# $\lim_{n\to \infty} \int^1_0 f_n(x)dx=0$ , is $f_n$ pointwise convergent?

Let $f_n(x)$ be a sequence of continuous non negative functions on $[0,1]$ such that $\lim_{n\to \infty} \int^1_0 f_n(x)dx=0$ . Then does $f_n$ necessarily converge pointwise?

I think it is not necessary , but not getting an example!

Think of boxes of height $1$ moving back and forth through the interval $[0, 1]$, but getting skinnier and skinnier. In particular, try thinking about how this relates to the sequence

$$\left\{0, 1, 0, \frac 1 2, 1, 0, \frac 1 3, \frac 2 3, 1, \dots\right\}$$

Then adjust the boxes to make the relevant functions continuous.

• not getting it completely – Bhauryal Dec 12 '13 at 4:00

Let $g_{n,k}$ be the characteristic function of the set $$[\frac{k-1}{2^n}, \frac{k}{2^n}], k=1,2,\dots , 2^n$$ Then define the sequence $f_n$ as

$f_1=g_{1,1}, f_2=g_{1,2}, f_{3}=g_{2,1},f_{4}=g_{2,2},f_5=g_{2,3},f_{6}=g_{2,4}$ and so on. Prove that $\int f_n\to 0$ but $f_n$ does not converge pointwise.

Of course, $f_n$ are not continuous, but you can make them by adding some linear parts right and left of the intervals $[\frac{k-1}{2^n}, \frac{k}{2^n}]$. The conclusion is still true.