Simple function

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 $$f: X \to \mathbb{C}$$ of the form


 * $$f(x)=\sum_{k=1}^n a_k {\mathbf 1}_{A_k}(x),$$

where $${\mathbf 1}_A$$ 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 $$\mathbb{C}$$.

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


 * $$\sum_{k=1}^na_k\mu(A_k),$$

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 $$f\colon X \to\mathbb{R}^{+}$$ 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 $$\mathbb{R}^{+}$$ is the restriction of the Borel σ-algebra $$\mathfrak{B}(\mathbb{R})$$ to $$\mathbb{R}^{+}$$. The proof proceeds as follows. Let $$f$$ be a non-negative measurable function defined over the measure space $$(X, \Sigma,\mu)$$. For each $$n\in\mathbb N$$, subdivide the co-domain of $$f$$ into $$2^{2n}+1$$ intervals, $$2^{2n}$$ of which have length $$2^{-n}$$. That is, for each $$n$$, define
 * $$I_{n,k}=\left[\frac{k-1}{2^n},\frac{k}{2^n}\right)$$ for $$k=1,2,\ldots,2^{2n}$$, and $$I_{n,2^{2n}+1}=[2^n,\infty)$$,

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

Now define the sets
 * $$A_{n,k}=f^{-1}(I_{n,k}) \,$$ for $$k=1,2,\ldots,2^{2n}+1,$$

which are measurable ($$A_{n,k}\in \Sigma$$) because $$f$$ is assumed to be measurable.

Then the increasing sequence of simple functions
 * $$f_n=\sum_{k=1}^{2^{2n}+1}\frac{k-1}{2^n}{\mathbf 1}_{A_{n,k}}$$

converges pointwise to $$f$$ as $$n\to\infty$$. Note that, when $$f$$ is bounded, the convergence is uniform.