Probability distribution

A probability distribution is defined by the tupel $$(\Omega,\mathcal{S},P)$$ where
 * $$\Omega$$ is the set of all possible results of an random experiment,
 * $$\mathcal{S}$$ is $\sigma$-algebra as special subset of power set of $$\Omega$$, to which a probility can be assigned.
 * $$P:\mathcal{S}\rightarrow [0,1]$$ is the probability measure on the $$\sigma$$-algebra.

σ-Algebra
The concept of the sigma-algebra had to introduced in the probability theory, because there exists a /measurment problem/.