Let $f\in L_{\text{loc.}}^1(\Bbb{R}^d)$ such that for some $p\in (0,1)$,
$$\left\vert\int f(x)\,g(x)\,\mathrm dx\right\vert\leq\left(\int\vert g(x)\vert^p\,\mathrm dx\right)^{\frac{1}{p}}$$
for all $g\in C_c^1(\Bbb{R}^d)$, i.e., all continuous functions of compact support. Then show $f=0$ a.e.
Attempt/Thoughts:
We know $f<\infty$ a.e. because it’s in $L^1$, so if
$$A_k:=\left\{x:f(x)>\frac{1}{k}\right\},$$
one would like to show for each $k$ that $m(A_k)=0$. Then I said for $k=1$, one has $A_1=\{x:f(x)>1\}$ and since $f$ may be negative,
$$\int_{\Bbb{R}^d}f(x)\,g(x)\,\mathrm dx=\int_{\Bbb{R}^d\setminus A_1}f(x)\,g(x)\,\mathrm dx+\int_{A_1}f(x)\,g(x)\,\mathrm dx.$$
Could I say, suppose $m(A_1)\neq 0$? Then since $A_1^c$ also has infinite Lebesgue measure, then does this somehow tell me $\int_{\Bbb{R}^d\setminus A_1} f(x)\,g(x)\,\mathrm dx$ is not finite? Or can I write $\Bbb{R}^d$ as (And if $B_k:=\left\{x:f(x)<-\frac{1}{k}\right\}$)
$$\bigcap_{k}A_k \cup \bigcap_k B_k\cup\left\{x:f(x)=0\right\}.$$
But we know the $A_k,B_k$ are compact since $g \in C_c^1(\Bbb{R}^d)$.
