I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is the average of $u(y)$ over $Q$ and the supremum is taken over all cubes of arbitrary diameter.

Questions: 1. I think what can be the definition of BMO spaces on a torus $T^n = S^1 \times ...\times S^1$? We cannot have cubes of arbitrary diameter. Maybe we can look at the supremum over cubes whose diameter is smaller than or equal to that of torus?

2 What will happen if we take the supremum over cubes $Q$ whose diameter is less than or equal $r$, say, r being very small? Will that give the same norm? Thanks please.

Edit: Question 3. Please also advise a little about non-quotient spaces?

Edit:Joonas Ilmavirta answers Q1 and 3 below. Someone please look at Question 2.