[FOM] a popular verstion-a question about L-measurable set

[Yu Liang]

Is there a model of ZFC say M so that there is  a model of ZFC say M' 
inculding M  so that the both real sets in M and M' have measure larger than 
0 in M'?

Is it possible  L=M?

Actually, I really want to know is whether every real set which is closed 
under 
Turing-eqivalent has measure 0 or 1.
 
I guess this question could not be answered under ZF (not sure under ZFC).
 
The known fact is "every cone of Turing degrees has measure 0 and the set of 
minimal Turing-degrees has measure 0".
 
Were there such a model, then it may mean the question can not been solved 
under ZFC(again not sure).

So much for that.
 
Thank you for your caring the question.

Liang Yu