$\sigma$-algebra of coin tosses until first head Suppose that a coin is tossed until a heads is obtained. For this situation, what would be the tripe $(\Omega,\mathcal F, P)$?
For the sample space, I suppose it would be
$$\Omega := \{H, TH, TTH, TTTH, ...\}$$
Now, to properly define a probability measure such that for $n$ tosses we have $P( TT...TH)= 1/2^n$. I suppose we need to define an algebra where this probability measure works, and then expand using Caratheodóry. Am I correct in assuming this? If so, what would be the algebra to which I'd expand in order to use Caratheodory? Or does the $\sigma$-algebra of the set of all combinations (i.e. $2^\Omega$) works as my $\mathcal F$?
 A: Your probability measure should be a function from your $\sigma$-algebra to $[0,1]$. For instance, you could take $\mathcal{F} = 2^\Omega$.
If you can just define the probability measure and $\sigma$-algebra right off the bat, I wouldn't bother thinking about caratheodory and premeasure nonsense. Here, your life is a bit easier because $\Omega$ is countable.
A: Another natural way I might suggest is to embed your situation inside the canonical coin toss space $\Omega = \{ H, T \}^{\mathbb{N}}$. Here the $\sigma$-algebra $\mathcal{F}$ is the one generated by the collection of cylinder sets of the sort $\mathcal{C}_x =\{ \omega \in \Omega : \omega_1 = x_1, \dots, \omega_n = x_n \}$ for each $n \geq 1$, where $x = (x_i)_{i = 1}^n$ with each $x_i \in \{ H, T \}$.
Then the classic argument is to build a probability measure on this space $(\Omega, \mathcal{F})$ using Carathéodory's extension theorem with $\mathbb{P}(\mathcal{C}_x) = 1/2^n$ for the same example given above. (Technically we have a "semi-algebra" here that can readily be extended to an algebra; but there is a version of Carathéodory's that works here too. I found a nice explanation in Section 6.5 of these notes on building the coin-tossing measure).
This way, you can still capture everything you need to know about the tosses until the first head (e.g. by the random variable $T$). For example, the event $\{ T = k \}$ corresponds exactly to the cylinder set $\{ \omega \in \Omega: \omega_1 = \dots = \omega_{k-1} = T, \, \omega_k = H \}$, which has probability $1/2^k$.
