I'm currently reading "Iterating Brownian motions, ad libitum" (available here).
Lemma 6 proves an equivalence criterion of vague convergence in the distribution of random measures. The statement and further ideas seem clear, but there is a construction that doesn't seem to make sense. The construction is as follows:
Let $(\lambda_n)_{n\in \mathbb{N}}$ be random probability measures on $\mathbb{R}$.
For each $p \in \mathbb{N}$ let $(X_p)_{n\in\mathbb{N}}$ be random variables such that for all $p \in \mathbb{N}$ ($A_1 , \dots , A_p$ are Borel sets):
$$\mathbb{P}(X_1 ^n \in A_1 , \dots , X_p ^n \in A_p|\lambda _n) = \lambda _n (A_1) \cdot \dots \cdot \lambda_n (A_p) \ \ \ \ a.s.$$.
Let $\xi _n ^p = p^{-1} \sum _{i=1}^p \delta _{X_i ^n}$ and let $f$ be a continuous, bounded function with compact support on $\mathbb{R}$. For notational purposes, let $\lambda _n f := \int _{\mathbb{R}} f \mathrm{d} \mu _n$.
Now a statement is made that does not seem to make sense, namely
$\mathbb{P}(|\lambda _n f - \xi _n f|\geq \varepsilon)$.
$= \mathbb{E}[\mathbb{P}(|\lambda _n f - \xi _n f|\geq \varepsilon | \lambda _n)]$
$\leq \frac{||f||_{\infty}}{\varepsilon ^2 p}$
As far as I understand, $\mathbb{P}(|\lambda _n f - \xi _n f|\geq \varepsilon)$ is a random variable, since, due to the fact that $\lambda _n$ and $\xi _n$ as random measures depend on $\omega$, so does $\mathbb{P}(|\lambda _n f - \xi _n f|\geq \varepsilon)$.
What is the idea behind $\mathbb{P}(|\lambda _n f - \xi _n f|\geq \varepsilon) = \mathbb{E}[\mathbb{P}(|\lambda _n f - \xi _n f|\geq \varepsilon | \lambda _n)]$? (the application of Chebyshev's inequality is clear, once the expecation is put)
Kind regards
