Complex Measures & Total Variation
Two of the function spaces we have dualized return their own kind: the dual of a Hilbert space is again
a space of vectors through the
Riesz Representation
Theorem, and the dual of \(L^p\) is the conjugate space \(L^q\). The continuous functions
on a compact space behave differently. A continuous linear functional on \(C(X)\) cannot in general be
written as integration against an \(L^q\) density, because \(C(X)\) carries the supremum norm and its
dual must account for evaluation-like behavior concentrated on small sets. The objects that do the work
are measures: the dual of \(C(X)\) turns out to be a space of measures on \(X\), with
the functional acting by integration \(f \mapsto \int f \, d\mu\). The goal of this page is to make that
identification precise and to prove it.
Establishing this requires measures that may take complex values, since the functionals on \(C(X)\) are
complex-linear. We therefore begin by extending the notion of a signed measure to the complex setting and
by constructing the total variation, the device that measures the size of such an object and ultimately
supplies the dual norm. We work over a compact (or, where noted, locally compact) Hausdorff space \(X\),
and we write \(\Omega\) for a \(\sigma\)-algebra of subsets of \(X\).
Recall the real-valued notion: a
signed
measure on a measurable space \((X, \Omega)\) is a countably additive set function
\(\nu : \Omega \to [-\infty, \infty]\) with \(\nu(\varnothing) = 0\), assuming at most one of the values
\(\pm\infty\). The complex case removes the infinite values entirely, since the sum of a convergent
complex series must be finite.
Definition: Complex Measure
Let \((X, \Omega)\) be a measurable space. A complex measure on \((X, \Omega)\) is a
function \(\mu : \Omega \to \mathbb{C}\) that is countably additive: for every sequence
\((A_n)_{n \geq 1}\) of pairwise disjoint sets in \(\Omega\),
\[
\mu\!\left(\bigsqcup_{n=1}^\infty A_n\right) \;=\; \sum_{n=1}^\infty \mu(A_n),
\]
the series converging in \(\mathbb{C}\). Taking all \(A_n = \varnothing\) for \(n \geq 2\) forces
\(\mu(\varnothing) = 0\). A complex measure takes only finite values by definition.
Writing \(\mu = \operatorname{Re}\mu + i \operatorname{Im}\mu\), the real and imaginary parts
\((\operatorname{Re}\mu)(\Delta) = \operatorname{Re}\bigl(\mu(\Delta)\bigr)\) and
\((\operatorname{Im}\mu)(\Delta) = \operatorname{Im}\bigl(\mu(\Delta)\bigr)\) are each real-valued and
countably additive, hence finite signed measures. Applying the
Jordan
decomposition to each gives four positive finite measures \(\mu_1, \mu_2, \mu_3, \mu_4\) with
\[
\mu \;=\; (\mu_1 - \mu_2) + i(\mu_3 - \mu_4),
\]
where \(\mu_1 \perp \mu_2\) and \(\mu_3 \perp \mu_4\) are mutually singular. A real-valued signed measure
that happens to be finite is the special case \(\operatorname{Im}\mu = 0\); the development below therefore
subsumes the real theory, with the four-measure decomposition collapsing to the two-measure Jordan form.
The Total Variation
For a finite signed measure the
total
variation is defined through the Jordan decomposition as \(|\nu| = \nu^+ + \nu^-\). A complex
measure has no such order structure, so we take instead the supremum-over-partitions description, which
agrees with \(\nu^+ + \nu^-\) in the real case and extends verbatim to the complex one.
Definition: Total Variation of a Complex Measure
Let \(\mu\) be a complex measure on \((X, \Omega)\). The total variation of \(\mu\)
is the set function \(|\mu| : \Omega \to [0, \infty]\) defined by
\[
|\mu|(\Delta) \;=\; \sup\left\{\, \sum_{j=1}^{m} |\mu(E_j)| \;:\; \{E_j\}_{j=1}^{m}
\text{ is a measurable partition of } \Delta \,\right\},
\]
the supremum running over all finite partitions of \(\Delta\) into pairwise disjoint measurable sets.
