[FOM] A question about the possible sizes of definable collections of non measurable sets of reals

[Ashutosh]

Is there a definable collection of non measurable sets of reals, in
the theory ZFC, whose size is not equal to the power set of reals?
That is, does there exist a formula \phi(x) in the language of ZFC
such that the following hold:

(1) ZFC proves  \phi(x) --> "x is a non measurable subset of R" and
(2) ZFC proves "cardinality of {x : \phi(x)}" < "cardinality of the
power set of reals"?

Any references to results that answer questions in similar directions
would be appreciated.

Thanks.
Ashutosh