I asked the following on MSE a few weeks ago but I did not get any answer : https://math.stackexchange.com/questions/1039593/carlsons-translatability-are-theses-characterisations-equivalent
Reference : "strong measure zero and strongly meager sets", Timothy Carlson (p.582).
Suppose $\kappa$ is a cardinal number and $\mathcal{U}$ is a set of subsets of a group $G$. $\mathcal{U}$ is $\kappa$-translatable if for every $A$ in $\mathcal{U}$ there is a $B$ in $\mathcal{U}$ such that the union of any $\kappa$ many translates of $B$ is contained in a translate of $A$.
Here is what Carlson says :
Notice that if $\mathcal{U}$ is a filter then $\mathcal{U}$ is $\kappa$-translatable iff for every $A$ in the dual ideal there is a $B$ in the dual ideal such that the intersection of any $\kappa$ translates of $B$ contains a translate of $A$. Hence, $\mathcal{U}$ is $\kappa$-translatable iff for every $A$ in the dual ideal there is $B$ in the dual ideal such that the union of any $\kappa$ translates of $A$ is contained in a translate of $B$.
The second iff is not transparently clear to me...
