Riemann integration cuts the domain into intervals. Lebesgue integration begins by deciding which sets can be measured and then organises the function by measurable values.
R25.03 built integration from finite interval partitions. That theory is powerful, but modern analysis needs a framework that handles limits of functions more flexibly, distinguishes negligible exceptional sets, and supports probability, Fourier analysis, partial differential equations and functional analysis.
Measure theory provides that framework. A σ-algebra identifies admissible measurable sets. A measure assigns size with countable additivity. Measurable functions interact correctly with those sets. The Lebesgue integral is then built in stages: indicator functions → simple functions → nonnegative measurable functions → integrable signed functions.
Level: advanced undergraduate / beginning graduate analysis enrichment. Prerequisites: limits, sequences, completeness, Riemann integration and basic set operations.
Reading route: σ-algebras → measures → measure spaces → null sets → almost everywhere → measurable functions → indicator and simple functions → nonnegative integral → signed integrable functions → Fatou → monotone convergence → dominated convergence → Tonelli/Fubini → modes of convergence → Lᵖ bridge → Riemann comparison → practice → solutions.
1. A σ-algebra is a collection of sets stable under countable logic
Let X be a set. A collection Σ of subsets of X is a σ-algebra if:
- X∈Σ;
- A∈Σ implies X\A∈Σ;
- A₁,A₂,…∈Σ implies ⋃_{n=1}^∞A_n∈Σ.
Countable intersections also belong to Σ by De Morgan’s laws.
The word “countable” is essential. Analysis repeatedly forms limits indexed by natural numbers, so the measurable-set structure must survive countable unions and intersections.
2. The Borel σ-algebra is generated by open sets
On R, the Borel σ-algebra B(R) is the smallest σ-algebra containing every open set.
It therefore contains open and closed intervals, countable unions of such sets, countable intersections, and many more complicated sets produced from these operations.
Lebesgue measurable sets form a larger σ-algebra than the Borel sets after completion by null subsets.
3. A measure assigns size countably additively
A measure μ on (X,Σ) is a function μ:Σ→[0,∞] satisfying μ(∅)=0 and countable additivity:
if A₁,A₂,… are pairwise disjoint, then μ(⋃A_n)=Σμ(A_n).
The value ∞ is allowed. Countable additivity is stronger than finite additivity and is the property that interacts correctly with monotone limiting processes.
4. Measure spaces separate the underlying world from its size rule
A measure space is a triple (X,Σ,μ).
- X is the underlying set.
- Σ says which subsets are measurable.
- μ says how large those measurable subsets are.
Different measures can live on the same measurable space. Geometry, probability and counting are therefore different interpretations of the same abstract structure.
5. Counting measure and point mass are simple examples
Counting measure assigns μ(A)=number of elements of A, allowing ∞ for infinite sets.
A Dirac measure at x₀ assigns δ_{x₀}(A)=1 if x₀∈A and 0 otherwise.
These examples show that “measure” is not synonymous with Euclidean length. It is any countably additive size assignment on a σ-algebra.
6. Lebesgue measure extends ordinary length
Lebesgue measure λ on R assigns intervals their expected length:
λ((a,b))=b−a.
It extends this consistently to a vast σ-algebra of subsets of R while preserving translation invariance and countable additivity.
On Rⁿ, Lebesgue measure extends ordinary n-dimensional volume.
7. Continuity from below and above are measure-limit theorems
If A₁⊂A₂⊂⋯ and A=⋃A_n, then
μ(A_n)→μ(A).
If A₁⊃A₂⊃⋯, A=⋂A_n, and μ(A₁)<∞, then
μ(A_n)→μ(A).
The finite-measure hypothesis in the decreasing case matters. These are early examples of exchanging a set limit with a measure operation under explicit conditions.
8. Null sets have measure zero
A measurable set N is null if μ(N)=0.
Every countable subset of R has Lebesgue measure zero because singletons have measure zero and measure is countably subadditive.
The converse is false: there are uncountable null sets. The standard middle-thirds Cantor set is uncountable and has Lebesgue measure zero.
9. “Almost everywhere” means outside a null exceptional set
A property holds almost everywhere, abbreviated a.e., if the set of points where it fails has measure zero.
