Measure Theory

The Riemann integral of riemann-integral.md and the Jordan content of multiple-integrals.md both measure sets by approximating them with finitely many boxes. That is enough for the path to Stokes' theorem, but it leaves two debts outstanding: the phrase "measure zero" used in Lebesgue's criterion (§3 of both pages) has never been defined, and no notion of length/area/volume in this spine is stable under countable operations.

This page pays the first debt. We build the general notion of a measure — a countably additive set function on a $\sigma$-algebra — construct Lebesgue measure on ${\mathbb{R}}^n$ via Carathéodory's extension theorem, and prove the uniqueness theorem ($\pi$–$\lambda$) that makes measures cheap to identify. The Lebesgue integral is built on top of it, and probability is the special case $\mu (X)=1$.

References: Rudin, Real and Complex Analysis, ch. 1–2; Folland, Real Analysis, ch. 1; Tao, An Introduction to Measure Theory.

1. Why Not Measure Every Set?

The naive goal is a function $\mu :\mathcal{P}({\mathbb{R}}^n)\to [0,\infty ]$ assigning a "volume" to every subset, with three properties one would not want to give up:

1. Countable additivity. $\mu \left({⨆}_{k=1}^{\infty }{A}_k\right)={\sum }_k\mu (A_k)$ for pairwise disjoint $A_k$.
2. Translation invariance. $\mu (A+x)=\mu (A)$.
3. Normalization. $\mu ([0,1{]}^n)=1$.

Theorem (no total measure). For $n\ge 1$ no such $\mu$ exists.

The proof — Vitali's construction of a non-measurable set — is deferred to non-measurable.md, because it is really a statement about the axiom of choice rather than about analysis. Its consequence, though, dictates the design of everything below: we must give up (0) "every subset". Measure theory is the discipline of specifying, in advance, a family of sets we promise to measure — and that family must be closed under exactly the operations we intend to perform.

Why countable additivity in particular? Finite additivity alone is achievable on all subsets (in dimensions 1 and 2, via Banach limits), but it is useless for analysis: every limit theorem — monotone convergence, dominated convergence, the continuity of $\mu$ along increasing sequences — is a countable statement. The whole point of the Lebesgue theory is to exchange limits and integrals, and that exchange is countable additivity in disguise.

2. $\sigma$-Algebras

Definition ($\sigma$-algebra). A collection $\mathcal{A}$ of subsets of a set $X$ is a $\sigma$-algebra if

1. $X\in \mathcal{A}$;
2. $A\in \mathcal{A}⟹A^c\in \mathcal{A}$ (complements);
3. $A_1,A_2,\dots \in \mathcal{A}⟹{\bigcup }_{k=1}^{\infty }{A}_k\in \mathcal{A}$ (countable unions).

The pair $(X,\mathcal{A})$ is a measurable space; members of $\mathcal{A}$ are measurable sets.

De Morgan gives countable intersections for free, and $\varnothing =X^c\in \mathcal{A}$. Note the asymmetry with a topology: a $\sigma$-algebra is closed under complement but only countable union, a topology under arbitrary union but not complement.

Two extremes always exist: $\{\varnothing ,X\}$ and $\mathcal{P}(X)$. Between them, the useful construction is generation.

Definition (generated $\sigma$-algebra). For $\mathcal{E}\subseteq >\mathcal{P}(X)$, $\sigma (\mathcal{E})$ is the intersection of all $\sigma$-algebras containing $\mathcal{E}$ — the smallest one containing $\mathcal{E}$.

The intersection of any family of $\sigma$-algebras is a $\sigma$-algebra (check the three axioms pointwise), and $\mathcal{P}(X)$ is always available, so $\sigma (\mathcal{E})$ is well defined. It is defined "from above", by intersection; there is in general no description of its members from below, as sets built from $\mathcal{E}$ in countably many steps. That is why §4 below (the $\pi$–$\lambda$ theorem) matters so much: it is the standard substitute for the induction one cannot do.

Definition (Borel $\sigma$-algebra). For a topological space $X$, $\mathcal{B}(X)=\sigma (\{\text{open sets}\})$. On $\mathbb{R}$, $\mathcal{B}(\mathbb{R})$ is generated equally by the open intervals, the closed intervals, or the rays $(-\infty ,a]$.

Proof of the last claim. Each ray is closed, hence Borel; conversely $(a,b)={\bigcup }_k((-\infty ,b-1/k]\setminus (-\infty ,a])$, and every open subset of $\mathbb{R}$ is a countable union of open intervals (metric-spaces.md §2), so $\sigma (\text{rays})\supseteq \mathcal{B}(\mathbb{R})$. $\square$

Borel sets include everything encountered in practice — open, closed, $F_{\sigma }$, $G_{\delta }$, and every set obtainable from them by countable Boolean operations — but the class is much larger than that hierarchy suggests, and its structure is a descriptive-set-theory question rather than an analytic one.

