Integral in termsof supremum

Question

I am facing difficulty to prove the following fact: If $w$ is a locally integrable positive function in $\Omega$, then $$ \sup_{B(x,R)}\lvert v\rvert=\lim_{p\to\infty}\lVert v\rVert_{L^p(B(x,R),w))}, $$ where $B(x,R)\subset \Omega$ and $\Omega$ is a bounded domain, and $\lVert v\rVert_{L^p(B(x,R),w))}$ is defined to be $$ \left(\int_{B(x,R)}\lvert v\rvert^pw(x)\,dx\right)^{1/p}. $$ For $w=1$, I know this is true. But for non-constant $w$, even for $A_p$ weights does the same hold true? I have sen this fact is applied in the paper attached in the link: see page 15 or 16. http://imar.ro/journals/Mathematical_Reports/Pdfs/2017/3/3.pdf