Phase 6 - Lesson 6.2
Measures and Sigma-Algebras
How to assign a consistent notion of size to sets - and why we cannot measure everything.
Builds on: Compactness/topology (6.1) supply the Borel sets we will measure.
Leads to: A probability measure (Phase 7) is exactly a measure of total mass 1.
Leads to: A probability measure (Phase 7) is exactly a measure of total mass 1.
Learning Objectives
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- Define a sigma-algebra and give the Borel sigma-algebra as the key example.
- Define a measure and prove countable additivity implies monotonicity and continuity.
- Explain why Lebesgue measure cannot be defined on all subsets (non-measurable sets).
- Compute the measure of simple and limiting sets, including sets of measure zero.
- Interpret ‘almost everywhere’ and its role in later integration theory.
Key Vocabulary
- Sigma-algebra
- A collection \(\mathcal{F}\) containing \(X\) and closed under complement and countable unions.
- Measure
- A function \(\mu:\mathcal{F}\to[0,\infty]\) with \(\mu(\emptyset)=0\) and countable additivity on disjoint sets.
- Borel sigma-algebra
- The smallest sigma-algebra containing all open sets; the natural domain for measuring on \(\mathbb{R}\).
- Lebesgue measure
- The translation-invariant measure on \(\mathbb{R}^n\) assigning length/area/volume, extending intervals \(\mu([a,b])=b-a\).
- Null set
- A set of measure zero; e.g. any countable set has Lebesgue measure 0.
- Almost everywhere
- A property holding except on a null set; written ‘a.e.’
Intuition & Motivation
Intuition
We want to say how ‘big’ a set is - its length, area, or (in probability) its likelihood. The naive hope is to assign a size to every subset, but that hope is provably impossible: the Vitali construction builds a set whose translates tile the line, so any translation-invariant, countably-additive size would have to be both 0 and positive. The fix is to restrict attention to a rich but not universal collection of ‘nice’ sets - a sigma-algebra - on which a consistent measure lives. Everything you will ever integrate is measurable; the pathological sets are simply excluded.
Sigma-algebras
Definition - Sigma-algebra
A collection \(\mathcal{F}\) of subsets of \(X\) with (i) \(X\in\mathcal{F}\), (ii) \(A\in\mathcal{F}\Rightarrow A^c\in\mathcal{F}\), and (iii) closure under countable unions \(\bigcup_n A_n\in\mathcal{F}\). De Morgan then gives closure under countable intersections.
The Borel sigma-algebra \(\mathcal{B}(\mathbb{R})\) is the smallest sigma-algebra containing every open interval; it contains all open, closed, countable, \(F_\sigma,G_\delta\) sets - essentially every set arising in practice.
Definition - Measure
A function \(\mu:\mathcal{F}\to[0,\infty]\) with \(\mu(\emptyset)=0\) and countable additivity: for disjoint \(A_1,A_2,\dots\), \(\mu\big(\bigcup_n A_n\big)=\sum_n\mu(A_n)\).
Theorem - Consequences of countable additivity
Any measure is monotone (\(A\subseteq B\Rightarrow\mu(A)\le\mu(B)\)), countably subadditive, and continuous from below/above: \(A_n\uparrow A\Rightarrow\mu(A_n)\to\mu(A)\) (and from above when some \(\mu(A_n)\lt \infty\)).
Proof
For monotonicity write \(B=A\cup(B\setminus A)\) disjointly; additivity gives \(\mu(B)=\mu(A)+\mu(B\setminus A)\ge\mu(A)\). For continuity from below, telescope \(A=\bigsqcup_n(A_n\setminus A_{n-1})\) and apply countable additivity to the partial sums. ∎
Why not measure everything?
Non-measurable sets exist
Using the axiom of choice, Vitali builds a subset of \([0,1]\) with no consistent Lebesgue measure. Hence Lebesgue measure is defined only on the (very large) sigma-algebra of measurable sets, not on the full power set. This is a feature: measurability is exactly the closure property that makes limits behave.