3. Measures

Definition (measure). A measure on $(X,\mathcal{A})$ is a function $\mu :\mathcal{A}\to [0,\infty ]$ with $\mu (\varnothing )=0$ that is countably additive: for pairwise disjoint $A_1,A_2,\dots \in \mathcal{A}$,

$$\mu \left(\underset{k=1}{\overset{\infty }{⨆}}{A}_k\right)=\sum _{k=1}^{\infty }\mu (A_k).$$

$(X,\mathcal{A},\mu )$ is a measure space. It is finite if $\mu (X)<>\infty$, $\sigma$-finite if $X={\bigcup }_k{X}_k$ with $\mu (X_k)<\infty$, and a probability space if $\mu (X)=1$.

Values in $[0,\infty ]$ are genuinely allowed; the arithmetic convention $0\cdot \infty =0$ is standard and makes the integral of §15 behave.

Proposition (elementary properties). For $A,B,A_k\in \mathcal{A}$:

• Monotonicity. $A\subseteq B⟹\mu (A)\le \mu (B)$.
• Countable subadditivity. $\mu \left({\bigcup }_k{A}_k\right)\le {\sum }_k\mu (A_k)$, without disjointness.
• Continuity from below. $A_1\subseteq A_2\subseteq \cdots ⟹>\mu \left({\bigcup }_k{A}_k\right)={\lim }_k\mu (A_k)$.
• Continuity from above. $A_1\supseteq A_2\supseteq \cdots$ and $\mu (A_1)<\infty$ $⟹\mu \left({\bigcap }_k{A}_k\right)={\lim }_k\mu (A_k)$.

Proof. Monotonicity: $\mu (B)=\mu (A)+\mu (B\setminus A)\ge \mu (A)$. Subadditivity: disjointify, $B_k=A_k\setminus (A_1\cup \cdots \cup A_{k-1})$, so $\bigcup A_k=⨆B_k$ with $B_k\subseteq A_k$. Continuity from below: apply countable additivity to the disjoint pieces $A_k\setminus A_{k-1}$ and recognize the partial sums as $\mu (A_k)$. Continuity from above: apply the previous item to $A_1\setminus A_k$. $\square$

The finiteness hypothesis in the last item is not decorative: on $\mathbb{R}$ with Lebesgue measure, $A_k=[k,\infty )$ has $\mu (A_k)=\infty$ for all $k$ while ${\bigcap }_k{A}_k=\varnothing$.

Definition (null set, a.e.). $N$ is $\mu$-null if $N\in \mathcal{A}$ and $\mu (N)=0$. A property holds almost everywhere (a.e.) if the set where it fails is contained in a null set.

This is the definition the Riemann theory borrowed on credit: the discontinuity set in Lebesgue's criterion (riemann-integral.md §3, multiple-integrals.md §2) is required to be null for ${\lambda }^n$, and the criterion is proved in lebesgue-integral.md §6.

A measure space is complete if every subset of a null set is measurable (and null). Countable additivity does not deliver this — a subset of a Borel null set need not be Borel — but every measure has a completion $\overline{\mathcal{A}}=\{A\cup Z:A\in \mathcal{A},Z\subseteq N\text{ for some null }N\}$, with $\bar{\mu }(A\cup Z)=\mu (A)$; the definition is forced and the check that it is well posed is routine.

4. Uniqueness: the $\pi$–$\lambda$ Theorem

Because $\sigma (\mathcal{E})$ has no constructive description, one cannot verify a claim about all its members by induction. Dynkin's theorem supplies the missing induction principle, and is used constantly (in probability, it is the tool for proving that two laws agree).

Definitions. $\mathcal{P}\subseteq \mathcal{P}(X)$ is a $\pi$-system if it is closed under finite intersection. $\mathcal{L}$ is a $\lambda$-system (Dynkin system) if $X\in \mathcal{L}$, it is closed under proper differences ($A\subseteq B$, both in $\mathcal{L}$ $⟹B\setminus A\in \mathcal{L}$), and closed under increasing countable unions.

Theorem (Dynkin $\pi$–$\lambda$). If a $\lambda$-system contains a $\pi$-system $\mathcal{P}$, it contains $\sigma (\mathcal{P})$.

