Proof Verification that if $A$ is a measurable set then $-A$ is a measurable set

Question

Let $A$ be a measurable subset of $(-1,1)$. For every $X \subset \mathbb{R}$, let $-X=\{-x : x\in X\}$. Prove that $-A$ is a measurable set.

My Attempt:

Since $A$ is measurable, then I define $g:A \rightarrow (-1,1)$ as $g(x)=-x$. Since $A$ is measurable, $-x$ is continuous, then $g$ is a measurable function. Then, I consider the inverse image:

$g^{-1}(-1,1)=\{x: g(x) \in (-1,1)\}= \{x: -x \in (-1,1)\}=\{-y: y \in (-1,1)\}$

Since this is the pre image of a measurable function then this set is measurable. However, we have that: $-A \subset (-1,1)$ hence:

$-A \subset g^{-1}(-1,1)$.

Since $g^{-1}(-1,1)$ is measurable, then $-A$ is also measurable.

Any problems with this proof?

Answer 1

Your idea is correct. But your final steps are wrong and you could write it down cleaner. Just because a set $C$ is subset of a measurable set, it doesn't have to be measurable.

$g:(-1,1)\rightarrow (-1,1) , x \mapsto -x$ is continuous, hence measurable, and bijective. So we have

$$- A = g^{-1}(A)$$

as preimage of the measurable function $g$. Since $A$ is measurable, $-A$ is measurable, too.