What does it mean when a number is subscripted with a truth statement?

Question

I have seen the following in several papers: $1_{\lvert r\rvert>1}$. What does this mean? Does this evaluate to 1 if $\lvert r\rvert>1$ and 0 otherwise? What would this evaluate to if instead of 1 we had a variable like $x$?

Thanks.

Answer 1

The usual notation is $1_A$ for some set $A$, which is a function that returns $1$ if its input is in $A$ and $0$ otherwise. Without context the notation you wrote sounds weird, but assuming that $r$ is a scalar variable I would assume it is $1_{[-1,1]^c}(r)$ in the notation I wrote.

This kind of function is called an indicator function. People also write $\chi_A$ (for "characteristic function") and $I_A$.

Something like $x_A$ would be very strange notation, but you could use it if you defined it.

Answer 2

Yes, it is an indicator function: $1$ when the condition is true, $0$ when it is not. I have not seen it used with $x$ instead of $1$.