For a finite signed measure \(\nu\), choosing the partition \(\{P \cap \Delta,\, N \cap \Delta\}\) given by
a Hahn decomposition \(X = P \sqcup N\) yields \(\sum_j |\nu(E_j)| = \nu^+(\Delta) + \nu^-(\Delta) =
|\nu|(\Delta)\). No partition can exceed this: for any measurable \(E_j\), the values \(\nu^+(E_j)\) and
\(\nu^-(E_j)\) are nonnegative, so
\[
|\nu(E_j)| \;=\; |\nu^+(E_j) - \nu^-(E_j)| \;\leq\; \nu^+(E_j) + \nu^-(E_j),
\]
and summing over any partition \(\{E_j\}\) of \(\Delta\) gives \(\sum_j |\nu(E_j)| \leq \nu^+(\Delta) +
\nu^-(\Delta)\) by additivity of \(\nu^+\) and \(\nu^-\). The supremum definition thus recovers
\(\nu^+ + \nu^-\), confirming the two notions coincide where both apply.
The central structural fact is that this supremum is itself a measure, and a finite one. The finiteness is
the substantive part: it is what makes the dual norm well defined.
Theorem: The Total Variation is a Finite Positive Measure
Let \(\mu\) be a complex measure on \((X, \Omega)\). Then \(|\mu|\) is a positive measure on
\((X, \Omega)\), and it is finite: \(|\mu|(X) < \infty\).
Proof
Countable additivity.
Let \(\Delta = \bigsqcup_{n=1}^\infty \Delta_n\) be a disjoint union of measurable sets. We show
\(|\mu|(\Delta) = \sum_n |\mu|(\Delta_n)\).
For the inequality \(|\mu|(\Delta) \leq \sum_n |\mu|(\Delta_n)\), let \(\{E_j\}_{j=1}^m\) be any finite
measurable partition of \(\Delta\). Each \(E_j = \bigsqcup_n (E_j \cap \Delta_n)\), so by countable
additivity of \(\mu\) and the triangle inequality,
\[
\sum_{j=1}^m |\mu(E_j)| \;=\; \sum_{j=1}^m \left| \sum_{n=1}^\infty \mu(E_j \cap \Delta_n) \right|
\;\leq\; \sum_{j=1}^m \sum_{n=1}^\infty |\mu(E_j \cap \Delta_n)|
\;=\; \sum_{n=1}^\infty \sum_{j=1}^m |\mu(E_j \cap \Delta_n)|.
\]
For each fixed \(n\), the sets \(\{E_j \cap \Delta_n\}_{j=1}^m\) form a measurable partition of
\(\Delta_n\), so \(\sum_{j=1}^m |\mu(E_j \cap \Delta_n)| \leq |\mu|(\Delta_n)\). Hence
\(\sum_{j=1}^m |\mu(E_j)| \leq \sum_{n=1}^\infty |\mu|(\Delta_n)\). Taking the supremum over all
partitions \(\{E_j\}\) of \(\Delta\) gives \(|\mu|(\Delta) \leq \sum_{n=1}^\infty |\mu|(\Delta_n)\).
For the reverse inequality \(|\mu|(\Delta) \geq \sum_n |\mu|(\Delta_n)\), fix \(N \geq 1\) and
\(\varepsilon > 0\). For each \(n \leq N\) choose a measurable partition \(\{E^{(n)}_j\}_j\) of
\(\Delta_n\) with \(\sum_j |\mu(E^{(n)}_j)| > |\mu|(\Delta_n) - \varepsilon / 2^n\). The collection
\(\{E^{(n)}_j : 1 \leq n \leq N,\, j\} \cup \{\Delta \setminus \bigsqcup_{n \leq N} \Delta_n\}\) is a
finite measurable partition of \(\Delta\), so
\[
|\mu|(\Delta) \;\geq\; \sum_{n=1}^N \sum_j |\mu(E^{(n)}_j)|
\;>\; \sum_{n=1}^N \left( |\mu|(\Delta_n) - \frac{\varepsilon}{2^n} \right)
\;>\; \sum_{n=1}^N |\mu|(\Delta_n) - \varepsilon.
\]
Letting \(\varepsilon \to 0\) gives \(|\mu|(\Delta) \geq \sum_{n=1}^N |\mu|(\Delta_n)\), and then
\(N \to \infty\) gives \(|\mu|(\Delta) \geq \sum_{n=1}^\infty |\mu|(\Delta_n)\). Combined with the first
inequality, \(|\mu|\) is countably additive. Since \(|\mu|(\varnothing) = 0\) and \(|\mu| \geq 0\), it
is a positive measure.
Finiteness.
Decompose \(\mu = (\mu_1 - \mu_2) + i(\mu_3 - \mu_4)\) into four positive finite measures as above. For
any measurable \(E\),
\[
|\mu(E)| \;=\; \bigl| (\mu_1 - \mu_2)(E) + i(\mu_3 - \mu_4)(E) \bigr|
\;\leq\; \mu_1(E) + \mu_2(E) + \mu_3(E) + \mu_4(E).
