It is my understanding that $\sigma $-algebras are necessary to satisfy certain desired properties of a measure, and that these conditions are mutually inconsistent if we consider arbitrary open sets of $\mathbb{R}$. However, I have also read that power set of a set is necessarily a $\sigma $-algebra, so how come we cannot use the power set of $\mathbb{R}$ to define a measure space? I feel I am missing something fundamental here.