Null Sets, Completeness, and the Cantor Set
An uncountable set with probability zero. A set of positive probability with no interior. A random variable equal to zero almost surely yet never actually zero. This is the chapter where "negligible" stops being a synonym for "small".
In this subchapter
3.3.1Null sets: sets the measure cannot see
By now (3.1–3.2) we own the Lebesgue measure λ — lowercase lambda, the mathematician's letter for length — on the interval [0,1]: a countably additive assignment λ([a,b]) = b − a, defined on a σ-algebra containing the Borel sets. A probability measure on the spinner's space is nothing but P(A) = λ(A). Some sets turn out to be beneath its resolution.
A set N ⊆ [0,1] is a null set (with respect to Lebesgue measure) if λ(N) = 0. Equivalently, from the outer-measure construction of 3.1: for every ε > 0, N can be covered by countably many intervals of total length less than ε. Probabilists say such events almost never happen; their complements happen almost surely, abbreviated a.s.
The covering formulation is the practical one — most null sets are not assigned zero directly but are outfitted: you exhibit a cheap cover, and negligibility follows. Watch it work on the most important null set in applied mathematics.
The rationals in the unit interval are countable: list them as q₁, q₂, q₃, …. Cover the n-th rational by an interval of length ε/2n. The total cover length is
Since ε was arbitrary, λ(ℚ ∩ [0,1]) = 0. Hold the conclusion up against Chapter 0: the rationals are dense — every interval contains one — yet their total probability is zero. Dense in space, empty in measure. The spinner lands on a rational number with probability zero, even though rationals are woven through every corner of the interval. (Read that sentence twice: intuition will keep insisting on the wrong answer.)
Every finite set is null (cover each point by an interval of length ε/N). Every countable set is null, by the argument above. But the converse fails catastrophically: the Cantor set of the next section is uncountable — the same size as the whole interval, in the sense of bijections — and still has measure zero. "Countable" and "null" are related but not the same relation, and the gap between them is exactly where the famous counterexamples live. Any intuition of the form "zero measure means very few points" dies here; measure counts weight, not points.
3.3.2The Cantor set: uncountable and invisible
Take [0,1] and remove the open middle third (1/3, 2/3). From each remaining interval, remove its open middle third: (1/9, 2/9) and (7/9, 8/9). Repeat forever. What survives the endless subtraction is the Cantor set C.
λ(C) = 0, yet there is no list c₁, c₂, c₃, … containing every element of C. The set has the cardinality of the continuum: as many points as [0,1] itself.
After stage n, what remains is Cn: 2ⁿ disjoint intervals each of length 3⁻ⁿ, and C ⊆ Cn. By the countable subadditivity of outer measure from 3.1,
so λ(C) = 0. This is the cover-outfitting pattern from 3.3.1 with the cover arranged in stages: C hides inside sets of arbitrarily small length, and measure cannot see it.
Write each x ∈ [0,1] in base three: x = 0.a₁a₂a₃… with digits an ∈ {0,1,2}. Removing the middle third deletes exactly the numbers with a₁ = 1; removing middle thirds again deletes those with a₂ = 1; and so on. What survives is
Now observe: the map sending 0.a₁a₂a₃… (digits 0/2) to the binary number 0.b₁b₂b₃… with bn = an/2 is a bijection from C onto all of [0,1]. Since [0,1] is uncountable, so is C. One caveat handled carefully in texts: base-3 expansions are not unique at endpoints (e.g. 0.1₃ = 0.0222…₃); both expansions of an endpoint use only 0s and 2s after some point, and one checks the bijection survives the ambiguity.
Test your instincts: "C is finite or countable, since we keep deleting" — false, uncountable. "C is dense somewhere, since it's so big" — false: C contains no interval whatsoever (any interval would survive some stage with positive length, but the surviving lengths shrink below any fixed positive number). "C is just the endpoints we see in pictures" — false: those are countable; the rest of C — the overwhelming majority of it, in cardinality — consists of numbers never touched by any stage's reasoning, like 1/4 = 0.020202…₃. The Cantor set is the first fully-worked instance of measure and cardinality disagreeing completely, and every subtlety of this course traces back to being at peace with that disagreement.