\]
Hence for any finite partition \(\{E_j\}\) of \(X\),
\[
\sum_j |\mu(E_j)| \;\leq\; \sum_j \sum_{k=1}^4 \mu_k(E_j) \;=\; \sum_{k=1}^4 \mu_k(X),
\]
the last equality by additivity of each \(\mu_k\). Taking the supremum over partitions,
\(|\mu|(X) \leq \mu_1(X) + \mu_2(X) + \mu_3(X) + \mu_4(X) < \infty\), since each \(\mu_k\) is finite.
The finiteness argument also shows \(|\mu(E)| \leq |\mu|(E)\) for every measurable \(E\), taking the
trivial partition \(\{E\}\) in the definition. The total variation therefore dominates \(\mu\) setwise
while being a genuine positive measure, and it assigns to all of \(X\) a finite number — the quantity that
will serve as the norm of \(\mu\).
Regular Borel Measures and M(X)
A complex measure as defined above lives on an abstract measurable space. To represent functionals on
\(C(X)\) we need measures that interact correctly with the topology of \(X\): the mass of a set must be
approximable by the mass of compact sets from inside and open sets from outside. These regularity
conditions tie the measure to the topology and are what make the representation unique. We now place the
measures on the natural \(\sigma\)-algebra and impose regularity.
Throughout, \(X\) is a locally compact Hausdorff space. The Borel \(\sigma\)-algebra
\(\Omega\) is the smallest \(\sigma\)-algebra of subsets of \(X\) containing all
open sets; its
members are the Borel sets. A compact subset and an open subset of \(X\) are Borel, so the
approximation conditions below are well posed.
Definition: Regular Borel Measure
Let \(X\) be a locally compact Hausdorff space with Borel \(\sigma\)-algebra \(\Omega\). A positive
measure \(\mu\) on \((X, \Omega)\) is a regular Borel measure if
(a) \(\mu(K) < \infty\) for every
compact
set \(K \subseteq X\);
(b) \(\mu(E) = \sup\{\, \mu(K) : K \subseteq E,\ K \text{ compact} \,\}\) for every
\(E \in \Omega\) (inner regularity);
(c) \(\mu(E) = \inf\{\, \mu(U) : U \supseteq E,\ U \text{ open} \,\}\) for every
\(E \in \Omega\) (outer regularity).
A complex measure \(\mu\) on \((X, \Omega)\) is a regular Borel measure if its total
variation \(|\mu|\) is a regular Borel measure in the sense above.
When \(X\) is compact, condition (a) is automatic, since \(\mu(X) < \infty\) for any finite measure and
every compact set has no greater measure. Conditions (b) and (c) remain the substance: they say a Borel
set is squeezed between the compact sets it contains and the open sets that contain it. This is the case
relevant to \(C(X)\) on a compact \(X\); the locally compact formulation is stated as primary because the
representation theorem and its corollary on \(C_0(X)\) hold at that generality, with the compact case as
the specialization \(C_0(X) = C(X)\).
The Space M(X)
The regular Borel measures of finite total variation form a vector space, on which the total variation of
the whole space supplies a norm.
Definition: The Space \(M(X)\)
Let \(X\) be a locally compact Hausdorff space. Denote by \(M(X)\) the set of all complex-valued
regular Borel measures on \(X\). It is a vector space over \(\mathbb{C}\) under the pointwise
operations \((\mu + \nu)(E) = \mu(E) + \nu(E)\) and \((\lambda\mu)(E) = \lambda\,\mu(E)\). For
\(\mu \in M(X)\), set
\[
\|\mu\| \;=\; |\mu|(X),
\]
the total variation of \(\mu\) over all of \(X\), which is finite by the preceding section.
That the pointwise operations preserve regularity, and that \(\|\cdot\|\) is a norm, must be checked. We
record both as a single proposition.
Proposition: \(M(X)\) is a Normed Space
\(M(X)\) is a complex vector space, and \(\|\mu\| = |\mu|(X)\) defines a norm on it.
Proof
Vector-space structure.
