When is a feature map continuous?

Question

Let (X,$\tau$) be a Topological space (V,A,$d \nu$) a measure space. We have a function $\varphi: X \times V \to \mathbb{R}$ and we define the kernel $k:X \times X \to \mathbb{R}$ by $k(x,y)= \int_V \varphi(x,v) \varphi(y,v) d\nu (v)$. The questions now is: What properties should $\varphi$ satisfy such that the kernel $k$ is continuous? What I figured so far was: It is enough to prove that the feature map is continuous. The feature space being $L^2(d\nu)$ and the feature map being $\Phi: X \to L^2(d\nu)$ such that $\Phi(x)=\varphi(x,\cdot)$. Also if $\varphi$ is continuous and we have a $g\in L^2(d\nu)$ such that $\forall x \in X$ $ || \varphi(x,\cdot) ||_{L^2(d\nu)} \leq || g ||_{L^2(d\nu)}$. I tried to prove it with sequence continuity and used the theorem of dominated convergence. The problem is this only works if $(X,\tau)$ has a first countable neighborhood basis. If I try to extend the proof with a net. I cannot use the theorem of dominated convergence.