$ \mathcal{F} $-Measurability and Examples

Motivation and Problem

Measurability is crucial in probability theory for defining random variables that allow probability assignments to events. For a random variable $ X: \Omega \to \mathbb{R} $ to be useful in a probability space $ (\Omega, \mathcal{F}, P) $, it must be compatible with the sigma-algebra $ \mathcal{F} $.

If $ X $ is not $ \mathcal{F} $-measurable, certain Borel sets in $ \mathbb{R} $ won’t correspond to measurable events in $ \Omega $, making it impossible to calculate probabilities for $ X $.

Informally, a random variable $ X $ is $ \mathcal{F} $-measurable if the information provided by $ \mathcal{F} $ is sufficient to determine the values of $ X $. This means that the events defined by $ \mathcal{F} $ are detailed enough to describe the outcomes of $ X $ in a way that allows us to assign probabilities to these outcomes.

Think of $ \mathcal{F} $ as describing the “level of detail” or “lens” through which you view the sample space $ \Omega $. If $ X $ outputs values that depend on details outside this lens, it’s not $ \mathcal{F} $-measurable.

In simpler terms, $ X $ is $ \mathcal{F} $-measurable if knowing the events in $ \mathcal{F} $ tells us everything we need to know about the possible values of $ X $.

Formal Definition of $ \mathcal{F} $-Measurability

A random variable $ X: \Omega \to \mathbb{R} $ is $ \mathcal{F} $-measurable if for every Borel set $ B \subseteq \mathbb{R} $, the preimage $ X^{-1}(B) $ is in $ \mathcal{F} $.

This all sounds very vague and abstract. Let’s see some examples.

Simple Examples: Coin Flip

Let’s take an example involving a coin flip, one of the simplest random phenomena, to illustrate measurability.

Setup

  1. Sample space: $ \Omega = {\text{H}, \text{T}} $, representing heads and tails.
  2. Sigma-algebra $ \mathcal{F} $: $ \mathcal{F} = {\emptyset, {\text{H}}, {\text{T}}, \Omega} $.
  3. Random variable $ X $: $$ X(\omega) = \begin{cases} 1, & \text{if } \omega = \text{H}, \\ -1, & \text{if } \omega = \text{T}. \end{cases} $$

Checking $ \mathcal{F} $-Measurability

To determine if $ X $ is $ \mathcal{F} $-measurable, we need to verify that for any Borel set $ B \subseteq \mathbb{R} $, the preimage $ X^{-1}(B) \in \mathcal{F} $.

  1. For $ B = {1} $:

    $$ X^{-1}(\{1\}) = \{\text{H}\} \in \mathcal{F}. $$

  2. For $ B = {-1} $:

    $$ X^{-1}(\{-1\}) = \{\text{T}\} \in \mathcal{F}. $$

  3. For $ B = {1, -1} $:

    $$ X^{-1}(\{1, -1\}) = \{\text{H}, \text{T}\} = \Omega \in \mathcal{F}. $$

Since $ X^{-1}(B) $ is in $ \mathcal{F} $ for all Borel sets $ B $, $ X $ is $ \mathcal{F} $-measurable.

Exercise

For this example, we have $ \mathcal{F} = \mathcal{P}(\Omega) $, the power set of $ \Omega $. Can you show that for any $ X: \Omega \to \mathbb{R} $, $ X $ is $ \mathcal{P}(\Omega) $-measurable?

Negative Example: Coin Flip with Insufficient $ \mathcal{F} $

Now consider the same random variable $ X $, but with a smaller sigma-algebra:

$$ \mathcal{F} = \{\emptyset, \Omega\}. $$

  1. For $ B = {1} $:

    $$ X^{-1}(\{1\}) = \{\text{H}\} \notin \mathcal{F}. $$

  2. For $ B = {-1} $:

    $$ X^{-1}(\{-1\}) = \{\text{T}\} \notin \mathcal{F}. $$

In this case, $ X $ is not $ \mathcal{F} $-measurable because $ \mathcal{F} $ doesn’t provide enough “detail” to distinguish between heads and tails.

