Kolmogorov laid the foundation of axiomatic probability theory. {Kolmogorov, 1956}

## Probability space

A probability space is a measurable (sample) space assigned with a probability measure. Symbolically, it's a triplet $(\Omega, \Sigma, P)$.

Sample space is the collection of elementary events, denoted as $\Omega$.

The $\sigma$-algebra of the sample space is all the entities that can be measured, using some instrument. It is denoted as $\Sigma$.

A probability measure is a normalized finite measure. That is, the entire domain set has measure one. Symbolically, $\mu$ is a probability measure on a set $X$ if $\mu$ is a measure over a sigma-algebra $\Sigma$ of $X$ and $\mu(X)=1$.

## Reference

1. Kolmogorov AN. Foundations of the Theory of Probability (2nd english edn). Chelsea Publishing Company: New York, NY, 1956.