  Let $\mathcal{X}$ be a subset of $L^1$ that contains the space of simple
random variables $\mathcal{L}$ and $\rho: \mathcal{X} \rightarrow
(-\infty,\infty]$ a dilatation monotone functional with the Fatou property. In
this note, we show that $\rho$ extends uniquely to a $\sigma(L^1,\mathcal{L})$
lower semicontinuous and dilatation monotone functional $\overline{\rho}: L^1
\rightarrow (-\infty,\infty]$. Moreover, $\overline{\rho}$ preserves
monotonicity, (quasi)convexity, and cash-additivity of $\rho$. Our findings
complement recent extension results for quasiconvex law-invariant functionals
proved in [17,20]. As an application of our results, we show that transformed
norm risk measures on Orlicz hearts admit a natural extension to $L^1$ that
retains the robust representations obtained in [4,6].
