Indicator function

In mathematics, the indicator function (sometimes also called characteristic function) is a particular function associated to a subset of a given set. Given any set X and any subset A of X we define the indicator function on A from X to {0,1} as follows:

<math>I_A(x) = \left\{\begin{matrix} 1 &\mbox{if}\ x\in A\\

0 &\mbox{if}\ x\not\in A\end{matrix}\right.</math>

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:

<math>E(I_A)= \int_{X} I_A(x) dP = \int_{A} dP = P(A).\; </math>

For discrete spaces the proof may be written more simply as

<math>E(I_A)= \sum_{x\in X} I_A(x)p(x) = \sum_{x\in A} p(x) = P(A). </math>

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

<math>I_{A\cap B} = \min\{I_A,I_B\} = I_A I_B \qquad \mbox{and} \qquad I_{A\cup B} = \max\{{I_A,I_B}\} = I_A + I_B - I_A I_B.</math>