Generalized Density Function

category	measurable space	distribution	measure	description
Continuous	$(\mathbb{R},\mathcal{R})$	${\mu }_X$	$\lambda$
Discrete	$(S,\mathcal{S})$	${\tilde{\mu }}_X:\mathcal{S}\to [0,1]$	$\#$: Counting measure	$\#$ is not sigma-finite on $(\mathbb{R},\mathcal{R})$
Discrete	$(\mathbb{R},\mathcal{R})$	${\mu }_X$	${\#}_X$	${\#}_X$ is sigma-finite on $(\mathbb{R},\mathcal{R})$

Where:

• $S$ is a support of a discrete random variable
• $\mathcal{S}=2^S$.
• ${\#}_X:={\sum }_{i\in X}{\delta }_i$: only counts the number of elements in the support of a discrete random variable $X$. It can be expressed as the sum of Dirac measure