Questions: Sigma-Algebras and Measurable Sets

Question 1

You want to assign a probability P to the event 'a randomly chosen real number lies in the Cantor set.' Why does this require the Cantor set to be in the sigma-algebra, and why can't we just assign probabilities to every subset of ℝ?

• A. We can assign probabilities to every subset of ℝ — sigma-algebras are just a convention for organizing calculations
• B. The sigma-algebra is needed only for uncountable sample spaces; for finite or countable spaces, every subset can receive a probability
• C. Probability must satisfy countable additivity: P(∪ Aₙ) = ΣP(Aₙ) for disjoint events. This forces the collection of 'events' to be closed under countable unions. Non-measurable sets exist for which no consistent probability assignment satisfies this requirement
• D. Sigma-algebras restrict which sets receive probability to make computation tractable, but every subset of ℝ technically has a probability

Answer: (C)

The key insight is that probability is defined by the requirement P(∪ Aₙ) = ΣP(Aₙ) for disjoint countable collections (countable additivity). For this equation to even make sense, the union ∪ Aₙ must itself be a set to which P is assigned — so the collection of measurable sets must be closed under countable unions. Non-measurable sets (provably existing under the axiom of choice) cannot be assigned a consistent probability without violating countable additivity. This is not a computational convenience — it is a logical necessity.

Question 2

Which of the following collections of subsets of Ω = {1, 2, 3, 4} is NOT a valid sigma-algebra?

• A. {∅, {1,2}, {3,4}, {1,2,3,4}}
• B. {∅, {1}, {2,3,4}, {1,2,3,4}}
• C. {∅, {1,2}, {2,3}, {1,2,3,4}}
• D. {∅, {1,2,3,4}}

Answer: (C)

Option C fails because it is not closed under union: {1,2} ∪ {2,3} = {1,2,3}, which is not in the collection. A sigma-algebra must contain the union of any of its members. Options A, B, and D each consist of a set, its complement, and ∅ and Ω — satisfying all three axioms. The closure-under-unions requirement is the most commonly overlooked when checking whether a collection is a sigma-algebra.

Question 3

A sigma-algebra that is closed under finite unions but not necessarily countable unions is sufficient for most of probability theory.

• T. True
• F. False

Answer: False

This is the core misconception flagged in the topic. Countable additivity — P(∪ Aₙ) = ΣP(Aₙ) — is the foundational property of probability measures, and it involves countable (possibly infinite) unions. If the collection of events is closed only under finite unions, countable unions of events might not be events, making countable additivity meaningless for those combinations. The upgrade from finite to countable closure is what makes modern probability theory work for limits, sequences, and all the machinery of analysis.

Question 4

A coarser sigma-algebra (one containing fewer measurable sets) represents less information about outcomes in a probability space.

• T. True
• F. False

Answer: True

This is a deep fact used throughout stochastic processes. If fewer sets are measurable, fewer events are distinguishable — you cannot ask 'what is the probability of event A?' if A is not in the sigma-algebra. In filtrations, the sigma-algebra Fₜ at time t represents exactly the information available up to time t: events that are 'known' by time t. A coarser Fₜ means less information; a finer one means more. The sigma-algebra is the mathematical language for what is knowable.

Question 5

Why must a sigma-algebra be closed under *countable* unions specifically, rather than just finite unions? What goes wrong if we only require finite closure?

Model answer: Probability theory requires countable additivity: P(A₁ ∪ A₂ ∪ ...) = ΣP(Aₙ) for disjoint events. This equation involves the countable union A₁ ∪ A₂ ∪ ..., which must itself be an event (in the sigma-algebra) for the equation to be meaningful. If only finite unions were required, a countable union of measurable sets might not be measurable, making countable additivity undefined for that combination. Since countable additivity is what distinguishes probability from finitely additive set functions and is needed for all limit theorems (law of large numbers, central limit theorem), finite closure is not sufficient.

The distinction between finite and countable closure mirrors the distinction between finitely additive and countably additive measures — a foundational fork in measure theory. Only countable additivity supports the full apparatus of real analysis (limits, convergence, integration), which is why the sigma-algebra condition matches it exactly.