For instance say I have random variable $X \sim \mathcal{N(\mu,\sigma)}$ Now the $\lim\limits_{\sigma \to \infty} f_{X} = 0$, so unfortunaltly it's not a pdf. However infinitesimally smaller values for $\sigma$ are a valid pdf. I haven't worked with nonstandard calculus, but would something like $\displaystyle st(f_X)$ be right? Is this the correct notation?
This came up when thinking about wavefunction. For example, if we have a particle that isn't constraint, then we know atleast it has a 50% of being found on one side of the universe $\int_{0}^{\infty} st(f_X(x))dx = 0.5$