Why countable unions, intersections etc.? I was just wondering why one always insists on countability when it comes to the definition of a $\sigma$-algebra in measure theory. I mean, measure theory works as it does, but is there a deeper reason why e.g. it is reasonable to demand that the countable union of measurable sets is measurable again? Why not an uncountable union? Thanks in advance!

Countable has this very peculiar property of infinity.
It can be indexed, such that every point is only a finite distance away. Namely, it can be indexed using $\Bbb N$. And $\Bbb N$ has that great property that given any $k\in\Bbb N$, you only need to travel finitely many steps to meet with $k$.
Why is that important? Because (unfortunately) mathematics and mathematicians take their intuition from the real world. This means that you develop probability and measure from finite experiments. Often finite experiments have "unimportant residue", which is to say that the more times you run something, the closer that residue gets to $0$. So when you take a limit, you get this squeaky clean result.
Well... sometimes, anyway.
But this is where countable unions come from. First we cared about running an experiment $n$ times, then we also noticed that when $n\to\infty$ things look nicer. But what does it mean $n\to\infty$ when running an experiment? It means that we have a countable sequence of events, and we want to know whether the limit of those events is an events. We sure want it to be. So why not make it? And so $\sigma$-algebras are born.
(All that being said, there are uses - interesting uses - to algebras which have closure of more than just countable unions. But that usually gets under set theory, rather than measure theory or probability.)
