Banach Limits and Finitely Additive Monsters
Countable additivity is the one axiom 101 never states. Today we build the model that refuses it — a "uniform distribution on the natural numbers" — and audit, item by item, which of our theorems survive.
In this subchapter
2.4.1The question the axioms never answer
Why countable additivity? We motivated it in 0.4: "a head eventually arrives" is a countable union, so a framework that only adds over finitely many events cannot weigh it. That is a positive reason. But there is a sharper way to understand any axiom: build the object that violates it, and watch what breaks. The Vitali construction of 0.3 played this role for the σ-algebra — not every subset can be an event. This subchapter plays it for the third Kolmogorov axiom: there are perfectly finitely additive set-functions, on perfectly good σ-algebras, that are not countably additive — and their existence is neither exotic nor avoidable.
The question comes from the most natural wish in probability, one a quant feels weekly. Toss a fair coin forever, or sample from a stream: what is the "probability" that a large random integer is even? One half, obviously — half the integers are even. But there is no countably additive probability on ℕ assigning each singleton zero and giving the evens 1/2: zero on singletons plus countable additivity forces zero on every subset (2.3: every set is a countable union of its singletons), contradicting P(ℕ) = 1. So the wish "uniform on an infinite countable set" is impossible for countably additive probability. The interesting fact: it is entirely possible if we only ask for finite additivity.
2.4.2What we would want from "uniform on ℕ"
A finitely additive probability on (ℕ, P(ℕ)) is a set function ν: P(ℕ) → [0,1] with ν(ℕ) = 1 and ν(A ∪ B) = ν(A) + ν(B) whenever A, B are disjoint — unions of two (hence any finite number of) pieces only. We would want such a ν to satisfy, at minimum:
- ν({n}) = 0 for every point — no integer is special;
- ν(A + 1) = ν(A) — shifting a set by one does not change its mass (shift-invariance);
- whenever the natural density of A exists — limN (1/N)·#(A ∩ {1,…,N}) — then ν(A) equals it.
The evens have density 1/2; the multiples of k have density 1/k; the primes have density 0 (a real theorem of number theory, not obvious); the finite sets have density 0. The density functional, where defined, is finitely additive, shift-tolerant in the limit sense, and matches every intuition a Monte Carlo practitioner has. The wish list asks: can this functional be extended from the density-having sets to all subsets of ℕ, keeping finite additivity? If yes, we have our uniform-on-ℕ.
First, the bad news that forces the heavy machinery: the density limit does not exist for all sets. A set whose running proportion oscillates between 1/4 and 3/4 forever (long blocks of membership, then longer blocks of absence) has no density — the limit genuinely fails to exist, it is not a matter of computation. And worse: two sets can each lack density while their union has one. So "just take the density of everything" is not a definition; it is a wish without an object. The extension, if it exists, must make choices on the density-less sets — and there is no canonical choice.
2.4.3The monster exists
There exists a finitely additive ν on P(ℕ) satisfying all three items of the wish list: zero on singletons, shift-invariant, and agreeing with natural density wherever density exists.
Write the wish list as a system of constraints. For each set A that has density d(A), constrain ν(A) = d(A); for each A and shift, constrain ν(A+1) = ν(A); for each n, constrain ν({n}) = 0; constrain finite additivity over each disjoint pair. Now the engine: every finite subsystem is satisfiable — given finitely many constraints, only finitely many sets are mentioned, and all their data is determined by the first N integers for some large N; the empirical frequency functional on {1, …, N}, scaled, satisfies every finite fragment exactly (densities are eventually approximated, shifts only move finitely many points of a finite window, and the errors are absorbed by taking N large). A topological principle — the finite-intersection property version of compactness, equivalent to the axiom of choice, the same species of tool flagged in 0.3 and 2.3 — then upgrades "every finite fragment is satisfiable" to "the whole system is satisfiable". That solution is the monster. ∎
Nothing in the proof names ν(A) for a density-less A. The compactness principle guarantees a solution exists; it does not compute one, and different choices in the principle give different monsters that disagree on density-less sets. This is exactly the Vitali situation re-run: an existence argument by choice, producing an object that cannot be written down. If you find this unsatisfying — good. The course's position: the axioms of choice are used sparingly, and every use is flagged. This subchapter is a reductio, not a construction we will ever compute with.
