For $k \in N$, let $X_k$ be the outcome of the $k$-th toss and let $T$ be the toss in which the first "Heads" appears and define $T_n = T$ if $T \leq n$ and $T_n = n+1$ otherwise. That is, $$T = \inf\{ k \in N; X_k = \text{Heads}\}$$ and $$T_n = \min(T, n+1)$$
Describe $F_n$, which is the $\sigma$-algebra generated by $T_n$.
My attempt:
First, the probability space $(\Omega, F, P)$ where $\Omega = \{H, TH, TTH, ...\}$ is the sample space of all finite sequences of Heads and Tails, where the first $k-1$ tosses are Tails and the $k$th toss is a Head. The probability measure is $P(\{\omega_k\}) = 2^{-k} $.
The stopping time T is an integer-valued non-negative random variable which represents the first moment $k$ in which the first "Head" appears.
Now, $T_n$ is also a random variable where the sigma-algebra $F_n = \sigma(T_n) $ and $\sigma(T_n) = \sigma(\{ T_n \leq k\}, k = 1,2,... ,n+1) = \sigma(\{ T_n = k\}, k = 1,2,... ,n+1)$ and so the event $ \{T_n = k\} = \{w_k\}$.
So for example, say that the first time a coin lands on a "Head" is at the 4th toss, then $T = 4$ which will mean that $X_1 = X_2 = X_3 = 0$ and so $T_1 = 2, T_2 = 3, T_3 = 4, T_4 = 4$.
Then, the set $\{ T_n = k\} = \{T_1 \neq 1\} \cap \{T_2 \neq 2\} \cap \{T_3 \neq 3\} \cap \{T_4 = 4\} = \{ 0\} \cap \{00\}\cap \{000\}\cap \{0001\} = \{0001\} = \{w_4\}$