Atoms and Discrete Spaces Done Properly
What is a probability measure made of? Answer: grains. Counting the grains takes one theorem, and the discrete/continuous divide turns out to be exactly the question of whether the grains exist.
In this subchapter
2.3.1The question behind "discrete"
Probability 101 says: a "discrete" random variable has a probability mass function — a list p(x) summing to one — while a "continuous" one has a density. The words are doing more work than the course admits. On a countable sample space everything is discrete: we proved in 1.2 that singletons generate the power set, so a measure is exactly a mass list. But on [0,1], the uniform measure assigns zero to every singleton yet one to the whole interval — no mass list can do that. And there are stranger objects in between: measures that put positive mass on uncountable sets yet still concentrate on "grains". To say precisely what a measure is made of, we need a definition of grain that does not mention points.
An event A ∈ F is an atom of the probability measure P if P(A) > 0 and every event B ⊆ A has P(B) = 0 or P(B) = P(A). The mass inside A is unsplittable: any measurable attempt to cut A either takes everything or nothing. A measure with no atoms is called atomless (or: P is nonatomic).
Dirac measure: P(A) = 1 if a ∈ A, else 0. The singleton {a} is an atom, and so is every event containing it — atoms can nest. For a grain that is not a point, take Ω = [0,1] with the small σ-algebra F = {∅, [0, 1/2], (1/2, 1], Ω} and P giving each half mass 1/2. The set [0, 1/2] is an atom of mass 1/2: it is uncountable, but F contains no way to cut it measurably. Atomhood is a partnership between the measure and the σ-algebra — coarsen the σ-algebra and grains appear; refine it and they split.
On ([0,1], ℬ, λ): every event A of positive length can be split — 3.2 builds the tool — and more elementarily, no singleton carries mass. In fact no event at all is an atom: given P(A) = p > 0, the bisection argument of §5 produces a subset of A with mass p/2 exactly. The uniform measure is pure continuum: infinitely divisible, zero grain size. Keep this example in mind — it is the negative space that makes the definition meaningful.
2.3.2At most countably many grains
The first real theorem about the anatomy of a measure: however rich the σ-algebra, the grains can be listed.
Any finite measure (in particular any probability) has at most countably many atoms. Moreover the atoms can be listed A₁, A₂, … with pairwise disjoint versions and ∑n P(An) ≤ 1.
For each k, let 𝒜k be the atoms with mass at least 1/k. Claim: 𝒜k is finite. If A₁, …, AN are distinct atoms, trim them to be disjoint: replace Ai by Ai ∖ ⋃j<i Aj — still an atom or null, since removing earlier grains leaves either all or none of the mass (each difference is a measurable subset of the atom Ai). Disjoint atoms have disjoint masses, so N/k ≤ ∑ P(Ai) ≤ 1, giving N ≤ k. Now every atom has some mass, hence belongs to some 𝒜k (mass ≥ 1/k for some k — Archimedes: some 1/k fits under it). The atom collection is ⋃k 𝒜k — a countable union of finite collections: countable. ∎
This is the standard "big grains are few" pattern: stratify by size, bound each stratum by total mass, take a countable union of finite sets. The same skeleton proves that a convergent series has few large terms, that a finite-variance random variable has few huge deviations, that a σ-finite measure has few heavy regions. When you meet "there are only countably many X of substantial size" anywhere in analysis, this is the proof wearing different clothes.
2.3.3Discrete spaces, done properly
A probability space (Ω, F, P) is discrete if there is a countable set C = {c₁, c₂, …} — the support — with P(Ω ∖ C) = 0 and P({ci}) = pi > 0, ∑pi = 1. The list (pi) is the probability mass function. On such a space every event's probability is forced: P(A) = ∑ci ∈ A pi.
Binomial, geometric, Poisson: all are mass functions on countable supports, so each is a genuine probability measure on a power set — no σ-algebra anxiety, no measure theory needed, and Theorem 1 is visible in the flesh: the atoms are exactly the points of positive mass, countably many, masses summing to one. The geometric mass list pn = 2−n is the measure behind the first-head bar chart of 0.4. This is the world 101 lives in, and it is a theorem, not a default.
Here is the trap Theorem 1 warns about. A countable atom list does not mean the measure lives on a countable set. Consider the mixture: uniform measure on [0,1] (no atoms at all, by Example 2) plus a Dirac mass of 1/2 at the point 0. Atoms: just the one at zero. But half the mass is spread over an uncountable atomless continuum. The measure splits into a discrete part and a continuous part, and the discrete part's total mass — 1/2 — can be any number in [0,1]. Doing this split in general is the Lebesgue decomposition, and the honest home for it is 8.2 with the Radon–Nikodym theorem. Until then: atoms measure the grain content, not the support.
2.3.4The atomless side: uniform measure splits forever
We close with the complementary structural theorem — the one that makes atomless measures feel like liquid rather than sand. Its proof opens with a fully honest halving lemma, and we flag precisely the one step that needs heavier machinery.
If P is atomless and P(A) = p, then for every t ∈ [0, p] there is an event B ⊆ A with P(B) = t. Atomless measures attain every mass up to what is available.
Step 1 (the halving lemma, in full). Any event D of positive mass contains a measurable piece of positive mass at most half: since P is atomless, D is not an atom, so it splits measurably into two pieces of positive mass; the lighter one has mass ≤ P(D)/2 and > 0. Iterating, D contains pieces of arbitrarily small positive mass — the measure has no smallest coin in its pocket.
Step 2 (attainment, flagged). Fix t ∈ [0, p]. Consider the family of events B ⊆ A with P(B) ≤ t, ordered by inclusion. A maximal such event — whose existence is a maximality principle of the same species as the axiom of choice that selected Vitali's delegates in 0.3 — must satisfy P(B) = t exactly: if P(B) < t, the remainder D = A ∖ B has positive excess mass p − P(B) > t − P(B) ≥ 0, and by Step 1 it contains a small positive piece that could be added while staying under t, contradicting maximality. We flag the maximality step rather than hide it: unlike the halving lemma, it is not constructive. This is the same honesty policy as the transfinite climb in 1.2: the course says which steps are algorithms and which are choice. ∎
Apply Theorem 2 to A = [0,1], p = 1: for any t there is an event of mass exactly t — of course, intervals already do this, which is the sanity check. The theorem's point is that this richness needs no geometry: any atomless probability, on any σ-algebra, no intervals anywhere in sight, still attains every mass in [0,1]. The bisection used nothing but the axioms and continuity — the same axioms doing continuous work in a purely combinatorial environment. (One honest flag: to run the argument on a named atomless measure we still need one to exist — constructing it is precisely Chapter 3's job.)
Measures need not be purely atomic or purely atomless — Example 4's mixture is both at once. The clean statements are: grain content is at most countable (Theorem 1); grainless measures are infinitely divisible (Theorem 2); and the general measure is a blend, sorted out properly in 8.2. Do not let the 101 vocabulary ("discrete vs continuous") harden into a false binary: the third category — mixed measures, like a default-with-jump bond price — is where practice actually lives.
A probability measure is grains plus liquid: at most countably many grains, and whatever is grainless can be poured to any level you like.
The axioms have shown us their anatomy: countable grains, divisible liquid. Next, a subchapter of monsters: what happens if we weaken countable additivity to finite additivity? A finitely additive "uniform distribution on the natural numbers" exists — it vanishes on every singleton yet assigns mass one to the whole space — and watching exactly which of our theorems it slaughters is the sharpest way to learn what the axioms were protecting.