THE GENERALIZED RIEMANN INTEGRAL ON LOCALLY COMPACT SPACES Extended Abstract

A. EDALAT, S. NEGRI Department of Computing, Imperial College, London SW7 2BZ.

We extend the basic results on the theory of the generalized Riemann integral to the setting of locally compact second countable Hausdorff spaces with bounded or locally finite measures, the latter being measures which are finite on compact subsets of the space. We show that the correspondence between Borel measures on X and continuous valuations on the upper space UX gives rise to a topological embedding between the space of locally finite measures and locally finite continuous valuations, both endowed with the Scott topology. We construct an approximating chain of simple valuations on the upper space of a locally compact space, whose least upper bound is the given locally finite measure. We define the generalized Riemann integral for bounded functions with respect to bounded and unbounded measures. Also in this setting generalized R-integrability for a bounded function is proved to be equivalent to the condition that the set of its discontinuities has measure zero. Finally we prove that if a bounded function is R-integrable then it is also Lebesgue integrable and the two integrals coincide.

1 Introduction

Domain theory was introduced by Dana Scott in 1970 ([11]) as a foundational basis for the semantics of computation.

In [3], a basic connection between domain theory and some main branches of mathematics has been established, giving rise, in particular, to a novel computational approach to measure theory and integration.

A domain-theoretic framework for measure and integration has been provided in [4], by proving that any bounded Borel measure on a compact metric space X can be obtained as the least upper bound of simple valuations on the dcpo given by the upper space UX.

Simple valuations approximating the given measure play the role of partitions in Riemann integration and are used to obtain increasingly better approximations of the integral of a bounded real-valued function on a compact metric space. This idea leads to a new notion of integration, called generalized Riemann integration, R-integration for short, similar in spirit to Riemann integration but more general.

All the basic results of the theory of Lebesgue integration can be extended to this setting. For instance, it is proved in [4] that a bounded real-valued function on a compact metric space is R-integrable w.r.t. a bounded Borel measure if and only if its set of discontinuities has measure zero and that if it is R-integrable then it is also Lebesgue integrable and the two integrals coincide.

This theory has had applications in exact computation of integrals [5], in the 1-dimensional random fields Ising model in statistical physics [6], in forgetful neural networks [7], in stochastic processes [2] and in chaos theory [5].

In order to apply the generalized Riemann integration to a wider range of problems, we look for a generalization of the results in [4] and [5] to the more general setting of locally compact second countable Hausdorff spaces with bounded or locally finite measures, the latter being measures which are finite on compact subsets of the space.

We stress the fact that these more general hypotheses cover some central fields of application. For instance, probability distributions on the real line are examples of bounded measures on a locally compact space, whereas the Lebesgue measure on the real line is an example of a locally finite measure on a locally compact space.

In this paper we face the problem ab initio, from the correspondence between Borel measures on X and continuous valuations on the upper space UX (as given in [3]). We start with showing, in section 2, that such correspondence also holds between locally finite measures and locally finite continuous valuations.

In the fourth section we show how to approximate a locally finite valuation on a locally compact space by means of an increasing sequence of simple valuations on the upper space, that is built up from any given presentation of the locally compact space as a union of increasing regular compact subsets.

We proceed with the definition of the generalized Riemann integral of a bounded real-valued function defined on a locally compact space with respect to a bounded Borel measure. As in the case of compact spaces [4], generalized Rintegrability for a bounded function f is seen to be equivalent to the condition that the set of discontinuities of f has measure zero.

Finally, we define the R-integral for a real-valued function on a locally compact space with respect to a locally finite measure. In the case of functions with compact support, R-integrability reduces to R-integrability in compact spaces with respect to the restriction of the measure to the support of the function. In the case of functions with unbounded support, R-integrability is defined in a way similar to improper Riemann integrability and it is proved, also in this more general case, that if a function is R-integrable then it is also Lebesgue integrable and the two integrals coincide.

All the proofs can be found in the full version of this paper ([8]).

