[FOM] Re: L-measurable sets

[E. Todd Eisworth]

[Reply to Enayat]

Many of the standard "tree forcings" [Sacks forcing, Miller forcing, Laver
Forcing] preserve the property of having positive outer measure,i.e., if

A is a set of reals of positive outer measure, then in the generic
extension, A still has positive outer measure.

I know that Miller forcing actually preserves outer measure, i.e., 

If A is a set of reals of outer measure c, then the outer measure of A is
still c in the generic extension.
[I believe this is due to Bartoszynski, Judah, and Shelah.]

I don't recall if the same is true for Sacks and Laver forcing, but the book
"Set theory: on the structure of the real line" by Bartoszynski and Judah is
a good reference for such questions.

Best,

Todd