Simple function

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.

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 "A"₁, ..., "A"_"n" ∈ Σ be a sequence of 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

:$f(x)=sum\_\{k=1\}^n\; a\_k\; \{mathbf\; I\}\_\{A\_k\}(x).$

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 $fcolon\; X\; omathbb\{R\}^\{+\}$ isthe pointwise limit of a monotonic increasing sequence of non-negative simple functions.Indeed, let $f$ be a non-negative measurable function defined over a measurespace $(Omega,\; \{mathcal\; F\},mu)$. For each $ninmathbb\; N$,we subdivide the range of $f$ into $2^\{2n\}+1$ intervalsof length $2^\{-n\}$. We set $I\_\{n,k\}=left\; [frac\{k-1\}\{2^n\},frac\{k\}\{2^n\}\; ight)$ 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$f\_n=sum\_\{k=1\}^\{2^\{2n\}+1\}frac\{k-1\}\{2^n\}\{mathbf\; I\}\_\{A\_\{n,k$converges pointwise to $f$ as $n\; oinfty$.

Note that when $f$ is bounded the convergence is uniform.

Integration of simple functions

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

:$sum\_\{k=1\}^na\_kmu(A\_k),$if all summands are finite.

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.