Simple function
Encyclopedia
In the mathematical
field of real analysis
, a simple function is a (sufficiently 'nice' - see below for the formal definition) real
-valued function over a subset of the real line
which attains 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 function
s 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 very easy to create a definition of an integral for a simple function, and also, it is straightforward to approximate more general functions by sequences of simple functions.
of indicator functions of measurable sets. More precisely, let (X, Σ) be a measurable space
. Let A1, ..., An ∈ Σ be a sequence
of measurable sets, and let a1, ..., an be a sequence of real
or complex number
s. A simple function is a function
of the form
where
is the indicator function of the set A.
.
μ is defined on the space (X,Σ), the integral of f with respect to μ is
if all summands are finite.
is the pointwise
limit of a monotonic increasing sequence of non-negative simple functions. Indeed, let
be a non-negative measurable function defined over the measure space 
as before. For each 
, subdivide the range of 
into 
intervals, 
of which have length 
. For each 
, set
for 
, and 
.
(Note that, for fixed
, the sets 
are disjoint and cover the non-negative real line.)
Now define the measurable sets
for 
.
Then the increasing sequence of simple functions
converges pointwise to
as 
. Note that, when 
is bounded, the convergence is uniform. This approximation of 
by simple functions (which are easily integrable) allows us to define an integral 
itself; see the article on Lebesgue integration
for more details.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
, a simple function is a (sufficiently 'nice' - see below for the formal definition) real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued function over a subset of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
which attains only a finite number of values. Some authors also require simple functions to be measurable
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
; as used in practice, they invariably are.
A basic example of a simple function is the floor function
Floor function
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...
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 function
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...
s are simple.
Simple functions are used as a first stage in the development of theories of integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, such as the Lebesgue integral, because it is very easy to create a definition of an integral 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 combinationLinear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of indicator functions of measurable sets. More precisely, let (X, Σ) be a measurable space
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
. Let A1, ..., An ∈ Σ be a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of measurable sets, and let a1, ..., an be a sequence of real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. A simple function is a function
of the formwhere
is the indicator function of the set A.Properties of simple functions
By definition, 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 measureMeasure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
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
Any non-negative measurable function is the pointwisePointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f of some function f. An important class of pointwise concepts are the pointwise operations — operations defined on functions by applying the operations to function values...
limit of a monotonic increasing sequence of non-negative simple functions. Indeed, let
be a non-negative measurable function defined over the measure space as before. For each , subdivide the range of into intervals, of which have length . For each , set for , and .(Note that, for fixed
, the sets are disjoint and cover the non-negative real line.)Now define the measurable sets
for .Then the increasing sequence of simple functions
converges pointwise to
as . Note that, when is bounded, the convergence is uniform. This approximation of by simple functions (which are easily integrable) allows us to define an integral itself; see the article on Lebesgue integrationLebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
for more details.