Question

Let (X,@) be a measurable space.



Statement 1:



it is given that for each a in R the set

{x in X | f(x) <= a } is measurable



Statement 2: f is measurable



We need to prove that statement 1 implies 2.



My approach



Since for each a in R the set



{x in X | f(x) <= a } is measurable = intersection of of n from 1 to infinity

complement of {x in X | f(x) > a } is measurable

Since
{x in X | f(x) > a } is measurable, its complement is also measurable


==> f is measurable





Is this a correct approach?



Many thanks for your feedback!

Answer 1

Yes, your approach is correct. You used the fact that the sets {x in X | f(x) <= a } and {x in X | f(x) > a } are measurable for every a in R, and then used set theory to show that f(x) is measurable. Well done!