Preserving Non-null with Suslin+ Forcings

We (i.e. I) present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is the ideal generated by a Suslin ccc forcing (e.g. Lebesque– null or meager), and P is a Suslin forcing, and P is I–preserving (i.e. it doesn’t make any positive Borel–set small), then P preserves generics over candidates and therefore is strongly I–preserving (i.e. doesn’t make any positive set small). This is also useful for preservation in limit–steps of iterations (Pα)α<δ: while it is not clear how one could argue directly that Pδ still is weakly I–preserving, the equivalent “preservation of generics” can often be shown to be iterable (see e.g. the chapter on preservation theorems for proper iterations in [BJ95] for the case of I=Lebesque–null or meager). For (short) iterations of Suslin forcings, see [GJ92]. For the ideal of meager sets the equivalence of weak preservation and preservation was done by Goldstern and Shelah [She98, Lem 3.11, p.920]. Pawlikowski [Paw95] showed the equivalence for P=Laver and I=Null, building on [JS90]. Shelah proved the theorem in the context of nep forcing [She]. The definition and basic properties of the ideal belonging to Q have been used for a long time, e.g. in works of Judah, Bartoszyński and Ros lanowski, cited in [BJ95], also related is [Sik64, §31].