A probability space is a triplet $(\Omega, F, P)$, where $\Omega$ is a set, $F$ is a $\sigma$-algebra on $\Omega$ and $P$ is a probability measure on $(\Omega, F)$.
However, do we need $\Omega$?
What if we define as follows:
A probability set $G$ is a set of propositions which satisfies
- $\bot \in G$
- If $p \in G$, then $\lnot p \in G$
- If $p_1, p_2, \dots \in G$, then $\lor_{i=1}^\infty p_i \in G$
A probalitiy function $P$ is a function from $G$ to $[0, 1]$ which satisfies
- $P(\top)=1$
- $P(\lor_{i=1}^\infty p_i) = \sum_{i=1}^\infty P(p_i)$ if $p_i \land p_j$ is false when $i \neq j$
Yes. If you think about it, you need $\Omega$ and $P$ first. Or else for a given $p\in\Omega$, $\neg p$ does not make sense. Since $$\neg p = \{q\in\Omega | q \neq p\}$$ The $\sigma$-algebra $F$ is in some sense "What events in $\Omega$ satisfy conditions of $P$"