This is a different question from my previous question Difference between a partial selector and a selector, however I am going to repeat the preamble...

In Kharazishvili's "Nonmeasurable Sets and Functions" there is the following theorem:

There exists a subset $X$ of $\mathbb{R}$ which is a Vitali set and a Bernstein set.

The proof begins as follows:

Let $\alpha$ denote the first ordinal of cardinality the continuum. Let {$x_{\xi} : \xi < \alpha$} be an injective family of all points of $\mathbb{R}$ and let {$F_{\xi} : \xi < \alpha$} denote an injctive family of all uncountable closed subsets of $\mathbb{R}$. Similarly to the classical Bernstein construction, we define, by applying the method of transfinite recursion, an injective family {$x_{\xi} : \xi < \alpha$} of points in $\mathbb{R}$. Suppose that, for an ordinal $\beta < \alpha$, the partial family of points {$x_{\xi} : \xi < \beta$} has already been constructed.

We are now going to depart from the proof a little...

Let $A \subseteq \mathbb{R}$ be uncountable. Consider the set

$Z_{\beta} = \cup${$x_{\xi} + A : \xi < \beta$}.

Take any element $z \in F_{\beta}\backslash Z_{\beta}$ and put $x_{\beta} = z$. In this way, we construct a family of points
{$x_{\xi} : \xi < \alpha$}. Now we define:

$X =$ {$x_{\xi} : \xi < \alpha$}.

My question is can $X$ be non-measurable?

This is a different question from my previous question Difference between a partial selector and a selector, however I am going to repeat the preamble...

In Kharazishvili's "Nonmeasurable Sets and Functions" there is the following theorem:

There exists a subset $X$ of $\mathbb{R}$ which is a Vitali set and a Bernstein set.

The proof begins as follows:

Let $\alpha$ denote the first ordinal of cardinality the continuum. Let {$x_{\xi} : \xi < \alpha$} be an injective family of all points of $\mathbb{R}$ and let {$F_{\xi} : \xi < \alpha$} denote an injctive family of all uncountable closed subsets of $\mathbb{R}$. Similarly to the classical Bernstein construction, we define, by applying the method of transfinite recursion, an injective family {$x_{\xi} : \xi < \alpha$} of points in $\mathbb{R}$. Suppose that, for an ordinal $\beta < \alpha$, the partial family of points {$x_{\xi} : \xi < \beta$} has already been constructed.

We are now going to depart from the proof a little...

Let $A \subseteq \mathbb{R}$ be uncountable. Consider the set

$Z_{\beta} = \cup${$x_{\xi} + A : \xi < \beta$}.

Take any element $z \in F_{\beta}\backslash Z_{\beta}$ and put $x_{\beta} = z$. In this way, we construct a family of points
{$x_{\xi} : \xi < \alpha$}. Now we define:

$X =$ {$x_{\xi} : \xi < \alpha$}.

My question is can $X$ be non-measurable?