If $X\cap (1-X)=\varnothing$ and $X\cup (1-X)=[0,1]$, can I conclude that $X$ has Lebesgue measure $1/2$?

Question

Let's say I have a set $X\subset[0,1]$, and $1-X:=\{1-x\mid x\in X\}$.

Question: If $X\cap (1-X)=\varnothing$ and $X\cup (1-X)=[0,1]$, can I conclude that $X$ has Lebesgue measure $1/2$?

I am pretty sure that this is true when $X$ is measurable, so I am worried that $X$ could also be non-measurable.

Answer 1

Consider any $A\subset[0,1/2)$. Let $\bar{A}=[0,1/2)\setminus A$.

(I think the question is strange with the element $1/2$, but we are not too much interested in this one point so I ignore)

Let $X=A\cup(1-\bar{A})$. Then, every condition you mentioned is satisfied. However, to make your expectation true, we should be able to say that every subset of $[0,1/2)$ is measurable in $\mathbb{R}$, which is false.