What does this symbol mean? (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.

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

Answer 2

It is the indicator function.

Answer 3

It's the indicator function, you can write $\chi_A(x)=\begin{cases}1, & x\in A\\0, & x\notin A\end{cases}$ instead. It's easier in $\text{LaTeX}$ and everybody knows that you mean the indicator function.