




















































CUBO A Mathematical Journal
Vol.20, No¯ 02, (01–12). June 2018

http: // dx. doi. org/ 10. 4067/ S0719-06462018000200001

An approach to F. Riesz representation Theorem

Rafael del Rio, Asaf L. Franco and Jose A. Lara

Departamento de F́ısica Matemática,

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas,

Universidad Nacional Autónoma de México

C.P. 04510, CDMX, México.

delrio@iimas.unam.mx, asaflevif@hotmail.com, nekrotzar.ligeti@gmail.com

ABSTRACT

In this note we give a direct proof of the F. Riesz representation theorem which charac-

terizes the linear functionals acting on the vector space of continuous functions defined

on a set K. Our start point is the original formulation of Riesz where K is a closed in-

terval. Using elementary measure theory, we give a proof for the case K is an arbitrary

compact set of real numbers. Our proof avoids complicated arguments commonly used

in the description of such functionals.

RESUMEN

En esta nota, damos una demostración directa del teorema de representación de F. Riesz

que caracteriza los funcionales lineales actuando en el espacio vectorial de funciones

continuas definidas en un conjunto K. Nuestro punto de partida es la formulación

original de Riesz, donde K es un intervalo cerrado. Usando teoŕıa elemental de la

medida, damos una demostración para el caso en que K es un conjunto arbitrario

compacto de números reales. Nuestra demostración evita argumentos complicados

comúnmente usados en la descripción de dichos funcionales.

Keywords and Phrases: Riesz representation theorem, positive linear functionals, Riemann-

Stieltjes integral.

2010 AMS Mathematics Subject Classification: 46E11,46E15,28A25.

http://dx.doi.org/10.4067/S0719-06462018000200001


2 Rafael del Rio, Asaf L. Franco and Jose A. Lara CUBO
20, 2 (2018)

1 Introduction

The Riesz representation theorem is a remarkable result which describes the continuous linear

functionals acting on the space of continuous functions defined on a set K. It is very surprising

that all these functionals are just integrals and vice versa. In case K is a closed interval of real

numbers, any such functional is represented by Riemann-Stieltjes integral, which is a generalization

of the usual Riemann integral. This was first announced by F. Riesz in 1909 [14]. In case K is

compact set (not necessarily a closed interval), then a more general concept of integral is needed,

because the Riemann-Stieltjes integral used by Riesz is defined only for functions on intervals. In

this work, we prove that there is a short path between the two cases.

Besides its aesthetic appeal, the above mentioned theorem has far-reaching applications. It

allows a short proof of the Kolmogoroff consistency theorem, see [3] thm 10.6.2., and can be used

to give an elegant proof of the spectral theorem for selfadjoint bounded operators, see section VII.2

of [13]. Both these theorems are main results in probability and functional analysis respectively.

Moreover, the entire theory of integration for general spaces can be recovered using the theorem

of Riesz. See for example [19], where the Lebesgue measure on Rn is constructed. More generally

it can also be used to show the existence of the Haar measure on a group, see [3] chap. 9.

In this note we give a short proof of the Riesz representation theorem for the case K is an

arbitrary compact set of real numbers, see Theorem 3.1 below. This is interesting because in many

situations we have a compact set which is not a closed interval. To prove the spectral theorem,

for example, one considers the set of continuous functions defined on the spectrum of selfadjoint

bounded operator, which is a compact set of R, but not necessarily a closed interval. We get

our result starting from the nondecreasing function that appears in the Riemann-Stieltjes integral

representation of Riesz original formulation. To this function we associate a measure which is used

to integrate over general compact sets. Then we show how this Lebesgue integral representation

can be seen as a Riemann-Stieltjes integral. Our proof is new, avoids technical arguments which

appear frequently in proofs of Riesz theorem, it is elementary, direct and quite simple.

2 Preliminaries

Let us introduce first some definitions and notations we shall use.

