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In many cases, such as order theory, the inverse of the indicator function may be defined. This is commonly called the generalized Möbius function, as a generalization of the inverse of the indicator function in elementary number theory, the Möbius function. (See paragraph below about the use of the inverse in classical recursion theory.)
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Such indicators have some special properties. For example, the following statements are all true for an indicator function that is trigonometrically convex at least on an interval (,): [1]: 55–57 [2]: 54–61
The strength of Lusin's theorem might not be readily apparent, as can be demonstrated by example. Consider Dirichlet function , that is the indicator function 1 Q : [ 0 , 1 ] → { 0 , 1 } {\displaystyle 1_{\mathbb {Q} }:[0,1]\to \{0,1\}} on the unit interval [ 0 , 1 ] {\displaystyle [0,1]} taking the value of one on the rationals, and zero ...
In mathematics, the Dirichlet function [1] [2] is the indicator function of the set of rational numbers, i.e. () = if x is a rational number and () = if x is not a rational number (i.e. is an irrational number).
Indicator function: maps x to either 1 or 0, depending on whether or not x belongs to some subset. Step function: A finite linear combination of indicator functions of half-open intervals. Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the Dirac delta function. Sawtooth wave; Square wave ...
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. An example of step functions (the red graph).
Monotone class theorem for functions — Let be a π-system that contains and let be a collection of functions from to with the following properties: If A ∈ A {\displaystyle A\in {\mathcal {A}}} then 1 A ∈ H {\displaystyle \mathbf {1} _{A}\in {\mathcal {H}}} where 1 A {\displaystyle \mathbf {1} _{A}} denotes the indicator function of A ...