Set of all Events

probability-theory

Definition

Set of All Events

Let $\Omega$ be a sample space. Its set of all admissible events is a family

$$\mathfrak{S}\subseteq \mathcal{P}(\Omega )\cong 2^{\Omega }$$

that forms a sigma-algebra on $\Omega$. Each $A\in \mathfrak{S}$ is an event to which a probability may be assigned. Together with a probability measure $P:\mathfrak{S}\to [0,1]$, it forms the probability space $(\Omega ,\mathfrak{S},P)$.

Why It May Be Smaller Than the Power Set

The phrase “set of all events” means all events admitted by the probabilistic model, not necessarily every subset of $\Omega$. In finite models one often chooses $\mathfrak{S}=\mathcal{P}(\Omega )$, whereas models on uncountable sample spaces commonly use a proper subset

$$\mathfrak{S}⊊\mathcal{P}(\Omega ).$$

The required closure properties are those of a [[Knowledge/Sigma Algebra|$\sigma$-algebra]].

Example

Recording only parity

For a six-sided die, let

$$\Omega =\{1,2,3,4,5,6\},E=\{2,4,6\}.$$

The event family

$$\mathfrak{S}=\{\varnothing ,E,E^c,\Omega \}$$

records whether the outcome is even or odd but cannot express an event such as $\{1\}$. Thus $\mathfrak{S}⊊\mathcal{P}(\Omega )$.