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Simple function

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Title: Simple function  
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Subject: Freudenthal spectral theorem, Ba space, Henri Lebesgue, Lebesgue integration, Bochner integral
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Simple function

In the mathematical field of real analysis, a simple function is a real-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.) Note also that 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.

Contents

  • Definition 1
  • Properties of simple functions 2
  • Integration of simple functions 3
  • Relation to Lebesgue integration 4
  • References 5

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, ..., An ∈ Σ be a sequence of disjoint measurable sets, and let a1, ..., an 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 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.

Relation to Lebesgue integration

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 the measure space (X, \Sigma,\mu) as before. For each n\in\mathbb N, subdivide the range of f into 2^{2n}+1 intervals, 2^{2n} of which have length 2^{-n}. For each n, 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).

(Note that, for fixed n, the sets I_{n,k} are disjoint and cover the non-negative real line.)

Now 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. This approximation of f by simple functions (which are easily integrable) allows us to define an integral f itself; see the article on Lebesgue integration for more details.

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.
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