2 Locally finite measures and locally finite continuous valuations

All throughout the paper X will denote a second countable locally compact Hausdorff space. We will use the decomposition X = ⋃_{i∈IN} X_{i}, where ⟨X_{i}⟩_{i∈IN} is an increasing sequence of relatively compact subsets of X such that \overline{X_{i}} ⊆ X_{i+1}.

In this section we will prove that the correspondence stated in [3] between Borel measures on X and valuations on the upper space supported in s(X) gives rise, when X is a locally compact topological space, to an equivalence between locally finite Borel measures on X and locally finite continuous valuations on the upper space supported in s(X) (defined below).

Definition 2.1 Let X be a second countable locally compact Hausdorff space. Then we define M^{l}(X) to be the set of locally finite Borel measures μ on X, i.e., of Borel measures with the property that μ(C) < ∞ for all compact subsets C of X.

We recall from [3] that the upper space UX of a topological space is the set of all non-empty compact subsets of X, with the base of the upper topology given by □a = {C ∈ UX | C ⊆ a} (a ∈ Ω(X)). When X is a second countable locally compact Hausdorff space, then the upper space UX of X is an ωcontinuous dcpo and the Scott topology of (UX, ⊇) coincides with the upper topology.

Definition 2.2 Let X be a second countable locally compact Hausdorff space. A locally finite continuous valuation on UX is a mapping

ν : Ω(UX) → [0, ∞]

satisfying, for all U, V ∈ Ω(UX) and for any directed family A in Ω(UX), the following conditions:

1. ν(U) + ν(V) = ν(U ∪ V) + ν(U ∩ V);

2. ν(∅) = 0;

3. U ⊆ V ⇒ ν(U) ≤ ν(V);

4. ν(⋃_{O∈A} O) = sup_{O∈A} ν(O);

5. ν(V) < ∞ for V ≪ UX.

We will denote with P^{l}(UX) the set of all such valuations.

Proposition 2.3 Let s : X → UX be the singleton map. Then the map

\begin{matrix}e : M^{l}(X) & → & P^{l}(UX) \\ μ & ↦ & μ ∘ s^{−1} \end{matrix}

is well defined, i.e., whenever μ is a locally finite Borel measure on X, then μ ∘ s^{−1} is a locally finite continuous valuation.

By a result from [10] (theorem 3.9), if P is a continuous poset with its Scott topology, then every continuous valuation extends uniquely to a measure. Thus, in particular, μ∘s^{−1} extends to a unique measure on UX, which we will denote by μ^{∗}. Such a measure satisfies μ^{∗}(UX − s(X)) = 0, that is we have: Proposition 2.4 The valuation μ ∘ s^{−1} is supported in s(X).

Adapting the notation from [3], we will denote with S^{l}(X) the locally finite valuations on UX which are supported in s(X). We have: Theorem 2.5 The map

\begin{matrix}e : M^{l}(X) & → & S^{l}(X) \\ μ & ↦ & μ ∘ s^{−1} \end{matrix}

is a bijection with inverse given by

\begin{matrix}j : S^{l}(X) & → & M^{l}(X) \\ ν & ↦ & ν^{∗} ∘ s . \end{matrix}

The following lemma is used in the proof and elsewhere in the paper: Lemma 2.6 Let X be a second countable locally compact Hausdorff space and suppose that μ and ν are locally finite Borel measures which have the same restriction to Ω(X). Then μ = ν.

We recall from [3] that, given a topological space Y, the point valuation based at b ∈ Y is the valuation

δ_{b} : Ω(Y) → [0, ∞)

defined by

δ_{b}(O) = \{ \begin{matrix}1 & if b ∈ O \\ 0 & otherwise . \end{matrix}

Any finite linear combination

∑_{i=1}^{n} r_{i}δ_{b_{i}} (r_{i} ∈ [0, ∞), i = 1, …, n)

of point valuations is called a simple valuation.

When considering locally finite valuations closure under directed joins does not hold anymore. However, it is still true that locally finite valuations can be approximated by simple valuations, since the results holds in general for (unbounded) valuations (cf. theorem 5.2, page 46 in [9]), and simple valuations are in particular locally finite. So we have: Proposition 2.7 For all μ ∈ P^{l}(UX), μ is the supremum of the (directed) set of simple valuations way-below it.

