$g_n = m(E_n)^{\frac{-1}{q}}\chi_{E_n}$. Show $\int fg_n \rightarrow 0$

Question

Let $f\in L^p(\mathbb{R})$. Let $E_n$ be measurable sets of finite measure, where $\lim_{n \to \infty}m(E_n)=0$. Let $p,q$ be such that $1<p,q<\infty$ and $\frac{1}{p}+\frac{1}{q} = 1$. And define $g_n = m(E_n)^{\frac{-1}{q}}\chi_{E_n}$. Show $\lim_{n\to\infty}\int f g_n = 0$.

I can't figure out how to get this one off the ground. I initially thought to use Holder's inequality and then take the limit of $||f||_p||g_n||_q$, but $||g_n||_q = 1$, so that's not getting me anywhere.

Any thoughts would be greatly appreciated.

Thanks in advance.

Answer 1

Hint: Hölder gives you $$\left\vert \int fg_n \right\vert \leq \int \vert fg_n \vert \leq \Vert f \chi_{E_n}\Vert_p \cdot \Vert g_n \Vert_q = \Vert f \chi_{E_n}\Vert_p,$$ which will allow you to conclude what you want.