# Indicator function

In

mathematics, the

**indicator function** (sometimes also called

**characteristic function**) of a

subset *A* of a

set *X*
is a

function from

*A* into to {0,1} defined as follows:

The term

*characteristic function* is potentially confusing becase it is also used to denote a quite different concept that is also prevalent in probability theory; see

characteristic function.

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 *I*_{A} becomes a random variable whose expected value is equal to the probability of *A*:

For discrete spaces the proof may be written more simply as

Furthermore, if

*A* and

*B* are two subsets of

*X*, then