3 The Scott topology on M^{l}(X)

This section is devoted to prove that the map

e : M^{l}(X) → P^{l}(UX)

is a topological embedding when both spaces are endowed with the Scott topology.

We can define a partial order on M^{l}(X) by putting, for μ_{1}, μ_{2} ∈ M^{l}(X)

μ_{1} ≤ μ_{2} ≡ (∀O ∈ Ω(X))(μ_{1}(O) ≤ μ_{2}(O)) .

The relation ≤ is clearly reflexive and transitive, and it is antisymmetric by lemma 2.6. Observe that the space (M^{l}(X), ≤) is not a dcpo since, for instance, the ⟨n · δ_{x}⟩_{n>0} is an increasing sequence in M^{l}X whose least upper bound is not a locally finite measure.

We proceed by characterizing those directed subsets of M^{l}X having least upper bound in M^{l}X. We have: Proposition 3.1 A directed subset D of M^{l}X has least upper bound in M^{l}X if and only if for all open sets O such that O ≪ X in Ω(X) the set {μ(O) : μ ∈ D} is bounded in IR.

It is immediate to check that the map e is continuous with respect to the Scott topologies on M^{l}X and P^{l}UX. The map e is a topological embedding since also its inverse j is continuous with respect to the Scott topologies on M^{l}(X) and S^{l}(X) (cf. [8] for the details).

4 Approximation of locally finite measures

Given a measure on a compact metric space it is possible to construct a chain of simple valuations on the upper space with the measure as its least upper bound (cf. [5]). In this section we will generalize this construction to the case of locally finite measures on a locally compact space.

The relatively compact subsets X_{i} will serve as a basis to construct a sequence of simple valuations on UX approximating any given locally finite measure on X, thus generalizing the procedure which is worked out in [5]. We assume as in [5] that the measure is given by its values on a given countable basis closed under finite unions and finite intersections.

Since the spaces X_{i} are relatively compact, for all i ∈ IN there exists an ordered finite covering C_{i} of X_{i} made of open sets of the basis with diameter ≤ \frac{1}{i}. Moreover such coverings can be chosen satisfying the additional requirement ∪C_{i} ⊆ ∪C_{i+1}. These hypotheses are needed in order to obtain an increasing sequence of simple valuations. We will assume in this section (and the rest of the paper) that open covers are always constructed from the given countable base satisfying the above property.

For each i ∈ IN we define inductively a finite open cover D_{i} for X_{i} in the following way:

D_{1} ≡ C_{1};

D_{2} ≡ D_{1} ∧ C_{2} ∪ {O ∈ C_{2} : O ⊆ / ∪ C_{1}};

D_{3} ≡ D_{2} ∧ C_{3} ∪ {O ∈ C_{3} : O ⊆ / ∪ C_{2}};

…

D_{i+1} ≡ D_{i} ∧ C_{i+1} ∪ {O ∈ C_{i+1} : O ⊆ / ∪ C_{i}};

…

where for given covers A and B, A ∧ B denotes the cover

{O_{1} ∩ O_{2} : O_{1} ∈ A, O_{2} ∈ B}.

Assume the covers D_{i} to be ordered with the lexicographic order for the first part (D_{i−1} ∧ C_{i}), and with the order induced by the order on C_{i} for the second part.

We recall from [5] that given an ordered cover A = {O_{1}, …, O_{n}}, the simple valuation μ_{A} associated with A is given by

μ_{A} ≡ ∑_{i=1}^{n} r_{i}δ\overline{O_{i}} , where r_{i} = μ(O_{i} − ⋃_{j<i} O_{j}).

We will denote with μ_{i} the simple valuation μ_{D_{i}} associated with D_{i}.

We are now going to prove that the μ_{i}’s constitute a chain of simple valuations with least upper bound μ.

The following lemma is proved as proposition 3.1 in [5].

