Empirical measures of iid euclidean samples converge weakly

Countable compactly supported tests determine euclidean weak convergence: For each finite $d\ge 1$ there is a countable uniformly dense subset $\mathcal{D}$ of $C_c({\mathbb{R}}^d;\mathbb{R})$ containing nonnegative compact cutoffs ${\chi }_m\uparrow 1$. If Borel probabilities ${\mu }_n,\mu$ have $\int hd{\mu }_n\to \int hd\mu$ for every $h\in \mathcal{D}$, then ${\mu }_n\Rightarrow \mu$.

Measurable coordinatewise functions preserve independence: Let $(X_i{)}_{i\in I}$ be an independent family of random elements $X_i:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S_i,{\mathrm{\Sigma }}_i)$. For each $i$, let $g_i:(S_i,{\mathrm{\Sigma }}_i)\to (T_i,{\mathcal{T}}_i)$ be measurable. Then the family $(g_i\circ X_i{)}_{i\in I}$ is independent.

Change of variables for expectation: Let $X:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S,\mathrm{\Sigma })$ be a random element, let ${\mathbb{P}}_X$ be its law, and let $g:(S,\mathrm{\Sigma })\to \mathbb{R}$ or $g:(S,\mathrm{\Sigma })\to \mathbb{C}$ be measurable.

1. If $g\ge 0$, then $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$
2. If $g(X)$ is integrable, then $g$ is integrable with respect to ${\mathbb{P}}_X$ and the same formula holds: $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

Kolmogorov iid l1 strong law: For IID real $(X_n{)}_{n\ge 1}$ with $\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$, $S_n\mathrm{/}n\to \mu =\mathbb{E}{X}_1$ almost surely.

Finite and countable subadditivity of measures: Let $\mu$ be a measure and let $(E_k{)}_{k\in \mathbb{N}}$ be measurable. Then

$\mu \left({\bigcup }_{k\in \mathbb{N}}{E}_k\right)\le {\sum }_{k=0}^{\mathrm{\infty }}\mu (E_k)\mathrm{.}$

For every $m\in \mathbb{N}$ one also has

$\mu \left({\bigcup }_{k<m}{E}_k\right)\le {\sum }_{k<m}\mu (E_k),$

including $m=0$, where both sides are $0$.