Indicator function

This article is about the characteristic function in set theory. For characteristic function in probability theory see characteristic function

In mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate or characterize a subset A of X.

Definition

The indicator function for a subset A of a set X is a function

1_A : X \to \lbrace 0,1 \rbrace

defined as

1_A(x) = \left\{\begin{matrix} 1 &\mbox{if}\ x \in A \\ 0 &\mbox{if}\ x \notin A \end{matrix}\right.

Notes

The indicator function is sometimes denoted

χA(x)

or

IA(x)

Probability theory

The indicator function is a basic tool in probability theory: if X is a probability space with probability measure P and A is a measurable set, then 1A becomes a random variable whose expected value is equal to the probability of A:

E(1_A)= \int_{X} 1_A(x)\,dP = \int_{A} dP = P(A).\;

It may be called an indicator variable, as a random variable returning a 0-1 data point.

For discrete spaces the proof may be written more simply as

E(1_A)= \sum_{x\in X} 1_A(x)p(x) = \sum_{x\in A} p(x) = P(A).

Furthermore, if A and B are two subsets of X, then

1_{A\cap B} = \min\{1_A,1_B\} = 1_A 1_B \qquad \mbox{and} \qquad 1_{A\cup B} = \max\{{1_A,1_B}\} = 1_A + 1_B - 1_A 1_B.


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