Let $(\Omega,\mathcal{F},P)$ be a probability space. If $A\in\cal F$ is an event with $P(A)=1$, then $ P_{\mid A}(B)=P(B\mid A)=\frac{P(B\cap A)}{P(A)}=P(B),\quad B\in\cal F. $ I wonder if something can be said about how "close" $P_{\mid A}$ and $P$ are, when $A\in\cal F$ is an event with probability close to $1$ and also what "close" should mean.
For example, if $P(A)=p$ and let's say that $p=0.99$, can we give a non-trivial upper bound on the maximal distance $ \sup_{B\in\cal F}|P_{\mid A}(B)-P(B)| $ in terms of $p$? And could other types of distances be interesting?
This is just me thinking, so anything you can add will be appreciated. Thanks.