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, 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.

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 disjoint measurable sets, and let a₁, ..., a_n be a sequence of real or complex numbers. A simple function is a function $f:X\to \mathbb {C}$ of the form

f(x)=\sum _{k=1}^{n}a_{k}{\mathbf {1} }_{A_{k}}(x),

where ${\mathbf {1} }_{A}$ 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 $\mathbb {C}$ .

Integration of simple functions

If a measure $\mu$ is defined on the space $(X,\Sigma )$ , the integral of a simple function $f:X\to \mathbb {R}$ with respect to $\mu$ is defined to be

\int _{X}fd\mu =\sum _{k=1}^{n}a_{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 {R} ^{+}

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 $\mathbb {R} ^{+}$ is the restriction of the Borel σ-algebra ${\mathfrak {B}}(\mathbb {R} )$ to $\mathbb {R} ^{+}$ . 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 $2^{2n}+1$ intervals, $2^{2n}$ of which have length $2^{-n}$ . That is, for each $n$ , define

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 )

,

which are disjoint and cover the non-negative real line ( $\mathbb {R} ^{+}\subseteq \cup _{k}I_{n,k},\forall n\in \mathbb {N}$ ).

Now define the sets

A_{n,k}=f^{-1}(I_{n,k})\,

for

k=1,2,\ldots ,2^{2n}+1,

which are measurable ( $A_{n,k}\in \Sigma$ ) because $f$ is assumed to be measurable.

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.

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.