Assigning Numbers to Events in Probability

So far, we’ve learned that in probability theory, the “events” we talk about must belong to:

• a σ-algebra (to allow set operations), and
• especially the Borel sets (the natural σ-algebra on ℝ)

The next big question is:

👉 How do we assign actual numbers (probabilities) to those sets?

To answer that, we need the idea of a measure.

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What Is a Measure? Start with Length and Area

Although the word “measure” may sound abstract, it’s something we use every day.

For example:

• The interval $[0, 3]$ has length 3
• The set $[0,1] \cup [2,3]$ has length 2 (1 + 1)
• The interval $[0,1.5]$ has length 1.5

→ Measures work not only for continuous intervals but also for disjoint unions of intervals.

For areas:

• The rectangle $[0,2] \times [0,5]$ has area 10
• The triangle $\{(x,y) \mid 0 \leq x \leq 2, 0 \leq y \leq 5 - 2.5x\}$ has area 5
• The unit disk $\{(x,y) \mid x^2 + y^2 \leq 1\}$ has area $\pi$

👉 A measure is simply a rule that assigns a size (non-negative number) to each set.

Length MeasuresArea Measures

Examples of Length Measures

Assigning lengths to intervals - the most familiar example of measures

[0, 3]

0

1

2

3

4

Length calculation: 3 - 0 = 3

[0,1] ∪ [2,3]

0

1

2

3

4

Interval decomposition:

[0, 1] has length = 1

[2, 3] has length = 1

Total length = 1 + 1 = 2

[0, 1.5]

0

1

2

3

4

Length calculation: 1.5 - 0 = 1.5

📏 Properties of Length Measure

• • Length of interval [a, b] = b - a
• • Sum of lengths of disjoint intervals = length of their union
• • Length of empty set = 0
• • All lengths are non-negative

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What Properties Does a Measure Have?

A measure $\mu$ is a function from sets to non-negative numbers satisfying:

1. Empty set has zero measure

   $$\mu(\varnothing) = 0$$

2. Non-negativity

   $$\mu(A) \geq 0$$

3. σ-additivity (countable additivity)
   For disjoint sets $A_1, A_2, \dots$:

   $$\mu\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \mu(A_i)$$

This justifies why breaking a shape into parts and adding their areas gives the correct total.

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What Is a Probability Measure?

A probability measure is just a measure that totals to 1:

• $P(\varnothing) = 0$
• $P(A) \geq 0$
• $P\left(\bigcup A_i\right) = \sum P(A_i)$
• $P(\Omega) = 1$

So it’s just a regular measure that’s normalized to 1.

Example:

• For a uniform distribution on $[0,1]$:
  • The measure is length
  • The set $[0,0.3]$ has length 0.3 → So probability is also 0.3

👉 Probability is just a size ratio in a world where the total size is 1.

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Random Variables and Measurability: Why It Matters

A random variable is a function from the sample space to ℝ:

But not every function qualifies — it must be measurable.

What is Measurability?

Measurability ensures that “events defined via $X$” can be assigned probabilities.

Specifically, it requires:

For any reasonable subset of ℝ (like $(-\infty,3]$), its preimage under $X$ must belong to $\mathcal{F}$ — the collection of sets we can assign probabilities to.

Why is this important?

• We can only assign probabilities to sets in $\mathcal{F}$
• So anything constructed via $X$ must land in $\mathcal{F}$ — and measurability guarantees this

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Formal Definition: Measurable Function

A function $X: (\Omega, \mathcal{F}) \to (\mathbb{R}, \mathcal{B})$ is $\mathcal{F}$-measurable if:

Here:

• $\mathcal{B}$ is the Borel σ-algebra on ℝ (intervals, open sets, etc.)

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Intuition and Examples

• $\mathcal{F}$ = the collection of measurable events (can assign probabilities)
• Measurability = ensures events defined by $X$ are measurable too

Example 1: Dice Roll

• $\Omega = \{1,2,3,4,5,6\}, \mathcal{F} = 2^\Omega$
• $X(\omega) = \omega$ (the die roll)
• Event “$X$ is even” = $\{2,4,6\} \in \mathcal{F}$ → measurable!

Example 2: Continuous Uniform

• $\Omega = [0,1], \mathcal{F} = \mathcal{B}([0,1])$
• $X(\omega) = \omega$
• Event “$X \leq 0.5$” = $[0,0.5] \in \mathcal{F}$ → measurable!

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Summary

• A measure assigns size to sets
• A probability measure is a measure with total size 1
• A random variable is a measurable function from $\Omega$ to ℝ
• Measurability ensures we can assign probabilities to events defined via $X$