probability space

A probability space consists of:

1. A set $\Omega$ of outcomes aka possible worlds; these represent all the ways the world might be. This is the 'sample space'.
2. A set of events $\mathcal{F}$, where each event $A\in\mathcal{F}$ is some subset of possible worlds. This is sometimes called the 'information set' since it encodes any limits on the information available for us to distinguish between possible worlds. This set must satisfy some technical conditions, namely that $\mathcal{F}$ is a sigma-algebra on the sample space $\Omega$:
   1. The information set must contain the sample space: $\Omega \in \mathcal{F}$.
   2. It must be closed under complements: for any event $A \in \mathcal{F}$, $\mathcal{F}$ must also contain the complement $(\Omega \backslash A)$.
   3. It must be closed under countable unions: for any $A, B\in \mathcal{F}$, we also have $(A \cup B) \in \mathcal{F}$.
3. A probability function $\mathbb{P}(A)$ that assigns values in $[0, 1]$ to each event $A\in\mathcal{F}$. These must be countably additive: the probability of the union of two disjoint events must equal the sum of their individual probabilities. Furthermore, we must have $\mathbb{P}(\Omega) = 1$, i.e., the total probability of all outcomes is 1.

Intuitively we would usually take $\mathcal{F} = 2^\Omega$ to include all elementary outcomes $\omega \in \Omega$, i.e., to assign a probability to each possible world, and thus (by countable additivity) to any collection of possible worlds. This is called a complete probability space.

The general reason to choose a different (incomplete) $\mathcal{F}$ is to model incomplete information. For example, the outcomes in $\Omega$ might correspond to realizations of a stochastic process. In reality, we only observe the values of the process up to the current time $t$, so at time $t$ it makes sense to work with $\mathcal{F}_t$ in which the events are equivalence classes of realizations ignoring any differences at times $> t$. An increasing sequence of such information sets $\mathcal{F}_t$ is called a filtration.