A
probability space on a
set X is a
measure space (which see) (X,
B,P) for which P(X)=1.
One oddity of probability theory is that all objects of Analysis have different names. In this case, the measure P is called a probability, and the sets A∈B (usually known as measurable sets) are called the events in the probability space. These are the sets for which a probability is defined.
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X={1,2,3,4,5,6}, B is all 64 subsets of X, and P(A)=|A|/6 (the probability of a set is its size divided by 6).
We can verify (X,B,P) is a probability space. It is commonly used to describe a single roll of a fair die. The event "the die comes up even" is the set {2,4,6}∈B; its probability is 1/2.
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X=R (the real numbers), B = the σ algebra of Lebesgue sets (a technical detail; apart from noting not every subset of R is in B, just ignore it), and
P(A) = ∫-∞+∞ 1A(x) 1/(2π)1/2 exp(-x2/2) dx,
where the characteristic function of A is 1A(x)=1 for x∈A and =0 otherwise.
This describes a random variable which follows the normal distribution.