Two functions equal almost everywhere can differ at infinitely many points, even uncountably many points, provided the disagreement set is null.
Lebesgue integration naturally identifies many such functions because changing values on a null set does not change the integral of an integrable function.
10. Measurable functions preserve measurable threshold questions
A function f:X→R is measurable if the preimage of every Borel set is measurable.
Equivalent practical tests include checking that sets {x:f(x)>a} are measurable for every real a.
Continuity implies Borel measurability. Measurability is much weaker than continuity and is designed to survive limits far more flexibly.
11. Pointwise limits of measurable functions are measurable
If f_n are measurable and f_n(x)→f(x) pointwise, then f is measurable.
Likewise sup_n f_n, inf_n f_n, limsup f_n and liminf f_n are measurable when defined in the extended-real sense.
This closure under countable limiting operations is one of the main reasons measure theory is well adapted to analysis.
12. Indicator functions turn sets into functions
For a measurable set A, the indicator function 1_A is 1 on A and 0 outside A.
The Lebesgue integral is designed so that
∫1_A dμ=μ(A).
This identity is the first bridge between measuring sets and integrating functions.
13. Simple functions are finite measurable step functions
A nonnegative simple function has the form
φ=Σ_{k=1}^m a_k 1_{A_k},
where a_k≥0 and A_k are measurable, commonly chosen disjoint.
Define
∫φ dμ=Σa_k μ(A_k).
This is exact, not an approximation formula.
14. The nonnegative Lebesgue integral is built from below
For a measurable f≥0, define
∫f dμ=sup{∫φ dμ: 0≤φ≤f, φ simple}.
The integral may equal ∞.
Instead of partitioning the domain into narrow intervals, this definition approximates the function from below by measurable simple functions.
15. Measurable nonnegative functions admit increasing simple approximations
For every nonnegative measurable f, there exists a sequence of nonnegative simple functions φ_n such that
φ_n(x)↑f(x)
pointwise.
This approximation theorem makes simple functions the finite building blocks of the Lebesgue integral, just as stepwise finite sums support Riemann integration.
16. Signed functions split into positive and negative parts
Define
f⁺=max(f,0), f⁻=max(−f,0).
Then f=f⁺−f⁻ and |f|=f⁺+f⁻.
A measurable real-valued f is Lebesgue integrable if ∫|f|dμ<∞. Then both ∫f⁺ and ∫f⁻ are finite and
∫f=∫f⁺−∫f⁻.
17. Integrability is absolute in L¹
For the Lebesgue integral of a signed real function, “integrable” normally means
∫|f|dμ<∞.
This is stronger than conditionally convergent improper integration based on cancellation.
The distinction matters when comparing Lebesgue integrability with improper Riemann integrals such as oscillatory tails.
18. Monotone Convergence passes increasing limits through the integral
If 0≤f₁≤f₂≤⋯ are measurable and f_n→f pointwise, then
∫f_n dμ→∫f dμ,
allowing ∞.
No finite dominating function is needed. Monotonicity and nonnegativity prevent hidden cancellation or escaped mass.
19. Worked Monotone Convergence example
On (0,∞), let f_n(x)=1_{(0,n)}(x)e^{-x}.
Then f_n↑e^{-x} pointwise and every f_n is nonnegative measurable.
Therefore
∫₀^∞e^{-x}dx=lim_{n→∞}∫₀^n e^{-x}dx=lim(1−e^{-n})=1.
The theorem justifies the limit passage structurally.
20. Fatou’s lemma gives a one-sided inequality under nonnegative limits
For nonnegative measurable f_n,
∫ liminf f_n dμ ≤ liminf ∫f_n dμ.
Mass can disappear in the limit from the viewpoint of pointwise convergence, but Fatou controls the direction in which the integral can fail.
Equality requires stronger hypotheses.
21. Dominated Convergence is the main bounded-exchange theorem
Suppose f_n→f almost everywhere, each f_n is measurable, and there exists an integrable g such that
|f_n|≤g almost everywhere for every n.
Then f is integrable and
∫f_n dμ→∫f dμ.
Moreover ∫|f_n−f|dμ→0.
The integrable dominating envelope prevents mass from concentrating in thinner regions while escaping pointwise detection.
