Simple function

Summary

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

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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   of the form

 

where   is the indicator function of the set A.

Properties of simple functions

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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  .

Integration of simple functions

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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.

Relation to Lebesgue integration

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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.

See also

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Bochner measurable function

References

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  • 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.