Chapter 2 Random Variable

Let $(\Omega,\mathcal{F})$ be a measurable space.

Definition 1 ($\sigma$-measurable function)

Suppose that $h:\Omega\to\mathbb{R}$. For $A\subseteq\mathbb{R}$, define $h^{-1}(A):=\lbrace \omega\in\Omega:h(w)\in A\rbrace$. We say that the function $h$ is $\mathcal{F}$-measurable ($h\in m\mathcal{F}$, where $m\mathcal{F}$ is the class of $\mathcal{F}$-measurable functions on $\Omega$), if

or to say that $h^{-1}:\mathcal{B}\to\mathcal{F}$.

The picture of $\mathcal{F}$-measurable function $h$ is

Theorem 1 (Generation)

Let $\mathcal{C}$ be a family of subsets. If $\sigma(\mathcal{C})=\mathcal{B}$, and $h^{-1}(\mathcal{C})\subset\mathcal{F}$, then $h\in m\mathcal{F}$.

Proof of theorem 1

Assert $\mathcal{A}:=\lbrace B\in\mathcal{B}: h^{-1}(B)\in\mathcal{F} \rbrace$ is a $\sigma$-algebra (Prove it!!). $\mathcal{A}\supset\mathcal{C}$, hence $\mathcal{A}\supset\sigma(\mathcal{C})=\mathcal{B}$, which means each Borel set is the subset of $\mathcal{A}$, that is $\forall B\in\mathcal{B}: h^{-1}(B)\in\mathcal{A}$.

Corollary of theorem 1

Suppose $\Omega$ is a topological space, and the function $h:\Omega\to\mathbb{R}$ is continuous. Then, $h\in\mathcal{F}$.

Proof: $\sigma(\text{all open sets})=\mathcal{B}$, since $h$ is continuous, $h^{-1}(\text{open set})\subset\Omega$, thus $h\in\mathcal{F}$.

Lemma 1

If $\lambda\in\mathbb{R}$ and $g_1,g_2\in\mathcal{F}$, then $g_1+g_2\in\mathcal{F}$, $\lambda g_1\in\mathcal{F}$.

Lemma 2

Let $(g_n)_{n\geq1}\in\mathcal{F}$, then

1. $\inf_ng_n\in\mathcal{F}$
2. $\lim\inf g_n\in\mathcal{F}$
3. $\lim\sup g_n\in\mathcal{F}$
4. $\lbrace \omega:\lim_ng_n(\omega)\text{ exists in }\mathbb{R} \rbrace\in\mathcal{F}$

Proof of Lemma 2

1. $\forall\alpha\in\mathbb{R}:\lbrace\inf_ng_n\geq\alpha\rbrace:=\lbrace\omega\in\Omega:\inf_ng_n(w)\geq\alpha\rbrace\in\mathcal{F}\implies\inf_ng_n\in\mathcal{F}$, since $\sigma([\alpha,\infty):\alpha\in\mathbb{R})=\mathcal{B}$, and $\inf_n g_n^{-1}([\alpha,\infty))=\lbrace\omega\in\Omega:\inf_ng_n(w)\geq\alpha\rbrace=\lbrace\inf_ng_n\geq\alpha\rbrace=\bigcap_n\lbrace g_n\geq\alpha\rbrace\in\mathcal{F}$
2. Prove it!
3. Prove it!
4. $\lbrace\omega:\lim_ng_n(\omega)\text{ exists in }\mathbb{R}\rbrace=\lbrace\omega\in\Omega:\lim\sup g_n(\omega)=\lim\inf g_n(\omega)\rbrace\cap\lbrace\omega\in\Omega:\lim\sup g_n(\omega)<\infty\rbrace\cap\lbrace\omega\in\Omega:\lim\inf g_n(\omega)>-\infty\rbrace\in\mathcal{F}$

Example

Let $\Omega=\lbrace H,T\rbrace^{\mathbb{N}}$, that is tossing a coin n times. $A_{n,H}:=\lbrace\omega:\omega_n=H\rbrace$, $A_{n,T}:=\lbrace\omega:\omega_n=T\rbrace$, $\mathcal{F}:=\sigma(A_{n,\omega};n\in\mathbb{N}$, $\omega\in\lbrace H,T\rbrace)$. $X_n=1$ if $\omega_n=H$, $X_n=0$ otherwise. $S_n=\sum_{i=1}^nX_i$. $\Lambda=\lbrace\omega:\lim_{n\to\infty}\frac{S_n}{n}=p\in(0,1)\rbrace\in\mathcal{F}$. $\mathcal{F}=\sigma(X_m\leq n)\in\mathcal{F}$.

$\sigma$-algebras generated by RVs

In the above example, $\mathcal{F}_n=\sigma(X_m:m\leq n)$ is the smallest $\sigma$-algebra that makes all $X_m$, $m=1,\dots,n$ are measurable. Any collection to $(X_i)_{i\in\mathcal{I}}$ can be extended.

Definition 2 (Distribution function and law of X)

Suppose $X$ is a r.v. on $(\Omega, \mathcal{F}, \mathbb{P})$. Define the law of $X$ by

Observe that $\mathcal{L}_X$ is a set function on $\mathcal{B}$, and it is a probability measure on $\mathcal{B}$. (Prove it!!)

Recall $\mathcal{B}=\sigma((-\infty,\alpha]:\alpha\in\mathbb{R})$. $\{(-\infty,\alpha]:\alpha\in\mathbb{R}\}$ is a $\pi$-system. If we want to characterize the probability measure $\mathcal{L}_X$, we know that it is enough to look at the behaviour of the probability measure on $\pi$-system.

Therefore, define the distribution function as follows:

Properties of $F_X$

• $F_X:\mathbb{R}\to[0,1]$
• $F_X(x)\leq F_X(y)$ if $x\leq y$, that is $F_X$ is non-decreasing
• $\lim_{x\to\infty}F_X(x)=1$ ($=\mathbb{P}(X<\infty)$)
• $\lim_{x\to-\infty}F_X(x)=0$ ($=\mathbb{P}(X>-\infty)$)
• $\mathcal{F}$ is right continuous, i.e. $F_X(x)=\lim_{y\downarrow x}F_X(y)$

Existence of a RV with a given distribution

Let $F$ be a distribution function with above properties. Then, exists a r.v. on $([0,1],\mathcal{B}[0,1],Leb)$, such that $F=\mathcal{L}_X$, is a probability space.

Theorem 2 (The Monotone-Class Theorem)

Analogue of $\pi$-$\lambda$ theorem for measurable functions.

Let $\mathcal{H}$ be a class of bouned functions from a set $\Omega$ into $\mathbb{R}$ satisfying the following conditions:

1. $\mathcal{H}$ is a vector space over $\mathbb{R}$
2. the constant function $\mathbb{1}$ is an element of $\mathcal{H}$
3. if $(f_n)$ is a sequence of non-negative functions in $\mathcal{H}$ such that $f_n\uparrow f$ where $f$ is a bounded functions on $\Omega$, then $f\in\mathcal{H}$.

Then, if $\mathcal{H}$ contains the indicator function of every set in some $\pi$-system $\mathcal{I}$, then $\mathcal{H}$ contains every bouned $\sigma(\mathcal{I})$-measurable function on $\Omega$.