22. Pointwise convergence alone does not control integrals
On [0,1], define f_n(x)=n·1_{(0,1/n)}(x).
For every fixed x>0, eventually x≥1/n, so f_n(x)→0. At x=0 the indicators are zero as defined here, so convergence is pointwise to zero.
Yet
∫₀¹ f_n dx = n(1/n)=1
for every n.
The integral does not converge to the integral of the pointwise limit. The functions develop a narrowing spike whose total mass stays fixed.
23. Why no integrable dominator exists for the spike sequence
If one function g dominated every f_n, then near zero it would need to be at least n on intervals of size about 1/n for arbitrarily large n.
The required envelope behaves too singularly near zero to be integrable.
This shows how the domination hypothesis diagnoses exactly what the counterexample is doing.
24. Tonelli handles nonnegative product-space functions
For a nonnegative measurable f on a product measure space, Tonelli’s theorem allows iterated integration and states that the integrals agree, possibly with value ∞.
Nonnegativity prevents cancellation, so no prior finite absolute integral is required.
Tonelli is especially useful when proving that a double integral is finite: compute or estimate an iterated nonnegative integral first.
25. Fubini handles integrable signed functions
If f is integrable on a product measure space, meaning ∫|f|<∞, then Fubini’s theorem permits iterated integration and exchange of order almost everywhere in the standard formulation.
The absolute-integrability condition matters. Without it, different iterated integrals can fail to exist or can disagree under conditional cancellation.
26. The Lebesgue integral extends the Riemann integral
If a bounded function on [a,b] is Riemann integrable, then it is Lebesgue integrable and both integrals agree.
Lebesgue’s criterion sharpens the boundary: a bounded function on [a,b] is Riemann integrable exactly when its discontinuity set has Lebesgue measure zero.
Thus Lebesgue theory does not discard Riemann integration. It places it inside a broader framework.
27. The Dirichlet function becomes Lebesgue integrable
On [0,1], let f=1_Q, equal to 1 on rationals and 0 on irrationals.
R25.03 showed this function is not Riemann integrable because every interval contains rationals and irrationals.
But Q∩[0,1] is countable and therefore has Lebesgue measure zero.
Hence
∫₀¹1_Q dλ=0.
Lebesgue integration recognises that the set carrying value 1 is negligible in measure even though it is dense.
28. Dense and large are different concepts
The rationals are dense in R but have Lebesgue measure zero.
The irrationals are also dense and have full measure in every finite interval after removing the null rational set.
Topology asks whether every neighbourhood meets a set. Measure asks how much size a set carries. Neither notion subsumes the other.
29. Different modes of function convergence answer different questions
- Pointwise: for each x, f_n(x)→f(x).
- Almost everywhere: pointwise convergence outside a null set.
- Uniform: one index threshold works for all x.
- In measure: for every ε>0, the measure of {|f_n−f|>ε} tends to zero.
- In L¹: ∫|f_n−f|→0.
These modes are related but not interchangeable without hypotheses. R25.08 will use counterexamples to map the implications carefully.
30. Lᵖ spaces measure average magnitude
For 1≤p<∞, Lᵖ consists of measurable functions with
∫|f|^p dμ<∞,
where functions equal almost everywhere are identified.
The norm is ||f||_p=(∫|f|^p)^{1/p}. For p=∞, the essential supremum replaces an ordinary maximum.
These spaces are complete: Lᵖ is a Banach space for 1≤p≤∞. That result forms a direct bridge from R25 completeness to R27 functional analysis.
31. Hölder and Minkowski control products and sums
If 1/p+1/q=1 with p,q>1, Hölder’s inequality gives
∫|fg|≤||f||_p||g||_q.
Minkowski’s inequality gives
||f+g||_p≤||f||_p+||g||_p.
These inequalities make Lᵖ spaces workable geometric objects, not merely sets of integrable functions.
32. Nonmeasurable sets mark a real boundary
Not every subset of R can be assigned a translation-invariant countably additive length extending interval length while preserving the desired consistency conditions.
Classical constructions such as Vitali sets exhibit non-Lebesgue-measurable subsets using a choice principle.
This is why the σ-algebra is part of the data of a measure space. “Every subset has a size” is not available under the standard Lebesgue requirements.
