Sigma-Algebras and Lebesgue Measure

Part 12, Chapter 12: Measure Theory and the Lebesgue Integral

Learning objectives

Define a $\sigma$ -algebra and identify which subsets of $\mathbb{R}$ are measurable
State the measure axioms (non-negativity, $\mu(\emptyset) = 0$ , countable additivity)
Construct the Lebesgue measure on $\mathbb{R}$ by extending the length function on intervals
Recognise null sets and explain why not every subset of $\mathbb{R}$ can be assigned a measure

What does it mean to ask "how big is this set?" For an interval $[a, b]$ the answer is obvious: the length $b - a$ . For a finite union of intervals the answer is still clear. But what is the length of the rationals inside $[0, 1]$ ? The Cantor set? A randomly chosen subset of $\mathbb{R}$ ? The startling answer, from the early twentieth century, is that no consistent notion of "length" can be defined for every subset of $\mathbb{R}$ , some sets must be left out. Measure theory is the careful framework that decides which sets can be measured, assigns lengths to them, and produces an integration theory that is robust enough to power probability, harmonic analysis, and quantum mechanics.

This section is prose-heavy; the underlying objects ( $\sigma$ -algebras, abstract measures) do not lend themselves to a single small interactive widget. We pause for a thought experiment instead.

The starting question: what should "length" measure?

You want a function $m : \mathcal{P}(\mathbb{R}) \to [0, \infty]$ from subsets of $\mathbb{R}$ to non-negative real numbers (or infinity), satisfying three reasonable properties:

$m([a, b]) = b - a$ , agrees with length on intervals.
Translation invariance: $m(A + x) = m(A)$ for any $x \in \mathbb{R}$ .
Countable additivity: if $\{A_n\}$ n are pairwise disjoint, $m\big(\bigcup_n A_n\big) = \sum_n m(A_n)$ .

The bad news, due to Vitali: no such function exists on all subsets of $\mathbb{R}$ . The remedy: restrict $m$ to a carefully chosen collection of "measurable" sets and accept that exotic subsets, built using the Axiom of Choice, fall outside the theory.

$\sigma$ -algebras

A $\sigma$ -algebra on a set $X$ is a collection $\mathcal{A}$ of subsets of $X$ that is:

Closed under complement: $A \in \mathcal{A} \Rightarrow A^c \in \mathcal{A}$ .
Closed under countable unions: if $\{A_n\}_{n=1}^\infty \subset \mathcal{A}$ , then $\bigcup_n A_n \in \mathcal{A}$ .
Contains $X$ (and hence $\emptyset$ ).

The triple $(X, \mathcal{A}, \mu)$ is a measure space when $\mu : \mathcal{A} \to [0, \infty]$ satisfies $\mu(\emptyset) = 0$ and is countably additive on disjoint families.

The Borel and Lebesgue $\sigma$ -algebras on $\mathbb{R}$

The Borel $\sigma$ -algebra $\mathcal{B}(\mathbb{R})$ is the smallest $\sigma$ -algebra containing every open subset of $\mathbb{R}$ . It contains every interval, every open set, every closed set, every countable union and intersection thereof, an enormous family, large enough for nearly all practical purposes.

The Lebesgue measure $\lambda$ extends the length function from intervals to a still-larger $\sigma$ -algebra $\mathcal{L}(\mathbb{R})$ , the Lebesgue measurable sets, by the formula

$\lambda^*(A) = \inf \Big\{ \sum_{n=1}^\infty (b_n - a_n) : A \subseteq \bigcup_{n=1}^\infty (a_n, b_n) \Big\}$

(the outer measure: the infimum of total lengths of open-interval covers). A set $A$ is Lebesgue measurable if it splits every test set additively: $\lambda^*(E) = \lambda^*(E \cap A) + \lambda^*(E \cap A^c)$ for all $E$ . This is the Carathéodory criterion.

Null sets and "almost everywhere"

A set $N$ is a null set if $\lambda(N) = 0$ . Every countable set is null: $\{q_1, q_2, \ldots\}$ has measure $\sum 0 = 0$ . The rationals $\mathbb{Q}$ are null inside $\mathbb{R}$ . The Cantor set (next section) is null but uncountable. A property holds almost everywhere (a.e.) if the set where it fails is null. This phrase is the workhorse of integration theory: "f equals g almost everywhere" means we can ignore their differences on a measure-zero set without changing any integral.