If \(\mu, \nu \in M(X)\) and \(\lambda \in \mathbb{C}\), then \(\mu + \nu\) and \(\lambda\mu\) are
complex measures, being pointwise combinations of countably additive set functions. We must check they
remain regular, i.e. that \(|\mu + \nu|\) and \(|\lambda\mu|\) are regular Borel measures. From the
partition definition, for any \(E\) and any partition \(\{E_j\}\),
\[
\sum_j |(\mu + \nu)(E_j)| \;\leq\; \sum_j |\mu(E_j)| + \sum_j |\nu(E_j)|
\;\leq\; |\mu|(E) + |\nu|(E),
\]
so taking the supremum gives \(|\mu + \nu|(E) \leq |\mu|(E) + |\nu|(E)\); that is,
\(|\mu + \nu| \leq |\mu| + |\nu|\) setwise. Likewise \(|\lambda\mu| = |\lambda|\,|\mu|\), since each
term \(|\lambda\mu(E_j)| = |\lambda|\,|\mu(E_j)|\) scales by \(|\lambda|\).
It remains to deduce that \(|\mu + \nu|\) is regular; the same argument applies to
\(|\lambda\mu| = |\lambda|\,|\mu|\), which is regular because a nonnegative scalar multiple of a
regular measure is regular. Write \(\rho = |\mu + \nu|\) and \(\sigma = |\mu| + |\nu|\), so that
\(\rho \leq \sigma\) setwise. Both are finite positive measures, and \(\sigma\) is regular, being a
sum of two regular measures (approximate each summand separately and add). The key elementary fact is
this: because \(\rho\) and \(\sigma\) are finite and \(\rho \leq \sigma\), for any two Borel sets
\(A \subseteq B\) we have
\[
\rho(B) - \rho(A) \;=\; \rho(B \setminus A) \;\leq\; \sigma(B \setminus A),
\]
so whenever \(\sigma\) approximates a set well, \(\rho\) does too.
Inner regularity. Fix \(E \in \Omega\) and \(\varepsilon > 0\). By inner regularity of
\(\sigma\) there is a compact \(K \subseteq E\) with \(\sigma(E \setminus K) < \varepsilon\). Applying
the fact above with \(A = K\), \(B = E\),
\[
\rho(E) - \rho(K) \;\leq\; \sigma(E \setminus K) \;<\; \varepsilon,
\]
so \(\rho(K)\) comes within \(\varepsilon\) of \(\rho(E)\). Since \(\varepsilon\) was arbitrary,
\(\rho(E) = \sup\{\rho(K) : K \subseteq E,\ K \text{ compact}\}\).
Outer regularity. Similarly, by outer regularity of \(\sigma\) there is an open
\(U \supseteq E\) with \(\sigma(U \setminus E) < \varepsilon\), and the fact above with \(A = E\),
\(B = U\) gives \(\rho(U) - \rho(E) \leq \sigma(U \setminus E) < \varepsilon\). Hence
\(\rho(E) = \inf\{\rho(U) : U \supseteq E,\ U \text{ open}\}\). Thus \(\rho = |\mu + \nu|\) is regular,
and \(\mu + \nu, \lambda\mu \in M(X)\).
Positivity and definiteness.
Clearly \(\|\mu\| = |\mu|(X) \geq 0\). If \(\|\mu\| = 0\), then \(|\mu|(X) = 0\), so \(|\mu|(E) = 0\)
for every \(E \subseteq X\) by monotonicity; since \(|\mu(E)| \leq |\mu|(E) = 0\), we get
\(\mu(E) = 0\) for all \(E\), i.e. \(\mu = 0\). Conversely \(\|0\| = 0\).
Homogeneity.
\(\|\lambda\mu\| = |\lambda\mu|(X) = |\lambda|\,|\mu|(X) = |\lambda|\,\|\mu\|\), using
\(|\lambda\mu| = |\lambda|\,|\mu|\) from above.
Triangle inequality.
\(\|\mu + \nu\| = |\mu + \nu|(X) \leq |\mu|(X) + |\nu|(X) = \|\mu\| + \|\nu\|\), using
\(|\mu + \nu| \leq |\mu| + |\nu|\) from above. Thus \(\|\cdot\|\) is a norm.
Support and Point Masses
Two further notions attached to a measure will be needed when the representation is applied: the set on
which a measure genuinely lives, and the simplest measures of all, concentrated at a single point.
Definition: Support of a Measure
Let \(\mu\) be a complex measure on a locally compact Hausdorff space \(X\) with Borel
\(\sigma\)-algebra \(\Omega\). The support of \(\mu\) is the set
\[
\operatorname{supp}\mu \;=\; X \setminus \bigcup\{\, V : V \text{ is open and } |\mu|(V) = 0 \,\}.
\]
Equivalently, \(\operatorname{supp}\mu\) is the smallest closed set whose complement is
\(|\mu|\)-null: it is the set of points every open neighborhood of which carries positive total
variation.
