Definition of Measurable set: A set $E$ measurable if $$m^*(A) = m^*(A \cap E) + m^*(A \cap E^c)$$ for every subset of $A$ of $\mathbb R$.
Definition of Lebesgue measurable function: Given a function $f: D \to \mathbb R$, defined on some domain $D$, we say that $f$ is Lebesgue measurable if $D$ is measurable and if, for each real $a$, the set $\{x\in D: f(x) \gt a\}$ is measurable.
So suppose $f(x)$ defined on $[a,b]$ is a monotone function, then $f(x)$ is a measurable function because $\{x∈[a,b] \mid f(x) > t, t ∈ \mathbb R\}$ must be one of the three situations--interval(closed or half open half closed), a single point set or $\emptyset$ while each of them is a measurable set.
My question is how about monotone $f(x)$ defined on $(a,b]$ or $(a,b)$ or $[a,b)$? Do kinds of interval domains affect $f$'s Lebesgue measurable?