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.

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\colon 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 \mu is defined on the space (X, \Sigma), the integral of a simple function f\colon X \to \mathbb R with respect to \mu is defined to be

\int_X f d \mu = \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.

See also

Bochner measurable function

References

  • . Introduction to Measure and Probability, 1966, Cambridge.
  • . Real and Functional Analysis, 1993, Springer-Verlag.
  • . Real and Complex Analysis, 1987, McGraw-Hill.
  • . Real Analysis, 1968, Collier Macmillan.