Well - order on reals and not Lebesgue measurable sets

Let $(x_t )_{t<\omega_1}$ be a well ordering of the set of real numbers of the type $\omega_1$ ( the first uncountable ordinal, we assume the continuum hypothesis ) and let $\leq$ be a natural order on reals. Does there exists $b\in\mathbb{R}$ such that the set $$\{x_t :x_{t+1} \leq b\}$$ is not Lebesgue measurable?

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You are assuming the continuum hypothesis? –  Asaf Karagila Nov 22 '13 at 13:11