$L_p$ space,convergence Let $1<p<\infty$ and $h\in L_p(\mathbb{R})$,that is,$\left(\displaystyle\int_{\mathbb{R}}|h|^p\right)^{1/p}<\infty$. Define a sequence $(f_n)_{n\in\mathbb{N}}$ by $f_n(x):=h(x-n)$. How to prove that $$\lim_{n\to\infty}\displaystyle\int_{\mathbb{R}}f_n g=0$$ for all $g\in L_q(\mathbb{R})$ with $\dfrac{1}{p}+\dfrac{1}{q}=1$?
 A: *

*The sequence $\{f_n\}$ is bounded in $L^p$, hence it's enough to prove the result for $g$ lying in a dense set of $L^q$? 

*We can choose simple functions, and by linearity, the characteristic functions of sets of finite measure. 

*Such sets can be approximated by bounded sets. So actually we just have to deal with the case $g=\chi_B$, where $B$ is a bounded set. 

*Assuming that $B\subset [-R,R]$ and $h\geqslant 0$, show that $\lim_{n\to +\infty}\int_{[-R+n,R+n]}h(x)dx=0$.

A: First, I think you are able to show that $\|f_n\|_p=\|h\|_p<\infty$. We need to use the following two points:


*

*$g\in L^q(\mathbb{R})\ \Rightarrow\ \exists\ M_1>0\ s.t.\ \int_{|x|>M_1} |g|^q \le \epsilon^{\ q}$

*$h\in L^p(\mathbb{R})\ \Rightarrow\ \exists\ M_2>0\ s.t.\ \int_{|x|>M_2} |h|^p \le \epsilon^{\ p}$


Now we are ready to show the result:
$$ |\int_{\mathbb{R}} f_n\cdot g|\le \int_{|x|\le M_1}|f_n\cdot g|+\int_{|x|> M_1}|f_n\cdot g|
$$


*

*$H\ddot{o}lder$'s inqequalty: $\int_{|x|>M_1}|f_n\cdot g|\le\|f_n\|_p\cdot (\int_{|x|>M_1} |g|^q)^{\frac{1}{q}}\le \epsilon\cdot \|h\|_p$

*\begin{eqnarray}
\int_{|x|\le M_1}|f_n\cdot g|&\le& \|g\|_q\cdot (\int_{|x|\le M_1}|f_n|^p)^{\frac{1}{p}}\\ &\le& \|g\|_q\cdot (\int_{|x|\le M_1}|h(x-n)|^p\mathrm{d}x)^{\frac{1}{p}}\\
&\le& \|g\|_q\cdot (\int_{x\in (-M_1-n,\ M_1-n)}|h|^p)^{\frac{1}{p}}
\end{eqnarray}
since $x-n\in(-M_1-n,\ M_1-n)$, we can choose large $n$ such that $ M_1-n<-M_2 $, then $\int_{|x|\le M_1}|f_n\cdot g|\le \epsilon\cdot \|g\|_q$


Form the above, we can conclude that $$ \int_{\mathbb{R}}f_n\cdot g\to 0$$
