In my textbook for Introduction to Probability, there is the following extract on the topic of sigma algebras:
If Ω is finite or countable then we can take F to be the power set P(Ω) of Ω, that is the set of all subsets of Ω. However, if Ω is uncountable, then taking P(Ω) for F could cause problems when trying to measure the likeliness, or probability, of events, so in that case we usually choose F to be a subset of P(Ω).
This seems a bit unclear to me as I'm new to the set-theoretic approach to probability so I was hoping I could get a concrete example of a sample space where we can't define probability measure and so have to consider a measurable subset.
What your book refers to is the existance of non-measurable sets. The existance of non measurable sets is really non-trivial and to give rigorous proof for their existance you have to develop measure theory to some extent.The most basic such set is called the Vitaly set (see here).
Coming from another directions, the Banach Tarski Paradox(see here) states that you can partition the solid 3-d sphere into 5 pieces and rearrange them to get two balls identical to the first one ! If this 5 pieces had a logical volume (which is exactly the measure of probability to pick them, in a sense) then by moving them in space you couldn't take something of double the volume ! Therefore the mustn't have a well defined notion of volume!
This might seem like handwaving but there is a deeper reason. It has been proven that to create such bad sets you need to use the Axiom of choise, therefore you can show that they exists but you can't really describe what they are made of!