Sequence of Events - Basic understanding I have an idea of the meaning of a sample space, and the events included in a sigma algebra. However, I am stuck in the definition of a sequence of events.

My difficulty is in the fact that in a sequence order matters, whereas in a set it does not. An event is a set of outcomes, and as such, it is not supposed to be ordered.

One way to look at it is to just to think of the real line, and Borel sets defined as open intervals, such that the intervals $(1,2), (2,3), (3,4)$, for example, would correspond to the sequence of events $\{(1,2)\},\{(2,3)\},\{(3,4)\}$. But I don't think this is the idea, and not all sets in a sigma algebra are defined on the real line.

Perhaps the idea is of nested "Russian dolls", whereby $E_1\subseteq E_2 \subseteq E_3 \subseteq E_4,\cdots$



So what is the definition and some valid examples of sequences of events (or sets)?
If you have any sample space, label it $\Omega$, then the Cartesian product $\Omega{\times}\Omega$ is also a sample space; and so in any $n$-order Cartesian exponential, $\Omega^n$ (for any positive integer $n$).

Any event drawn from the sample space $\Omega^n$ is also a sequence of $n$ events drawn from the sample space $\Omega$.



For instance, take the sample space of a fair coin toss, $\Omega=\{{\small\rm H},{\small\rm T}\}$. &nbsp; A sequence of the events resulting from two fair coin tosses, such as $({\small\rm T},{\small\rm H})$ , is a event from the sample space of $\{({\small\rm H},{\small\rm H}),({\small\rm H},{\small\rm T}),({\small\rm T},{\small\rm H}),({\small\rm T},{\small\rm T})\}$; which is $\{{\small\rm H},{\small\rm T}\}^2$
If you have any sample space, label it $\Omega$, then the Cartesian product $\Omega{\times}\Omega$ is also a sample space; and so in any $n$-order Cartesian exponential, $\Omega^n$ (for any positive integer $n$).

Any event drawn from the sample space $\Omega^n$ is also a sequence of $n$ events drawn from the sample space $\Omega$.



For instance, take the sample space of a fair coin toss, $\Omega=\{{\small\rm H},{\small\rm T}\}$. &nbsp; A sequence of the events resulting from two fair coin tosses, such as $({\small\rm T},{\small\rm H})$ , is a event from the sample space of $\{({\small\rm H},{\small\rm H}),({\small\rm H},{\small\rm T}),({\small\rm T},{\small\rm H}),({\small\rm T},{\small\rm T})\}$; which is $\{{\small\rm H},{\small\rm T}\}^2$