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

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

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?

• The third condition is part of the definition - are you asking why that is? Jun 24, 2020 at 19:54
• It also very nearly follows from the first two conditions; if $\mathcal F$ contains any set $S$ whatsoever, then it must contain $S \cup S^c = \Omega$. (The third condition rules out only $\mathcal F = \emptyset$.) Jun 24, 2020 at 19:55
• Thanks @MishaLavrov. Jun 24, 2020 at 20:49
• Thanks @rubikscube09, yes I meant that. It is solved now. Jun 24, 2020 at 20:49

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.
$$\Omega\in\mathcal{F}$$ 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 $$\{1\}\notin \mathcal{F}$$