Let $K$ be a $\sigma$-finite measure on $\mathbb{R}^n\times\mathbb{R}_+\times\mathbb{R}^n$, with the Borel $\sigma$-algebra. Suppose that $K$ is symmetric ($K(A\times C\times B)=K(B\times C\times A)$) and positive definite ($\iiint_{\mathbb{R}^n\times[0,T]\times\mathbb{R}^n} f(x,t)f(y,t)K(dx,dt,dy)\geq0$ for each bounded measurable function $f:\mathbb{R}^n\times [0,T]\rightarrow\mathbb{R}$).
Define the norm $\|f\|$ of a measurable function $f:\mathbb{R}^n\times [0,T]\rightarrow\mathbb{R}$ as $$ \|f\|^2=\iiint_{\mathbb{R}^n\times[0,T]\times\mathbb{R}^n} |f(x,t)f(y,t)|K(dx,dt,dy).$$ Let $P$ be the space of such functions with finite norm. I want to prove that $P$ is complete.
First, given any $f$, consider $B=\{(x,s)\in\mathbb{R}^n \times[0,T]:|f(x,s)|\geq\epsilon\}$, $\epsilon>0$. Then $$\epsilon K(B\times\mathbb{R}^n)\leq \|f\| (K(B\times\mathbb{R}^n))^{1/2}.$$ This is easily proved as follows: $$ \epsilon K(B\times\mathbb{R}^n)=\iiint_{B\times\mathbb{R}^n}\epsilon K(dx,dt,dy)\leq (f,1_{B\times\mathbb{R}^n})_K\leq \|f\| (K(B\times\mathbb{R}^n))^{1/2}, $$ where Cauchy-Schwarz inequality has been applied.
Let $\{f_m\}$ be a Cauchy sequence in $\|\cdot\|$. By the inequality derived, $\{f_m\}$ is Cauchy in measure with respect to $K(\cdot\times\mathbb{R}^n)$, so there exists the limit in measure $f$: $$ B_m=\{(x,s)\in\mathbb{R}^n \times[0,T]:|f(x,s)-f_m(x,s)|\geq\epsilon\},$$ $$ \lim_{m\rightarrow\infty}K(B_m\times\mathbb{R}^n)=0. $$ Now I estimate $\|f_m-f\|^2$ by decomposing the integral into $B_m^c\times\mathbb{R}^n$ and $B_m\times\mathbb{R}^n$: \begin{align*} \|f_m-f\|^2= & \iiint_{B_m^c\times\mathbb{R}^n} |f(x,t)f(y,t)|K(dx,dt,dy)+\iiint_{B_m\times\mathbb{R}^n} |f(x,t)f(y,t)|K(dx,dt,dy) \\ \leq & \epsilon \iiint_{\mathbb{R}^n\times [0,T]\times\mathbb{R}^n} |f(y,t)|K(dx,dt,dy)+\|f\|_\infty^2 K(B_m\times\mathbb{R}^n). \end{align*} Finally, let $m\rightarrow\infty$.
The main issue is that $K(B_m\times\mathbb{R}^n)$ or $\|f\|_\infty$ may not be finite. I do not know if the $\sigma$-finite condition has some use here. Any help would be greatly appreciated.