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 , 9), whose only values are . 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">integral">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 sigma-algebra, 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 $f\colon X \to \mathbb$ of the form

:$f(x)=\sum_^n a_k _(x),$

where $_A$ 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 $\mathbb$.

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

:$\int_X f d \mu = \sum_^na_k\mu(A_k),$
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 $f\colon X \to\mathbb^$ 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 $\mathbb^$ is the restriction of the Borel σ-algebra $\mathfrak(\mathbb)$ to $\mathbb^$. 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 $2^+1$ intervals, $2^$ of which have length $2^$. That is, for each $n$, define
:$I_=\left[\frac,\frac\right)$ for $k=1,2,\ldots,2^$, and $I_=[2^n,\infty)$,

which are disjoint and cover the non-negative real line ($\mathbb^ \subseteq \cup_I_, \forall n \in \mathbb$).

Now define the sets
:$A_=f^(I_) \,$ for $k=1,2,\ldots,2^+1,$
which are measurable ($A_\in \Sigma$) because $f$ is assumed to be measurable.

Then the increasing sequence of simple functions
:$f_n=\sum_^\frac_$

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.

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Real analysis
Measure theory
Types of functions