Is normalizing $P(\Omega)=1$ in probability theory arbitrary?

Question

When I first learned probability theory I was told that under the Kolmogorov axioms, normalizing the probability of the whole sample space to $1$ was arbitrary, and any other positive real number would have worked. I later came across Cox's theorem and alternative Bayesian constructions of probability theory, where I seem to remember a proof that the probability of the sample space must sum to $1$, but now I can't find that proof. Is this true or is my memory failing me? If it is, could someone provide a reference?

Answer 1

Yes it is.

Probability measures are defined to be positive non-trivial measures with $P(\Omega)=1$.

If you would define them as measures with e.g. $P(\Omega)=2$ then there is only one obstacle: inconvenience.

For instance two events $A, B$ will be defined to be independent iff $$\frac12P(A\cap B)=\frac12P(A)\frac12P(B)$$ or equivalently: $$2P(A\cap B)=P(A)P(B)$$

I do not believe in the existence of a proof that $P(\Omega)=1$ is somehow necessary.