For the filtration $\{\mathcal{F}_t\} $ we defined for $t \geq 0$ the filtration $\mathcal{F}_t^+:=\cap_n~ \mathcal{F}_{t+\frac{1}{n}}$.
There is one equality I don't understand. For $A \in \mathcal{F}_0^+$ we say $E_x[1_A] = E_x[1_A|\mathcal{F}_0^+]$
Why does this equality hold? Is $1_A$ independent of $\mathcal{F}_0^+$?