$A^c\in\mathcal{F}$.

    3.  If $(A_i)_{i\in I}$ is a countable collection of set in $\mathcal{F}$, then $\cup_{i\in I} A_i\in \mathcal{F}$.

2.  Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two $\sigma$-algebra on $\Omega$. $\mathcal{F}_1$ is said to be a sub-$\sigma$-algebra of $\mathcal{F}_2$ if $\mathcal{F}_1\subset\mathcal{F}_2$.

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1.  Given any family $\mathcal{B}$ of subset of $\Omega$, we denote by $$\begin{aligned}
    \sigma(\mathcal{B}):=\cap\{\mathcal{C}: \hspace{0.2cm}\mathcal{C}, \hspace{0.2cm}\sigma- \text{algebra of}\hspace{0.2cm} \Omega, \hspace{0.2cm}\mathcal{B}\subset \mathcal{C}\}\end{aligned}$$ the smallest $\sigma$-field of $\Omega$ containing $\mathcal{B}$, $\sigma(\mathcal{B})$ is called the $\sigma$-field generated by $\mathcal{B}$.

    When $\mathcal{B}$ is a collection of all open sets of a topological space $\Omega$, $\sigma(\mathcal{B})$ is called the Borel $\sigma$-algebra on $\Omega$ and the elements of $\sigma(\mathcal{B})$ are called Borel sets.

2.  If $ X: \Omega \longrightarrow\mathbb{R}^n$ is a function, then the $\sigma$-algebra generated by $X$ is the smallest $\sigma$-algebra on $\Omega$ containing all the sets of the form $$\begin{aligned}
    \{X^{-1}(U) : U\subset \mathbb{R}^n,  \hspace{0.5cm}\text{open}\}.\end{aligned}$$

**\[Probability measure\]**.

Let $\mathcal{F}$ be a $\sigma$-field on $\Omega$. A probability measure is an application $\mathbb{P} : \mathcal{F}\longrightarrow [0,1]$ satisfying

1.  $\mathbb{P}(\Omega)=1-\mathbb{P}(\emptyset)=1$.

2.  If $(A_i)_{i\in I}$ is a countable collection of elements of $\mathcal{F}$ pairwise disjoints, then

    $\mathbb{P}(\cup_{i\in I}A_i)=\sum\limits_{i\in I}\mathbb{P}(A_i)$.

**\[Probability space\]**.

Let $\Omega$ be a non-empty set, $\mathcal{F}$ a $\sigma$-field on $\Omega$ and $\mathbb{P}$ a probability measure 