$\sigma$-algebra intuition So, i am trying to learn measure theory, for applications in probability theory. However, i am having some issues fully understanding the definition of a $\sigma$-algebra.
I am working with the following definition of a $\sigma$-algebra:

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*Definition: Let $\Omega \ne \emptyset$. Then $\mathcal{B} \subseteq \mathcal{P}(\Omega)$ is a $\sigma$-algebra if

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*(i) $ \Omega \in \mathcal{B} $

*(ii) if $B \in \mathcal{B}$, then $B^c \in \mathcal{B}$

*(iii) If $B_n \in \mathcal{B}$ for all $n \in \mathbb{N}$ then $\cup_{n=1}^{\infty}B_n \in \mathcal{B}$
I do not fully understand what is going on in (iii). My intuition would say, that it should mean "If a set is in the sigma-algebra, then every possible union you could make with other sets in the sigma-algebra, is also in the sigma-algebra", is this correct? Just looking at the notation i would think that it means that the union of all sets in the sigma-algebra, should also be in the sigma-algebra, but that would just return $\Omega$ as in (i), so that cannot be correct?

As pointed out by @J. W. Tanner in the comments, the third criterion states that any countable union of sets in the $\sigma$-algebra is included. However, it is possible that no countable union of sets includes $\Omega$ itself. That is why we need the first criterion.