Null sets are closed under countable unions: cover the n-th set with total length ε/2ⁿ, add up (this is countable subadditivity, workhorse of 3.1). But allow uncountable unions and everything collapses: every point is a null set, and [0,1] is the union of its points — total length 1. The step from "countably many negligible things are negligible" to "any many" is precisely the step the theory forbids. This is not a technical footnote: it is the boundary stone of the entire theory of integration in Chapter 5, and the reason the word countable appears in every clause of every definition since Chapter 1.
3.3.3Reading the Cantor set in base three
The digit description deserves a section of its own, because it is how you compute with C rather than merely admire it. Base three: digits are 0, 1, 2 and place values are thirds, ninths, twenty-sevenths…
1/3 = 0.1₃ = 0.02222…₃ — member of C (endpoint; use the all-2 tail expansion). 1/4 = 0.020202…₃ — member, and not an endpoint of anything: an invisible point of the invisible set. 1/2 = 0.111…₃ — removed at stage one (first digit 1). 0.7 — first digit 2, second digit 0, third digit 1: 0.2101…₃ — removed at stage three. Membership in C is exactly the statement "some base-3 expansion avoids the digit 1": a membership test you can run by long division.
Here is a random variable to keep for the rest of the course. Flip a fair coin forever; from the n-th flip write digit 2 if heads, 0 if tails; read the result as a base-3 fraction. The outcome X always lies in C — so P(X ∈ C) = 1 — while λ(C) = 0. Its distribution function, the Cantor function of 3.3.7, is continuous, rises from 0 to 1, and yet has zero derivative almost everywhere: a "continuous random variable" with no density. If you had believed "continuous CDF implies a density exists", this is the exhibit that retires the belief — a quant model with a Cantor-distributed factor passes every smoothness check at the CDF level while having no density to differentiate. The lesson: existence of a density is a real hypothesis, not a formality.
3.3.4Fat Cantor sets: nowhere dense yet heavy
The Cantor set's measure is zero because each stage keeps 2/3 of the length. Nothing forces that choice — and the variants are where the negative examples get interesting.
Modify the construction: at stage n, remove from each remaining interval an open middle segment of relative length 1/4ⁿ instead of the middle third. The total removed is 1/4 + 2/16 + 4/64 + … = 1/2, so the surviving set — the Smith–Volterra–Cantor set — has λ = 1/2. It is closed, has no intervals in it (nowhere dense, like the Cantor set), yet carries half the probability mass of the interval. Every intuition formed on the thin Cantor set fails: "closed and interval-free means negligible" is simply false. Fat Cantor sets can be built with measure arbitrarily close to 1 — sets that are topologically nowhere and probabilistically everywhere.
Practitioners say "the support of the distribution is [0,1]" and picture mass spread across the interval. A distribution can put full mass on a closed, nowhere-dense set of measure zero inside [0,1] — the topological closure of the support says nothing about the measure's weight. This is the mathematical fact behind the practical observation that a distribution can be concentrated on a set that is, in the naive geometric sense, "thin".
3.3.5Completeness: why Lebesgue beats Borel by one move
The Borel σ-algebra ℬ (1.3) is generated by intervals — every set reachable by countable set-operations on intervals. It is the natural home for defining measures. But it has a hole, and the hole is exactly the size of the null sets.
Take the Cantor set C (Borel: it is a countable intersection of finite unions of intervals) and take any subset M ⊆ C. Since λ(C) = 0 and M ⊆ C, whatever length M has should be 0 — smaller than a zero-length set is zero length. The obvious thing to do is declare M measurable with λ(M) = 0. And here is the surprise: some subsets of C are not Borel.
The Cantor set is in bijection with [0,1], so it has as many subsets as [0,1] does — the cardinality 2^𝔠, an exponential tower over the continuum. On the other hand, a counting induction (done properly in 1.3) shows the Borel σ-algebra has only the cardinality of the continuum: 𝔠, the size of [0,1] itself. Since 2^𝔠 > 𝔠 (Cantor's theorem, base case of all of set theory), most subsets of C are not Borel. The Borel family is generated by countable operations from intervals — and countably many countable operations, iterated countably often, still only reach 𝔠 many sets. The construction cannot keep up with the subsets.
The Lebesgue σ-algebra ℒ is the Borel σ-algebra completed with respect to null sets: A ∈ ℒ exactly when A = B ∪ M where B is Borel and M is contained in some Borel null set. Every such A gets λ(A) = λ(B) — well-defined, since the added M contributes nothing. Lebesgue measure is λ extended to ℒ. A measure with this property — every subset of a null set is measurable (with measure zero) — is called complete.
