Tow problems in measure need to be solve

Aug 2009
What is your definition of measurable? I assume you mean that
\(\displaystyle f^{-1}(B) \in \mathcal{F}\)
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
\{f \geq n\}\)

For 3 consider sets of the form \(\displaystyle [a-1/n,a+1/n]\).

Q2 is wrong, you need the function to be surjective. You should attempt it first (and post here).
  • Like
Reactions: amro05
May 2010
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}.
  • Like
Reactions: amro05