I am having an issue when trying to prove the following theorem:
Let $(\Omega,S,p)$ be a space of probability. Fix $B\in S$ where we define $S_B=\{A\cap B:A\in S\}$, then $(B, S_B, P_B)$ is a space of probability.
I know that for this we have to show that $S_B$ is a sigma algebra. After proving this, we need to show that the Kolmogorov's axioms apply for $P_B$ in order to conclude that $(B, S_B, P_B)$ is a space of probability. What I don't get, is how can we define $P_B$, so that we don't have any issues when trying to find the probabilities. In a previous theorem, my professor defined $P_B=P(A\cap B)/P(B)$ where we did require for $P(B)$ to be greater than $0$. For this theorem that has not been required, and I am not sure what to do in case $P(B)=0$.
Any help given is appreciated. I feel stuck when trying to prove $P_B$ is a probability function thru Kolmogorov's axioms because I don't believe we can apply the conditional probability when $P(B)=0$. In case we can, what would be a way to avoid the issue I am concerned about?