# Showing some random process is predictable

I'm asked to show that given $$\tau$$ is a stopping time and if $$I_n = \begin{cases} 1 & \text{if}\ n\leq\tau \\ 0 & \text{if}\ n>\tau \end{cases}$$ then $$(I_n)_{n\geq1}$$ is a predictable process.

I know that $$I_n$$ is predictable if it's $$\mathbb{F}_{n-1}$$-measurable but I could really need some help getting through this. I was wondering if I should look at three different possibilities for the time $$n$$.

If $$n-1\leq n\leq \tau$$ it holds that $$I_{n-1}=I_n=1$$ and if $$n>n-1>\tau$$ it holds that $$I_{n-1}=I_n=0$$ and lastly if $$n-1\leq \tau$$ and $$n>\tau$$ it holds that $$0=I_n. But is this enough?

Hint: Show that $$I_n = 1-1\{\tau \leq n-1\}$$, and show this is $$\mathbb{F}_{n-1}$$-measurable.
Comparing the value of $$I_n$$ to $$I_{n-1}$$ is not enough to show that $$I_n$$ is $$\mathbb{F}_{n-1}$$-measurable (after all, how do we even know $$I_{n-1}$$ is $$\mathbb{F}_{n-1}$$-measurable?). Instead you have to show that at time $$n-1$$ (given the information in the sigma algebra $$\mathbb{F}_{n-1}$$) you can determine the value of $$I_n$$.
• Just to make sure; when you're writing $1\{\tau\leq n-1\}$ is that the indicator function $1_{(\tau \leq n-1)}$?