Are $\sigma$ algebras of multiple coin tosses subsets?

Question

If we consider the $\sigma$-algebra of a single coin toss, it is denoted by $\Delta_1=\{\{\},\{H\},\{T\},\{H,T\}\}$.

Similarly for a two coin toss it would be $\Delta_2=\{\{\},\{HH\},\{TT\},........,\{HH,HT,TT,TH\}\}$.

Then can it be said that $\Delta_1 \subset \Delta_2$?

Answer 1

Clearly not. $\{H\}\in\Delta_1,\{H\}\notin\Delta_2 $

EDIT

In answer to the OP's question, $\mathscr F_n$ is the collection of subsets of all sequences of tosses, where membership can be decided on the basis of the first $n$ tosses. For example, members of $\mathscr F_1$ include the set of sequences that start with heads, the set of sequences that start with tails and so on. Clearly, if membership can be decided on the basis of the first toss, then it can be decided on the basis of the first two tosses. The set of sequences that start with two heads is a subset of the set of sequences that start with one head.

Not every sequence belongs to $\mathscr F_n$ for some $n.$ Consider for example, the set of sequences with only finitely many heads., or the set of sequences where the cumulative number of tails is always greater than the cumulative number of heads.

Answer 2

Of course not, outcome $\{H\}$ is not in $\Delta _2$.

Answer 3

Example on request (see comments on question).

Going for $2$ tosses we can take as sample space the set $\Omega=\{HH,HT,TH,TT\}$ and as $\sigma$-algebra the collection $\Delta_2=\wp(\Omega)$.

In this model we focus on the first toss.

It induces the following sub-$\sigma$-algebra: $$\Delta_1=\{\varnothing,\{HH,HT\},\{TH,TT\},\Omega\}\subset\Delta_2$$

Note that e.g. $\{HH,HT\}$ is exactly the event that the first toss shows a head.