Negative Example: Non-$ \mathcal{F} $-Measurable Random Variable

Setup:

  1. Sample space: $ \Omega = [0, 1] $.
  2. Sigma-algebra $ \mathcal{F} $: $ \mathcal{F} = {\emptyset, \Omega} $.
  3. Random variable $ X $: $ X(\omega) = \omega $.

Checking $ \mathcal{F} $-Measurability:

  • For $ X $ to be $ \mathcal{F} $-measurable, $ X^{-1}(B) $ must be in $ \mathcal{F} $ for every Borel set $ B $.
  • Consider $ B = [0, 0.5] $: $$ X^{-1}([0, 0.5]) = [0, 0.5] \notin \mathcal{F} $$
  • Thus, $ X $ is not $ \mathcal{F} $-measurable.

Fixing the Problem: Restricting $ X $

New Random Variable $ X’ $:

$$ X'(\omega) = \begin{cases} 1, & \text{if } \omega \in \Omega = [0, 1] \\ 0, & \text{otherwise} \end{cases} $$

Checking $ \mathcal{F} $-Measurability of $ X’ $:

  • If $ 1 \in B $: $$ X'^{-1}(B) = \Omega \in \mathcal{F} $$
  • If $ 1 \notin B $: $$ X'^{-1}(B) = \emptyset \in \mathcal{F} $$
  • Thus, $ X’ $ is $ \mathcal{F} $-measurable.

Positive Example: $ \mathcal{F} $-Measurable Random Variable

Setup:

  1. Sample space: $ \Omega = {1, 2, 3} $.
  2. Sigma-algebra $ \mathcal{F} $: $ \mathcal{F} = {\emptyset, {1}, {2, 3}, \Omega} $.
  3. Random variable $ X $: $$ X(1) = 0, \quad X(2) = 1, \quad X(3) = 1 $$

Checking $ \mathcal{F} $-Measurability:

  • For $ B = {0} $: $$ X^{-1}(\{0\}) = \{1\} \in \mathcal{F} $$
  • For $ B = {1} $: $$ X^{-1}(\{1\}) = \{2, 3\} \in \mathcal{F} $$
  • Thus, $ X $ is $ \mathcal{F} $-measurable.

Sigma-Algebra Generated by a Random Variable

What if instead of fitting the random variable to the sigma-algebra, we wanted to find the smallest sigma-algebra that fits the random variable?

Given a random variable $ X: \Omega \to \mathbb{R} $ on a probability space $ (\Omega, \mathcal{F}, P) $, the sigma-algebra generated by $ X $, denoted $ \sigma(X) $, is the smallest sigma-algebra making $ X $ measurable. It represents the information content of $ X $.

Definition

$$ \sigma(X) = \{ X^{-1}(B) : B \in \mathcal{B}(\mathbb{R}) \} $$

where $ \mathcal{B}(\mathbb{R}) $ is the Borel sigma-algebra on $ \mathbb{R} $.

Properties

  1. Subset of $ \mathcal{F} $: $ \sigma(X) \subseteq \mathcal{F} $.
  2. Generating Events: Includes events like $ { X \leq a } $, $ { X > b } $.
  3. Measurability: Any $ Y = g(X) $ for measurable $ g $ is $ \sigma(X) $-measurable.

Example 1: Identity Random Variable

Consider $ \Omega = [0, 1] $ and $ X(\omega) = \omega $.

Key Events in $ \sigma(X) $:

  1. $ { \omega : X(\omega) \leq 0.5 } = [0, 0.5] $
  2. $ { \omega : X(\omega) > 0.7 } = (0.7, 1] $
  3. $ { \omega : 0.3 \leq X(\omega) \leq 0.6 } = [0.3, 0.6] $

These intervals and their complements, unions, and intersections form $ \sigma(X) $.

