Simple function

In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently 'nice' that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are. In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently 'nice' that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are. A basic example of a simple function is the floor function over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the 'simple' of 'simple function' has a technical meaning somewhat at odds with common language.) Note also that all step functions are simple. Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because it is easy to define integration for a simple function and also it is straightforward to approximate more general functions by sequences of simple functions. 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 A1, ..., An ∈ Σ be a sequence of disjoint measurable sets, and let a1, ..., an be a sequence of real or complex numbers. A simple function is a function f : X → C {displaystyle f:X o mathbb {C} } of the form where 1 A {displaystyle {mathbf {1} }_{A}} is the indicator function of the set A. The sum, difference, and product of two simple functions are again simple functions, and multiplication by constant keeps a simple function simple; hence it follows that the collection of all simple functions on a given measurable space forms a commutative algebra over C {displaystyle mathbb {C} } . If a measure μ is defined on the space (X,Σ), the integral of f with respect to μ is