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In mathematics, the indicator function (sometimes also
called characteristic function) of a subset A of a set X is a function from X into {0,1} defined as follows:
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The term characteristic function is potentially confusing because 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 IA becomes a random variable whose
expected value is equal to the probability of A:
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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
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Furthermore, if A and B are two subsets of X, then
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