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How Do We Assign Probability to Real Numbers?

This article introduces the Borel sigma-algebra, a fundamental concept when dealing with continuous random variables.

The term Borel set might sound intimidating, but it simply refers to the collection of “well-behaved” subsets of the real line that allow us to define probability naturally.

🔍 The Borel sigma-algebra is a sigma-algebra generated from open intervals.
It’s unnecessary for discrete variables (like dice or coins), but essential for continuous ones.

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1. Why Do We Need the Borel Sigma-Algebra?

When working with real-valued outcomes (e.g., $\mathbb{R}$),
we must decide which subsets of $\mathbb{R}$ can be assigned probabilities.

Intuitively, we want to assign probability to intervals:

• $[0, 1]$ has length 1 → probability 1
• $[0, 0.5]$ has length 0.5 → probability 0.5

But in reality, we also want to handle:

• Open intervals, closed intervals, half-open intervals, infinite intervals
• Sets like the rational numbers, irrational numbers, and others

✅ The Borel sigma-algebra includes all sets that can be built from open intervals using set operations.

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2. Definition and Construction of the Borel Sigma-Algebra

The Borel sigma-algebra $\mathcal{B}$ is defined as:

This means:
Start with open intervals, and build the smallest sigma-algebra that contains them.

Sigma-algebras are closed under:

• Complements
• Countable unions
• Countable intersections

So applying these operations repeatedly yields the full Borel sigma-algebra.

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3. Examples of Borel Sets

✅ Initially included

• Open intervals $(a,b)$

✅ Built via set operations

• Closed intervals $[a,b]$
• Half-open intervals $[a,b)$ and $(a,b]$
• Infinite intervals $(-\infty,a],\ (b,\infty)$
• Rational numbers $\mathbb{Q}$
• Irrational numbers $\mathbb{R} \setminus \mathbb{Q}$
• Sets like $[0,1] \setminus \mathbb{Q}$
• $G_\delta$, $F_\sigma$ sets, and more

💡 Nearly every “natural” set you can imagine on the real line is a Borel set!

To get an intuitive feel for Borel sets, let’s explore this visual demo.

What Are Borel Sets?

A visual journey through the construction of Borel σ-algebra

Previous

Next

Step 1: Start with Open Intervals

0

0.25

0.5

0.75

1

(0.2, 0.8)

Begin with simple open intervals like (0.2, 0.8)

Common Sets: Are They Borel?

Closed Interval

[a, b]

✓

Half-open Interval

[a, b)

✓

Single Point

{x}

✓

Rational Numbers

ℚ

✓

Irrational Numbers

ℝ \ ℚ

✓

Weird Set

Vitali Set

✗

Key Insight

Borel sets include virtually every set you can naturally describe or construct in real analysis. They're built systematically from open intervals using standard set operations.

The Big Picture

Borel sets are the collection of "well-behaved" sets on the real line. Starting from simple open intervals, we systematically build up a rich family of sets by taking complements, unions, and intersections. This gives us exactly the sets we need for probability theory on continuous spaces.

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4. Are There Sets Not in the Borel Sigma-Algebra?

Yes. There exist subsets of $\mathbb{R}$ not in $\mathcal{B}$.

For example:

• Vitali sets, which require advanced set-theoretic construction
• Sets that rely on the Axiom of Choice

📌 However, such sets never appear in practical probability or analysis.
For defining probability, Borel sets are more than sufficient.

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5. Probability and the Borel Sigma-Algebra

📏 Based on length (Lebesgue measure)

We can define probability on Borel sets using length as a base:

• Uniform distribution on $[0,1]$:
  • $P([0,0.3]) = 0.3$
  • $P((0.2,0.7)) = 0.5$

This naturally forms a probability space:

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Summary

• The Borel sigma-algebra is a sigma-algebra generated from open intervals.
• It includes almost every familiar set on $\mathbb{R}$.
• It is closed under complements, countable unions, and intersections.
• Borel sets form the standard foundation for defining probability on the real line.
• Discrete spaces don’t need it—but for continuous variables, it’s essential.

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In the next article, we’ll explore how to define a probability measure on $\mathcal{B}$,
and fully construct the probability space $(\mathbb{R}, \mathcal{B}, P)$.