Is $L^1(\mathbb R^n)\cap L^2_{loc}(\mathbb R^n)$ included in $L^2(\mathbb R^n)\,$?

Question

This might be trivial, but I can't find the answer.

For example if $|x|^{-r}$ is integrable over $(-\infty,-1]\cup[1,\infty)$, so does $|x|^{-2r}$. Of course one can extend $|x|^{-r}$ by a continuous function on $[-1,1]$ in order to remove problems at $x=0$. I might guess the answer is positive also in general.

Answer 1

Define $f$ by : $$\forall x\in\mathbb R, f(x) = \sum_{n =2}^{+\infty}n1_{[n,n+1/n^3]}(x)$$

Then, $f$ is locally bounded and therefore locally square integrable. Furthermore, we have : $$\int_{\mathbb R}|f| \text d\lambda = \sum_{n=2}^{+\infty}\frac{1}{n^2} <+\infty$$ and $$\int_{\mathbb R}|f|^2 \text d\lambda = \sum_{n=2}^{+\infty}\frac{1}{n} =+\infty$$