The $L^p_\text{loc}(\mathbb{R})$ topology

Question

I was reading some lecture notes that I found on internet and I saw something that I cannot completely understand. For $p\geq 1$, let's consider the $L^p_\text{loc}(\mathbb{R})$ space, which is given by $$
L^p_\text{loc}:=\{f:\mathbb{R}\to\mathbb{R}: \, \hbox{ for all compact interval } \ I=[a,b], \ a<b, \ \Vert f\Vert_{L^p(I)}<\infty\}.
$$
I was wondering what is the "natural" $L^p_\text{loc}$-topology? What are open or closed sets in the "standard topology" given by $L^p_\text{loc}$. This question is trivial in the case of $L^p$ since we can use the $L^p$-norm to define the "natural" topology in $L^p$. However, in this case the $L^p$-norm makes no sense, since functions in $L^p_\text{loc}$ don't belong to $L^p$. Am I missing something? Maybe the natural topology is related to some weak or weak-* topology? If that is the case, is it true that $(L^p_\text{loc})^*=L^q_\text{loc}$ with $\tfrac{1}{p}+\tfrac{1}{q}=1$ as in the standard case $L^p$ and $L^q$ with $p\neq \infty$.

Answer 1

A natural metric on this space is defined by $d(f,g)=\sum_n \frac 1 {2^{n}} \min\{(\int_{-n}^{n} |f(x)-g(x)|^{p} dx)^{1/p} ,1\}$.

In this topology $f_k \to f$ iff $\int_{-n}^{n} |f_k(x)-f(x)|^{p}dx\to 0$ for each $n$ iff $\int_{a}^{b} |f_k(x)-f(x)|^{p}dx\to 0$ whenever $a <b$ iff $\int_K |f_k(x)-f(x)|^{p}dx\to 0$ for any compact set $K$.