurable functions $(f_n)_n$ such that $\lim_{n \rightarrow \infty } || f_n ||_1=0$ but $(f_n)_n$ does not converge to zero a.e.

For this one, I can't think of a sequence that fits this criteria. What would this sequence look like? I can't think of one right now.

**Laurent** · December 13th 2009, 01:37 PM

Originally Posted by xianghu21

Give an example in $[0, 1]$ of a sequence of measurable functions $(f_n)_n$ such that $\lim_{n \rightarrow \infty } || f_n ||_1=0$ but $(f_n)_n$ does not converge to zero a.e.

Have a look at this:

Choose a sequence $(a_n)_n$ in $[0,1]$ and real numbers $r_n\to_n 0$ such that every $x\in[0,1]$ belong to infinitely many intervals $[a_n,a_n+r_n]$ .

For instance, $a_0=0, a_1=\frac{1}{2}, a_2=0, a_3 =\frac{1}{4}, a_4=\frac{1}{2}, a_5=\frac{3}{4}, a_6=0$ , etc. (dyadic points) and $r_0=r_1=\frac{1}{2}, r_2=r_3=r_4=r_5=\frac{1}{4}, r_6=\cdots = r_{13} = \frac{1}{8}$ ,...

Then let $f_n(x)={\bf 1}_{[a_n,a_n+r_n]}(x)$ .

There may be simpler, that's what came to my mind. Since $(f_n)_n$ can't be monotonic, it must look like my example anyway.

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