I asked the following on MSE a few weeks ago but I did not get any answer :

Given a translation-invariant ideal $\mathcal{I}$ on a commutative group $G$ and it's dual filter $\mathcal{I}^*$, why is $$ (\forall I \in \mathcal{I})(\exists I' \in \mathcal{I})(\forall g_1, g_2 \in G)(\exists h \in G)\bigg((I+g_1)\cup( I+g_2) \subseteq ( I' + h)\bigg)$$ equivalent to $$(\forall F \in \mathcal{I}^*)(\exists F' \in \mathcal{I}^*)(\forall g_1, g_2 \in G)(\exists h \in G)\bigg((F'+g_1)\cup( F'+g_2) \subseteq ( F + h)\bigg).$$

Reference : "strong measure zero and strongly meager sets", Timothy Carlson (p.582).