User:NikelsenH/P-trivial σ-algebra

Distinction:Trivial sigam algebra

P-trivial σ-algebra P-trivial sigma-algebra P-trivial σ-field P-trivial sigma-field

A P-trivial σ-algebra, also called a P-trivial σ-field is a special type of σ-algebra in probability theory, a branch of mathematics.

Definition
Let $$ (\Omega, \mathcal A, P) $$ be a probability space.

Then the σ-algebra $$ \mathcal A $$ is called P-trivial if it only contains sets that have probability zero or one, e.g. if for all $$ A \in \mathcal A $$ it is either $$ P(A) =0 $$ or $$ P(A)=1 $$.

Examples
Showing that a σ-algebra is P-trivial is often difficult. Theorems that do so are often called 0-1-laws
 * Kolmogorov's zero–one law: The terminal σ-algebra of a series of i.i.d. random variables is P-trivial.
 * Hewitt–Savage zero–one law