In general, if $X$ is a random variable defined on a probability space $(Ω, Σ, P)$, then the expected value of $X$ is defined as
\begin{align}
\int_\Omega X \, \mathrm{d}P = \int_\Omega X(\omega) P(\mathrm{d}\omega)
\end{align}
Let $X_1, X_2, \dots$ be a sequence of independent random variables identically distributed with probability measure $P$.
The empirical measure $P_n$ is given by
\begin{align}
P_n =\frac{1}{n}\sum_{i=1}^n \delta_{X_i}
\end{align}
Is the following the correct notation for the expectation with respect of the empirical measure?
\begin{align}
\int_\Omega X \, \mathrm{d}P_n = \int_\Omega X(\omega) P_n(\mathrm{d}\omega) = \frac{1}{n}\sum_{i=1}^n X_i
\end{align}
Edit:
What is important for me, is that $P_n$ is an approximation for $P$.
I think $X$ and $X_1,\dotsc,X_n$ should map from $(Ω, Σ)$ to some measureable space $(\mathcal{F},\mathscr{F})$.
Best Answer
It's not correct.
The empirical measure isn't a measure on the sample space $\Omega$, it's a (random) measure on $\mathbb{R}$. Notationally, I think most people reserve letters like $P, P_n$, etc, for measures on $\Omega$, using letters like $\mu, \nu$ for measures on other spaces.
So I'd call your empirical measure $\mu_n$ and then write its mean as $$\int_{\mathbb{R}} x\,\mu_n(dx) = \frac{1}{n} \sum_{i=1}^n X_i.$$ Note that the left-hand side denotes the integral over $\mathbb{R}$, with respect to the measure $\mu_n$, of the identity function $f : \mathbb{R} \to \mathbb{R}$ given by $f(x) = x$. The lower-case $x$ is intentional and not a typo.