Indicatorensets

Indicatorensets, or indicator sets, denote subsets of a given universal set that are studied together with their indicator functions. Given a universal set Ω, an indicator set A ⊆ Ω represents the property of belonging to A. The indicator function χ_A: Ω → {0,1} is defined by χ_A(x)=1 if x∈A and 0 otherwise. Every indicator set corresponds uniquely to its indicator function, and vice versa; thus the collection of indicator sets is in bijection with the set {0,1}^Ω.

Operations on indicator sets translate into arithmetic on their indicator functions: χ_{A∪B}=max(χ_A,χ_B) (or χ_A ∨ χ_B), χ_{A∩B}=χ_A χ_B, and χ_{A^c}=1−χ_A. In probability theory, if (Ω, F, P) is a probability space and A ∈ F, then P(A)=E[χ_A]. Indicator sets thus provide a simple way to model events and to build simple (step) functions.

In measure theory and analysis, indicator functions are used to approximate more general functions by simple functions, where the value on each set A is given by a constant. Indicator sets are also used in statistics and data encoding to convert categorical features into binary indicators (binary presence, one-hot like representations for individual attributes).

Examples: Let Ω={a,b,c,d} and A={a,c}. Then χ_A=(1,0,1,0). The union with B={c,d} yields χ_{A∪B}=(1,0,1,1).

See also: indicator function, characteristic function, simple function, sigma-algebra, measurable space, random variable.