Does $f_n = n \chi_{[0,\frac{1}{n}]} \to 0$ almost uniformly?

Question

Does $f_n = n \chi_{[0,\frac{1}{n}]} \to 0$ almost uniformly? (On $\mathbb{R}$)

I’ve got this function as an example of a sequence of functions which converges to 0 almost uniformly but not uniformly.

It’ll be thankful if someone can help me with the proof.

——————————— By almost uniformly convergence, I meant

$\forall \delta > 0$, there exist a subset $A \subset E$ such that $m(A) < \delta$ and $f_n$ converges to $f$ in $E | A^c$.

Answer 1

Let $\delta>0$. We show that $\{f_j\}_{j\in\mathbb{Z}^+}$ converges uniformly to zero on $\mathbb{R}\setminus[0,\delta)$. Take $N>\frac{1}{\delta}$. Then $\left[0,\frac{1}{n}\right]\subseteq[0,\delta)$ for all $n\geq N$, and so $f_j(x)=0$ for all $j\geq N$ and all $x\in\mathbb{R}\setminus[0,\delta)$. It follows that $f_j\to0$ as $j\to\infty$ uniformly on $\mathbb{R}\setminus[0,\delta)$. As $m([0,\delta))=\delta$ follows that $\{f_j\}_{j\in\mathbb{Z}^+}$ converges almost uniformly on $\mathbb{R}$.

To see that it does not converge uniformly on $\mathbb{R}$, simply observe that

$$\sup_{x\in\mathbb{R}}\lvert f_j(x)\rvert=j\to\infty$$

as $j\to\infty$.