TITLE: Probability that a Turing machine will nontrivially reduce a real
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: Not sure whether the following answers your question, but they might be helpful.
Fix any number $n\geq 2012$.
1 For any $e_0$ so that $\Phi_{e_0}^X=X_0$ where $X_0$ is the unique real so that $X=X_0\oplus X_1$.  Then for such $e_0$, $m(N_{e_0})=1$. 
2 For any $e_1$ so that $\Phi_{e_1}^X=0$ if $X(0)=0$ and $\Phi_{e_1}^X=X_0$ if $X(0)=1$.   Then for such $e_1$, $m(N_{e_1})=\frac{1}{2}$.
3 $m(N_e)$ must be $\Delta^0_n$. Just note that  $N_e$ is a $\Delta^0_{n-1}$ set.
4 $m(N_e)>0$ if and only if $N_e$ contains an $n$-random real.
\begin{proof}
If  $m(N_e)>0$, then obviously  $N_e$ contains an $n$-random real.
$N_e$ is a $\Delta^0_{n-1}$ set. So if it is null, then it does not contain any $n$-random real.
\end{proof}
5 $m(N_e)=1$ if and only if $N_e$ contains all $n$-random reals. 
The proof is similar to 4.
The lower bound $2012$ can be certainly significantly smaller. 
For randomness notions, you may refer Downey and Hirschfeldt (2010) or Nies (2009).