For a fixed Turing machine $\Phi_e$, what is the probability that it will reduce a given real to some less complex, yet still non-computable real?

More precisely: It is known that the set of reals with minimal Turing degree has measure zero. Since $N_e:=\lbrace X: \Phi_e^X\text{ is total and }X>_T\Phi_e^X>_T\emptyset\rbrace$ is Borel, it is Lebesgue measurable. But each non-minimal $X$ is in some $N_e$, and hence by the result quoted above not every $N_e$ has measure zero (since the set of non-minimal reals, with measure 1, is the union of the countably many measurable $N_e$). My question is: what is known about the possible values of $m(N_e)$ for $e\in\omega$? (I am also interested in a characterization of the set of $e$ such that $N_e$ has measure zero (or one).)

One thing that is easy to show: just by examining the definition, it is clear that $m(N_e)$ is $\Sigma^1_2$ (I think) for each $e\in\omega$. I presume much more can be said (perhaps $\forall e, m(N_e)\in\lbrace 0, 1\rbrace$?), yet I cannot seem to prove anything nontrivial.