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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.)
By the very definition of the indicator function, we have that the indicator of the product of two functions does not exceed the sum of the indicators: [2]: 51–52 () + (). Similarly, the indicator of the sum of two functions does not exceed the larger of the two indicators: h f + g ( θ ) ≤ max { h f ( θ ) , h g ( θ ) } . {\displaystyle h ...
When a function is periodic, with ... is the indicator function of . This agrees ... Convolution and related operations are found in many applications in science ...
Indicator function – Mathematical function characterizing set membership; Linear discriminant function – Method used in statistics, pattern recognition, and other fields; Multicollinearity – Linear dependency situation in a regression model; One-hot – Bit-vector representation where only one bit can be set at a time
A simple function can be written in different ways as a linear combination of indicator functions, but the integral will be the same by the additivity of measures. Some care is needed when defining the integral of a real-valued simple function, to avoid the undefined expression ∞ − ∞ : one assumes that the representation
The Dirichlet function, which is the indicator function for rationals, is a bounded function that is not Riemann integrable. The Cantor function is a monotonic continuous surjective function that maps [ 0 , 1 ] {\displaystyle [0,1]} onto [ 0 , 1 ] {\displaystyle [0,1]} , but has zero derivative almost everywhere .
Ecological indicator of ecosystem health (ecological processes) Health indicator, which is used to describe the health of a population; Honeyguides, also known as "indicator birds", a family of Old World tropical birds Indicator, a genus of birds in the honeyguide family; Indicator species, a species that defines a characteristic of an environment
Formally, a simple function is a finite linear combination of indicator functions of measurable sets.More precisely, let (X, Σ) be a measurable space.Let A 1, ..., A n ∈ Σ be a sequence of disjoint measurable sets, and let a 1, ..., a n be a sequence of real or complex numbers.