Indicator function

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

Remark. The term "characteristic function" has an unrelated meaning in probability theory; see characteristic function.

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

defined as

The indicator function of A is sometimes denoted

or or even

(The Greek letter χ because it is the initial letter of the Greek etymon of the word characteristic.)

The Iverson bracket allows the notation .

Warning. The notation may signify the identity function.

Basic properties

The mapping which associates a subset A of X to its indicator function 1A is injective; its range is the set of functions f:X →{0,1}.

If A and B are two subsets of X, then

and

More generally, suppose A1, ..., An is a collection of subsets of X. For any xX,

is clearly a product of 0s and 1s. This product has the value 1 at precisely those xX which belong to none of the sets Ak and is 0 otherwise. That is

Expanding the product on the left hand side,

where |F| is the cardinality of F. This is one form of the principle of inclusion-exclusion.

As suggested by the previous example, the indicator function is a useful notational device in combinatorics. The notation is used in other places as well, for instance 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:

This identity is used in a simple proof of Markov's inequality.

References

  • Folland, G.B.; Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.

See also


This article incorporates material from Characteristic function on PlanetMath, which is licensed under the GFDL.

de:Charakteristische Funktion (Mathematik) it:Funzione indicatrice zh:指示函数