My book defines a set $E \subseteq \Bbb{R}$ as Lebesgue measurable provided $m^*(A) = m^*(A \cap E) + m^*(A \cap E^C)$ for $A \subseteq \Bbb{R}$. What I am currently working on is trying to prove the equivalence of the following statemnets:
(i) $\forall c \in \Bbb{R}$, $\{x \in E ~|~ f(x) > c \}$ is Borel measurable
(ii) $\forall c \in \Bbb{R}$, $\{x \in E ~|~ f(x) \ge c \}$ is Borel measurable
(iii) $\forall c \in \Bbb{R}$, $\{x \in E ~|~ f(x) < c \}$ is Borel measurable
(iv) $\forall c \in \Bbb{R}$, $\{x \in E ~|~ f(x) \le c \}$ is Borel measurable
My book defines what it means for a function to be Borel measurable (i.e., $f: E \to \Bbb{R}$ is Borel measurable provided $E$ is a Borel set, and $\{x \in E ~|~ f(x) > c \}$ is a Borel set for every $c \in \Bbb{R}$), but it furnishes no corresponding definition for what it means for a set to be Borel measurable. What does it mean for a set to be Borel measurable?
It just means the set is a Borel set.
By definition, any Borel measure acts on the Borel $\sigma$-algebra, so only sets in that $\sigma$-algebra are Borel measurable. Thus, a set being Borel measurable is equivalent to it being a Borel set.