Bayes theorem with infinitesimal evidence The Bayes theorem is often stated as 
$$\mathrm{posterior} = \frac{\mathrm{likelihood}\times\mathrm{prior}}{\mathrm{evidence}}$$
So what happens to the posterior probability when the evidence for an event is very small/infinitesimal? Mathematically, it becomes large but what does it mean in real terms?

If its a true posterior probability, then it should always be bounded between $0$ and $1$.
Also, your formula is not really a formula, but a conceptual framework. The actual formula is instructive:
$$P(B\mid A)=\frac{P(B)P(A\mid B)}{P(A)}=\frac{P(A\cap B)}{P(A)}$$
From the axioms of probability, we know that $\lim\limits_{P(B)\rightarrow 0} P(A\cap B) = 0$ and $$\lim_{P(B)\rightarrow 1} P(A\cap B) = P(A)\implies 0\leq \lim\limits_{P(B)\rightarrow 0} \frac{P(A\cap B)}{P(B)}\leq 1$$
So this should always be between 0 and 1. This is not true if you are looking at the posterior likelihood, where you use teh densities in place of the probability measures.
