Let $X_1, \ldots, X_n$ be $\mathbb{R}^d$-valued random variables on the probability space $(\Omega, \mathcal{F}, P)$ and let $\delta_c$, $c \in \mathbb{R}^d$, denote the Dirac measure. Consider the emirical measure defined by
$$
\mu := \frac{1}{n} \sum_{k=1}^n \delta_{X_{k}},
$$
so that for every fixed $\omega \in \Omega$ and a measurable set $A$ we get
$$
\mu(A) = \frac{1}{n} \sum_{k=1}^n \delta_{X_k(\omega)} ( A ) = \frac{1}{n} \sum_{k=1}^n 1_A (X_k(\omega)).
$$
Now assume that an $\mathbb{R}^d$-valued function $G$ is known to depend on $\mu$ which allows for the notation $G(\mu)$.
Why is it justified to have $$G(\mu)= \frac{1}{n} \sum_{k=1}^n X_k, \tag{1}$$ i.e. can $(1)$ be expressed via the measure $\mu$ (for example, through some integrals w.r.t. $\mu$ for every fixed $\omega \in \Omega$)?
If $d=1$, then for every fixed $\omega$ we can write
$$
\int_{\mathbb{R}} x \, d \left( \frac{1}{n} \sum_{k=1}^n \delta_{X_k(\omega)} \right) = \frac{1}{n} \sum_{k=1}^n \int_{\mathbb{R}} x \, \delta_{X_k(\omega)} = \frac{1}{n} \sum_{k=1}^n X_k(\omega) = G(\mu(\omega)).
$$
Can something similar be done for general $d$? Is there maybe an alternative representation?
Best Answer
Well, I'm rather confused by your notation. So I assume that you want to know why can we find (roughly speaking) a function $G$ defined on space of measures such that $G(\mu) = \frac{1}{N} \left( \sum_{n} X_n \right)$.
And to be honest, it is not anything mysterious and I guess you also knew it. It is just linear functional on space of measures.
To be more precise, for any bounded function $g$ , we define:
$G: \mathcal{M}_f\left( \mathbb{R}^d \right) \longrightarrow \mathbb{R}$
$ \mu \mapsto \int g d\mu$
where $\mathcal{M}_f\left( \mathbb{R}^d \right)$ is the space of all finite measures on $\mathbb{R}^d$
Remark: There are some useful properties of $G$ that you can see. $G$ is clearly continuous wrt to strong norm in $\mathcal{M}_f$, and if $g$ is continuous, $G$ is also continuous wrt weak topology on $M_f$.etc.
If $g$ is not bounded, $G$ is just a linear operator defined only on some subspace of $\mathcal{M}_f$, but you can always adjust $g$ for your needs.