Probability Axioms (Kolmogorov Axioms)ΒΆ

Let the set of all possible outcomes of an experiment be \(\Omega\), and let events be defined as measurable subsets, \(\omega\subset\Omega\). Then a measure \(\mu:2^{|\Omega|}\mapsto\mathbb{R}\) is called a probability measure iff

  1. Non-negativity: \(\mu(\omega)\ge 0\) for any \(\omega\subset\Omega\).

  2. Unitarity: \(\mu(\Omega)=1\).

  3. \(\sigma\)-Additivity: For \(A_1,A_2,\cdots\subset\Omega\) such that \(A_i\cap A_j=\emptyset\) for \(i\neq j\)

\[\mu(\bigcup_{i=1}^\infty A_i)=\sum_{i=1}^\infty \mu(A_i).\]

Tip

It is customary to represent probability measure as \(\mathbb{P}\).