I am currently reading a paper by Goldstern, Kellner and Shelah, in which they, pretty nonchalantly, state *"Amoeba forcing will add a null set covering all old null sets"*, without proving this fact or giving a reference. The only thing I could find that would prove this statement was in the Bartoszynski book *"Set Theory: On the structure of the Real Line"*, where he states that any two Amoeba forcings are equivalent, which would mean that, even when forcing with just one Amoeba forcing, for any $n\in\omega$, there would be a set of measure $<\frac{1}{n}$ (by a standard density argument), covering all old null sets, which would imply the statement. But I cannot seem to understand his proof, therefore I am wondering if perhaps there is an easier way, i.e. simply constructing an $\mathbb{A}\_{1/2^{n+1}}$-generic Filter from a $\mathbb{A}\_{1/2^n}$-generic filter.



Amoeba forcing adds a null set covering all old null sets
The conditions in Amoeba forcing are open sets of measure less than $1/2$. (Say, in $2^\omega$.) 


A condition $q$ is stronger than $p$ iff $q \supseteq p$. (Alternatively, use closed sets of measure greater than $1/2$. Then stronger conditions will be smaller.) 


For a generic filter $G$ let $U\_G$ be the union of all sets in $G$. An easy density argument shows that $U$ is an open set of measure $1/2$. (I can elaborate if necessary)


Another easy density argument shows that every ground model null set is contained in $U\_G$. More precisely, whenever $c$ is a (natural) code for a $G\_\delta$ null set, then the $G\_\delta$ null set $B\_c$ described by $c$ (in $V[G]$) is a subset of $U\_G$. 


Every "rational" infinite $01$-sequence (i.e., with only finitely many $1$'s) defines a measure-preserving translation map $x \mapsto x+t$ on $2^\omega$. 
Every rational translate of $U\_G+t $ has the same property as $U\_G$: measure $1/2$, and it covers every ground model $G\_\delta$ null set. 


(Proof: given such a null set $N$, also the set $N-t$ is null, hence covered by $U\_G$, so $N$ is covered by $U\_G+t$.) 


Now we get to the main point: **How to get from $1/2$ down to $0$?**


Let $U'\_G:= \bigcap\_t (U\_G + t)$, where $t$ ranges over all rational sequences. Then $U'\_G$ is a $G\_\delta$ set, its measure is at most $1/2$, but by Kolmogorov's 0-1-law, its measure must be $0$. But $U'\_G$ still covers all ground model null sets. 


Remark: If you do an iteration of Amoeba forcing (of limit length) rather than a single forcing, you could (but why would you?) replace the above argument by the following: the first Amoeba forcing gives you an open set of measure 1/2, the second a new open set of measure 1/3, etc. Now take the intersection of all these sets. 


