I have posted this question on MSE but haven't received an answer yet. I rephrase it here.
Let $(X,B)$ be a klt pair where $K_X+B$ is $\mathbb{R}$-Cartier. Let $\pi:X\rightarrow U$ be a projective morphism of quasi-projective varieties. Assume either that $B$ is $\pi$-big and $K_X+B$ is $\pi$-pseudo-effective, or $K_X+B$ is $\pi$-big, then the work of Birkar-Cascini-Hacon-McKernan says that the log canonical ring $$\mathfrak{R}(\pi,K_X+B)=\bigoplus_{m\in\mathbb{N}}\pi_*\mathcal{O}_X(\left\lfloor m(K_X+B)\right\rfloor )$$ is a finitely generated $\mathcal{O}_U$-algebra under the assumption that $K_X+B$ is $\mathbb{Q}$-Cartier.
I wonder why the $\mathbb{Q}$-Cartier assumption is necessary for the proof? What changes to the ring $\mathfrak{R}(\pi,K_X+B)$ when we allow $K_X+B$ to be $\mathbb{R}$-Cartier? It looks to me that it should changes nothing since all that matter is the round down of $K_X+B$. What am I missing?
