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In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, a simple function is a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
(or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
)-valued function over a subset of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
, similar to a
step function In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having onl ...
. 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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
; as used in practice, they invariably are. A basic example of a simple function is the
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
over the half-open interval ,_9),_whose_only_values_are_._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_function_ In_mathematics,_a__function_on_the_real_numbers_is_called_a_step_function_if_it_can_be_written_as_a__finite__linear_combination_of_indicator_functions_of__intervals._Informally_speaking,_a_step_function_is_a_piecewise_constant_function_having_onl_...
s_are_simple. Simple_functions_are_used_as_a_first_stage_in_the_development_of_theories_of_integral.html" ;"title="Dirichlet_function.html" ;"title=", 9), whose only values are . A more advanced example is the Dirichlet function">, 9), whose only values are . 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 function In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having onl ...
s are simple. Simple functions are used as a first stage in the development of theories of integral">integration, such as the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
, 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 function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
s of measurable sets. More precisely, let (''X'', Σ) be a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...
. Let ''A''1, ..., ''A''''n'' ∈ Σ be a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of disjoint measurable sets, and let ''a''1, ..., ''a''''n'' be a sequence of
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. A ''simple function'' is a function f: X \to \mathbb of the form :f(x)=\sum_^n a_k _(x), where _A is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
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 Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
over \mathbb.


Integration of simple functions

If a measure μ is defined on the space (''X'',Σ), the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of ''f'' with respect to μ is :\sum_^na_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 In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
is defined. This extension is based on the following fact. : Theorem. Any non-negative
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
f\colon X \to\mathbb^ is the
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
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^ is the restriction of the
Borel σ-algebra In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
\mathfrak(\mathbb) to \mathbb^. 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^+1 intervals, 2^ of which have length 2^. That is, for each n, define :I_=\left[\frac,\frac\right) for k=1,2,\ldots,2^, and I_=[2^n,\infty), which are disjoint and cover the non-negative real line (\mathbb^ \subseteq \cup_I_, \forall n \in \mathbb). Now define the sets :A_=f^(I_) \, for k=1,2,\ldots,2^+1, which are measurable (A_\in \Sigma) because f is assumed to be measurable. Then the increasing sequence of simple functions :f_n=\sum_^\frac_ converges pointwise to f as n\to\infty. Note that, when f is bounded, the convergence is uniform.


References

*. ''Introduction to Measure and Probability'', 1966, Cambridge. *. ''Real and Functional Analysis'', 1993, Springer-Verlag. *. ''Real and Complex Analysis'', 1987, McGraw-Hill. *. ''Real Analysis'', 1968, Collier Macmillan. {{DEFAULTSORT:Simple Function Real analysis Measure theory Types of functions