Zero on singletons, one on the whole space, yet ℕ = ⋃n {n} is a disjoint countable union: if ν were countably additive we would get 1 = ν(ℕ) = ∑n ν({n}) = 0. Contradiction — so the monster, which we just proved exists, is a set-function on a perfectly good σ-algebra (the full power set!) that is finitely but not countably additive. Note the shape of this argument: we did not construct the failure; we constrained the object into it. Non-countably-additive set-functions are not rare bugs — they are what you get the moment you insist on the full power set of an infinite space.
2.4.4The audit: what the monster cannot do
Now the payoff: run the monster through the theorems of this chapter and record the casualties. This audit is the real content of the subchapter — each casualty is a theorem whose proof used countable additivity in a way you can now point to.
- Atoms (2.3, Theorem 1) dies. Every singleton has mass 0, so there are no atoms of positive mass at all; yet the space has total mass 1. Theorem 1 said "finite measure ⟹ at most countably many atoms, and the rest is splittable liquid" — the monster is all liquid and no liquid: no singleton grain, and no way to split ℕ into countably many null pieces summing right. Its "anatomy" has no description at all.
- Continuity from above (2.1) dies. Let Am = {m, m+1, m+2, …} — the marching tails. Shift-invariance plus finite additivity gives ν(Am) = ν(A1) = 1 for every m (the first m−1 integers form a finite, hence null, set, and Am is the complement of that null set — finite additivity at work). Yet ⋂m Am = ∅, mass 0. So ν(Am) → 1 ≠ 0: the descending continuity theorem fails while every hypothesis except countable additivity holds. There is no "eventually" for the monster.
- Limits of events (2.2) die wholesale. "Infinitely often", "eventually always", Borel–Cantelli — every one of them is defined by countable sandwiches, and weighed by continuity. With continuity dead, the limit events cannot even be computed from their slices: the grammar of 2.2 becomes unwritable. For the monster, the question "does the coin eventually show heads" is not merely unanswerable — it is unaskable.
Here is the deepest casualty, and the one the course is secretly about. The frequency interpretation of probability — "the share of heads converges to 1/2" — needs the event {frequency → 1/2} to be an event, needs continuity to weigh it, and needs countable additivity for both (2.2, Proposition). Under the monster, each individual long-run frequency statement can be massaged, but the limiting statement cannot be formulated as a set with a mass. The frequency foundation of probability, which 0.6 promised to repair, is repaired by the axiom the monster rejects. Countable additivity is not a technical convenience bolted on; it is the load-bearing wall between probability and an inert list of bets.
2.4.5Why the axioms are a package, not a menu
There is a serious intellectual tradition — de Finetti's, and the "Dutch book" school — that holds finite additivity to be the honest core of probability: the axioms of coherence for a bookmaker who must survive any finite set of bets. The monster is coherent in that sense; it cannot be Dutch-booked. What the audit shows is the price of coherence-only: without countable additivity you keep the right to quote odds, and lose the ability to make statements about the long run. Kolmogorov's third axiom is precisely the choice to trade a little ontology (every subset an event, all weights finitely additive) for the entire theory of limits — the LLN, the CLT, martingales, everything in Chapters 5 through 8. The monsters are the receipt for that trade.
Fairness demands the positive case: finitely additive objects are not useless. Banach limits — the translation-invariant positive extensions of limit to all bounded sequences, constructed exactly like the monster but for sequences — give rigorous meaning to "long-run average" for sequences with no convergent Cesàro mean, and appear in ergodic theory and economics (discounting with no discount). And asymptotic density itself, on the sets where it exists, is a perfectly good finitely additive probability used constantly in number theory. The moral is not "monsters are junk" but "know which axioms your theorems consumed": a result about densities needs only finite additivity; a result about almost-sure convergence has countable additivity in its bones.
The axioms are a package: keep countable additivity and you get the long run; drop it and you keep only the odds.
Chapter 2 is complete: the axioms, their bookkeeping consequences, the language of limit events, the anatomy of measures, and now the audit showing the third axiom is load-bearing. What is still missing: a single serious countably additive measure on an uncountable space — the one whose failure in Chapter 0 started everything. Chapter 3 builds Lebesgue measure: outer measure as the over-estimate, Carathéodory's criterion as the tollgate, and finally the set V of 0.3 shown, honestly, to have no length at all.