Simple function

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In mathematics, especially in mathematical analysis, a simple function is a measurable function whose range is finite.

Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, due to the fact that it is very easy to create a definition of an integral for a simple function, and also, it is straightforward to approximate more general functions by sequences of simple functions.

1 Examples
2 Definition
3 Properties of simple functions
4 Integration of simple functions
5 References

[edit] Examples

Consider a function of a real variable x. Define f(x) = 0 when x is 0, f(x)=−1 when x is negative, and f(x)=1 when x is positive. Then f is a simple function, since its range is {-1, 0, 1}, which is a finite set, and one can check that this function is measurable on the usual space of Lebesgue measurable sets.

Another example is the indicator function of the rational numbers, which takes the value 1 on the measurable set $\mathbb{Q}$ and the value 0 on the measurable set $\mathbb{R} \setminus \mathbb{Q}.$

[edit] 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 measurable sets, and let a₁, ..., a_n be a sequence of real or complex numbers. A simple function is a function of the form

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

[edit] Properties of simple functions

By definition, sum, difference, and product of two simple functions is again a simple function, as well multiplication by constant, hence it follows that the collection of all simple functions forms a commutative algebra over the complex field.

For the development of a theory of integration, the following result is important. 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. Indeed, let $f$ be a non-negative measurable function defined over a measure space $(\Omega, {\mathcal F},\mu)$ . For each $n\in\mathbb N$ , we subdivide the range of $f$ into $2 2 n + 1$ intervals of length $2 - n$ . We set $I_{n,k}=[\frac{k-1}{2^n},\frac{k}{2^n})$ for $k=1,2,\ldots,2^{2n}$ and $I_{n,2^{2n}+1}=[2^n,\infty]$ . We define the measurable sets $A n, k = f - 1 (I n, k)$ for $k=1,2,\ldots,2^{2n}+1$ . 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.

[edit] 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.

[edit] 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.