I'm using this version of Caratheodory's Theorem:
If $(Ω,R,μ)$ is a pre-measure space, the there is $μ′$ that extends $μ$ to $σ(R)$ such that $(Ω,σ(R),μ′)$ is a measure space.
I have already proven that if $μ$ is $σ$-finite, then $μ'$ is too and it's unique.
On the other hand in class it was mentioned that if $μ$ is finite (just finite) then $μ'$ is finite too, and unique.
I understand that $μ'$ is finite because $μ'(Ω)=μ(Ω)<\infty$ where Ω is in $σ(R)$ because it's a $σ$-algebra over $Ω$.
However I'm stuck with the uniqueness part, I've already taken another $v$ that extends $μ$ but I'm completely clueless as to how to prove $μ=v$.
Any help, hints, etc?
Thanks in advance.