I have an example of an unmeasurable set in the measure space defined as follows: If $\Omega$ is the real line, we can consider a $\sigma$-algebra consisting of all the closed intervals of the real line and a measure that is just the length of the interval. This $\sigma$-algebra does not contain all the subsets of the real line and thus some sets are not measurable in this measure space.
I am confused as to, what are the subsets that are not contained in this measure space? I am guessing the subset of open intervals is missing, but can't all open intervals be expressed as closed intervals?
Not everything is a closed interval or a union of closed intervals. For example, $(0,1)$, $\mathbb{Q}$, $\mathbb{R\backslash Q}$, and the Cantor Set. Later on these can assign a measure to these sets, but in the way you defined measure they are non-measurable.