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Simple function

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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.

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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.

Definition

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 of the form

f ( x ) = ∑ k = 1 n a k 1 A k ( x ) ,

where 1 A is the indicator function of the set A.

Properties of simple functions

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 .

Integration of simple functions

If a measure μ is defined on the space (X,Σ), the integral of f with respect to μ is

∑ k = 1 n a k μ ( A k ) ,

if all summands are finite.

Relation to Lebesgue integration

Any non-negative measurable function f : X → R + is the pointwise limit of a monotonic increasing sequence of non-negative simple functions. Indeed, let f be a non-negative measurable function defined over the measure space ( X , Σ , μ ) as before. For each n ∈ N , subdivide the range of f into 2 2 n + 1 intervals, 2 2 n of which have length 2 − n . For each n , set

I n , k = [ k − 1 2 n , k 2 n ) for k = 1 , 2 , … , 2 2 n , and I n , 2 2 n + 1 = [ 2 n , ∞ ) .

(Note that, for fixed n , the sets I n , k are disjoint and cover the non-negative real line.)

Now define the measurable sets

A n , k = f − 1 ( I n , k ) for k = 1 , 2 , … , 2 2 n + 1 .

Then the increasing sequence of simple functions

f n = ∑ k = 1 2 2 n + 1 k − 1 2 n 1 A n , k

converges pointwise to f as n → ∞ . Note that, when f is bounded, the convergence is uniform. This approximation of f by simple functions (which are easily integrable) allows us to define an integral f itself; see the article on Lebesgue integration for more details.

References

Simple function Wikipedia
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