Let's $\mu$ a probability measure on $(\Omega, \mathcal{F})$. Let's $\mathcal{J}_1\supset\dots\supset\mathcal{J}_n\supset\dots$ a sequence of sub-$\sigma$-fields of $\mathcal{F}$. If $A\in\mathcal{J}_n$ for all $n\in\mathbb{N}$ is well known that $ \mu(A|\mathcal{J}_n)=1_{A} $ Here $1_{A}$ is the caracteristic of $A$.
Question. How to prove that if $A\in\displaystyle\bigcap_{n\in\mathbb{N}}\mathcal{J}_n$ then
$ \mu(A|\displaystyle\bigcap_{n\in\mathbb{N}}\mathcal{J}_n)=1_A\;? $