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:
- Definition: Let $\Omega \ne \emptyset$. Then $\mathcal{B} \subseteq \mathcal{P}(\Omega)$ is a $\sigma$-algebra if
- (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?
Best Answer
The key-word which is not written in the definition is countable. A $\sigma-$algebra is intuitively a subset of $\mathcal{P}(\Omega)$ which is stable by countable "operations" (such as union, intersection and complementary) on its elements.