The complement of \(\operatorname{supp}\mu\) is a union of \(|\mu|\)-null open sets, hence open, so
\(\operatorname{supp}\mu\) is closed. Moreover this null complement is itself \(|\mu|\)-null, and crucially
no countability of \(X\) is needed: writing \(W = X \setminus \operatorname{supp}\mu\), every compact
\(C \subseteq W\) is covered by finitely many of the null open sets whose union is \(W\), so
\(|\mu|(C) = 0\); inner regularity of \(|\mu|\) then gives
\(|\mu|(W) = \sup\{|\mu|(C) : C \subseteq W \text{ compact}\} = 0\). Consequently
\(\int_X f \, d\mu = \int_{\operatorname{supp}\mu} f \, d\mu\) for every integrable \(f\), since the
integrand contributes nothing off the support. This is exactly the step that lets one replace \(X\) by the
support of a measure without losing any of its mass.
Definition: Dirac Measure
For a point \(x \in X\), the Dirac measure (or point mass)
\(\delta_x\) is the positive measure defined by
\[
\delta_x(E) \;=\; \begin{cases} 1 & x \in E, \\ 0 & x \notin E, \end{cases}
\]
for \(E \in \Omega\). It is a regular Borel measure with \(\|\delta_x\| = 1\) and
\(\operatorname{supp}\delta_x = \{x\}\), and integration against it is evaluation:
\(\int_X f \, d\delta_x = f(x)\) for every \(f \in C(X)\). A scalar multiple \(\alpha\delta_x\) with
\(\alpha \in \mathbb{C}\) is the complex measure \(E \mapsto \alpha\,\delta_x(E)\), with
\(\|\alpha\delta_x\| = |\alpha|\).
Countable additivity of \(\delta_x\) holds because at most one set in a disjoint family contains \(x\);
regularity holds because \(\{x\}\) is compact and the value on any Borel set is determined by whether it
contains \(x\). These point masses are the extreme points of the unit ball of \(M(X)\), a fact that
surfaces once the representation identifies \(M(X)\) with a dual space.
From Functionals to Measures
We now connect the two sides. One direction is easy and explicit: every measure produces a functional by
integration. The other direction — that every functional arises this way — is the substance of the
representation theorem and occupies the rest of this section. Throughout, \(C_0(X)\) is the space of
continuous functions \(f : X \to \mathbb{C}\) that vanish at infinity, meaning that for
each \(\varepsilon > 0\) the set \(\{|f| \geq \varepsilon\}\) is compact, equipped with the supremum norm
\(\|f\|_\infty = \sup_x |f(x)|\). When \(X\) is compact this is simply \(C(X)\), since every continuous
function then has compact support automatically.
Each Measure Defines a Functional
Lemma: Integration Against a Measure is a Functional
Let \(\mu \in M(X)\) and define \(F_\mu : C_0(X) \to \mathbb{C}\) by \(F_\mu(f) = \int f \, d\mu\). Then
\(F_\mu\) is a bounded linear functional, \(F_\mu \in C_0(X)^*\), and
\[
\|F_\mu\| \;=\; \|\mu\| \;=\; |\mu|(X).
\]
Proof Sketch
Boundedness and the upper estimate (full).
Linearity of \(F_\mu\) is the linearity of the integral. For \(f \in C_0(X)\), the bound
\(|f(x)| \leq \|f\|_\infty\) and the inequality \(\left| \int f \, d\mu \right| \leq \int |f| \, d|\mu|\)
— which holds because, decomposing into real and imaginary signed parts and then into positive and
negative pieces, the integral against \(\mu\) is dominated termwise by the integral of \(|f|\) against
\(|\mu|\) — give
\[
|F_\mu(f)| \;=\; \left| \int f \, d\mu \right| \;\leq\; \int |f| \, d|\mu|
\;\leq\; \|f\|_\infty \, |\mu|(X) \;=\; \|f\|_\infty \, \|\mu\|.
\]
Hence \(F_\mu\) is bounded with \(\|F_\mu\| \leq \|\mu\|\).
The lower estimate (sketch — it rests on the approximation principle stated after
the proof).