Lemma 4.1 Let A = {O_{1}, …, O_{n}} be a covering of X_{i} and let μ be a locally finite measure on X. Then for all open subsets O of X_{i} we have

∑_{O_{i}⊆O} r_{i} ≤ μ(O) ≤ ∑_{\overline{O_{i}}∩O≠∅} r_{i} .

Then we have: Proposition 4.2 For all i ∈ IN the following hold

1. μ_{i} ⊑ μ in P^{l}(UX);

2. μ_{i} ⊑ μ_{i+1} in P^{l}(UX);

3. μ = ⊔_{i∈IN}μ_{i}.

5 The generalized Riemann integral

Let X be a second countable locally compact Hausdorff space. Let μ be a Borel measure on X such that 0 < μ(X) < ∞. By rescaling we can suppose that μ is normalized, i.e. that μ(X) = 1. The continuous dcpo UX does not have a bottom element, so that for practical purposes we will adjoin a bottom element ⊥ and denote the dcpo with bottom thus obtained with (UX)_{⊥}. By corollary 3.3 in [4], P^{1}(UX)_{⊥}, the set of normalized valuations of the probabilistic power domain of (UX)_{⊥}, is an ω-continuous dcpo with a basis of normalized simple valuations. There exists an embedding

\begin{matrix}M^{1}X & → & P^{1}(UX)_{⊥} \\ μ & ↦ & μ ∘ s^{−1} \end{matrix}

and by the result of the previous section we also know that there exists an increasing chain of simple valuations ⟨μ_{i}⟩_{i≥0} in P (UX)_{⊥} such that μ ∘ s^{−1} = sup_{i≥0} μ_{i},

If we consider the valuations

μ_{i}^{+} ≡ μ_{i} + (1 − μ_{i}((UX)_{⊥}))δ_{⊥}

then the μ_{i}^{+}’s are in P^{1}(UX)_{⊥}. Moreover, for any proper open subset O of (UX)_{⊥}, we have μ_{i}^{+}(O) = μ_{i}(O) and μ ∘ s^{−1} = sup_{i≥0} μ_{i}^{+}, where ⟨μ_{i}^{+}⟩_{i≥0} is again an increasing chain of simple valuations.

Let f be a bounded real-valued function defined on X and let μ be a bounded Borel measure on X. For a simple valuation ν ≡ ∑_{b∈B} r_{b}δ_{b}, define, as in [4],

S_{X}^{l}(f, ν) ≡ ∑_{b∈B} r_{b} inf f[b], S_{X}^{u}(f, ν) ≡ ∑_{b∈B} r_{b} sup f[b]

and observe that, since f is bounded, S_{X}^{l}(f, ν) and S_{X}^{u}(f, ν) are well-defined real numbers. By proposition 4.2 in [4] we have: Proposition 5.1 Let μ_{1}, μ_{2} ∈ P^{1}(UX)_{⊥} be simple valuations with μ_{1} ⊑ μ_{2}. Then S_{X}^{l}(f, μ_{1}) ≤ S_{X}^{l}(f, μ_{2}) and S_{X}^{u}(f, μ_{2}) ≤ S_{X}^{u}(f, μ_{1}). Corollary 5.2 If μ_{1}, μ_{2} ∈ P^{1}(UX)_{⊥} are simple valuations with μ_{1}, μ_{2} ≪ μ, then S_{X}^{l}(f, μ_{1}) ≤ S_{X}^{u}(f, μ_{2}).

Therefore we have, for all i, j ∈ IN,

1. S_{X}^{l}(f, μ_{i}^{+}) ≤ S_{X}^{l}(f, μ_{i+1}^{+});

2. S_{X}^{u}(f, μ_{i+1}^{+}) ≤ S_{X}^{u}(f, μ_{i}^{+});

3. S_{X}^{l}(f, μ_{i}^{+}) ≤ S_{X}^{u}(f, μ_{j}^{+}).

Using the notation introduced above we define the lower and upper Rintegral as follows: Definition 5.3 R\underline{∫} fdμ ≡ sup_{ν≪μ} S_{X}^{l}(f, ν) (lower R-integral);

R\overline{∫} fdμ ≡ inf_{ν≪μ} S_{X}^{u}(f, ν) (upper R-integral). By proposition 5.2 we have

R\underline{∫} fdμ ≤ R\overline{∫} fdμ .

