Questions: Probability Spaces (Measure-Theoretic Definition)

Question 1

Why can't we assign probabilities to ALL subsets of ℝ when defining a continuous probability distribution?

• A. Because ℝ is uncountably infinite, individual subsets are too large to measure
• B. Because non-measurable sets exist (e.g., Vitali sets) that cannot be consistently assigned a probability
• C. Because the axioms of probability only allow finite sample spaces
• D. Because probability must sum to 1, and infinitely many subsets would each receive zero probability

Answer: (B)

Vitali sets and similar constructions show that if you try to assign a translation-invariant measure (like Lebesgue measure or a uniform probability) to ALL subsets of ℝ, you reach a contradiction. These non-measurable sets cannot be consistently assigned a probability value. The sigma-algebra ℱ solves this by restricting attention to the 'Borel-measurable' subsets, which include all open intervals, closed sets, and countable combinations thereof, while excluding the paradoxical sets.

Question 2

A student claims that finite additivity is sufficient for probability theory on continuous spaces because 'you can just add up infinitely many zeros.' What is wrong with this reasoning?

• A. Nothing — finite additivity is equivalent to countable additivity for probability measures
• B. Finite additivity permits inconsistencies when summing over countably infinite collections; countable additivity must be stated explicitly
• C. The student is correct that a sum of zeros can equal any value; the error is in the 'infinite' part
• D. Probability theory doesn't apply to continuous spaces at all, so the argument is moot

Answer: (B)

Finite additivity only guarantees that P(A ∪ B) = P(A) + P(B) for finitely many disjoint events. It says nothing about infinite collections. For continuous distributions, we need P([a,b]) = ∫ᵃᵇ f(x)dx to be consistent with the axioms, which requires countable additivity — the ability to take limits of sums. Without it, even basic results like P(ℝ) = 1 cannot be proved from the behavior on individual points. Countable additivity is an independent axiom, not derivable from finite additivity.

Question 3

In a probability space (Ω, ℱ, P), nearly every subset of Ω is an event to which P assigns a probability.

• T. True
• F. False

Answer: False

Only subsets of Ω that belong to the sigma-algebra ℱ are events. ℱ is a carefully chosen subcollection of P(Ω) — the set of all subsets — that excludes non-measurable sets. This is a critical distinction: you cannot ask 'what is the probability of this subset?' unless that subset is in ℱ. The entire purpose of the sigma-algebra component is to restrict which subsets count as legitimate events.

Question 4

Countable additivity (σ-additivity) is strictly stronger than finite additivity, in the sense that countable additivity implies finite additivity but not vice versa.

• T. True
• F. False

Answer: True

Countable additivity states that P(∪ₙAₙ) = ΣₙP(Aₙ) for any countable collection of disjoint events — this includes finite collections as a special case (by setting all but finitely many Aₙ to ∅). So countable additivity implies finite additivity. The converse fails: there exist finitely additive set functions on algebras that are not countably additive. This is why countable additivity must be stated as an explicit axiom — it is not derivable from the other axioms.

Question 5

Why is the sigma-algebra ℱ a necessary component of the probability space triple (Ω, ℱ, P), rather than simply using all subsets of Ω as events?

Model answer: Non-measurable sets exist — subsets of Ω that cannot be consistently assigned a probability without producing contradictions. The sigma-algebra restricts attention to measurable subsets: those closed under complementation and countable unions, which can be assigned probabilities consistently. On continuous spaces like ℝ, using all subsets leads to paradoxes (Vitali sets, Banach-Tarski). ℱ is the collection of sets we CAN measure, and the probability measure P is only defined on that collection.

This is the foundational reason the measure-theoretic framework exists. The axioms of probability look simple — non-negativity, total probability 1, additivity — but on infinite spaces they require a domain restriction to be consistent. The sigma-algebra formalizes 'which questions can we ask?' Random variables are then defined as measurable functions from (Ω, ℱ) to (ℝ, Borel sets) — functions that are compatible with the measurable structure on both sides.