Notation – Meaning of Symbol That Looks Like a 1 with Double Vertical Line

Question

I'm studying a course on probability and statistics and at some point this symbol comes up without introduction. It looks like the number one, but slightly bigger and with a double vertical line.

First time it comes up is when discussing stochastic/random variables that are neither continuous nor discreet in an example:

And somewhat later in a proof:

Anyone got an idea what this symbol represents?

EDIT: Thanks for the fast answers, I was still editing the question for better (larger) images 🙂

Answer 1

It's the characteristic function (or indicator function) of the set in the subscript.

$$\mathbb 1_A(x) = \begin{cases} 1\,, & x\in A \\ 0\,, & x\notin A\end{cases}.$$