# Convergence of integral by using a Lebesgue point

I need some help with this problem:
let $$\lambda>0$$ and $$w(x)=Ce^{-|x|}$$ a function defined on $$\mathbb{R}^{N}$$, where $$C>0$$ is a constant such that $$\int_{\mathbb{R}^{N}}w(x)dx=1$$.
Then we define a new function as $$w_{\lambda}(x)=C\lambda^{N}e^{-\lambda|x|}$$.
Moreover, let $$V\in L^{q}(\mathbb{R}^{N})$$, with $$q\geq\frac{N}{2}$$, such that $$V^{-}\not\equiv0$$ on a set of positive measure and let $$0$$ a Lebesgue point of V belonging to this set (note that with $$V^{-}$$ I state the negative part of $$V$$).
I shall prove that $$\int_{\mathbb{R}^{N}}V(x)w_{\lambda}(x)dx\xrightarrow[\lambda\rightarrow+\infty]{}V(0).$$
I took the ball $$B_{\lambda^{-1}}(0)$$ and I wrote $$\bigg|\int_{\mathbb{R}^{N}}V(x)w_{\lambda}(x)dx-V(0)\bigg|\leq C\lambda^{N}\int_{\mathbb{R}^{N}}|V(x)-V(0)|e^{-\lambda|x|}dx$$$$=C\lambda^{N}\int_{B_{\lambda^{-1}(0)}}|V(x)-V(0)|e^{-\lambda|x|}dx+C\lambda^{N}\int_{\mathbb{R}^{N}\setminus B_{\lambda^{-1}(0)}}|V(x)-V(0)|e^{-\lambda|x|}dx$$ and I already proved the convergence to $$0$$ of the first integral, so I need help with the second one. I assume I should use the Hoelder Inequality in some way, but I really don't understand how.
I hope someone could help me and I already thank you.

• I'm sorry I made a mistake, the function $w(x)=Ce^{-|x|}$ without $\lambda$. – Francesca May 25 at 10:15
• You can fix errors like that with the "edit" link. You meant $\int w=1$, not $C\int w=1$. – David C. Ullrich May 25 at 10:23