Question:

Question

Let (X,@) be a measurable space.

Prove that if for each a in R, {x in X | f(x) <= a } is measurable, then f is measurable.

My approach

Let a belong to R.

Then,
{x in X | f(x) <= a } = {x in X | f(x) > a }c
since the right hand side is measurable if f is measurable, left hand side implies that f is measurable.

Is this a correct approach?

Answer 1

No, your approach is not correct. The statement "the right hand side is measurable if f is measurable" is not necessarily true. In fact, it is possible for a function to be measurable but for the set {x in X | f(x) > a }c to be non-measurable.

Here is an alternative approach:

Let A be any measurable subset of R. We want to show that the set {x in X | f(x) ∈ A} is measurable. We can write this set as the union of the sets {x in X | f(x) <= a} and {x in X | f(x) > b}, where a is the infimum of A and b is the supremum of A. Both of these sets are measurable by assumption, so their union is also measurable. Therefore, f is measurable.