Today I am reading David Williams's Probability with Martingales. In chapter one, He introduce the notion of Measurable space:
A pair $(S,\Sigma)$,where $S$ is a set and $\Sigma$ is a $\sigma$-algebra on $S$, is called a measurable space. An element of $\Sigma$ is called a $\Sigma$-measurable set of $S$.
I know, all the measurable sets form a $\sigma$-algebra. But for an arbitrary $\sigma$-algebra. for example , all the subsets of $S$ form a $\sigma$-algebra.But in this $\sigma$-algebra $2^S$, there exist a non-measurable set. Why author called elements in $\Sigma$ a $\Sigma$-measurable set of $S$? Does this make sense?