I am studying Steven E. Shreve's Stochastic Calculus book. Example 1.1.4 (p.4-6) constructs a probability measure on the space of infinely many coin tosses $\Omega_\infty$. In the example the $\sigma$-algebras for $n$ coin tosses are defined like this $\mathcal{F}_0 = \{\emptyset, \Omega\}$, $\mathcal{F}_1 = \{\emptyset, \Omega , A_{H}, A_{T}\}$, where $A_H$ is the set of all sequences beginning with head. For three coin tosses we get $\mathcal{F}_3 = \{\emptyset, \Omega ,A_H, A_T, A_{HH}, A_{HT}, \ldots\}$
Now the authors state
By continuing this process, we can define the probability of every set that can be described in terms of finitely many tosses.
And later:
We create a $\sigma$-algebra, called $\mathcal{F}_\infty$ by putting in every set that can be described in terms of finitely many coin tosses and then adding all other sets required in order to have a $\sigma$-algebra. It turns out that once we specify the probability of every set that can be described in terms of finitely many coin tosses, the probability of every set in $\mathcal{F}_\infty$ is determined.
I find this puzzling! Why do finite descriptions suffice? For example I don't understand how the probability of the event "infinitely many heads" is determined. I would guess it has probability 1 but how can I conclude this from the finite cases? How is this done for general elements $A\in\mathcal{F}_\infty$?
Edit 2: Can I argue like this: for $A\in (\mathcal{F}_\infty\setminus (\bigcup_{n=1}^\infty F_n))$, the complement $A^C$ is in some $\mathcal{F}_m$ and therefore $\mathbb{P}(A) = 1-\mathbb{P}(A^C)$?
Best Answer
No real answer but too much for a comment:
$$\left\{ \text{number of heads infinite}\right\} =\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}\left\{ \text{head at }k\text{-th toss}\right\} $$
showing that this event can be described by means of sets in $\bigcup_{n\in\mathbb{N}}\mathcal{F}_{n}$.
I am not sure, but suspect that $\bigcup_{n\in\mathbb{N}}\mathcal{F}_{n}$ is an algebra and that a probability 'premeasure' on it determines uniquely a probability measure on $\sigma$-algebra $\sigma\left(\bigcup_{n\in\mathbb{N}}\mathcal{F}_{n}\right)$.