We want : $ \int g$ d$\mathbb{P}_n = \frac{1}{n} \sum_{i=1}^n g(y_i) $ .
Important for this exercise : $\mathbb{P}_n (A) = \frac{1}{n} \sum_{i=1}^n 1_{y_i}(A) $ , with $1_{y_i}$ point measures. So $1_{y_i}(A) =1 $, if $ y_i \in A $ and $1_{y_i}(A) = 0$ otherwise.
I already show this claim for $g : \mathbb{R} \rightarrow \mathbb{R} $ with $g(a) = [a \in B]$ ( indicator function ) for a set B $\in \mathbb{B}$ ( Borel). Here:
$ \int g$ d$\mathbb{P}_n = \mathbb{P}_n(B) = \frac{1}{n} \sum_{i=1}^n 1_{y_i}(B) =\frac{1}{n} \sum_{i=1}^n [y_i \in B] = \frac{1}{n} \sum_{i=1}^n g(y_i) $. With the help of integration techniques I want to generalize this now for any measurable function $g$.
Here is my start: We know $g$ is measurable. So $g^{+}$, and $g^{-}$ are integrable (non-negative and measurable ). Now we know there are increasing sequences $(h_j)_{j \in \mathbb{N}}$ and $(k_j)_{j \in \mathbb{N}}$ of nonnegative simple functions with $(h_j) \nearrow g^{+}$ and $(k_j) \nearrow g^{-}$. So $ \int g^{+}$ d$\mathbb{P}_n = \lim_{j \rightarrow \infty} \int h_j $ d$\mathbb{P}_n$ and $ \int g^{-}$ d$\mathbb{P}_n = \lim_{j \rightarrow \infty} \int k_j $ d$\mathbb{P}_n$.