Sample Space and Probability Space
Probability Measure
A probability measure
on a measurable space is a measure that .
Probability Space
A probability space
is a measure space where
is called a sample space, which is the set of all possible outcomes. An event is a subset of ; is called an event space, which is a -algebra over . is a a probability measure, which assigns, to each event a probability.
Random Variable
Random Variable and Probability
Let
be a probability space, and a measurable space. Then an -valued random variable is a measurable function . Then the probability that takes on a value in a measurable set is