# nLab zero-one law

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

In probability theory, zero-one laws are conditions for which a certain probability measure can only assume the values zero and one (i.e. it is a zero-one measure).

Sometimes they can be seen as conditions for which a probability measure, usually an iid one, is ergodic (that is, it is zero-one on the invariant sets of some action).

## Statements

### Hewitt-Savage zero-one law

Let $(X,p)$ be a probability space. Consider the infinite product $X^\mathbb{N}$, and form the iid measure $\tilde{p}$ on $X^\mathbb{N}$ whose marginals are given by $p$.

Then for every exchangeable event $A\subseteq X^\mathbb{N}$ we have that either $\tilde{p}(A)=0$ or $\tilde{p}(A)=1$. That is, $\tilde{p}$ restricted to the exchangeable sigma-algebra is a zero-one measure.

(The equivalent characterizations of zero-one measures give equivalent formulations of this statement.)

### Kolmogorov’s zero-one law

Let $(X,p)$ be a probability space. Consider the infinite product $X^\mathbb{N}$, and form the iid measure $\tilde{p}$ on $X^\mathbb{N}$ whose marginals are given by $p$.

Then for every tail event $A\subseteq X^\mathbb{N}$ we have that either $\tilde{p}(A)=0$ or $\tilde{p}(A)=1$. That is, $\tilde{p}$ restricted to the tail sigma-algebra is a zero-one measure.

(The equivalent characterizations of zero-one measures give equivalent formulations of this statement.)

### Levy’s zero-one law

See martingale convergence theorem?.

## In categorical probability

### Markov categories

(For now, see Fritz-Rischel’20.)

### Dagger categories

(For now, see Ensarguet-Perrone’23.)