Let $(\Omega,\mu)$ be a measure space.
The weighted Lebesgue space $L^1(\Omega,w)$, where $w : \Omega \rightarrow \mathbb R$ is a $\mu$-measurable weight function, is defined as the set of all $\mu$-measurable functions $f : \Omega \rightarrow \mathbb R$ such that $$ \|f\|_{L^1(\Omega,w)} = \int_\Omega |f(x)| w(x) \, d\mu(x) < \infty. $$
Q: what are sufficient and necessary conditions for $L^1(\Omega,w) = L^1(\Omega,1)$ as Banach spaces?
It seems sufficient that $w(x) \geq w_0 > 0$ for almost all $x \in X$, where $w_0 > 0$ is a positive lower bound. I am wondering whether that is necessary.