Let be a nonempty set, a -algebra on , and a measure defined on such that . The triple is called a **probability space**; is the **sample space**, the set of **events**, and the **probability measure**. For an event , we call the **probability** of .

If is a probability space and a measurable space, a measurable function is called a **random element**. In particular, when , is called a **random variable**. For any , we define

The **expectation** of is defined by

The **cumulative distribution function** of is defined by for .

Theorem 1Let the distribution function of a random variable . Then is nondecreasing, is right-continuous, , and .

*Proof:* If , then , so .

If , let be a sequence of real numbers such that , , and . Then

Let be a sequence of real numbers such that and . Then

Let be a sequence of real numbers such that and . Then

Theorem 2If is a random variable with and , then

*Proof:* Using Tonelli’s theorem to justify the interchange in order of integration, we have

Theorem 3If is a random variable with and , then

*Proof:* Similar computation as above.

Theorem 4If is a random variable with then

*Proof:* Using integration by parts, we have

Note that

and similarly

so the limits used in the above computation are justified.