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: First example

And somewhat later in a proof:

Second example

Anyone got an idea what this symbol represents?

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

For those wondering, the symbol ( in Unicode; fonts may lack it) is the blackboard bold / open face / double-struck digit one — the same style as ℕ, ℝ, ℤ, etc. — deltab, May 02 '14 at 01:13

score 27 · Accepted Answer · answered May 01 '14 at 15:37

27

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

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

answered May 01 '14 at 15:37

Ted Shifrin

115,160

1

I think you mis-wrote that? Your $1_A$ does not appear to be the outline version in the OP. – KRyan May 02 '14 at 02:49
1

@KRyan So? It's still the same object. – Pierre Arlaud May 02 '14 at 10:57
1

Interestingly, it shows us in "blackboard" font just fine on my desktop, but not on my iPad. Perhaps you should try a different computer, @KRyan. – Ted Shifrin May 02 '14 at 11:25

score 7 · Answer 2 · answered May 01 '14 at 15:38

7

It is the indicator function.

answered May 01 '14 at 15:38

npisinp

652

score 3 · Answer 3 · answered May 01 '14 at 15:40

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.

answered May 01 '14 at 15:40

3

"everybody knows that you mean the indicator function" -> except for the OP, apparently. – User1000547 May 01 '14 at 20:47
@mikeTheLiar, except for who? – Glinka May 01 '14 at 22:39
@Milena the OP (original poster) - the person who asked this question exactly because they didn't know what it meant. – User1000547 May 01 '14 at 22:41
I just wanted to give an alternative possibility to write it in case the OP finds this different notification in context of his literature. Surely the OP does not know it if he did not know the other way to write it. – May 01 '14 at 22:45
6

I knew the double-struck 1 notation but I haven't seen the chi-notation often and would not immediately think of this. So "everybody knows that you mean ... " is a bit of an over-generalization. – CompuChip May 02 '14 at 10:43
In probability, $\chi$ usually denotes the characteristic function (which is called the Fourier transform in the rest of mathematics). I prefer the 1 notation, since it is more similar to the right side of the definition. – Teepeemm May 02 '14 at 14:04

What does this symbol mean? (looks like a 1 with double vertical line)

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