Worked Example - Sets of measure zero
1
A single point has measure 0: cover it by \([a-\varepsilon,a+\varepsilon]\), so \(\mu(\{a\})\le2\varepsilon\) for all \(\varepsilon\gt 0\).
2
By countable additivity any countable set is null: \(\mu(\mathbb{Q}\cap[0,1])=\sum_n0=0\).
3
The Cantor set is uncountable yet has measure 0: at step \(k\) a fraction \((2/3)^k\) remains, and \((2/3)^k\to0\).
4
Null sets are ‘invisible’ to integration - changing a function on one changes no integral, which is what ‘almost everywhere’ captures.
Interactive: the Cantor set has measure zero
Track the remaining Lebesgue measure of the Cantor construction as you remove middle thirds - an uncountable set shrinking to size 0.
Common Mistakes to Avoid
- Assuming every subset of \(\mathbb{R}\) has a length - non-measurable sets show it cannot.
- Confusing countable additivity with finite additivity; the ‘countable’ is what powers limit theorems.
- Believing measure zero means countable - the Cantor set is uncountable yet null.
- Forgetting a sigma-algebra must be closed under complements, not just unions.
Quant Practitioner Tips
- ‘Almost everywhere’ lets you ignore null sets; most integration theorems only hold a.e.
- To show a set is Borel, build it from open sets via countable unions/intersections and complements.
- Countable additivity + continuity from below is the standard route to compute measures of limits.
- In probability, replace ‘a.e.’ with ‘almost surely’ - same concept, mass-1 measure.
Knowledge Check
Q1 Medium
Which property must a sigma-algebra satisfy that a mere algebra of sets need not?
Q2 Medium
Countable additivity of a measure directly implies all of the following EXCEPT:
Q3 Medium
The Cantor set is an example of a set that is:
Practical Exercise
Prove that a countable set \(A=\{a_1,a_2,\dots\}\subset\mathbb{R}\) has Lebesgue measure zero, using countable subadditivity.
▶ Show full solution
Fix \(\varepsilon\gt 0\). Cover \(a_n\) by the interval \(I_n=(a_n-\varepsilon 2^{-n-1},\,a_n+\varepsilon 2^{-n-1})\), of length \(\varepsilon 2^{-n}\).
By countable subadditivity, \(\mu(A)\le\sum_{n=1}^\infty\mu(I_n)=\sum_{n=1}^\infty\varepsilon 2^{-n}=\varepsilon\). Since this holds for every \(\varepsilon\gt 0\), \(\mu(A)=0\). In particular \(\mu(\mathbb{Q})=0\).
After the reveal, answer for yourself: Why does the same trick fail for an uncountable set like \([0,1]\)?
Lesson Summary
A sigma-algebra is a collection of sets closed under complements and countable unions; a measure assigns them sizes with countable additivity, from which monotonicity and continuity follow. Not every set is measurable (Vitali), so we work on the Borel/Lebesgue sigma-algebra. Null sets - even uncountable ones like the Cantor set - are the ‘almost everywhere’ we may ignore.
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
▶ Show retrieval prompts & answers
Q: What are the two closure axioms of a sigma-algebra beyond containing \(X\)?
A: Closure under complement and under countable unions (hence, by De Morgan, countable intersections).
A: Closure under complement and under countable unions (hence, by De Morgan, countable intersections).
Q: Give an uncountable set of measure zero and why it is null.
A: The Cantor set: after \(k\) middle-third removals a fraction \((2/3)^k\to0\) remains, so its Lebesgue measure is 0, yet it is uncountable.
A: The Cantor set: after \(k\) middle-third removals a fraction \((2/3)^k\to0\) remains, so its Lebesgue measure is 0, yet it is uncountable.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Confidence / mastery rating
Personal notes
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Measure, Integration & Real Analysis (Sheldon Axler, GTM 282) current - Ch. 2 - Ch. 2: sigma-algebras, measures, Borel sets, outer/Lebesgue measure.
- Probability: Theory and Examples (Rick Durrett, 5th ed.) current - Ch. 1 - Ch. 1: measures, sigma-fields, and the probability-space setup.