In Real Analysis written by Royden, the definition of measurable function is as follows.
An extended real-valued function f defined on E is said to be Lebesgue measurable, or simply measurable, provided its domain E is measurable and it satisfies one of the four statements of Proposition1.
Proposition 1 Let the function f have a measurable domain E. Then the following statements are equivalent:
(1) For each real number c, the set $\{x \in E | f(x) \gt c\}$ is measurable
(2) For each real number c, the set $\{x \in E | f(x) \ge c\}$ is measurable
(3) For each real number c, the set $\{x \in E | f(x) \lt c\}$ is measurable
(4) For each real number c, the set $\{x \in E | f(x) \le c\}$ is measurable
But I don't understand why mathematicians define LMS function as above. In my thought, a condition that a domain of a function is a measurable set is enough to define that a function is measurable.
What a property is added if the four statements is included in the definition?