Statement (1) is obvious. For the statement (2), write $f_{i}=g_{i}-h_{i}$ with $g_{i},h_{i}$ in $\mathscr{P}(\Lambda)$ (respectively, in ${\overline{\mathscr{P}}}(\Lambda)$). Then the upper envelope admits the DC decomposition $\max_{i}\{f_{i}\}=g-h$ with

$$g:=\sum_{j}g_{j},\quad h:=\min_{i}\bigg(h_{i}+\sum_{j\neq i}g_{j}\bigg),$$

which are both concave functions in $\mathscr{P}(\Lambda)$ (respectively, in ${\overline{\mathscr{P}}}(\Lambda)$). This shows that $\max_{i}\{f_{i}\}$ is in $\mathscr{D}(\Lambda)$ (respectively, in ${\overline{\mathscr{D}}}(\Lambda)$). The statement for the lower envelope follows similarly. ∎