Example 2: Piecewise Random Variable

Consider $ \Omega = [0, 1] $ and $ X $ defined as:

$$ X(\omega) = \begin{cases} 0, & \text{if } 0 \leq \omega < 0.5 \\ 1, & \text{if } 0.5 \leq \omega \leq 1 \end{cases} $$

Key Events in $ \sigma(X) $:

  1. $ { \omega : X(\omega) = 0 } = [0, 0.5) $
  2. $ { \omega : X(\omega) = 1 } = [0.5, 1] $
  3. $ { \omega : X(\omega) \leq 0.5 } = [0, 0.5) $

These events capture the information provided by $ X $, illustrating how $ \sigma(X) $ includes all necessary events to describe $ X $’s outcomes.

Conclusion

The sigma-algebra $ \sigma(X) $ generated by a random variable $ X $ encapsulates the information content of $ X $, including all events describable by $ X $’s values. Understanding $ \sigma(X) $ is crucial for analyzing probabilistic models and dependencies.

Exercises

Exercise 0: Showing Measurability

Can you show that for any $ X: \Omega \to \mathbb{R} $, $ X $ is $ \mathcal{P}(\Omega) $-measurable, where $ \mathcal{P}(\Omega) $ is the power set of $ \Omega $ (collection of all subsets of $ \Omega $)?

Exercise 1: Checking Measurability

Let $ \Omega = {a, b, c, d} $, and define a sigma-algebra $ \mathcal{F} $ as:

$$ \mathcal{F} = \{\emptyset, \{a, b\}, \{c, d\}, \Omega\}. $$

Define the random variable $ X: \Omega \to \mathbb{R} $ as:

$$ X(a) = 1, \quad X(b) = 1, \quad X(c) = -1, \quad X(d) = -1. $$
  1. Verify whether $ X $ is $ \mathcal{F} $-measurable.
  2. Identify the preimages of the sets $ B = {1} $ and $ B = {-1} $, and check if they belong to $ \mathcal{F} $.

Exercise 2: Constructing a Sigma-Algebra

Let $ \Omega = [0, 1] $ and define a random variable $ X(\omega) = \lfloor 10\omega \rfloor $.

  1. Construct the sigma-algebra $ \sigma(X) $ generated by $ X $.
  2. Identify the events in $ \sigma(X) $ that correspond to:
    • $ {X \leq 5} $,
    • $ {X = 3} $,
    • $ {X \geq 7} $.

Exercise 3: Measurability with Coarse Sigma-Algebra

Consider $ \Omega = [0, 1] $ with the trivial sigma-algebra $ \mathcal{F} = {\emptyset, \Omega} $. Let $ X(\omega) = \sin(2\pi\omega) $.

  1. Show that $ X $ is not $ \mathcal{F} $-measurable.
  2. Suggest a sigma-algebra that would make $ X $ measurable.

Exercise 4: Measurability with Piecewise Random Variables

Let $ \Omega = [0, 1] $, and define:

$$ X(\omega) = \begin{cases} 0, & \text{if } 0 \leq \omega < 0.25, \\ 1, & \text{if } 0.25 \leq \omega < 0.75, \\ 2, & \text{if } 0.75 \leq \omega \leq 1. \end{cases} $$
  1. Construct the sigma-algebra $ \sigma(X) $ generated by $ X $.
  2. Write down the preimages $ X^{-1}(B) $ for $ B = {0} $, $ B = {1} $, and $ B = {0, 2} $.

Exercise 5: Non-Measurable Functions (Advanced)

Let $ \Omega = [0, 1] $ with the Lebesgue sigma-algebra, and consider the characteristic function of the rationals in $ [0, 1] $:

$$ X(\omega) = \begin{cases} 1, & \text{if } \omega \in \mathbb{Q}, \\ 0, & \text{if } \omega \notin \mathbb{Q}. \end{cases} $$
  1. Show that $ X $ is measurable with respect to the Lebesgue sigma-algebra.
  2. What happens if $ \mathcal{F} $ is the trivial sigma-algebra $ {\emptyset, \Omega} $? Is $ X $ still measurable?