Simple function

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Short description: Function that attains finitely many values

In the mathematical field of real analysis, a simple function is a real (or complex)-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.) 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 [math]\displaystyle{ f: X \to \mathbb{C} }[/math] of the form

[math]\displaystyle{ f(x)=\sum_{k=1}^n a_k {\mathbf 1}_{A_k}(x), }[/math]

where [math]\displaystyle{ {\mathbf 1}_A }[/math] 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 [math]\displaystyle{ \mathbb{C} }[/math].

Integration of simple functions

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

[math]\displaystyle{ \sum_{k=1}^na_k\mu(A_k), }[/math]

if all summands are finite.

Relation to Lebesgue integration

The above integral of simple functions can be extended to a more general class of functions, which is how the Lebesgue integral is defined. This extension is based on the following fact.

Theorem. Any non-negative measurable function [math]\displaystyle{ f\colon X \to\mathbb{R}^{+} }[/math] is the pointwise limit of a monotonic increasing sequence of non-negative simple functions.

It is implied in the statement that the sigma-algebra in the co-domain [math]\displaystyle{ \mathbb{R}^{+} }[/math] is the restriction of the Borel σ-algebra [math]\displaystyle{ \mathfrak{B}(\mathbb{R}) }[/math] to [math]\displaystyle{ \mathbb{R}^{+} }[/math]. The proof proceeds as follows. Let [math]\displaystyle{ f }[/math] be a non-negative measurable function defined over the measure space [math]\displaystyle{ (X, \Sigma,\mu) }[/math]. For each [math]\displaystyle{ n\in\mathbb N }[/math], subdivide the co-domain of [math]\displaystyle{ f }[/math] into [math]\displaystyle{ 2^{2n}+1 }[/math] intervals, [math]\displaystyle{ 2^{2n} }[/math] of which have length [math]\displaystyle{ 2^{-n} }[/math]. That is, for each [math]\displaystyle{ n }[/math], define

[math]\displaystyle{ I_{n,k}=\left[\frac{k-1}{2^n},\frac{k}{2^n}\right) }[/math] for [math]\displaystyle{ k=1,2,\ldots,2^{2n} }[/math], and [math]\displaystyle{ I_{n,2^{2n}+1}=[2^n,\infty) }[/math],

which are disjoint and cover the non-negative real line ([math]\displaystyle{ \mathbb{R}^{+} \subseteq \cup_{k}I_{n,k}, \forall n \in \mathbb{N} }[/math]).

Now define the sets

[math]\displaystyle{ A_{n,k}=f^{-1}(I_{n,k}) \, }[/math] for [math]\displaystyle{ k=1,2,\ldots,2^{2n}+1, }[/math]

which are measurable ([math]\displaystyle{ A_{n,k}\in \Sigma }[/math]) because [math]\displaystyle{ f }[/math] is assumed to be measurable.

Then the increasing sequence of simple functions

[math]\displaystyle{ f_n=\sum_{k=1}^{2^{2n}+1}\frac{k-1}{2^n}{\mathbf 1}_{A_{n,k}} }[/math]

converges pointwise to [math]\displaystyle{ f }[/math] as [math]\displaystyle{ n\to\infty }[/math]. Note that, when [math]\displaystyle{ f }[/math] is bounded, the convergence is uniform.

References

  • J. F. C. Kingman, S. J. Taylor. Introduction to Measure and Probability, 1966, Cambridge.
  • S. Lang. Real and Functional Analysis, 1993, Springer-Verlag.
  • W. Rudin. Real and Complex Analysis, 1987, McGraw-Hill.
  • H. L. Royden. Real Analysis, 1968, Collier Macmillan.