2.1 Definitions and notation.

Let C(K) := {f : K → R : f continuous} where K is a compact subset of R, the real numbers. A
functional is an assignment L : C(K) → R. The functional is linear if L(c1f+c2g) = c1L(f)+c2L(g)
for all f, g ∈ C(K), c1, c2 ∈ R. It is continuous if there exists a fixed M > 0 such that |Lf| ≤ M‖f‖∞



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An approach to F. Riesz representation Theorem 3

for all f ∈ C(K), where ‖ · ‖
∞

denotes the uniform norm, that is, ‖f‖
∞

= sup{|f(x)| : x ∈ K}. We

define the norm of such functional as

‖L‖C(K) = ‖L‖ = sup{|L(f)| : f ∈ C(K) and ‖f‖∞ ≤ 1},

We denote the set of the linear continuous functionals on C(K) by C(K)∗. It is called the dual

space. In general, the dual of normed linear space X is denoted by X∗. A functional L on C(K) is

said to be a positive if L(f) ≥ 0 whenever f(x) ≥ 0 for every x ∈ R. We use the notation C(K)∗+ for

the set of positive linear functionals on C(K).

The function α : [a, b] −→ R is said to be normalized, if α(a) = 0 and α(t) = α(t+),
a < t < b, that is, α is continuous from the right inside the interval (not at a! If it were right

continuous at a, theorem (2.1) would not hold for the functional L(f) = f(a)). The total variation

of a monotone increasing function α is defined as V(α) = α(b)−α(a). We denote the characteristic

function of a set A ⊂ [a, b] by 1A where 1A(x) = 1 if x ∈ A and 0 if x ∈ [a, b] \ A.

2.2 Representation theorem for functionals on C[a, b].

We formulate the above-mentioned result by F. Riesz as follows:

Theorem 2.1. Let L : C[a, b] −→ R be a positive linear functional. There exists a unique normal-
ized monotone function α : [a, b] −→ R such that

Lf =

∫b

a

f(x)dα(x). (2.1)

The integral is understood in the sense of Riemann-Stieltjes. Moreover ‖L‖ = V(α).

The Riemann-Stieltjes integral is a generalization of the Riemann integral, where instead of

taking the length of the intervals, a α-weighted length is taken. For an interval I the α-length

is given by α(I) = α(y) − α(x), where x, y are the end points of I and α is a function of finite

variation. The integral of a continuous function f on [a, b] is defined as the limit, when it exists,

of the sum
∑

i
f(ci)α(Ii) where {Ii} is a finite collection of subintervals whose endpoints form a

partition of [a, b] and ci ∈ Ii. See [17] p.122.

There are different proofs of the above theorem, see for example [22]. Here we will give a

sketch of the proof which uses the following result about extensions of functionals known as the

Hahn-Banach theorem:

Let X a normed linear space, Y a subspace of X, and λ an element of Y∗. Then there exists a

Λ ∈ X∗ extending λ with the same norm. See [13] for a proof.

Proof of theorem 2.1. . We may assume that [a, b] = [0, 1]. Since L ∈ C[0, 1]∗ we use Hahn-Banach

theorem to conclude the existence of Λ ∈ B[0, 1]∗ such that ‖Λ‖ = ‖L‖ and L = Λ on C[0, 1] and



4 Rafael del Rio, Asaf L. Franco and Jose A. Lara CUBO
20, 2 (2018)

where B[0, 1] is the set of bounded functions on [0, 1].

Let us define the functions 1x := 1[0,x], that is 1x(t) = 1 when t ∈ [0, x] and zero otherwise.

Set α(x) = Λ(1x) for all x ∈ [0, 1].

Now for f ∈ C[0, 1], define

fn =

n∑

j=1

f(j/n)(1j/n − 1(j−1)/n).

Since f is continuous, it is uniformly continuous on [0, 1] and so ‖fn − f‖∞ → 0. Thus

lim
n

Λ(fn) = Λ(f) = L(f).

Using the definition of α we get

Λ(fn) =

n∑

j=1

f(j/n)(α(j/n) − α((j − 1)/n)).

