Mastering Probability Spaces: A Step-by-Step Tutorial for MATH154 Homework 2

Introduction to Probability Spaces

Probability spaces form the foundation of modern probability theory. In this tutorial, we'll walk through the key concepts from MATH154 Homework 2, including verifying axioms, understanding π-systems and λ-systems, and applying conditional probability. Whether you're studying for an exam or building a foundation for data science, these concepts are essential. Think of a probability space as the rules of a game — just like in the 2026 FIFA World Cup, where each match outcome has a certain probability based on team form, the axioms ensure consistency.

1. Verifying Basic Probability Axioms

Problem 2.1 asks you to verify six properties from the axioms. Let's break them down.

a) P[∅] = 0

The probability of the empty set is zero. This follows from the axiom that P[Ω] = 1 and countable additivity. Since ∅ and Ω are disjoint, P[Ω] = P[Ω ∪ ∅] = P[Ω] + P[∅] ⇒ P[∅] = 0.

b) A ⊂ B ⇒ P[A] ≤ P[B]

If A is a subset of B, then B = A ∪ (B \ A), disjoint union. So P[B] = P[A] + P[B \ A] ≥ P[A] because probabilities are non-negative.

c) P[∪ An] ≤ Σ P[An]

This is subadditivity, a direct consequence of countable additivity and non-negativity. For a finite or countable collection, the probability of the union is at most the sum of individual probabilities.

d) P[Ac] = 1 − P[A]

Since A and Ac partition Ω, we have P[A] + P[Ac] = P[Ω] = 1.

e) 0 ≤ P[A] ≤ 1

From axioms, probabilities are between 0 and 1 inclusive.

f) A1 ⊂ A2 ⊂ ... ⇒ P[∪ An] = lim P[An]

This is continuity from below, a key property derived from countable additivity.

2. Countable/Cocountable σ-Algebra

Problem 2.2 explores the collection A of subsets of Ω that are either countable or cocountable (complement is countable). This is a classic example of a σ-algebra.

a) Ring axioms

A ring must be closed under union and difference. For countable sets: union of two countable sets is countable; difference of countable sets is countable. For cocountable sets: complements are countable, so union/difference properties follow similarly.

b) π-system

A π-system is closed under finite intersections. If both sets are countable, intersection is countable. If one is cocountable, intersection is either cocountable (if both are cocountable) or countable (if one is countable and the other cocountable? Actually, if A is cocountable and B is countable, A∩B is countable. So it's closed under intersection.)

c) λ-system

A λ-system contains Ω, is closed under complement, and under countable disjoint unions. Check: Ω is cocountable (since its complement is empty, countable). If A is in A, its complement is also in A by definition. For disjoint countable union of sets in A, if all are countable, union is countable. If at least one is cocountable, the union is cocountable (since complement is intersection of countable sets, which is countable). So A is a λ-system.

e) Smallest σ-algebra containing cofinite topology

The cofinite topology consists of sets whose complement is finite or all of Ω. The σ-algebra generated by it must include countable unions of cofinite sets, which are exactly cocountable sets. So the generated σ-algebra is the countable/cocountable σ-algebra.

3. Probability on the Unit Square

Problem 2.3 deals with Ω = [0,1]² and rectangles of the form [a,b) × [c,d).

a) π-system

The intersection of two such rectangles is again a rectangle of the same type (or empty). So it's a π-system.

b) Probability measure

Define P([a,b)×[c,d)) = (b−a)(d−c). This is the area. It is finitely additive on the π-system, and by Carathéodory's extension theorem, it extends uniquely to the generated σ-algebra (the Borel σ-algebra on [0,1]²).

c) Extension

The extension is possible because the measure is σ-finite and the π-system generates the σ-algebra. The Carathéodory extension theorem guarantees a unique extension.

d) Independence

Two rectangles are independent if P(A∩B) = P(A)P(B). For rectangles [a,b)×[c,d) and [a',b')×[c',d'), this holds iff the intervals are independent in each coordinate, i.e., the product of lengths equals the length of intersection? Actually, independence requires that the product of areas equals area of intersection, which is a special condition.

4. Conditional Probability and Bayes' Theorem

Problem 2.4 covers the Keynes postulates and Bayes' theorem. These are fundamental for updating beliefs — similar to how AI models update predictions based on new data, like a recommendation system adjusting after you watch a trending show on Netflix in 2026.

Keynes Postulates

1. P[A|B] ≥ 0
2. P[A|A] = 1
3. P[A|B] + P[Ac|B] = 1

These follow directly from the definition P[A|B] = P[A∩B]/P[B] and the axioms.

Bayes' Theorem

Bayes' theorem states: P[A|B] = P[B|A] P[A] / P[B]. This is used everywhere — from spam filters to medical testing. For example, in the 2026 stock market, you might update the probability of a stock price increase given new earnings data using Bayes.

5. The ΠΣΛ Sorority Theorem

Problem 2.5 asks you to prove that the smallest λ-system containing a π-system is the smallest σ-algebra containing it. This is a key result in measure theory. The proof involves showing that a λ-system that is also a π-system is a σ-algebra. Since the generated λ-system is closed under complements and countable disjoint unions, and the π-system ensures closure under finite intersections, you can show it's a σ-algebra. Think of it like a sorority where members (sets) must satisfy certain rules — the ΠΣΛ theorem shows that if you have a group that is both a π-system (closed under intersections) and a λ-system (closed under complements and disjoint unions), then it's actually a full σ-algebra, just like a well-organized club that meets all requirements.

Conclusion

Understanding probability spaces is crucial for advanced probability and statistics. By mastering these exercises, you'll build a solid foundation for more complex topics like stochastic processes and machine learning. Remember, every probability problem starts with a clear definition of the sample space, σ-algebra, and probability measure — the three pillars of any probability space.