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?
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.
No, your approach is not correct.
To prove that f is measurable, you need to show that for any Borel set B in R, the set {x in X | f(x) ∈ B} is measurable.
Here's the correct approach:
1. Let B be a Borel set in R. We want to show that the set A = {x in X | f(x) ∈ B} is measurable.
2. Since B is a Borel set, it is generated by the intervals of the form (-∞, a] (for a in R). In other words, B can be written as the countable union of sets of the form (-∞, a] or (-∞, a).
3. For each a in R, let A_a = {x in X | f(x) <= a}. Given that A_a is measurable for every a in R, we can conclude that the set A' = {x in X | f(x) < a} is also measurable (by taking complements).
4. Now, let's consider the set A = {x in X | f(x) ∈ B}. We can write A as the union of three disjoint sets: A = (A - A') ∪ (A' - A) ∪ A'.
5. Note that (A - A') = {x in X | f(x) > a}, (A' - A) = {x in X | f(x) < a}, and A' = {x in X | f(x) <= a}.
6. Since each of these sets is measurable (by assumption and by step 3), and measurable sets are closed under unions and complements, we conclude that A is measurable.
Therefore, we have shown that for any Borel set B in R, the set {x in X | f(x) ∈ B} is measurable, which means that f is measurable.