Definition 5.4 We say that f is R-integrable w.r.t. μ (and write f ∈ R_{X}(μ)) if R\underline{∫} f = R\overline{∫} f.

If f is R-integrable then the integral of f can be calculated by using the increasing sequence of simple valuations ⟨μ_{i}^{+}⟩_{i≥0} (cf. proposition 4.9 in [4]): Proposition 5.5 If f is R-integrable w.r.t. μ, and ⟨μ_{i}⟩_{i≥0} is an increasing sequence of simple valuations on UX with least upper bound μ, then

∫_{X} fdμ = sup_{i≥0} S_{X}^{l}(f, μ_{i}^{+}) = inf_{i≥0} S_{X}^{u}(f, μ_{i}^{+}).

6 Lebesgue criterion for R-integrability

We are now going to generalize the Lebesgue criterion for Riemann integration of a bounded function on a compact real interval to R-integration of a bounded real-valued function on a locally compact space, by generalizing the result in [4], that is the analogous criterion for R-integration on compact spaces.

We first recall some definitions.

Definition 6.1 Let T ⊆ X and r > 0 and define

Ω_{f}(T) ≡ sup{f(x) − f(y) | x, y ∈ T }, called oscillation of f at T;

ω_{f}(x) ≡ lim_{h→0^{+}} Ω_{f}(B(x, h)), where B(x, h) is the open ball of radius h centred at x;

D_{r} ≡ \{x ∈ X | ω_{f}(x) ≥ \frac{1}{r}\}.

Then we have (cf. [1], p. 170 and [4]): Proposition 6.2 The following statements hold:

(i) f is continuous at x iff ω_{f}(x) = 0.

(ii) If X is compact and ω_{f}(x) < ϵ for all x ∈ X, then there exists δ > 0 such that for all compact subsets b ⊆ X with |b| < δ we have Ω_{f}(b) < ϵ.

(iii) For any r > 0, the set D_{r} is closed.

If D is the set of discontinuities of f, we have D = ⋃_{n≥1} D_{n} where D_{1} ⊆ D_{2} ⊆ D_{3} ⊆ … is an increasing chain of closed sets. Hence D is an F_{δ} and therefore a Borel set. In the following, X will be a second countable locally compact metric space and μ ∈ M^{1}(X). We have (cf. lemma 6.4 in [4] for the proof): Lemma 6.3 Let d be a compact subset of X and let ν = ∑_{b∈B} r_{b}δ_{b} ∈ P^{1}UX. Then we have:

1. If ν ⊑ μ then ∑_{b∩d≠∅} r_{b} ≥ μ(d).

2. If ν ≪ μ then ∑_{b^{∘}∩d≠∅} r_{b} ≥ μ(d), where b^{∘} denotes the interior of b.

Then Lebesgue criterion follows: Theorem 6.4 A bounded real-valued function on a locally compact second countable Hausdorff space is R-integrable with respect to a bounded Borel measure μ on X iff its set of discontinuities has measure zero.

7 R-integration and Lebesgue integration

In this section we will prove that the result in [4] connecting R-integration and Lebesgue integration for bounded real-valued functions defined on a compact space extends to the case of bounded functions defined on a locally compact second countable Hausdorff space. Theorem 7.1 If a bounded real-valued function f is R-integrable with respect to a bounded Borel measure μ on a locally compact second countable Hausdorff space, then it is also Lebesgue integrable and the two integrals coincide.

8 The improper R-integral

8.1 Functions with compact support

In this section we define the R-integral for a real-valued function with compact support defined on a locally compact space w.r.t. a locally finite measure.

Basically, our definition will be obtained by cutting down the construction of the chain of simple valuations approximating a given locally finite measure that we gave in section 4 to the compact support of the function.

