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.