Pasting of measures

Question

Let $\left(\Omega,\mathcal{A},P\right)$ be a probability space and $(\mathcal{F}_t)_{t\in\{0,\ldots,T\}}$ be a filtration.

Definition 1: Let $Q_1$ and $Q_2$ be two equivalent measures and $\tau:\Omega\rightarrow\{0,\ldots,T\}$ a stopping time. Then the pasting of $Q_1$ and $Q_2$ in $\tau$ is the measure $\tilde Q$ defined by $$\tilde Q(A)=E_{Q_1}[Q_2(A|\mathcal{F}_\tau)]$$ for $A\in\mathcal{F}_T$.

I am wondering if we have $E_{Q_1}[Q_2(A|\mathcal{F}_\tau)]=E_{Q_2}[Q_1(A|\mathcal{F}_\tau)]$?

Answer 1

Notice that $\tilde Q$ coincides with $Q_1$ on $\mathcal F_\tau$, and likewise when the roles of $Q_1$ and $Q_2$ are exchanged. Thus if the commutation you seek holds, then you must have $Q_1=Q_2$ on $\mathcal F_\tau$.