I'm reading this book [1] and met the definition of a measurable space:
A measurable space is an ordered pair $(X,\mathcal S)$, where $X$ is a set and $\mathcal S$ is a $\sigma$-algebra on $X$.
My question is, why does the measurable space need to be defined as an ordered pair $(X,\mathcal S)$ instead of just $\mathcal S$ itself since $\mathcal S$ itself contains the whole set $X$ as one element?
For example, let's consider the set $X=\{1,2,3,4\}$ and $\mathcal S=\{\{1,2\},\{3,4\},\emptyset,\{1,2,3,4\}\}$. I think the information of $X$ itself is also contained in $\mathcal S$. Hence the $X$ in the definition of a measurable space is redundant?
[1]: Measure, Integration & Real Analysis by Sheldon Axler