Simple function

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In mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line which attains 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.)

Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because 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 Definition
2 Properties of simple functions
3 Integration of simple functions
4 References

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