I have spent the last couple of days reading up on measure theory and absolute continuity to get my head around the definitions of discrete/empirical measures, absolute continuity and singularity. I would love to have someone check that I got everything right.

Let's assume $\nu$ is an empirical measure, defined as

$$ \nu = \sum_{i=1}^n a_i \delta_{x_i} $$

where $a=(a_1, \dots, a_n) \in \{u \in \mathbb{R}^n | \sum a_i=1\}$ and $X=(x_1, \dots,x_n) \in \mathbb{R}^n$.

If I understood correctly, I could view $\nu$ in two different settings.

  1. View $\nu$ as a measure on its countable support $X$
    In this case $\nu$ is absolutely continuous with respect to the counting measure on $X$. Let $\tau$ be the counting measure on $(X, \mathcal{P})$, which is positive sigma-finite. Then I can write $\nu$ as $$ \forall A \subset X, \nu(A)= \sum_{x_i \in A} a_i \tau(\{x_i\}) $$ Also, by Radon-Nikodym, $\nu$ admits a density $f$ wrt $\tau$ defined by $\forall x_i \in X, f(x)=a_i$ if $x=x_i$.
  2. View $\nu$ as a measure on the measurable space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$. In this case $\nu$ is a radon measure on $\mathbb{R}$. Also, the following line of argument is valid. $\nu$ is absolutely continuous to the measure $\tau:= \delta_{x_1}+\dots+\delta_{x_n}$.
    Let $\lambda$ denote the Lebesgue measure on $\mathbb{R}$. Since $\delta_{x_i} \perp \lambda$ we have $\tau \perp \lambda$. And since $\nu \ll \tau$ we have $\nu \perp \lambda$. Hence, $\nu$ is singular with respect to the Lebesgue measure on $\mathbb{R}$.
    Lastly, by Radon Nikodym $\nu$ admits a density with respect to $\tau$ given by $$ f(x) = \begin{cases} a_i & \text{if } x=x_i \\ 0 & \text{else} \end{cases} $$

Are both cases correct? Did I describe them correctly?


Both interpretations (and your descriptions) are correct, yes. I will point out that the second is typically more useful, especially in the setting where you want to prove some convergence properties of random empirical measures to some underlying measure, as one tends to do (Poisson point processes are a perfect example). Your set $X$ is then random, so isolating $\nu$ to be an element of the set of measures on the set $X$ is restrictive - the next realization will have a different $X$.


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