I am currently trying to define a Grothendieck Topology on the category **Prob** which consists of finite probability spaces with measure preserving maps between them.

I declared the covering sieves of an object $U$ to be the sieves generated the projections i.e. the covering sieves $S_V$ are each generated by set  $\Pi_V=\{\pi_U:U\times V\to U\}$ where $V$ is another finite probability space. I have already proved that the maximal sieve is covering by considering $V$ to be the terminal object and the stability action follows pretty easily too.

However for the transitivity axiom, I have tried a lot of strategies but none of them work. Note that the inclusion maps $i_U:U\to U\times V$ are not measure preserving so that fact cannot be used.

I also realise that this might not even be a Grothendieck Topology so a counterexample would also be helpful.