Let $t^*\in(0,1)$ and $(\Omega,\mathcal{F},P)$ be a probability space. Suppose I have set $A\in\mathcal{F}$ and an uncountable family of sets $(B_t : t\in[0,1])\subset\mathcal{F}$ with the following properties:
- $P(A)>0$ and $P(B_t)=p$ where $p>0$ is a constant.
- For each $\omega\in A$ there exists a $\delta>0$ such that $\omega\notin B_t$ for all $t\in (t^*-\delta, t^*+\delta)$.
Is it true that $\limsup_{t\rightarrow t^*}P(A\cap B_t) = 0$? Is the limit well defined?