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

**Sample space:**A nonempty set called the*sample space*, which represents all possible outcomes.**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.**Probability function:**A function, , that assigns probabilities to the events in . This will sometimes be referred to as a*probability distribution*over .

- for all .
- .
- if , for all .

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