Probability Theory

<ul>
    <li>
        Sample Space $\Omega$: the SET of possible outcomes or observations.
        Note the 'type' of $\Omega$ - it is a set of uniterpreted items $SET_0$.
        elements (or observations) of this set are labeled $\omega \in \Omega$
    </li>
    <li>
        Event Space $\mathcal{F}$: the SET of events $A \in \mathcal{F}$ and $A \subseteq \Omega$.
        Note the 'type' of $\mathcal{F}$ - it is a set of sets (a set of subsets of $\Omega$) $SET_1$.
        The Event Space must have the following properties:
        <ol>
          <li>
            $\emptyset \in \mathcal{F}$ (the empty set is in $\mathcal{F}$)
          </li>
          <li>
            $A \in \mathcal{F} \Rightarrow \Omega \setminus A \in \mathcal{F}$
            (if $A$ is an event in $\mathcal{F}$ then the relative complement of $A$ with respect to $\Omega$ is also in $\mathcal{F}$)
          </li>
        </ol>
    </li>
    <li>
        Probability Measure $P : \mathcal{F} \to \Bbb{R}$: maps an event in the event space to a real number
    </li>

</ul>