I am trying to study probability space, and so far I have come to point that probability space is defined as $( \Omega, \mathcal{F}, P )$ where $\mathcal{F}$ is the $\sigma-algebra$. I know what $\sigma-algebra$ is, but I am confused that the $\sigma-algebra$ can be easily obtained by the power set of $\Omega$ .i.e. $2^\Omega$. Since a set can have many $\sigma-algebras$, are we only interested in the smallest $\sigma-algebra$ generated by events of interest?
For example, consider tossing a coin twice, and we are interested in finding the probability such that there is at least one head. So, surely $\Omega = (HH, HT, TH, TT)$. But what about $\mathcal{F}?$ One straight forward option is power set, but I am interested in "smallest" part of $\sigma-algebra$ which seems to come very handy in probability theory? Suppose I take $\mathcal{F} = (\emptyset, \Omega, \{HH, HT\}, \{TH, TT\})$. Now with this $\mathcal{F}$, I cannot measure "at least one head" because $(HH,HT, TH)$ is not in $\mathcal{F}$. It definitely means that my $\mathcal{F}$ has to be chosen smartly. How can I help myself here?
Best Answer
A $\sigma$-algebra is a collection of sets that needs to satisfy certain conditions. Yours violates one of the most important ones, namely, the fact that it should be closed under countable unions. (If you have a finite $\Omega$ and you put all the single elements into your algebra, then it necessarily has to be the power set by this axiom.)