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.



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


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


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


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


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


Bochner measurable function


  • 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.