Why is this not a stopping time?

Question

I am going through Shreve's Stochastic Calculus book and he has a random time $\rho$ defined as follows: $$\rho(HH) = 0, \quad \rho (HT) = 0, \quad \rho(TH) = 1, \quad \rho(TT) = 2$$

which he claims is not a stopping time. I was under the impression that this is, as $\rho(HX) = 0$, regardless of time outcome of $X$. Would anyone be able to clear up where I have misunderstood? Thanks.

Answer 1

It is not a stopping time because $\{\omega : \rho(\omega) \le 1 \} = \{HH,HT,TH\}$ is not in $\mathcal F_1 = \{\emptyset,\Omega,\{HH,HT\},\{TH,TT\}\}.$