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Random Variables & Measurable Functions

Basics of Probability

A measurable space consists of a set and a $\sigma$-algebra on that set. If $(E, \mathcal E)$ and $(F, \mathcal F)$ are measurable spaces, then we say $f: E \rightarrow F$ is measurable if for each $B \in \mathcal F$, we have $f^{-1}(B) \in \mathcal E$.

If $\Omega$ is the sample space of a random experiment, and $\mathcal H$ a $\sigma$-algebra on $\Omega$, then if $X : \Omega \rightarrow \mathbb R$ is measurable with respect to $(\mathbb R, \mathcal B(\mathbb R))$ we call it a (real-valued) random variable. Likewise, if $X: \Omega \rightarrow E$ is measurable with respect to $(E, \mathcal E)$ then we call $X$ an $E$-valued random variable. Most of the time we will deal with real valued random variables, and in this situation we can decompose $X = X_+ – X_-$ into positive (non-negative) and negative parts. We will denote non-negative real valued random variables by $\mathcal H_+$.

If the set of values taken by $X$ is finite, we call $X$ a simple random variable. If the set of values of $X$ is discrete, then we say $X$ is a discrete random variable. The simplest simple random variables are indicator random variables: if $A \in \mathcal H$ then the indicator for $A$ is $\boldsymbol 1_A(\omega)$ takes the value 1 for $\omega \in A$ and is zero otherwise. Every simple random variable can be written as $$X = \sum a_n \boldsymbol 1_{A_n},$$ where $A_n = \{\omega \in \Omega : X(\omega) = a_n\}$ is the level set of $a_n$. This decomposition is unique up to reordering.

Exercise: Given any positive random variable $X$ produce an increasing sequence of simple random variables $(X_n)$ such that for any $\omega \in \Omega$, $\lim X_n(\omega) = X(\omega)$.

It is sometimes useful to think of positive random variables as limits of increasing sequences. This idea is captured in the notion of a monotone class.

Definition: A set $\mathcal M$ be a set of positive random variables is called a (positive) monotone class if

  • $1 \in \mathcal M$;
  • If $X, Y \in \mathcal M$ and $a, b \in [0, \infty)$ then $a X + b Y \in \mathcal M$;
  • If $(X_n) \in \mathcal M$ and $X_n \nearrow X$, then $X \in \mathcal M$.

Theorem: If $\boldsymbol 1_A \in \mathcal M$ for all $A \in \mathcal H$, then $\mathcal M = \mathcal H_+$. That is, every positive measurable random variable is the increasing limit of simple random variables.

The preceding theorem is a version of the Monotone Class Theorem, the general case of which deserves a brief discussion. First we introduce the notion of a $\pi$-system.

Definition: $\mathcal E \subseteq \mathcal H$ is called a $\pi$-system if $\mathcal E$ is closed under intersections. That is if $A, B \in \mathcal E$ then $A \cap B \in \mathcal E$.

Monotone Class Theorem: If $\mathcal E$ is a $\pi$-system that generates $\mathcal H$, and if $\mathcal M$ is a monotone class containing $\boldsymbol 1_A$ for all $A \in \mathcal E$, then $\mathcal M = \mathcal H_+$.

The Borel $\sigma$-algebra on $\mathbb R$ is generated by the collection of all open intervals $\mathcal E = \{(a, b) : a < b \}$. That is $\mathcal B(\mathbb R) = \sigma(\mathcal E)$ and it is easily seen that $\mathcal E$ is a $\pi$-system. The Monotone Class Theorem implies that every positive measurable function on $(\mathbb R, \mathcal B(\mathbb R) )$ is the increasing limit of simple functions whose level sets are finite unions of open intervals.

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