Is conditional probability unique?

Question

Suppose that we are given some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and some event $B$ with $\mathbb{P}[B]>0$. Then one can construct a new probability space $(\Omega, \mathcal{F}',\mathbb{P}')$ where every set $A\in \mathcal{F}$ is replaced with $A\cap B\in\mathcal{F}'$. Sure, there are many possible probability measures on this new $\sigma$-algebra: for any $C\in\mathcal{F}'$ there is $A\in\mathcal{F}$, such that $C = A\cap B$, so $\mathbb{P}'[C] = \mathbb{P}'[A\cap B] = \alpha(A)\mathbb{P}[A\cap B]$. We want $\alpha(\cdot)$ to obey some properties, so that $\mathbb{P}'$ is a probability measure. In particular, $\alpha(B) = 1/\mathbb{P}[B]$. The easiest way is to take $\alpha \equiv 1/\mathbb{P}[B]$, which makes $\mathbb{P}'$ a probability measure. I wonder if there is any kind of "natural" requirements, under which the conditional probability is unique (with $\alpha \equiv 1/\mathbb{P}[B]$)?