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Simple function

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

Formally, a simple function is a finite linear combinationofindicator functionsofmeasurable sets. More precisely, let (X, Σ) be a measurable space. Let A₁, ..., A_n ∈ Σ be a sequence of disjoint measurable sets, and let a₁, ..., a_n be a sequence of realorcomplex numbers. A simple function is a function ${\displaystyle f:X\to \mathbb {C} }$ of the form

${\displaystyle f($x)=\sum _{k=1}^{n}a_{k}{\mathbf {1} }_{A_{k}}(x),}

where ${\displaystyle {\mathbf {1} }_{A}}$ 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 ${\displaystyle \mathbb {C} }$.

Integration of simple functions

If a measure μ is defined on the space (X,Σ), the integraloff with respect to μ is

${\displaystyle \sum _{k=1}^{n}a_{k}\mu (A_{k}),}$

if all summands are finite.

Relation to Lebesgue integration

Any non-negative measurable function ${\displaystyle f\colon X\to \mathbb {R} ^{+}}$ is the pointwise limit of a monotonic increasing sequence of non-negative simple functions. Indeed, let ${\displaystyle f}$ be a non-negative measurable function defined over the measure space ${\displaystyle (X,\Sigma ,\mu )}$ as before. For each ${\displaystyle n\in \mathbb {N} }$, subdivide the range of ${\displaystyle f}$ into ${\displaystyle 2^{2n}+1}$ intervals, ${\displaystyle 2^{2n}}$ of which have length ${\displaystyle 2^{-n}}$. For each ${\displaystyle n}$, set

${\displaystyle I_{n,k}=\left[{\frac {k-1}{2^{n}}},{\frac {k}{2^{n}}}\right)}$ for ${\displaystyle k=1,2,\ldots ,2^{2n}}$, and ${\displaystyle I_{n,2^{2n}+1}=[2^{n},\infty )}$.

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

Now define the measurable sets

${\displaystyle A_{n,k}=f^{-1}(I_{n,k})\,}$ for ${\displaystyle k=1,2,\ldots ,2^{2n}+1}$.

Then the increasing sequence of simple functions

${\displaystyle f_{n}=\sum _{k=1}^{2^{2n}+1}{\frac {k-1}{2^{n}}}{\mathbf {1} }_{A_{n,k}}}$

converges pointwise to ${\displaystyle f}$as${\displaystyle n\to \infty }$. Note that, when ${\displaystyle f}$ is bounded, the convergence is uniform. This approximation of ${\displaystyle f}$ by simple functions (which are easily integrable) allows us to define an integral ${\displaystyle f}$ 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.