This in turn implies

Λ(f) = lim
n

Λ(fn) =

∫1

0

f dα.

Now to see that ‖L‖ = V(α) :

Let ε > 0 and choose f ∈ C[0, 1] such that ‖f‖
∞

≤ 1 and ‖L‖ ≤ |L(f)| + ε, we apply (2.1) and

we get

‖L‖ ≤ |L(f)| + ε =

∣

∣

∣

∣

∣

∫1

0

f(x)dα(x)

∣

∣

∣

∣

∣

+ ε ≤ α(1) − α(0) + ε = V(α) + ε.

It is possible to normalize α and in this case we easily have the other inequality, that is,

V(α) = α(1) − α(0) = α(1) = Λ(11) ≤ ‖Λ‖ = ‖L‖.

Remarks.

(1) The standard textbook’s proof uses Hahn-Banach’s theorem ([10],[22]), but the original proof

of F. Riesz does not use it. See [17] section 50 and [15],[16].



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An approach to F. Riesz representation Theorem 5

(2) E. Helly [8] should have similar results. J. Radon extended theorem 2.1 to compact subsets

K ⊂ Rn [12]. S. Banach and S. Saks extended the result to compact metric spaces, see

appendix of [21] and [20]. The proof by S. Saks is particularly elegant and clean. For

compact Hausdorff spaces the theorem was proven by S. Kakutani [9] and for normal spaces

by A. Markoff [11]. Nowadays this theorem is also known as Riesz-Markoff or Riesz-Markoff-

Kakutani theorem. There is a great variety of proofs of F. Riesz theorem using different

methods and even category theory see [7]. Our proof only uses basic knowledge of measure

theory. More information on the history of this theorem can be found in [5] p. 231, the

references therein, [23] p.238 and [6].

(3) Positivity of a linear functional L implies continuity of L. To see it, we take the function

1(x) = 1 for all x ∈ K, then 1 ∈ C(K) and |f(x)| ≤ ‖f‖
∞
1(x), therefore

‖f‖
∞
1(x) ± f(x) ≥ 0 implies ‖f‖

∞
L(1) ± L(f) ≥ 0

so |L(f)| ≤ L(1)‖f‖
∞
. See [5] Prop. 7.1.

3 Main Result

Next theorem is our main result. It is a generalization of Theorem 2.1 to continuous functions

defined on arbitrary compact sets K ⊂ R. Since an ordinary Riemann-Stieltjes integral is not

defined for functions on general compact K, we shall introduce the Lebesgue integral which makes

sense for such functions. In the Appendix, we collect the basic facts and definitions of measure

theory we need.

Theorem 3.1. Let K a compact subset of R and let ℓ : C(K) → R be a positive linear functional.
Then, there is a unique finite Borel measure µ such that µ(K) = ‖ℓ‖ C(K)∗ and

ℓf =

∫

K

fdµ. (3.1)

Proof. The proof proceeds in stages.

i) Integral representation. Let [a, b] be a closed and bounded interval containing K. Note that

the technique used in what follows is independent of this interval. Let r : C[a, b] −→ C(K)
be the restriction operator, that is, for every f ∈ C[a, b], r(f)(x) = f(x) for x ∈ K. It is

clear that r is a bounded linear operator, so we can define its transpose operator, see [23]

p.11, also known as adjoint, see [22]. Recall rt is defined as follows rt : C(K)∗ → C[a, b]∗,
rt(ℓ)(f) = ℓ(r(f)) for f ∈ C[a, b]; the expression ℓ(r(f)) assigns a scalar to each function

f ∈ C[a, b].



6 Rafael del Rio, Asaf L. Franco and Jose A. Lara CUBO
20, 2 (2018)

Let ℓ be a positive linear functional in C(K) and we define Lf = rt(ℓ)(f) = ℓ(rf). Since ℓ and r

are positive linear functionals, so is L and we can apply theorem 2.1 and (c) in the Appendix

to find a monotone increasing function α and an associated Borel measure µ such that

Lf = rt(ℓ)(f) =

∫b

a

fdα =

∫b

a

fdµ (3.2)

for every f ∈ C[a, b].

