Among the three axioms of the probability theory, the following is said to be the normalization axiom
$$p(\Omega) = 1$$
It states that the probability of the entire sample space ($\Omega$) is equal to one.
I want to know the rationale behind selecting the number "1". Why not some other constant $c$?
If we take $\mathsf{P}(\Omega)=c>0$, then we can equivalently work with $\mathsf{P}'$ defined as $\mathsf{P}'(A):=\mathsf{P}(A)/c$. Besides that, taking $c=1$ is useful when defining product spaces because $$ \mathsf{P}_2(\Omega\times \Omega)=\mathsf{P}(\Omega)\mathsf{P}(\Omega)=1. $$ (Consider the issue when defining probabilities on the infinite product $\Omega^{\mathbb{N}}$ if $c\ne 1$.)