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, ..., A_n ∈ Σ be a sequence of disjoint measurable sets, and let a₁, ..., a_n be a sequence of real or complex 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.

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 ${\displaystyle \mathbb {C} }$ .

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

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

if all summands are finite.

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 ${\displaystyle f\colon X\to \mathbb {R} ^{+}}$  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 ${\displaystyle \mathbb {R} ^{+}}$  is the restriction of the Borel σ-algebra ${\displaystyle {\mathfrak {B}}(\mathbb {R} )}$  to ${\displaystyle \mathbb {R} ^{+}}$ . The proof proceeds as follows. Let ${\displaystyle f}$  be a non-negative measurable function defined over the measure space ${\displaystyle (X,\Sigma ,\mu )}$ . For each ${\displaystyle n\in \mathbb {N} }$ , subdivide the co-domain of ${\displaystyle f}$  into ${\displaystyle 2^{2n}+1}$  intervals, ${\displaystyle 2^{2n}}$  of which have length ${\displaystyle 2^{-n}}$ . That is, for each ${\displaystyle n}$ , define

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

which are disjoint and cover the non-negative real line (${\displaystyle \mathbb {R} ^{+}\subseteq \cup _{k}I_{n,k},\forall n\in \mathbb {N} }$ ).

Now define the sets

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

which are measurable (${\displaystyle A_{n,k}\in \Sigma }$ ) because ${\displaystyle f}$  is assumed to be measurable.

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.

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.