Let f be a real-valued function on a locally compact space X with compact support C and let μ be a locally finite measure on X. Recall from section 4 that μ = ⊔μ_{i}, where μ_{i} is the simple valuation, associated with the covering D_{i} of X_{i}, given by ∑_{j=1}^{n} r_{j}δ\overline{O_{j}}, with r_{j} = μ(O_{j} − ⋃_{k<j} O_{k}). Since C is compact, there exists k ∈ IN such that C ⊆ X_{i} for all i > k. For i > k, define

μ_{i}^{C} = ∑_{j=1}^{n} r_{j}^{C}δ_{\overline{O_{j}}∩C}, where r_{j}^{C} = μ(O_{j} ∩ C − ⋃_{k<j} O_{k} ∩ C) .

Moreover define, for all open subsets O of X, μ^{C}(O) = μ(O ∩ C).

In order to prove that the integral of f can be defined by means of the chain of simple valuations ⟨μ_{i}^{C}⟩_{i}, some properties which are easily derived as in section 4 are needed. We observe that it is enough to know the value of μ on the induced topology of C. Lemma 8.1 ∑_{O_{j}∩C⊆O} r_{j}^{C} ≤ μ(O ∩ C) ≤ ∑_{\overline{O_{j}}∩C∩O≠∅} r_{j}^{C} .

Then we have Proposition 8.2 Let μ^{C} and μ_{i}^{C} be defined as above. Then:

1. For all i > k, μ_{i}^{C} ⊑ μ^{C};

2. For all i > k, μ_{i}^{C} ⊑ μ_{i+1}^{C};

3. μ^{C} = ⊔_{i>k}μ_{i}^{C}.

The R-integrability of f w.r.t. μ thus reduces to the R-integrability of f|_{C} w.r.t. μ^{C}.

8.2 Functions with unbounded support

Let X be a locally compact second countable Hausdorff space, μ a locally finite measure on X and f : X → IR a bounded positive function. Given a subset A of X let χ_{A} denote the characteristic function of A. In order to extend the theory of generalized Riemann integration to this setting we make the following:

Definition 8.3 The function f is improper R-integrable on X w.r.t. μ if for all i ∈ IN the function with compact support f · χ_{X_{i}} is R-integrable and there exists

lim_{i→∞} R ∫_{X} f · χ_{X_{i}}dμ .

In this case we put

R ∫_{X} fdμ = lim_{i→∞} R ∫_{X} f · χ_{X_{i}}dμ .

The above definition of improper R-integrability does not depend on the choice of the increasing sequence of relatively compact subsets X_{i}’s such that X = ⋃_{i} X_{i}. Indeed, if ⟨Y_{i}⟩_{i∈IN} is another such sequence then it is clear that for all i, f · χ_{X_{i}} is R-integrable if and only if for all i, f · χ_{Y_{i}} is R-integrable and

lim_{i→∞} R ∫_{X} f · χ_{X_{i}}dμ = lim_{i→∞} R ∫_{X} f · χ_{Y_{i}}dμ

since for all i there exists j such that X_{i} ⊆ Y_{j}, hence

R ∫_{X} f · χ_{X_{i}}dμ ≤ R ∫_{X} f · χ_{Y_{i}}dμ

and conversely.

If we release the assumption on f being positive, then we can use the decomposition

f = f^{+} − f^{−}

where f^{+} = max(f, 0) and f^{−} = max(−f, 0), and say that f is improper R-integrable if both f^{+} and f^{−} are improper R-integrable and put

R ∫_{X} fdμ = R ∫_{X} f^{+}dμ − R ∫_{X} f^{−}dμ .

The following result connects improper R-integration with Lebesgue integration, generalizing theorem 7.1. Theorem 8.4 If a bounded real-valued function f is improper R-integrable with respect to a locally finite Borel measure μ on a locally compact second countable Hausdorff space, then it is also Lebesgue integrable and the two integrals coincide.

Acknowledgments

This work has been funded by EPSRC project “Foundational Structures in Computer Science”.

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