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.

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, ..., A_n ∈ Σ be a sequence of disjoint measurable sets, and let a₁, ..., a_n be a sequence of real or complex numbers. A simple function is a function $f:X\to ℂ$ of the form

$f(x)=\sum _{k=1}^n{a}_k{\mathrm{𝟏}}_{A_k}(x),$

where ${\mathrm{𝟏}}_A$ 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 $\mu$ is defined on the space $(X,\Sigma )$, the integral of a simple function $f:X\to ℝ$ with respect to $\mu$ is defined to be

$\underset{X}{\int }fd\mu =\sum _{k=1}^n{a}_k\mu (A_k),$

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 $f:X\to {ℝ}^{+}$ 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 ${ℝ}^{+}$ is the restriction of the Borel σ-algebra $𝔅(ℝ)$ to ${ℝ}^{+}$. 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

$I_{n,k}=[\frac{k-1}{2^n},\frac{k}{2^n})$ for $k=1,2,\dots ,2^{2n}$, and $I_{n,2^{2n}+1}=[2^n,\mathrm{\infty })$,

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

Now define the sets

$A_{n,k}=f^{-1}(I_{n,k})$ for $k=1,2,\dots ,2^{2n}+1,$

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

Then the increasing sequence of simple functions

$f_n=\sum _{k=1}^{2^{2n}+1}\frac{k-1}{2^n}{\mathrm{𝟏}}_{A_{n,k}}$

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

See also

Bochner measurable function

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.