on $\mathcal{F}$.

The triple $(\Omega, \mathcal{F}, \mathbb{P})$ is called a probability space.

**\[Negligeable set\]**

1.  Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $A\subset\Omega$ is said to be $\mathbb{P}$-null or negligeable if $\mathbb{P}(A)=0$

2.  A property is said to be true almost surely (a.s) if the set on which this property is not true is negligeable.

**\[Measurability and random variable\]**

1.  Let $(\Omega, \mathcal{F}, \mathbb{P})$ and $(\Omega', \mathcal{F}', \mathbb{P}')$ be two probability spaces. A function $X : \Omega \longrightarrow \Omega'$ is said to be $\mathcal{F}$-measurable if and only if $$\begin{aligned}
    X^{-1}(U) := \{ \omega\in \Omega : X(\omega) \in U\}\subset \mathcal{F},  \hspace{0.5cm} \forall\hspace{0.2cm} U\in \mathcal{F}'\end{aligned}$$

2.  A random variable $X$ is a function $X : \Omega \longrightarrow \Omega'$ $\mathcal{F}$-measurable.

3.  If $\Omega'=\mathbb{R}$, then $X$ is called a real random variable.

4.  If $\Omega'=\mathbb{R}^n$, $n>1$ then $X$ is called a vector random variable.

In the following, unless otherwise state, $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space and $X$ a random variable, $X :\Omega \longrightarrow \mathbb{R}^n$.

.

Every random variable induces a probability measure on $\mathbb{R}^n$ denoted $\mu_X$ and define by

$\mu_X(B) :=\mathbb{P}(X^{-1}(B))$, $\forall\, B$ open set of $\mathbb{R}^n$. $\mu_X$ is called the distribution function of $X$.

**\[Expected value\]**

1.  If $X$ is a 