For the reverse inequality \(\|F_\mu\| \geq \|\mu\|\) we must produce functions \(f\) with
\(\|f\|_\infty \leq 1\) and \(|F_\mu(f)|\) close to \(|\mu|(X)\). The polar decomposition of \(\mu\)
writes \(\mu = h \, |\mu|\) for a Borel function \(h\) with \(|h| = 1\) almost everywhere with respect to
\(|\mu|\); this is the measure-theoretic statement that \(\mu\) and \(|\mu|\) differ only by a unimodular
phase. The natural candidate is \(f = \bar{h}\), for which
\[
\int \bar{h} \, d\mu \;=\; \int \bar{h} \, h \, d|\mu| \;=\; \int |h|^2 \, d|\mu| \;=\; |\mu|(X),
\]
but \(\bar{h}\) is only Borel, not continuous, and need not lie in \(C_0(X)\). The remedy is to
approximate \(\bar{h}\) by a continuous function. There is a classical approximation principle: a Borel
function that is bounded by \(1\) can, off a set of arbitrarily small \(|\mu|\)-measure, be matched by a
continuous function of supremum norm at most \(1\). Granting such an approximant \(\phi\) with
\(\|\phi\|_\infty \leq 1\) and \(\int |\phi - \bar{h}| \, d|\mu| < \varepsilon\), we get
\[
|F_\mu(\phi)| \;=\; \left| \int \phi \, d\mu \right|
\;\geq\; \left| \int \bar{h} \, d\mu \right| - \left| \int (\phi - \bar{h}) \, d\mu \right|
\;\geq\; |\mu|(X) - \int |\phi - \bar{h}| \, d|\mu| \;>\; |\mu|(X) - \varepsilon.
\]
Since \(\|\phi\|_\infty \leq 1\), this forces \(\|F_\mu\| \geq |\mu|(X) - \varepsilon\), and letting
\(\varepsilon \to 0\) gives \(\|F_\mu\| \geq \|\mu\| = |\mu|(X)\). With the upper estimate,
\(\|F_\mu\| = \|\mu\|\).
The approximation principle used here expresses the regularity of \(|\mu|\) at the level of functions: just
as a Borel set is approximable by compact and open sets, a bounded Borel function is approximable, in the
mean, by a continuous one of the same supremum bound. We use it as a stated principle; its proof builds the
continuous approximant on the compact sets supplied by inner regularity.
The map \(\mu \mapsto F_\mu\) is therefore a norm-preserving linear injection of \(M(X)\) into
\(C_0(X)^*\): linear because the integral is linear in the measure, injective because
\(\|F_\mu\| = \|\mu\|\) forces \(F_\mu = 0 \Rightarrow \mu = 0\). Surjectivity — recovering a measure from an
arbitrary functional — we approach in two steps: first reduce to a positive functional, then realize
a positive functional by a positive measure.
Reduction to a Positive Functional
Call a linear functional \(I : C_0(X) \to \mathbb{C}\) positive if \(I(f) \geq 0\)
whenever \(f \geq 0\). Positivity is the order-theoretic analogue of being represented by a positive
measure, and it is the form in which a measure is easiest to extract. The first step shows every bounded
functional dominates a positive one of the same norm.
Lemma: A Bounded Functional Yields a Positive Functional
Let \(F : C_0(X) \to \mathbb{C}\) be a bounded linear functional. For \(f \in C_0(X)\) with
\(f \geq 0\), define
\[
I(f) \;=\; \sup\{\, |F(g)| : g \in C_0(X),\ |g| \leq f \,\}.
\]
Then \(I\) extends to a positive linear functional on \(C_0(X)\) with \(\|I\| = \|F\|\) and
\(|F(g)| \leq I(|g|)\) for all \(g\).
Proof Sketch
For \(f \geq 0\), the defining supremum is finite because \(|F(g)| \leq \|F\| \, \|g\|_\infty \leq
\|F\| \, \|f\|_\infty\) whenever \(|g| \leq f\), so \(0 \leq I(f) \leq \|F\| \, \|f\|_\infty\). Taking
\(g = f\) gives \(I(f) \geq |F(f)|\), and the case \(g\) ranging over multiples of \(f\) gives
\(I(f) \geq \|F\|\,\|f\|_\infty\) in the limit, so in fact \(\|I\| = \|F\|\). One checks additivity on
nonnegative functions: \(I(f_1 + f_2) = I(f_1) + I(f_2)\) for \(f_1, f_2 \geq 0\), the inequality
\(\geq\) by combining near-optimal \(g_1, g_2\) into \(g_1 + g_2\) after adjusting phases so that
\(|F(g_1)| + |F(g_2)| = |F(g_1 + g_2)|\), and \(\leq\) by splitting any \(g\) with \(|g| \leq f_1 + f_2\)
as \(g = g_1 + g_2\) with \(|g_i| \leq f_i\) (set \(g_i = g f_i / (f_1 + f_2)\) where the denominator is
positive, \(0\) elsewhere). Positive homogeneity \(I(t f) = t I(f)\) for \(t \geq 0\) is immediate from
the definition. An additive, positively homogeneous functional on the nonnegative cone extends uniquely
to a linear functional on \(C_0(X)\) by writing a real function as a difference of its positive and
negative parts and a complex function through real and imaginary parts; the extension is positive by
construction and satisfies \(|F(g)| \leq I(|g|)\) since \(|g| \leq |g|\) places \(g\) among the
competitors defining \(I(|g|)\).
