A measurable space is a triple $(M, \Sigma, \mu)$.
I am confused by one thing, to my knowledge almost everything you think of is measurable. So why do we need $\Sigma$? Why is that a measurable set is an element of the sigma algebra instead of the whole space?
In other words, take $M = \mathbb{R}^2$. Draw a star. That's measurable. Draw a curve. That's measurable. Draw a dot. That's measurable. Draw a giraffe. That's measurable.
Take arbitrary union of open sets, that is an open set, which is measurable. Take arbitrary intersection of open sets, that may not be an open set, in fact it could be a dot, but a dot is still measurable. Same with closed sets. If it comes out to be an empty set, no worries, that is also measurable. Take the whole space to be the real line, that measure is infinity, still measurable.
So why don't we just write $(M, \mu)$ instead of $(M, \Sigma, \mu)$?
If you allow the axiom of choice, then there are subsets of $\mathbb{R}$ which are not Lebesgue measurable.
More generally, $\sigma$-algebras can be useful ways of keeping track of information. This is often the case when studying stochastic processes, for example.