#### Probability space

A probability space is a three-tuple, , in which the three components are

1. Sample space: A nonempty set called the sample space, which represents all possible outcomes.
2. Event space: A collection of subsets of , called the event space. If is discrete, then usually . If is continuous, then is usually a sigma-algebra on , as defined in Section 5.1.3.
3. Probability function: A function, , that assigns probabilities to the events in . This will sometimes be referred to as a probability distribution over .
The probability function, , must satisfy several basic axioms:
1. for all .
2. .
3. if , for all .
If is discrete, then the definition of over all of can be inferred from its definition on single elements of by using the axioms. It is common in this case to write for some , which is slightly abusive because is not an event. It technically should be for some .

Example 9..4 (Tossing a Die)   Consider tossing a six-sided cube or die that has numbers to painted on its sides. When the die comes to rest, it will always show one number. In this case, is the sample space. The event space is , which is all subsets of . Suppose that the probability function is assigned to indicate that all numbers are equally likely. For any individual , . The events include all subsets so that any probability statement can be formulated. For example, what is the probability that an even number is obtained? The event has probability of occurring. The third probability axiom looks similar to the last axiom in the definition of a measure space in Section 5.1.3. In fact, is technically a special kind of measure space as mentioned in Example 5.12. If is continuous, however, this measure cannot be captured by defining probabilities over the singleton sets. The probabilities of singleton sets are usually zero. Instead, a probability density function, , is used to define the probability measure. The probability function, , for any event can then be determined via integration: (9.3)

in which is the variable of integration. Intuitively, indicates the total probability mass that accumulates over .

Steven M LaValle 2012-04-20