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 $\mu(X)=1$.

## Conditional probability

## Total probability equation, Bayes formular

## Independence

## Reference

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

🏷 Category=Probability