A Positive Functional Comes From a Measure
The second step is the heart of the representation: a positive linear functional on \(C_0(X)\) is
integration against a positive regular Borel measure. This is the classically named Riesz representation
for positive functionals; its full construction of the measure is long, and we give the core idea.
Theorem: Positive Functionals are Positive Measures
Let \(X\) be a locally compact Hausdorff space and let \(I : C_0(X) \to \mathbb{C}\) be a positive
bounded linear functional. Then there is a unique positive regular Borel measure \(\nu\) on \(X\) such
that
\[
I(f) \;=\; \int f \, d\nu \qquad \text{for every } f \in C_0(X),
\]
and \(\|I\| = \nu(X)\).
Proof Sketch
The measure is built outside-in, starting from open sets. For an open \(U \subseteq X\), define
\[
\nu(U) \;=\; \sup\{\, I(\phi) : \phi \in C_0(X),\ 0 \leq \phi \leq 1,\ \operatorname{supp}\phi
\subseteq U \,\},
\]
the supremum of the functional over continuous "bump" functions trapped inside \(U\). This is monotone
in \(U\) and, using that a bump subordinate to a union can be split into bumps subordinate to the
pieces, countably subadditive. For an arbitrary set \(E\) one then sets
\[
\nu(E) \;=\; \inf\{\, \nu(U) : U \supseteq E,\ U \text{ open} \,\},
\]
an outer measure by construction. The technical core is to show that this outer measure is countably
additive on the Borel sets — the Carathéodory measurability of open sets — which is where the bulk of
the work lies and where the existence of continuous bumps separating compact sets from closed sets
(a separation property of locally compact Hausdorff spaces) is used repeatedly. Outer regularity holds
by definition; inner regularity follows because the bumps witnessing \(\nu(U)\) have compact support.
Once \(\nu\) is a measure, the identity \(I(f) = \int f \, d\nu\) is verified first for \(0 \leq f
\leq 1\) by sandwiching \(f\) between bumps adapted to the level sets \(\{f > t\}\) and integrating in
\(t\), then for general \(f\) by linearity. Uniqueness follows because two regular Borel measures
agreeing as functionals agree on open sets through the bump-supremum formula, hence on all Borel sets by
outer regularity. Finally \(\|I\| = \nu(X)\): the bound \(|I(f)| \leq \nu(X)\|f\|_\infty\) comes from
\(I(f) = \int f \, d\nu\), and equality is approached by bumps increasing to \(1\).
The two steps compose: an arbitrary bounded functional \(F\) yields a positive functional \(I\) with
\(|F(g)| \leq I(|g|)\), and \(I\) is integration against a positive measure \(\nu\). The remaining task is to
recover \(F\) itself — not just its positive envelope — as integration against a complex measure built from
\(\nu\). Assembling this is the content of the representation theorem in the next section.
The Riesz Representation Theorem
Everything is now in place. The previous section produced, from a bounded functional \(F\), a positive
functional \(I\) dominating it, and realized \(I\) as integration against a positive measure \(\nu\). We
complete the circle by recovering \(F\) itself as integration against a complex measure, and we record the
resulting identification of \(C_0(X)^*\) with \(M(X)\).
Theorem: Riesz Representation for \(C_0(X)\)
Let \(X\) be a locally compact Hausdorff space. For \(\mu \in M(X)\), let \(F_\mu \in C_0(X)^*\) be the
functional \(F_\mu(f) = \int f \, d\mu\). Then the map
\[
M(X) \longrightarrow C_0(X)^*, \qquad \mu \longmapsto F_\mu,
\]
is an isometric isomorphism: it is linear, \(\|F_\mu\| = \|\mu\|\) for every \(\mu\), and every
bounded linear functional on \(C_0(X)\) equals \(F_\mu\) for a unique \(\mu \in M(X)\). In words, the
dual of \(C_0(X)\) is the space of measures \(M(X)\).
Proof
Linearity of \(\mu \mapsto F_\mu\) and the isometry \(\|F_\mu\| = \|\mu\|\) were established earlier,
and the isometry makes the map injective: \(F_\mu = 0\) forces \(\|\mu\| = 0\), hence \(\mu = 0\). Only
surjectivity remains: given \(F \in C_0(X)^*\), we construct \(\mu \in M(X)\) with \(F = F_\mu\).