Sketch. Let ${\mathcal{L}}_0$ be the smallest $\lambda$-system containing $\mathcal{P}$. One shows ${\mathcal{L}}_0$ is closed under intersection — first that $\{B:B\cap P\in {\mathcal{L}}_0\forall P\in \mathcal{P}\}$ is a $\lambda$-system containing $\mathcal{P}$, hence all of ${\mathcal{L}}_0$; then the same trick with $P$ replaced by an arbitrary element of ${\mathcal{L}}_0$. A $\lambda$-system closed under intersection is a $\sigma$-algebra, so ${\mathcal{L}}_0\supseteq \sigma (\mathcal{P})$. $\square$

Corollary (uniqueness of measures). Let $\mathcal{P}$ be a $\pi$-system with $\sigma (\mathcal{P})=\mathcal{A}$, and suppose $X={\bigcup }_k{P}_k$ with $P_k\in >\mathcal{P}$ and $\mu (P_k)=\nu (P_k)<\infty$. If $\mu =\nu$ on $\mathcal{P}$, then $\mu =\nu$ on $\mathcal{A}$.

Proof. For fixed $P\in \mathcal{P}$ of finite measure, $\{A:\mu (A\cap P)=\nu (A\cap P)\}$ is a $\lambda$-system containing $\mathcal{P}$; apply the theorem, then let $P\uparrow X$ using continuity from below. $\square$

So a measure on $\mathbb{R}$ is pinned down by its values on intervals, and a probability distribution by its CDF — the fact that makes distribution functions a legitimate substitute for measures in random-variables.md.

5. Construction: Outer Measures and Carathéodory

Uniqueness is settled; existence is the harder half. The standard route builds a crude, everywhere-defined but merely subadditive object, then carves out the sets on which it behaves.

Definition (outer measure). ${\mu }^{\ast }:\mathcal{P}(X)\to [0,\infty ]$ with ${\mu }^{\ast }(\varnothing )=0$, monotone, and countably subadditive.

Outer measures are easy to produce: given any family $\mathcal{E}$ covering $X$ and any $\rho :\mathcal{E}\to [0,\infty ]$,

$${\mu }^{\ast }(A)=\inf \{\sum _{k=1}^{\infty }\rho (E_k):A\subseteq \bigcup _k{E}_k,E_k\in \mathcal{E}\}$$

is one. The cost is that ${\mu }^{\ast }$ is not additive. Carathéodory's insight was that additivity can be imposed as a definition of which sets are measurable.

Definition (Carathéodory measurable). $A$ is ${\mu }^{\ast }$-measurable if for every test set $E\subseteq X$,

$${\mu }^{\ast }(E)={\mu }^{\ast }(E\cap A)+{\mu }^{\ast }(E\cap A^c).$$

$A$ is measurable iff it splits every set additively — "$\le$" always holds by subadditivity, so only "$\ge$" is a condition.

Theorem (Carathéodory). The ${\mu }^{\ast }$-measurable sets form a $\sigma$-algebra ${\mathcal{A}}^{\ast }$, and ${\mu }^{\ast }$ restricted to ${\mathcal{A}}^{\ast }$ is a complete measure.

Sketch. Complementation is symmetric in the definition; finite unions follow by applying the splitting condition twice with well-chosen test sets; countable additivity comes from applying the finite case to $E={\bigcup }_{k\le n}{A}_k$ and passing to the limit with subadditivity. Any set of outer measure zero splits everything trivially, giving completeness. $\square$

Theorem (Carathéodory extension). Let ${\mathcal{A}}_0$ be an algebra (closed under complement and finite union) and ${\mu }_0:{\mathcal{A}}_0\to >[0,\infty ]$ a premeasure ($\sigma$-additive whenever a countable disjoint union happens to lie in ${\mathcal{A}}_0$). Then ${\mu }_0$ extends to a measure on $\sigma ({\mathcal{A}}_0)$; if ${\mu }_0$ is $\sigma$-finite the extension is unique.

Existence is the outer measure generated by ${\mu }_0$; uniqueness is §4 applied to the $\pi$-system ${\mathcal{A}}_0$.

6. Lebesgue Measure on ${\mathbb{R}}^n$

Take $\mathcal{E}$ = half-open boxes ${\prod }_{i=1}^n(a_i,b_i]$ with $\rho (\text{box})={\prod }_i(b_i-a_i)$, the elementary volume already used for Jordan content (multiple-integrals.md §1). Finite disjoint unions of boxes form an algebra, $\rho$ is a premeasure on it (the only real work is countable additivity, which follows from compactness of closed boxes plus a Heine–Borel argument, metric-spaces.md §4), and Carathéodory delivers:

Definition/Theorem (Lebesgue measure). There is a unique complete, translation-invariant measure ${\lambda }^n$ on a $\sigma$-algebra ${\mathcal{L}}^n\supseteq \mathcal{B}({\mathbb{R}}^n)$ with ${\lambda }^n(\text{box})=>{\prod }_i(b_i-a_i)$. Its restriction to $\mathcal{B}({\mathbb{R}}^n)$ is the unique Borel measure with that property, and ${\mathcal{L}}^n$ is the completion of $\mathcal{B}({\mathbb{R}}^n)$.

