Question

Show that E is measurable if and only if kE is measurable.

Answer 1

We need to show two things:

1. If E is measurable, then kE is measurable.
2. If kE is measurable, then E is measurable.

Proof of 1:

Assume E is measurable. We want to show that kE is measurable. Let A be an arbitrary subset of the real line. We have to show that kE intersects A in a measurable set if and only if E intersects k^{-1}(A) in a measurable set.

Let's first assume that kE intersects A in a measurable set. Then there exists a Borel set B such that kE ∩ A = B. Since E and k^{-1}(A) are related by the equation E = (1/k)kE and k^{-1}(A) = k^{-1}(1/k)kE, we have:

E ∩ k^{-1}(A) = (1/k)kE ∩ k^{-1}(1/k)kE = (1/k)kE ∩ E = (1/k)B

Since B is Borel, (1/k)B is also Borel, which means E intersects k^{-1}(A) in a measurable set. Therefore, kE is measurable.

Proof of 2:

Assume kE is measurable. We want to show that E is measurable. Let A be an arbitrary subset of the real line. We have to show that E intersects A in a measurable set if and only if kE intersects k(A) in a measurable set.

Let's first assume that E intersects A in a measurable set. Then there exists a Borel set B such that E ∩ A = B. Since kE and k(A) are related by the equation kE = k(k^{-1}(A) ∩ E), we have:

kE ∩ k(A) = k(k^{-1}(A) ∩ E) ∩ k(A) = k((k^{-1}(A) ∩ E) ∩ k^{-1}(k(A))) = k((k^{-1}(A) ∩ E) ∩ A)

Since B is Borel, (k^{-1}(A) ∩ E) ∩ A is also Borel, which means kE intersects k(A) in a measurable set. Therefore, E is measurable.

Hence, we have shown that E is measurable if and only if kE is measurable.