Let X be the Cantor-distributed random variable of 3.3.3 and let M be any subset of C. The event {X ∈ M} deserves a probability: it is contained in {X ∈ C}, which has probability 1·0 = 0 (a null event), so its probability should be 0 — and in the completed σ-algebra it is an event and P(X ∈ M) = 0. Completion is the difference between "we can prove this event has probability zero" and "we can prove this event has probability zero, and it is an event at all". A pedantic distinction — until the day you need to condition on it (Chapter 8 makes heavy use of completeness for exactly this reason).
Completion is generous but not boundless: the Vitali delegate set from Chapter 0 is not Lebesgue measurable either (the translation argument of 0.3 applies verbatim to any measure extending length to any family containing the delegates). The club of events is genuinely, permanently smaller than the universe of subsets — not because of a lazy definition, but because the Vitali argument forbids anything bigger. The chain ℬ ⊂ ℒ ⊂ 2[0,1] has two strict inclusions, and each strictness is a theorem, not a convention.
3.3.6Almost surely: the probabilist's "equals"
Two random quantities are equal almost surely (a.s.) if the set of outcomes where they differ is null: P({ω : X(ω) ≠ Y(ω)}) = 0. More generally, a property holds almost surely if the exceptional set is null. With respect to Lebesgue measure on [0,1], "almost everywhere" (a.e.) is the mathematician's identical phrasing.
On the spinner space Ω = [0,1], let Y(ω) = 0 identically and let X(ω) = 1 if ω is rational, 0 otherwise. Then X = Y a.s. (the rationals are null, 3.3.1), yet X and Y disagree at infinitely many points — a dense cloud of disagreements, in fact. For every probabilistic purpose — expectation (Chapter 5), convergence (Chapter 7), distribution — X and Y are the same object. The rationals occupy zero probabilistic space: they are visible to set theory, invisible to measure.
Almost-sure equality is not equality, and the difference can bite outside probability. Pointwise: if Xₙ → X a.s. and each Xₙ is, say, continuous, nothing forces X continuous — the exceptions can conspire. If a strategy is profitable a.s., it can still lose on an uncountable, dense set of scenarios, and those scenarios are exactly as "real" as any others; they just carry no probability. Risk that is invisible to a model is not absent risk — a lesson worth the entire apparatus of this chapter. The theorems of Chapters 5–7 are all statements of the form "…a.s."; keeping track of what that qualifier does and does not promise is a professional skill, not a formality.
3.3.7The devil's staircase: a preview of Chapter 5
One last exhibit — the most useful one for what is coming. Define a function G on [0,1] as follows: for x ∈ [1/3, 2/3], set G(x) = 1/2; on [1/9, 2/9], G(x) = 1/4; on [7/9, 8/9], G(x) = 3/4; continue, assigning constant dyadic values on every removed middle third. Extend to all of [0,1] by continuity.
Continuity: G is continuous on [0,1] — the jumps of neighboring plateaus shrink with the stages. Flatness: G is constant on every interval removed in the construction; the union of these intervals has full measure 1. So G′ = 0 wherever G′ exists, which is almost everywhere — the set where G actually climbs, the Cantor set, has measure zero. Yet G(1) − G(0) = 1 ≠ 0 — the integral of the derivative does not recover the function. The classical calculus slogan "a function is the integral of its derivative" fails here in the most vivid way possible: all the climbing happens on a set of measure zero, invisible to any integral that only sees the derivative.
For the Riemann integral, the standard example of a badly-behaved integrand is Dirichlet's function — 1 on the rationals, 0 elsewhere — non-integrable because it oscillates everywhere. The Lebesgue theory integrates it effortlessly: 1 on a null set, 0 elsewhere, integral 0. But the deeper comparison is subtler, and G is the emblem: the question "can we integrate all the derivatives and recover all the functions" is the question Riemann's theory answers badly and Lebesgue's answers structurally — by changing what an integral is. That story — simple functions, monotone convergence, the payoff theorems — is Chapter 5, and every idea in this subchapter (null sets, a.s. equality, completion) is used there within the first few pages.
The toolkit is now honest: we know which sets are negligible, which sets are measurable, and how far the club of events extends. In 3.4 we finally face the set the club excludes — the Vitali delegates — with the full construction and the full proof that no translation-invariant, countably additive extension of length to all subsets can exist. With that, the spinner of Chapter 0 gets its complete, closed-form answer.