Notation seen in "awfully sophisticated proof..." I don't understand

Question

I want to understand what the definition of $f_n$ given here means? I tried to seek on the net but I not succeeded. I precise I do chemistry, maths are "just" a curiosity for me.

I should be glad, thanks

Answer 1

$1_{[0,n]}$ is the function defined on the interval $[0,n]$ which takes the constant value $1.$ Hence $\frac{1}{n}1_{[0,n]}$ is the function defined on the interval $[0,n]$ which takes the constant value $1/n.$

EDIT: As pointed out in the comments, we wish to view each $f_n$ as a function defined on $\mathbb{R}$ and so we define $f_n(x) = 0$ when $x$ is outside the interval $[0,n].$

Answer 2

$f_n:\mathbb R\rightarrow\mathbb R$ is prescribed by $x\mapsto\frac1{n}$ if $x\in[0,n]$ and $x\mapsto0$ otherwise. This for $n\in\mathbb Z_{>0}$.

The function is the same as $\frac1{n}1_{[0,n]}$ where $1_{[0,n]}$ is the characteristic function of set $[0,n]\subset\mathbb R$.

Its codomain is less strict. You can also take e.g. $[0,\infty)$.