I am searching for references about the concept I tried to capture in the following definition.
Let $(X, \tau, \Sigma, \mu)$ be a topological measure space, that is, $(X, \tau)$ is a topological space and $(X, \Sigma, \mu)$ a measure space such that $\sigma(\tau) \subseteq \Sigma$.
Definition. An element $x \in X$ is called an infinitely (measure-theoretically) dense point of $(X, \tau, \Sigma, \mu)$ if for every measurable neighborhood $V \in \Sigma$ of $x$, it follows that $\mu(V) = \infty$.
As an example, consider the usual one-dimensional Lebesgue topological measure space $(\mathbb{R}, \tau_\mathbb{R}, \mathcal{L}_\mathbb{R}, \lambda)$, the density function $$f : \mathbb{R} \to \left[0, \infty \right[ : x \mapsto \begin{cases} \frac{1}{|x|}, & x \neq 0 \\ 0 \end{cases} $$ and the measure given by $$\mu : \mathcal{L}_\mathbb{R} \to \left[0, \infty\right] : E \mapsto \int_E f \,\mathrm{d}\lambda$$
Then $0$ is a infinitely dense point of $(\mathbb{R}, \tau_\mathbb{R}, \mathcal{L}_\mathbb{R}, \mu)$.
(In this case, it seems related to the fact that $0$ is an essential discontinuity of infinite type of $f$.)
Any reference on this concept would be greatly appreciated, especially the ones giving some kind of characterization (even if partial) of the spaces having this property.