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

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