Is $ \Omega \in \sigma$-algebra necessary true?

Question

In a probability space, it is said that a set of events should be $\sigma$-algebra, meaning:

This is from Probability and Statistics for Data Science by Prof. Fernandez-Grandza

But, $\sigma$-algebra does not necessarily contain all possible sets of outcomes and is not always equal to the power set of a sample space, how the third condition holds?

Answer 1

You are right, a $\sigma$-algebra does not contain necessarily all the possible outcomes as events but all outcomes are necessarily contained in $\Omega$. Ideally, a $\sigma$-algebra contains all the things that can happen for which you want to be able to give a probability (i.e. events), but this is not necessarily the case.

For example, we consider a dice throw. Let $\Omega = \{1,2,3,4,5,6\}$ and $\sigma$-algebra $\mathcal{F} = \{ \emptyset,\{1,2,3\},\{4,5,6\},\Omega\}$ with measure $\mathbb{P}$ defined by : $\mathbb{P}(\{1,2,3\})=1/2=\mathbb{P}(\{4,5,6\})$. The space $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space. In this space, we are only able to speak about two proper events : we draw a number in $\{1,2,3\}$ or in $\{4,5,6\}$. Note that the event "the draw is in $\Omega$", still always happen, as required by the axiom $\mathbb{P}(\Omega) =1$. Nothing can be said about the probability to draw a $1$ or a $2$ in this space.

Yet this is not problematic from a mathematical point of view, but it is not convenient at all to study further such experiment so we implictely always chose the power set in this case.

Note that the situation is much more complex for non-discrete sets, such as real random variables, because the power-set of the real numbers, for example, is the home of very weird "beasts" (you can search for Vitali sets if you are interested).

Answer 2

$\Omega\in\mathcal{F}$ does not imply $A\in\mathcal{F}$ if $A \subset \Omega$. An algebra is a set of sets. For example, it can happen, that $\{1,2\}\in \mathcal{F}$ but $\{1\}\notin \mathcal{F}$