Where this shows up

Probability theory: A probability space is a measure space $(\Omega, \mathcal{F}, P)$ with $P(\Omega) = 1$ . Every theorem about expected values, conditional probability, and convergence of random variables is a theorem about measures. Kolmogorov's 1933 axiomatisation of probability is the same axiomatisation as measure theory.
Quantum mechanics: Observables in quantum mechanics are self-adjoint operators on a Hilbert space; their spectral decomposition is given by a projection-valued measure. The probability of measuring an observable in an interval is the measure of that interval under the state.
Machine learning: The Radon-Nikodym derivative $d\mu / d\nu$ , a measure-theoretic likelihood ratio, is the foundation of importance sampling, density estimation, and the score function in classification. Cross-entropy loss is literally a KL divergence between measures.
Ergodic theory and dynamical systems: Long-run averages of dynamical systems are integrals against an invariant measure. The Birkhoff ergodic theorem says time averages equal space averages almost everywhere, "almost everywhere" being measure-theoretic.

Pause and think: The set $\mathbb{Q} \cap [0, 1]$ is countable. What is its Lebesgue measure? (Hint: use that countable unions of null sets are null.) Now: the set $[0, 1] \setminus \mathbb{Q}$ of irrationals in $[0, 1]$ has measure $1$ . So "most" of $[0, 1]$ is irrational in the measure-theoretic sense.

Try it

Predict first: is $\{\emptyset, \{1\}, \{2\}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}\}$ a $\sigma$ -algebra on $\{1, 2, 3\}$ ? Check the three closure properties. (Spoiler: it fails on complement of $\{2\}$ (which is $\{1,3\}$ ) or complement of $\{1,2\}$ (which is $\{3\}$ ).)
Compute $\lambda\big(\bigcup_{n=1}^\infty [n, n + 1/3^n]\big)$ . (Hint: the intervals are disjoint; sum the lengths of a geometric series.)
True or false: any subset of a null set is null. (Hint: measure is monotone, $A \subseteq B \Rightarrow \lambda(A) \leq \lambda(B)$ , assuming completeness of the measure, which Lebesgue measure has.)
Show that the smallest $\sigma$ -algebra containing all singletons in $\mathbb{R}$ is properly smaller than $\mathcal{B}(\mathbb{R})$ . (Hint: it contains only countable sets and their complements.)
If $\mu$ is a measure with $\mu(\mathbb{R}) = 1$ , prove that there can be at most countably many points $x \in \mathbb{R}$ with $\mu(\{x\}) > 0$ . (Hint: how many points can have $\mu(\{x\}) > 1/n$ ?)

A trap to watch for

Beginners often assume measurability is automatic, that every set you can describe in words is measurable. It is not. The Vitali construction, which uses the Axiom of Choice to pick one representative from each coset of $\mathbb{Q}$ inside $\mathbb{R}$ , produces a subset of $[0, 1]$ that is provably non-measurable under any translation-invariant countably additive measure. The good news is that the non-measurable sets are essentially unreachable: no Borel set is non-measurable, no set you can build with a recursive construction is non-measurable, no set in any specific science problem you will ever encounter is non-measurable. The technical layer "is this set measurable?" is conceptually important but operationally invisible, in practice, you assume measurability and proceed.

What you now know

You can state the $\sigma$ -algebra axioms, identify Borel sets, compute Lebesgue measure of countable unions and basic constructions, and articulate why measurability is a non-trivial restriction. The next section turns to a single pivotal example, the Cantor set, which is closed, uncountable, and has Lebesgue measure zero, showing how counterintuitive the geometry of measurable sets can be.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 12.
Royden, H. L., Fitzpatrick, P. M. (2010). Real Analysis (4th ed.). Pearson, ch. 1-2.
Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications (2nd ed.). Wiley, ch. 1.
Rudin, W. (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill, ch. 1-2.
Stein, E. M., Shakarchi, R. (2005). Real Analysis. Princeton UP, ch. 1.