What is your definition of measurable? I assume you mean that
for each Borel set.
So here are some hints;
First show 2) by considering sets of the form [a,\infty) then show 1 by using the 2 complement and sets
For 3 consider sets of the form .
Q2 is wrong, you need the function to be surjective. You should attempt it first (and post here).
Let f be measurable function if for each real no. a the set {x in E : f(x)>a} is measurable.
For all real a, the set { x in E :f(x)≥a} is measurable since,
{x in E : f(x)≥a } = intersection { x in E∶ f(x)>a-1/n }
= a measurable set and hence {x in E : f(x)=a}
= {x in E : f(x)≥a }-{x in E : f(x)>a} is measurable.
Also {x in E : f(x)=∞} = intersection { x in E: f(x)>n} is measurable.
{x in E : f(x)<a} is measurable as it is the complement of 1. Clearly the set {x in E : |f(x)|< a} is measurable for two measurable sets {x in E : f(x)>a} and {x in E : f(x)<a}.