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Building the Foundation of Probability with Sets

In this article, we’ll explore what an event and a σ-algebra (sigma-algebra) are—core concepts that form the mathematical foundation of probability theory.

This is especially helpful for those curious about how probability deals with:

• Infinite repetitions (like flipping a coin forever)
• Continuous distributions (like height or temperature)

🔰 If you’re new to probability, don’t worry about the details too much!
But if you’re wondering how probability is defined mathematically, you’re in the right place.

This topic draws heavily from set theory and measure theory.

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1. What is an “Event”?

In probability theory, we begin by defining a sample space $\Omega$,
which represents the set of all possible outcomes of an experiment.

An event is simply a subset of the sample space.

🎲 Example: Rolling a die once

• Sample space: $\Omega = \{1, 2, 3, 4, 5, 6\}$
• Event “rolling a 1”: $\{1\}$
• Event “not rolling a 1”: $\{2, 3, 4, 5, 6\}$
• Event “rolling an even number”: $\{2, 4, 6\}$

🔍 Event = a group of outcomes (a subset of $\Omega$)

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2. Why Do We Need a Collection of Sets?

You can’t just assign probabilities to any subset of $\Omega$ without rules.
To avoid contradictions, we must carefully choose which subsets are allowed to receive probabilities.

That’s where structures like algebras and σ-algebras come in:

📦 They define which sets are valid events in our probability system.

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3. Types of Set Collections: Algebras vs. σ-Algebras

To define probabilities properly, we must choose a well-behaved collection of subsets.

There are two main types to consider:

🎯 Algebra (finite additive field)

A set collection $\mathcal{A}$ is called an algebra if it satisfies:

1. Contains the whole space: $\Omega \in \mathcal{A}$
2. Closed under complements: If $A \in \mathcal{A}$, then $A^c \in \mathcal{A}$
3. Closed under finite unions: If $A_1, A_2 \in \mathcal{A}$, then $A_1 \cup A_2 \in \mathcal{A}$

✨ σ-Algebra (sigma-algebra)

A σ-algebra satisfies everything above, plus:

3’. Closed under countable unions:
If $A_1, A_2, A_3, \dots \in \mathcal{F}$, then $\bigcup_{i=1}^{\infty} A_i \in \mathcal{F}$

💡 What does “closed under” mean?
It means that when you perform certain operations (like union or complement),
the result is still inside the same set collection.

So once you’ve chosen a collection of sets, any allowed operation keeps you within that collection.
This consistency is crucial for defining probability.

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🎲 Visualizing Closure with a Die

Let’s use a die with sample space $\Omega = \{1, 2, 3, 4, 5, 6\}$, and consider the following set collection:

Let’s test whether $\mathcal{A}$ is closed under complements and unions.

🎲 Dice σ-Algebra Builder

Build a collection of sets from dice outcomes and see if it forms an algebra!

Sample Space Ω = Ω

All possible outcomes when rolling a six-sided die

Current Family ℱ

Collection Container

∅

Ω

Algebra Properties Check:

✓Contains Ω (full set)

✓Contains ∅ (empty set)

✓Closed under complement

✓Closed under finite union

🎉 This is an Algebra!

Add Sets to Family

{1}

⚃1

Even {2,4,6}

⚃2 ⚃4 ⚃6

Odd {1,3,5}

⚃1 ⚃3 ⚃5

{1,2}

⚃1 ⚃2

{3,4,5,6}

⚃3 ⚃4 ⚃5 ⚃6

Controls

Reset

How to Use

1. Add sets from the predefined list

2. Select sets by clicking on them (purple border)

3. Watch the properties check to see if your collection forms an algebra

💡 Key insight: An algebra must be "closed" under operations - all results must stay within the collection!

✅ Complements

• $\{1,3,5\}^c = \{2,4,6\}$ → ✅ in $\mathcal{A}$
• $\{1,2\}^c = \{3,4,5,6\}$ → ✅ in $\mathcal{A}$
• $\emptyset^c = \Omega$ → ✅ in $\mathcal{A}$

So, $\mathcal{A}$ is closed under complements.

❓ Unions

• $\{1,2\} \cup \{3,4,5,6\} = \Omega$ → ✅
• $\{1,3,5\} \cup \{1,2\} = \{1,2,3,5\}$ → ❌ not in $\mathcal{A}$
• $\{1,2\} \cup \{2,4,6\} = \{1,2,4,6\}$ → ❌ not in $\mathcal{A}$

So, $\mathcal{A}$ is not closed under unions.

📉 This shows how “closure” works and why it’s important in defining valid event collections.

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4. Why Closure Under Countable Unions Matters

📊 In real-world probability…

• We often model infinite repetitions (flipping a coin forever)
• Or use continuous distributions (like normal distributions)

In these situations, we must assign probabilities to infinitely many combined events.

🧠 Example: Countable additivity

Suppose $A_1, A_2, A_3, \dots$ are mutually exclusive events.

To define:

the union $\bigcup_{i=1}^{\infty} A_i$ must be a valid event.

→ That’s why we need σ-algebras: they are closed under countable unions.

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5. The Big Picture

• $\Omega$ (sample space): all possible outcomes
• $\mathcal{F}$ (σ-algebra): well-structured collection of subsets where probabilities can be assigned

Once you have $(\Omega, \mathcal{F})$, you have the foundation for defining probability.

🛠️ $(\Omega, \mathcal{F})$ is the stage on which probability lives.

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6. Summary

• Event = a subset of the sample space $\Omega$
• Algebra = closed under complements and finite unions
• σ-Algebra = closed under complements and countable (infinite) unions
• σ-algebras are necessary for dealing with infinite processes or continuous distributions
• Once we have a σ-algebra, we’re ready to define a probability measure $P$

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In the next article, we’ll learn how to define the probability measure $P$ on this structure,
and complete the probability space $(\Omega, \mathcal{F}, P)$.

Let me know when you’re ready to continue!