I am in a game theoretic framework in which I need to allow an agent to be able to conditionalize on events with probability zero. This means that I can not use the classical definition of conditional probability: $\mathbb{P}(A|B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}$ since this is too restrictive, as it demands that $\mathbb{P}(B)>0$.
Therefore, I am using an "alternative" definition of conditional probability, which is essentially a derivation of the multiplication axiom, which is: If $A\subseteq B\subseteq C$, then: $P(A|B)P(B|C)=P(A|C)$. Also, for some $B$ the function $P(\cdot |B)$ is a probability measure with all the usual axioms.
My problem now is that I wish to derive the following: $P(X\cap Z|Z)=P(X|Z)$ by the definition I mentioned above. Any help would be greatly appreciated.
You usually assume in the Renyi formulation of conditional probability that $$1=P(Z|Z).$$
Then $P(Z^C|Z)=0$ and therefore $P(X\cap Z^C|Z)=0$. So $$P(X|Z)=P((X\cap Z)\cup (X\cap Z^C)|Z)=P(X\cap Z|Z)+P(X\cap Z^C|Z)=P(X\cap Z|Z).$$