Sigma Field

Motivation

We can not define a uniform distribution (Borel Measure) on $2^{[0,1]}$. Thus, we compromise by assigning probabilities to only a certain collection of subsets of $\Omega$.

Actually, it is not possible to construct any probability space on $2^{[0,1]}$, with the property that $\mathbb{P}(\{x\})=0$ for every $x\in (0,1]$.

Definition

A set $\mathcal{F}\in 2^{\Omega }$ is a $\sigma$-field if:

1. $\varnothing \in \mathcal{F}$
2. $\forall A\in \mathcal{F},A^C\in \mathcal{F}$
3. $\forall$ countable sequence $(A_n{)}_{n\ge 1}$, ${\cup }_{n\ge 1}{A}_n\in \mathcal{F}$.

Notation

A set $A$ that belongs to $\mathcal{F}$ is called an event, an $\mathcal{F}$-measurable set, or simply a measurable set. The pair $(\Omega ,\mathcal{F})$ is called a measurable space.

• A measurable space does not require a measure, in contrast to a measure space.

Examples

1. $2^{\Omega }$
   • This is the conventional $\sigma$-field for discrete Probability Space
2. $\{\varnothing ,\Omega \}$
3. Borel Sigma Field

Properties

• For any collection of $\sigma$-fields $\{{\mathcal{F}}_i{\}}_{i\in \mathcal{I}}$, ${\cap }_{i\in \mathcal{I}}{\mathcal{F}}_i$ is a $\sigma$-field.
  • Thus, for any $C\in 2^{\Omega }$, $\sigma (C):={\cap }_{C\in {\mathcal{F}}_i,{\mathcal{F}}_i\text{ is a }\sigma \text{-field}}{\mathcal{F}}_i$ is well-defined. It is the smallest $\sigma$-field containing $C$.