Denote Kc := [a, b] \ K. We will show that µ(Kc) = 0. Let ε > 0 and choose Fε as a closed

subset of Kc such that

µ(Kc \ Fε) < ε, (3.3)

see (a) in the Appendix.

Let f̃ ∈ C[a, b] be a continuous function such that f̃(x) = 1 if x ∈ K, f̃(x) = 0 if x ∈ Fε and

‖f̃‖
∞

≤ 1. One can take for instance

f̃(x) =
d(x, Fε)

d(x, Fε) + d(x, K)
,

where d(x, A) = infy∈A |x − y|. Note that since |d(x, A) − d(y, A)| ≤ |x − y| the function

d(x, A) is even uniformly continuous, (cf. Urysohn’s Lemma. [5], 4.15.). Therefore

L(f̃) =

∫b

a

f̃dµ =

∫

K

dµ +

∫

Kc\Fε

f̃dµ +

∫

Fε

f̃dµ

The third integral on the right is equal zero, by definition of f̃. We can estimate the second

integral as follows,

0 ≤

∫

Kc\Fǫ

f̃dµ ≤

∫

Kc\Fε

dµ = µ(Kc\Fε) < ε,

since f̃ ≤ 1 and using (3.3). Then

L(f̃) <

∫

K

dµ + ε = µ(K) + ε.

We have that

µ(K) + µ(Kc) =

∫b

a

dµ = L(1[a,b]) = L(f̃) < µ(K) + ε,

The third equality follows from r(f̃) = r(1[a,b]). Thus 0 ≤ µ(K
c) < ε, since µ(K) < ∞.

To conclude, let f ∈ C(K) and f∗ a continuous extension of f to the closed interval [a, b].

We can do this extension taking, for example, straight lines as follows: since Kc is an open

subset of [a, b], it is at most a countable union of pairwise disjoint open intervals (αi, βi)



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An approach to F. Riesz representation Theorem 7

intersected with the interval [a, b], (see Lindeloef’s thm., [18] Prop.9. p.40). For x ∈ (αi, βi)

we define

f∗(x) = (1 − t)f(αi) + tf(βi)

if x = αi(1−t)+tβi for t ∈ (0, 1). The function f
∗ is continuous on the interval [a, b] since on

K coincides with the continuous function f and on Kc consists of straight lines, (cf. Tietze’s

Theorem [5], 4.16).

Then we have

ℓ(f) = ℓ(r(f∗)) = Lf∗ =

∫b

a

f∗dα =

∫b

a

f∗dµ =

∫

K

f∗dµ =

∫

K

fdµ. (3.4)

as was to be shown.

ii) Conservation of norm. Take f ∈ C(K) such that ‖f‖
∞

≤ 1. Since (3.1) holds we have,

|ℓ(f)| =

∣

∣

∣

∣

∫

K

fdµ

∣

∣

∣

∣

≤ ‖f‖
∞
µ(K) ≤ µ(K).

For the reverse inequality, let 1(x) = 1 for all x, as defined in remark (3), so

‖ℓ‖ ≥ |ℓ(1)| =

∣

∣

∣

∣

∫

K

1 dµ

∣

∣

∣

∣

= µ(K),

we can conclude that µ(K) = ‖ℓ‖.

iii) Uniqueness. Suppose µ and ν are finite measures that satisfy (3.1). Since µ and ν are regular

measures, from (a) in the Appendix, it is enough to show that µ(C) = ν(C) for any closed

set C of K. Let C a nonempty closed set of K and set fk(x) := max{0, 1 − kd(x, C)} for all

k and x ∈ K, where d(x, C) = infy∈C |x − y|. These functions are bounded, by 0 and 1,

and continuous. Thus fk belongs to C(K) for all k. Notice that they form a sequence that

decreases to the indicator of C, i.e., fk ↓ 1C, where 1C(x) = 1 if x ∈ C and 1C(x) = 0
if x /∈ C. Thus, for all k we must have that

∫
K
fkdµ =

∫
K
fkdν, and so we can use the

dominated convergence theorem, see (b) in the Appendix, to conclude that

µ(C) = lim
k

∫

K

fkdµ = lim
k

∫

K

fkdν = ν(C).

