Riemann vs Lebesgue: Which Integral, and When
You arrive owning an integral. The course built you a second one. This subchapter establishes the relationship: Lebesgue extends Riemann, they agree wherever both exist, and the cases where only Lebesgue exists are precisely the cases probability cannot live without.
In this subchapter
5.3.1Two ways to slice a function
The Riemann integral — the one from calculus — slices the domain: partition [0,1] into intervals, approximate the function on each by a constant (the height of some sample point), sum areas, take the limit as the mesh refines. The Lebesgue integral — the one built in 5.1 — slices the range: partition the values into bands, ask how much probability mass sits under each band (P((k−1)2−n ≤ X < k 2−n)), sum band-price times mass, refine.
Why does the axis matter? Riemann needs the function to be nearly constant on each domain piece — fine for continuous functions, hopeless for functions that oscillate at every scale. Lebesgue needs only that each band's level set is measurable — a question about sets, which is exactly what σ-algebras answer. A function can jump around wildly in the domain and stay tame in the range.
5.3.2Theorem: agreement, and extension
Let f: [0,1] → ℝ be bounded. If f is Riemann integrable (Riemann sense), then f is Lebesgue-integrable on ([0,1], ℒ, λ) and the two integrals are equal. If f is not Riemann integrable but bounded and Borel/Lebesgue-measurable, the Lebesgue integral still exists. In the other direction, there are bounded measurable functions that are not Riemann integrable — so the extension is strict.
For a partition Π of [0,1] into intervals, define the lower and upper Riemann sums via mI = infI f, MI = supI f per interval I: the simple functions LΠ = Σ mI 1I and UΠ = Σ MI 1I satisfy LΠ ≤ f ≤ UΠ, hence E[LΠ] ≤ E[f] ≤ E[UΠ] by monotonicity (5.1 Theorem 2) — the Lebesgue expectation is squeezed between the lower and upper Riemann sums for every partition. Riemann integrability says the supremum of lower sums equals the infimum of upper sums (that common value is the Riemann integral, by definition). The squeeze forces E[f] to equal that common value. Note the honest assumption: boundedness of f — the Riemann theory is a bounded-function theory, and the comparison lives there. ∎
Everything a quant computes by calculus — ∫₀¹ x² dx = 1/3, Gaussian integrals by polar tricks, the moment formulas E[Xk] for continuous densities — remains valid: these are Riemann integrals of (piecewise) continuous functions, Theorem 1 certifies the Lebesgue expectation agrees, and no previously computed number changes. The course's framework is an expansion of your toolkit, not a revocation of it. What changes is which manipulations are licensed (limit theorems of 5.2, far stronger than anything Riemann's theory offers) and which functions are in scope (§3).
5.3.3The separator: 1ℚ
Let f = 1ℚ∩[0,1]: 1 on rationals, 0 on irrationals. Every interval of positive length contains both rational and irrational points (0.1's density-of-ℚ, now with teeth), so for every partition: mI = 0 and MI = 1 on every interval. Lower sums are 0, upper sums are 1, always, at every mesh. The squeeze never closes: not Riemann integrable. Meanwhile the Lebesgue analysis is two lines: f is a simple function — 1 · 1ℚ∩[0,1], and ℚ∩[0,1] is countable hence measurable with λ = 0 (countable additivity over singletons, 3.1). So E[f] = 1 · λ(ℚ∩[0,1]) = 0. The function that calculus cannot integrate, the measure theory prices instantly. And it is not a pathology invented for the occasion: this is the P&L of a contract paying 1 if a uniform rate lands exactly on a rational — a silly contract, but the same shape as any contract keyed to a zero-probability level. Riemann's theory cannot state the fair value; Lebesgue's computes it.
5.3.4When is Riemann integrability available?
A bounded f: [0,1] → ℝ is Riemann integrable if and only if the set of its discontinuity points has Lebesgue measure zero. The proof is a beautiful ε-management exercise (build partitions that isolate the rare bad points into cheap strips); we state it and prove the easy direction: a function discontinuous on a null set — e.g. any piecewise-continuous payoff with finitely many jumps — is Riemann integrable, because on the complement of a null set one can refine the partition so the oscillation per interval is small.
A digital option's payoff 1{S≥K}(S) as a function of the underlying S: one jump at K, discontinuity set of measure zero — Riemann integrable, so Monte Carlo estimates of its expectation converge in the classical senses (this claim becomes a theorem in 7.2). A payoff with countably many jumps (a barrier ladder) — still a null discontinuity set, still fine. The criterion quietly underwrites a piece of market practice: practitioners integrate discontinuous payoffs by ordinary quadrature without anxiety, and the mathematics says their confidence is justified exactly up to "jumps on a null set". The moment the discontinuity set carries positive measure — a payoff oscillating on a positive-measure Cantor-like set — Riemann dies and only the measure-theoretic expectation remains.
Modify the digital payoff to depend on whether the underlying is rational — a contract settled on an exchange rule that reads a decimal expansion. The payoff is now 1ℚ-shaped: discontinuous at every point (any neighborhood of any point sees both values), discontinuity set of measure one. Not Riemann integrable; Lebesgue expectation still exists and equals (under a uniform model) 0. Contrived? Slightly — but its practical shadow is real: indicators of dense information (parity of the ten-thousandth decimal of a rate; membership in any given dense null set) are exactly the objects that decide whether a hedging argument can be run by classical tools or only by measure-theoretic ones.
5.3.5Practical consequences
- Every Riemann integral you computed is still right (Theorem 1). Nothing needs recomputing.
- Limits are the new power: MCT/Fatou/DCT (5.2) hold for the Lebesgue integral with minimal hypotheses; the Riemann theory's limit theorems are, by comparison, nearly unusable (uniform convergence required).
- Null sets are free: changing a payoff on a null set (a single level, countably many levels) changes no expectation — the a.s. license of 4.1 §4, now with integrals attached.
- Densities are optional: the expectation exists for any measurable variable — atoms, singular laws like the Cantor distribution (4.2 §5), mixed types. The formula ∫ x f(x) dx is a special case (made precise in 5.4), not the definition.
5.3.6Why probability had to choose Lebesgue
Assemble the case. Probability needs: (1) expectations of arbitrary measurable variables — its objects are functions defined by σ-algebra membership, and those routinely lack densities or continuity; (2) limit theorems — its fundamental results (LLN, CLT, martingale convergence) are statements about limits of expectations; (3) null-set tolerance — its equalities are a.s., and Riemann integrability is not stable under redefining a function on a null set (Example 2's function is a one-point change of the zero function on... a countable set of points, and it destroys Riemann integrability entirely). The Riemann integral fails all three; Lebesgue delivers all three. This is not a historical accident to be politely mentioned and ignored — the entire architecture of Chapters 1–3 exists because the objects of probability demand it. The course's answer to "why measure theory?" is now complete, and the remaining chapters simply live in the house it built.
Lebesgue does not replace your calculus; it secures it — and takes over precisely where the functions probability actually needs stop being Riemann's.
Expectation is defined, limit-safe, and reconciled with calculus. Two computational tools remain: how to compute expectations through change of variables (the law-only formula E[g(X)] = ∫g dμ, densities as Radon–Nikodym previews, the Jacobian rule), and what happens when the law has no density at all. The next subchapter builds the change-of-variables machinery, proves LOTUS, and includes the Jacobian counterexample that punishes mechanical computation.