I am reading the recent book by Kawamata, *Algebraic Varieties: Minimal Models and Finite Generation*. There is an English translation here .

In the bottom of page 16 he says that an $\mathbb{R}$-Cartier divisor can be written as $D = \sum d_i D_i$ with $D_i$ Cartier and $d_i \in \mathbb{R}$. Then for $f: X \rightarrow Y$ between normal varieties we can define $f^* D $ as $\sum d_i f^*D_i$. He says that this doesn't depends on the expression of $D$ (There are many ways to decompose $D$ as linear combination of Cartier divisors). How can I see that?

A similar fact(?) mentioned in this book is that an $\mathbb{R}$-Cartier divisor that is a $\mathbb{Q}$-Weil divisor is a $\mathbb{Q}$-Cartier divisor. I don't think this is obvious also, since we may write an $\mathbb{R}$-Cartier divisor as $\sum d_i \sum k_jD_{ij}$ with $d_i \in \mathbb{R}$ but after summing up the coefficient it becomes a $\mathbb{Q}$ divisor.