Let $w:\mathbb{R}^n\rightarrow \mathbb{R}$ be a non-negative, measurable function.
And we define a weighted integrable space $L^1(w)$ whose norm is defined by \begin{equation} \|f\|_{L^1(w)}:=\int_{\mathbb{R}^n}|f(x)|w(x)dx \end{equation}
My question: if $f\in L^1(w)$, is $f\cdot\chi_{B}$ (where $B$ is bounded ball centered at $0$) integrable in the usual sence? That is, if \begin{equation} \int_{\mathbb{R}^n}|f(x)|w(x)dx<\infty, \end{equation} can we show \begin{equation} \int_{B}|f(x)|dx<\infty\ ? \end{equation}
Please help me!
Thank you.
Without any more information on w, not a lot can be said. For example, the obvious case $w=0$ springs to mind. We always have $\frac{1}{||\cdot ||} \in L^1(w)$ but it is never in $L^1(B)$.