Notation $\mathcal{F}_M$ where $M$ is a stopping time

Question

The question arise from the question 4.4.3 on Durrett probability.

Suppose $M\leq N$ are stopping times. If $A\in \mathcal{F}_M$, then \begin{align*} L = \begin{cases} M & \text{on $A$}\\ N & \text{on $A^c$} \end{cases} \end{align*} Then $L$ is a stopping time.

I don't get the notation $\mathcal{F}_M$. Moreover, in my opinion, $M,N$ and $L$ should correspond to the same filtration, but the question seems not. Does anyone have any idea on this?

Thanks in advanced!