Let be a nonempty set. A collection of subsets is said to be a -algebra if

- implies
- If is a countable sequence of sets in , then .

If , the **-algebra generated by ** is the intersection of all -algebras containing :

If is a nonempty set and is a -algebra, the pair is called a **measurable space**. A function is said to be a **measure** if the following conditions hold:

- If is a countable sequence of
*disjoint*sets in , then .

If is a measurable space and is a measure defined on , the triple is called a **measure space**.

Theorem 1If and , then .

*Proof:*

Theorem 2If then

*Proof:* Put and for . Then for any , , and for any , . Hence

as .

Theorem 3If and for then

*Proof:* Put and for . Then for any , , and for any , . Hence

Let be a measure space, and be a measurable space. A function is said to be **measurable** if for all , .

Theorem 4If , then is measurable if and only if for all .

*Proof:* The set is a -algebra that contains .

The **Borel -algebra** on is the -algebra generated by the open sets. If , the **characteristic function** of is the function defined by

A **simple function** is a function of the form

where and . For a simple function , define the **Lebesgue integral** by

Theorem 5If is Borel-measurable, there exists a sequence of simple functions such that and converges pointwise to .

*Proof:* For each nonnegative integer and each nonnegative integer , define

Define

Observe that for each ,

and this union is disjoint. If then

so that

If then

for some , so that

If , then

If , then . Otherwise, for sufficiently large , , from which it follows that .

For , define the integral of by the supremum of simple functions less than :

For define the **positive** and **negative parts** of by

Then where and , so we can define the integral of by

For , the integral is defined as