Proper Turing cones are null?

From: Mike Oliver (mike_lists_at_verizon.net)
Date: 02/23/05 

Date: Tue, 22 Feb 2005 20:57:31 -0600

I *think* the following is true -- if x is a noncomputable
real, then the set of y such that x is computable wrt y has
measure 0. Or equivalently, a random real (in the sense of
forcing) does not compute any noncomputable real from the
ground model.

Anyone have a simple argument (or counterexample)? I
tried to come up with some sort of forcing homogeneity
or ergodicity argument, but they bogged down and didn't
look like they'd really get the result anyway.