Remarks

(a) It is possible to represent the linear positive functionals acting on C(K) as Riemann-Stieltjes

integrals, similar to the original work of F. Riesz. This follows immediately from the chain of



8 Rafael del Rio, Asaf L. Franco and Jose A. Lara CUBO
20, 2 (2018)

equalities (3.4). The caveat is that we cannot use f directly in order to define the Riemann-

Stieltjes integral, but any continuous extension of f works, cf. theorem 3.2 below. This

integral is independent of the extension of f.

(b) As just seen, the use of compact set K above allows us to extend the continuous functions

to the entire interval [a, b], using an elementary version of the Tietze’s theorem. This con-

struction is in general not possible if K is an arbitrary subset of the real line.

3.1 Isomorphic spaces

As a consequence of the previous results, we shall see that two spaces of functionals are

practically the same. One of the spaces consists of Lebesgue integrals on compact subsets of [a, b]

and the other of Riemann-Stieltjes integrals over the whole interval [a, b]. In this way we show

how the Lebesgue integral representation, that was introduced to represent functionals in the case

of general compact sets, can be seen as a Riemann-Stieltjes integral. To state this precisely we

need to introduce the terms isomorphic and constant in Kc.

A transformation T which preserves the norm, that is ‖Tx‖ = ‖x‖, is called an isometry. Two

normed vector spaces X and Y are said to be isomorphic if there is a linear, bijective, isometry

T : X → Y. Such functions are called isomorphisms. Since an isomorphism preserves the linear as
well as the metric structure of the spaces, two isomorphic spaces can be considered identical, the

isomorphism corresponding just to a labeling of the elements. We say that the monotone function

α is constant in Kc if it is constant in each interval of Kc.

Recall that C(X)∗+ denotes the set of positive linear functionals on C(X). Let Lα denote the

functional with corresponding monotone function α as introduced in (2.1).

The result mentioned above can be then stated as follows:

Theorem 3.2. The normed spaces
{
Lα ∈ C[a, b]

∗
+ : α is constant in K

c
}
and C(K)∗+ are isomor-

phic.

Before we prove this theorem we need two preparatory results.

Proposition 3.3. rt : C(K)∗+ → C[a, b]∗+ is an isometry.

Proof. ‖rtℓ‖C[a,b]∗
+
= V(α) = α(b) − α(a) = µ([a, b]) = µ(K) + µ([a, b] \ K) = µ(K) = ‖ℓ‖C(K)∗

+
.

The first equality follows from Theorem 2.1. The function α depends on ℓ. The second is the

definition of the total variation of α and the third is the definition of µ. The last two equalities

follow from the construction of Theorem 3.1.



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An approach to F. Riesz representation Theorem 9

We denote the range of rt by Rang rt =
{
L ∈ C[a, b]∗+ : ∃l ∈ C(K)

∗
+ s.t. L = r

tl
}

Proposition 3.4.

Rang rt =
{
Lα ∈ C[a, b]

∗
+ : α is constant in K

c
}

Proof. ”⊂”

Let L ∈ Rang rt ⊂ C[a, b]∗+. Then there exists ℓ ∈ C(K)
∗
+ such that as in (3.2)

rt(ℓ)(f) = Lf = Lαf =

∫b

a

fdα =

∫b

a

fdµ

As was shown in the proof of Theorem 3.1 i), µ(Kc) = 0. Since Kc is a countable union of intervals,

these have µ measure zero. By the relation which is given in (4.1) below, between the measure µ

and the monotone function α we conclude that α is constant in each one of the intervals of Kc.

