In Royden's book, he defines a function $f$ defined on $E$ to be measurable if $E$ is measurable and at least one (and therefore all) of the following sets are measurable:
$\{x \in E : f(x) > c\}$
$\{x \in E : f(x) \geq c\}$
$\{x \in E : f(x) > c\}$
$\{x \in E : f(x) \leq c\}$
where $c$ is any real number. My question is, are the 4 above sets measurable even if the domain $E$ is not measurable? Why do we need $E$ to be measurable?