Basic consequences:

• Points and countable sets are null: ${\lambda }^n(\{x\})=0$, so ${\lambda }^1(\mathbb{Q})=0$. Hence "$\mathbb{Q}$ is small" in measure while being topologically dense — the two notions of smallness are independent.
• Regularity. ${\lambda }^n(A)=\inf \{{\lambda }^n(U):U\supseteq A\text{ open}\}=\sup \{{\lambda }^n(K):K\subseteq A\text{ compact}\}$ for $A\in {\mathcal{L}}^n$. This is what lets one transfer topological arguments to measurable sets.
• Behaviour under affine maps. ${\lambda }^n(TA)=|\det T|{\lambda }^n(A)$ for linear $T$ — the measure-theoretic form of the Jacobian factor in the change-of-variables theorem (multiple-integrals.md §4).
• The Cantor set is uncountable, compact, and null: "measure zero" is strictly weaker than "countable". It is also nowhere dense, but there are fat Cantor sets which are nowhere dense with positive measure, so the two smallness notions dissociate in both directions.

Jordan vs. Lebesgue. Jordan content approximates from inside and outside with finitely many boxes; a set is Jordan measurable iff its topological boundary is null. $\mathbb{Q}\cap [0,1]$ has Jordan inner content $0$ and outer content $1$ — not Jordan measurable — but is Lebesgue-null. Every Jordan-measurable set is Lebesgue measurable with the same content, so nothing proved in multiple-integrals.md is disturbed; the theory is strictly extended, not revised.

7. Measurable Functions

Integration needs maps that pull measurable questions back to measurable sets.

Definition. $f:(X,\mathcal{A})\to (Y,\mathcal{B})$ is measurable if $f^{-1}(B)\in \mathcal{A}$ for all $B\in \mathcal{B}$. For real-valued $f$ we always take $\mathcal{B}=\mathcal{B}(\mathbb{R})$ (or $\mathcal{B}(\overline{\mathbb{R}})$ for extended values).

Proposition (generator criterion). If $\mathcal{B}=\sigma (\mathcal{E})$ it suffices that $f^{-1}(E)\in \mathcal{A}$ for $E\in \mathcal{E}$ — because $\{B:f^{-1}(B)\in \mathcal{A}\}$ is a $\sigma$-algebra.

Hence $f:X\to \mathbb{R}$ is measurable iff $\{f>a\}\in \mathcal{A}$ for all $a$, and every continuous function on a topological space is Borel measurable.

The class is closed under everything analysis does:

• $f+g$, $fg$, $cf$, $|f|$, $\max (f,g)$ are measurable;
• ${\sup }_k{f}_k$, ${\inf }_k{f}_k$, ${\mathrm{limsup}}_k{f}_k$, ${\mathrm{liminf}}_k{f}_k$ are measurable (this is exactly where countable unions are consumed: $\{{\sup }_k{f}_k>a\}={\bigcup }_k\{f_k>a\}$);
• consequently a pointwise limit of measurable functions is measurable.

That last item is the decisive advantage over the Riemann theory, where a pointwise limit of integrable functions need not be integrable (riemann-integral.md §7).

Theorem (simple approximation). For measurable $f\ge 0$ there are simple functions $0\le s_1\le s_2\le \cdots$ (finite-valued measurable, $s=>{\sum }_i{c}_i{1}_{A_i}$) with $s_k\uparrow f$ pointwise, uniformly on any set where $f$ is bounded.

Construction. $s_k=\min (k,2^{-k}\lfloor 2^{k}f\rfloor )$: chop the range into dyadic layers. That the layers are cut in the range rather than the domain is the one-sentence summary of the whole Lebesgue theory, and is the starting point of lebesgue-integral.md.

8. Where This Is Used

• The Lebesgue integral — $\int fd\mu$ built from §7's simple functions, with the convergence theorems.
• Product measures — $\mu \otimes \nu$ and Fubini, i.e. when iterated integrals may be exchanged.
• Non-measurable sets — the theorem promised in §1, and the role of the axiom of choice.
• Probability — a probability space is a measure space with $\mu (X)=1$; $\sigma$-algebras there carry the extra reading "information available".
• Functional analysis — the $L^p$ spaces, whose completeness (Riesz–Fischer) needs the convergence theorems.