” ⊃ ”

Let Lα ∈ C[a, b]
∗
+ with α constant in each interval of K

c and µ be the measure associated with

this α, as in Appendix (c). Define ℓ ∈ C(K)∗+ as ℓh =
∫
K
hdµ. We shall show that rt(ℓ)(f) = Lαf

for every f ∈ C[a, b]. Since α constant in each interval of Kc this implies, using again (4.1), that

µ(Kc) = 0. Then we have

Lαf =

∫b

a

fdα =

∫b

a

fdµ =

∫

K

fdµ =

∫

K

r(f)dµ = ℓ(r(f)) = rt(ℓ)(f)

where r(f) denotes, as in Theorem (3.1) i) above, the restriction of f to K.

Proof of Theorem 3.2. .

From Proposition 3.3 and Proposition 3.4 it follows that rt is a bijective isometry. Since rt is

linear as follows from its definition, then it is an isomorphism.

Acknowledgments We thank C. Bosch and Ma. C. Arrillaga for useful comments. We are

grateful to Ma. R. Sanchez for her help in the search of bibliographical information.

4 Appendix

A collection of subsets A of X is called an σ-algebra if it is closed under finite (countable)

union, complements and X ∈ A. If our space is R, the Borel σ-algebra, BR, is the smallest σ-algebra

containing all the open intervals. A function µ : A → [0, ∞], where A is a σ-algebra, it is called
a measure if it is countable additive, that is µ(

⋃

An) =
∑

µ(An) whenever {An} is a disjoint

sequence of elements in A, and µ(∅) = 0. A Borel measure is a measure defined on BR. We say



10 Rafael del Rio, Asaf L. Franco and Jose A. Lara CUBO
20, 2 (2018)

that a measure is regular if every measurable set can be approximated from above by open mea-

surable sets and from below by compact measurable sets. A function f from (X, A, µ) to (R, BR) is

A-measurable if {x : f(x) ≤ t} ∈ A for all t ∈ R.

The following results are used in the proof of Theorem 3.1.

(a) Every Borel measure in a metric space is regular. We will only use inner regularity, that is,

for every Borel set A and every ε > 0 there exist a compact set Fε such that Fε ⊂ A and

µ(A \ Fǫ) < ǫ . [2] Thm 7.1.7. or [3] Lemma 1.5.7.

(b) (Dominated convergence theorem) Let (X, A, µ) a measure spaces. Let g be a [0, ∞]-valued
integrable function on X, that is,

∫
gdµ < ∞, and let f, f1, f2, . . . real-valued A-measurable

functions on X such that f(x) = limn fn(x) and |fn(x)| ≤ g(x). Then f and {fn} are integrable

and
∫
fdµ = limn

∫
fndµ.

(c) Given a normalized monotone function α in the closed interval [a, b], there is a unique

Borel measure µ associated with it. This can be seen as follows (see for example [4]): for

a ≤ s ≤ t ≤ b let define 〈s, t] where

Let

F0 =

{
⋃

finite

〈sk, tk] : 〈sk, tk] ⊂ [a, b] pairwise disjoint

}

Then F0 is an algebra of subsets of [a, b] and therefore we can define a set function as

µ0

(

⋃

finite

〈sk, tk]

)

=
∑

finite

α(tk) − α(sk). (4.1)

Moreover, µ0 has a unique extension to a measure in the smallest σ-algebra containing F0 (Caratheodory’s

Theorem). See [1] . Moreover, for any continuous function f it happens that

∫b

a

fdα =

∫b

a

fdµ (4.2)

where the integral on the left is a Riemann-Stieltjes integral, whereas the integral on the right is

an integral in the sense of Lebesgue.

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An approach to F. Riesz representation Theorem 11

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	Introduction
	Preliminaries 
	Definitions and notation.
	Representation theorem for functionals on C[a,b].

	Main Result
	Isomorphic spaces

	Appendix