33. A dependable measure-theory workflow
First state the measure space (X,Σ,μ). A formula without its measurable structure can hide the actual domain of the theorem.
Check measurability before integration. For limits of functions, identify the convergence mode and whether nonnegativity, monotonicity, domination or absolute integrability is available.
Use Tonelli for nonnegative product-space functions and Fubini for integrable signed functions. Keep “almost everywhere” separate from “everywhere”.
If an interchange fails, look for concentration of mass, lack of an integrable bound, conditional cancellation, or an exceptional set whose measure is not negligible.
34. Common misconceptions
- Measure means length. Length is one measure; counting and probability are others.
- Measure zero means empty or countable. False; uncountable null sets exist.
- Dense means large in measure. False; Q is dense and null.
- Pointwise convergence permits limit-integral exchange. False.
- Lebesgue integrable means an improper integral converges by cancellation. Signed Lebesgue integrability requires absolute integrability.
- Fubini and Tonelli are the same theorem with interchangeable hypotheses. They solve related but distinct product-integration jobs.
35. Independent practice: twenty questions
- State the three σ-algebra closure properties.
- What is the Borel σ-algebra on R?
- Define a measure.
- Give two measures other than Lebesgue measure.
- What is a null set?
- Give an uncountable null set.
- Define “almost everywhere”.
- Give a practical threshold-set criterion for measurability of a real function.
- What is an indicator function?
- Define the integral of a nonnegative simple function.
- How is the integral of a nonnegative measurable function defined from simple functions?
- Define f⁺ and f⁻.
- When is a signed measurable function Lebesgue integrable?
- State the Monotone Convergence Theorem.
- State Fatou’s lemma.
- State the Dominated Convergence Theorem.
- Why does f_n=n1_(0,1/n) defeat pointwise limit-integral exchange?
- Distinguish Tonelli and Fubini by their typical hypotheses.
- Why is 1_Q on [0,1] Lebesgue integrable but not Riemann integrable?
- Name three modes of convergence for measurable functions.
36. Worked solutions and checks
1. Contain X; closed under complements; closed under countable unions.
2. The smallest σ-algebra containing all open subsets of R.
3. A nonnegative extended-real set function on a σ-algebra with μ(∅)=0 and countable additivity over pairwise disjoint sets.
4. Counting measure and a Dirac point mass.
5. A measurable set of measure zero.
6. The middle-thirds Cantor set.
7. The property fails only on a measurable null set.
8. For every real a, sets such as {x:f(x)>a} must be measurable.
9. 1_A equals 1 on A and 0 outside A.
10. If φ=Σa_k1_{A_k} on disjoint measurable A_k, then ∫φ=Σa_kμ(A_k).
11. As the supremum of integrals of nonnegative simple functions lying below f.
12. f⁺=max(f,0), f⁻=max(−f,0), so f=f⁺−f⁻.
13. When ∫|f|dμ<∞.
14. If 0≤f_n↑f pointwise, then ∫f_n→∫f.
15. For nonnegative measurable f_n, ∫liminf f_n≤liminf∫f_n.
16. If f_n→f a.e. and |f_n|≤g for one integrable g, then ∫f_n→∫f and f is integrable.
17. It converges pointwise to zero but each integral equals 1; mass concentrates into a shrinking neighbourhood.
18. Tonelli handles nonnegative measurable functions, possibly with infinite integrals; Fubini handles integrable signed functions, typically requiring ∫|f|<∞.
19. Q∩[0,1] has measure zero, so its indicator has Lebesgue integral zero, but every interval contains both rationals and irrationals, preventing Riemann integrability.
20. Examples: pointwise, almost everywhere, uniform, in measure, and L¹ convergence.
37. Where this guide hands off
R25.08 turns these theorems into a systematic exchange-of-limits map. It asks when a limit may cross an integral, derivative, infinite sum or supremum—and constructs counterexamples when the needed uniformity, domination, monotonicity or compactness is missing.
Sources and further study
For a complete academic route, see MIT OpenCourseWare 18.125 Measure and Integration. Its published course materials cover abstract measure, the Lebesgue integral, the main convergence theorems, Lebesgue measure on Rⁿ, Lᵖ spaces and further integration results.
Return to Riemann Integration and Accumulation · Return to the BTT Mathematics Hub.