Weight function of weighted $\mathcal{L}_{2}$ space

Question

Let us define the space: \begin{equation}\label{space} \mathcal{L}_{\mathit{2}, \varpi} \! \left(\mathcal{K}; \mathbb{R} \right) : = \left\{ g(\cdot) \;\textrm{is measurable}: \int_{\mathcal{K}} \varpi(\tau) g^2(\tau) \mathsf{d} \tau < \infty \right\}, \quad \mathcal{K} \subseteq\mathbb{R} \end{equation} where $\varpi(\cdot)$ is measurable and the integration $\int_{\mathcal{K}} \varpi(\tau) g^2(\tau) \mathsf{d} \tau$ is Lebesgue integral. My question is what is the proper condition for $\varpi(\cdot)$ to make sure that $ \mathcal{L}_{\mathit{2}, \varpi} \! \left(\mathcal{K}; \mathbb{R} \right)$ is a normed space? I believe that as long as $\varpi(\tau)>0$ for almost all $\tau \in \mathcal{K}$, (namely a countable zero values of $\varpi(\tau)$ is acceptable), then $ \mathcal{L}_{\mathit{2}, \varpi} \! \left(\mathcal{K}; \mathbb{R} \right)$ is a normed space. Is the above condition an answer to my question? Can I make such condition weaker to make $\mathcal{L}_{\mathit{2}, \varpi} \! \left(\mathcal{K}; \mathbb{R} \right)$ to be a normed space?