KNOWPIA
WELCOME TO KNOWPIA

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.

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 *A*_{1}, ..., *A*_{n} ∈ Σ be a sequence of disjoint measurable sets, and let *a*_{1}, ..., *a*_{n} be a sequence of real or complex numbers. A *simple function* is a function of the form

where is the indicator function of the set *A*.

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 .

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

if all summands are finite.

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 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 is the restriction of the Borel σ-algebra to . The proof proceeds as follows. Let be a non-negative measurable function defined over the measure space . For each , subdivide the co-domain of into intervals, of which have length . That is, for each , define

- for , and ,

which are disjoint and cover the non-negative real line ( ).

Now define the sets

- for

which are measurable ( ) because is assumed to be measurable.

Then the increasing sequence of simple functions

converges pointwise to as . Note that, when is bounded, the convergence is uniform.