Typical $\mathbb Z^n$-actions can be inserted only in injective $\mathbb R^n$-actions

We study actions of the groups $\mathbb Z^n$ and $\mathbb R^n$ by Lebesgue space automorphisms. We prove that a typical $\mathbb Z^n$-action can be inserted only in injective actions of $\mathbb R^n$, $n\in\mathbb N$. We give a simple proof of the fact that a typical $\mathbb Z^2$-action cannot be inserted in an $\mathbb R$-action.