Clarification on the notation $L^\infty(w(y) dy)$

Question

I'm trying to understand Besov spaces at the moment and in one characterisation, the space $L^r(dy/|y|^\alpha)$ is used. The $y$ denotes an element of $\mathbb R^n$. I know what this is for $r<\infty$, but for $r=\infty$, is it: $f\in L^\infty(dy/|y|^\alpha)$ iff

1. $f \in L^\infty(\mathbb R^n)$ (i.e. with standard Lebesgue measure) ? or
2. $|y|^{-\alpha}f\in L^\infty (\mathbb R^n)$?

(For $p<\infty$, people write $f\in L^p(d\mu)$ iff $\int |f|^p d\mu < \infty$.)

Does this hold more generally for other weights $w(y)dy$; and in general for arbitrary measures $\mu,\nu$, is $L^\infty(\mu)= L^\infty(\nu)$ if their nullsets are the same? Or is it defined to be something else?

Answer 1

1) is right, not 2). Yes, if $\mu$ and $\nu$ have the same null sets then $L^{\infty} (\mu) $ is same as $L^{\infty} (\nu) $.