Measure Theory for Dummies says
First, we need something to measure. So we define a “measurable space.” A measurable space is a collection of events B, and the set of all outcomes Ω, which is sometimes called the sample space. Given a collection of possible events B, why do you need to state Ω? For one, having a sample space makes it possible to define complements of sets; if the event F ∈ B, then the event $F^C$ is the set of outcomes in Ω that are disjoint from F. A measurable space is written (Ω, B).
I see that they want to measure B within Ω but I still do not get why do they need both B and Ω. I could measure and complement any subset within Ω: if F ∈ Ω, then event $F^C$ is the set of outcomes in Ω. I therefore need Ω alone and find the textbook explanation unsatisfactory. What do I miss/overlook?
if you have $\Omega$ alone, you can say "an event is any subset of $\Omega$", and this works fine for finite sets. But when you get to the real numbers, it turns out that if you declare every subset measurable, you find out that the measurable sets don't satisfy some axioms that seem very natural, and things basically grind to a halt.
So: we restrict to a smaller collection $E$ of subsets that's (a) big enough to be useful in practice and (b) small enough not to cause any problems with the statements that we want to use as "axioms." These are called the "measurable sets".
(In the case of $\mathbb R$, this usually means "measurable in a 'standard' measure, in which the measure of an interval is the length of the interval.")