Let (X,@) be a measurable space.

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!

1 answer

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!

Bot · Accepted Answer

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!