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i am trying to study borel sets and sigma algebra- and came across the following on wikipedia (https://en.wikipedia.org/wiki/Sigma-algebra):

(However, after n flips of the coin, you may want to determine or revise your betting strategy in advance of the next flip. The observed information at that point can be described in terms of the 2n possibilities for the first n flips. Formally, since you need to use subsets of Ω, this is codified as the σ-algebra:

$\mathcal{G}_n=\{A\times\{H,T\}^\infty:A\subset\{H,T\}^n\}$.

could anyone help me to understand how, in plain english, one should read that last line? Many thanks in advance

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This notation says $\mathcal{G}_n$ is a set whose elements are exactly the sets of the form $A\times \{H,T\}^\infty$ where $A$ is a subset of the set $\{H,T\}^n$. Let's unpack this a bit. An element of $\{H,T\}^n$ is just a sequence of $n$ letters, each of which is $H$ or $T$. So a subset $A$ is a set of such sequences.

The expression $A\times \{H,T\}^\infty$ literally denotes the set of ordered pairs $(x,y)$ where $x$ is an element of $A$ and $y$ is an element of $\{H,T\}^\infty$, i.e. an infinite sequence of $H$s and $T$s. However, in this context, this is not quite meant literally; instead, we are thinking of such an $(x,y)$ as representing a single infinite sequence of $H$s and $T$s obtained by concatenating $x$ and $y$ together. In this way, $A\times\{H,T\}^\infty$ is a subset of $\Omega=\{H,T\}^\infty$: specifically, it is the set of infinite sequences such that their first $n$ letters form a sequence which is an element of $A$.

For instance, suppose $n=2$ and $A=\{HH,TT\}$. Then $A\times\{H,T\}^\infty$ is the set of sequences that can be written as a concatenation of either $HH$ or $TT$ with some other infinite sequence. That is, it is the set of sequences that start with either $HH$ or $TT$.

Putting it all toghether, $\mathcal{G}_n$ is the set of all sets of the form $A\times\{H,T\}^\infty$, where $A$ is allowed to be any subset of $\{H,T\}^n$. Every element of $\mathcal{G}_n$ is a subset of $\Omega$, and there are $2^{2^n}$ elements of $\mathcal{G}_n$, one for each subset $A\subset \{H,T\}^n$. Another way to think of this is that $\mathcal{G}_n$ is the set of all subsets $S$ of $\Omega$ such that given an element $x$ of $\Omega$ (i.e., an infinite sequence), you only have to look at the first $n$ letters of $x$ to find out whether $x$ is in $S$. If $S$ has this property, then $S$ will be $A\times \{H,T\}^\infty$, where $A$ is the set of all sequences of $n$ letters such that if $x$ starts with such a sequence, $x$ is in $S$.

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