Why is the focus on half open intervals?

Question

I am studying measure theory, more specifically the Lebesgue measure on $\mathbb{R}$. Why is there such a focus on half open intervals rather than just closed or open intervals?

Answer 1

Because the collection of finite unions of half-open intervals is an algebra (closed under complements, finite unions and finite intersections).

(That's assuming of course we're talking about only one flavor of half-open intervals, for example $[a,b)$ but not $(a,b]$, and assuming we allow $\pm\infty$ as endpoints.)

To state what I thought was obvious but evidently isn't: On the other hand, the collection of finite unions of closed intervals (for example) is not an algebra. We need an algebra in order to be able to apply a certain theorem of Caratheodory. The algebra consisting of finite unions of half-open intervals has a particularly simple description, making the application simple.