Form the positive functional \(I\) dominating \(F\), and let \(\nu\) be the positive regular Borel
measure with \(I(g) = \int g \, d\nu\). The domination \(|F(g)| \leq I(|g|) = \int |g| \, d\nu\) says
that \(F\), viewed on \(C_0(X)\) sitting inside \(L^1(\nu)\), is bounded for the \(L^1(\nu)\)-norm:
\[
|F(g)| \;\leq\; \int |g| \, d\nu \;=\; \|g\|_{L^1(\nu)}.
\]
Since \(C_0(X)\) is dense in \(L^1(\nu)\) — continuous functions of compact support approximate
\(L^1\) functions in mean, a consequence of the regularity of \(\nu\) — the functional \(F\) extends
uniquely to a bounded linear functional on \(L^1(\nu)\) of norm at most \(1\). The dual of \(L^1\) is
\(L^\infty\) (the duality of \(L^p\) spaces in the endpoint case \(p = 1\), treated on the page on
dual spaces), so that extension is given by a function \(\phi \in
L^\infty(\nu)\) with \(\|\phi\|_\infty \leq 1\):
\[
F(g) \;=\; \int g \, \phi \, d\nu \qquad (g \in C_0(X)).
\]
Now define \(\mu\) by \(\mu(E) = \int_E \phi \, d\nu\), the measure with density \(\phi\) against
\(\nu\). It is a complex measure, and its total variation is \(|\mu|(E) = \int_E |\phi| \, d\nu\), so
\(|\mu| \leq \nu\); since \(\nu\) is a finite regular Borel measure, so is the dominated \(|\mu|\),
whence \(\mu \in M(X)\). For every \(g \in C_0(X)\),
\[
F_\mu(g) \;=\; \int g \, d\mu \;=\; \int g \, \phi \, d\nu \;=\; F(g),
\]
the middle equality being the change of variables for a density. Thus \(F = F_\mu\). Uniqueness is the
injectivity already shown. Therefore \(\mu \mapsto F_\mu\) is an isometric isomorphism of \(M(X)\) onto
\(C_0(X)^*\).
The harder direction was routed through the duality of \(L^p\) at \(p = 1\): the regularity of \(\nu\) makes
\(C_0(X)\) dense in \(L^1(\nu)\), and that density is what lets a density \(\phi\) represent \(F\). Regularity
is therefore essential to the construction, not cosmetic.
The Compact Case and Annihilators
When \(X\) is compact, \(C_0(X) = C(X)\) and the theorem reads \(C(X)^* = M(X)\): every continuous linear
functional on \(C(X)\) is integration against a unique complex regular Borel measure, with the dual norm
equal to the total variation. This is the form used when \(C(X)\) is the ambient algebra. In particular, if
\(\mathcal{A} \subseteq C(X)\) is a closed subspace, its
annihilator
\[
\mathcal{A}^\perp \;=\; \{\, \varphi \in C(X)^* : \varphi(f) = 0 \text{ for all } f \in \mathcal{A} \,\}
\]
becomes, under the identification, the space of measures that integrate to zero against every member of
\(\mathcal{A}\):
\[
\mathcal{A}^\perp \;=\; \Bigl\{\, \mu \in M(X) : \int f \, d\mu = 0 \text{ for all } f \in \mathcal{A}
\,\Bigr\}.
\]
An element of \(\mathcal{A}^\perp\) is now a concrete object — a measure on \(X\) — to which the full
machinery of support, total variation, and integration applies. This is precisely what turns an abstract
functional-analytic hypothesis about \(\mathcal{A}\) into a question about measures one can localize and
manipulate.
The extreme points of the unit ball of \(M(X)\) make the same identification vivid. For a compact \(X\),
those extreme points are exactly the unimodular point masses \(\alpha\delta_x\) with \(|\alpha| = 1\) and
\(x \in X\): among all measures of total variation \(1\), the indecomposable ones are concentrated at a
single point. A measure that lives on more than one point is an average of measures supported on smaller
sets and so cannot be extreme, while a point mass admits no such splitting. Reading this through the Riesz
isomorphism, the extreme points of the dual ball of \(C(X)\) are the scaled evaluations \(f \mapsto \alpha
f(x)\) — the functionals that read off a single value. The structure of the dual ball is thereby reduced to
the points of \(X\) themselves, the starting point for extracting a distinguished measure from an extremal
functional and the bridge by which compactness arguments on the dual ball return statements about
\(C(X)\).