How to show that $\int_{\mathbb{R}^d}I(\lVert u \rVert_\infty\leq C)du\leq M<\infty$?
My intuition says that $$\int_{\mathbb{R}^d}I(\lVert u \rVert_\infty\leq C)du\leq\int_{[-C,C]^d}1 du=(2C)^d:=M,$$
since the maximum coordinate must be less than $C$, and so the others.
Is it right?