Repeated quasi-integration on locally compact spaces

When X is locally compact, a quasi-integral (also called a quasi-linear functional) on Cc(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_c(X)$$\end{document} is a homogeneous, positive functional that is only assumed to be linear on singly-generated subalgebras. We study simple and almost simple quasi-integrals, i.e., quasi-integrals whose corresponding compact-finite topological measures assume exactly two values. We present equivalent conditions for a quasi-integral to be simple or almost simple. We give a criterion for repeated quasi-integration (i.e., iterated integration with respect to topological measures) to yield a quasi-linear functional. We find a criterion for a double quasi-integral to be simple or almost simple. We describe how a product of topological measures acts on open and compact sets. We show that different orders of integration in repeated quasi-integrals give the same quasi-integral if and only if the corresponding topological measures are both measures or one of the corresponding topological measures is a positive scalar multiple of a point mass.

Hausdorff space X and corresponding set functions, generalizing measures (initially called quasi-measures, now topological measures). He connected the two by establishing a representation theorem thereby giving an impetus to the field that has already resulted in a substantial body of work.
In [12] Entov and Polterovich first linked the theory of quasi-linear functionals to symplectic topology. They established that quasi-linear functionals can be viewed as an algebraic way of packing certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms, and proved many new results. Paper [12] has been cited over 100 times, and quasi-linear functionals and topological measures have been studied and used in many subsequent papers, as well as in a monograph ( [18]).
Quasi-linear functionals are functionals that are linear on singly-generated subalgebras. Such functionals are obtained by integration of continuous functions with respect to topological measures. Topological measures lack subadditivity and other properties typical for measures but, on the other hand, many properties of measures still hold for topological measures. Likewise, quasi-linear functionals in some respects are similar to linear functionals, and in other respects they are strikingly different. Some of these differences and similarities will be demonstrated in the current paper.
The vast majority of papers devoted to quasi-linear functionals and topological measures deal with compact spaces. The author has written several papers extending the theory to the locally compact setting. The current paper, devoted to repeated integration with respect to topological measures, is an important part of this series. First, the results of this paper are necessary for future research on topics that involve repeated integration. Second, topological measures are a subclass of deficient topological measures, which also correspond via integration to certain non-linear functionals. Knowledge about repeated integration with respect to topological measures is necessary for understanding (repeated) integration with respect to deficient topological measures.
In [13] Grubb gives a nice treatment of repeated quasi-integration on a product of compact spaces. In this paper we generalize all results from [13] and some results from [2] to a locally compact setting. We also obtain new results, see Theorems 4.3,4.4,4.5,and Proposition 4.1. In proving results about quasi-linear functionals and topological measures in a locally compact setting one loses certain convenient features present in the compact case: quasi-integrals and topological measures are not finite in general, subalgebras do not contain constants, quasi-integrals are no longer states, etc. On another level, to tackle the problems related to repeated integration on locally compact spaces, one needs a variety of results about quasi-liner functionals and topological measures. These include a Representation Theorem showing that quasi-linear functionals are obtained by integration with respect to compact-finite topological measures, continuity of quasi-integrals with respect to the topology of uniform convergence on compacta, and others. These results are obtained in [9,10,19], and [8].
In this paper X is a Hausdorff, locally compact space. In Sect. 2 we give necessary definitions and facts. We define quasi-integrals and topological measures on a locally compact space and outline the correspondence between them. In Sect. 3 we study simple and almost simple quasi-integrals and topological measures, which are necessary for investigating repeated quasi-integration. We present equivalent conditions for a quasi-integral to be simple or almost simple. In Sect. 4 we define and study repeated quasi-integrals. In particular, we give criteria for repeated quasi-integration to yield a quasi-linear functional, and for a double quasi-integral to be simple or almost simple. In Sect. 5 we give formulas that describe how a product of compact-finite topological measures acts on open and compact sets. We show that different orders of integration in repeated quasi-integrals give the same quasi-integral if and only if corresponding compact-finite topological measures are both measures or one of the corresponding topological measures is a positive scalar multiple of a point mass.

Preliminaries
We use continuous functions with the uniform norm; C c (X ) is the set of real-valued continuous functions on X with compact support, and C + c (X ) is the set of nonnegative functions from C c (X ). By supp f we mean {x : f (x) = 0}. We denote by 1 the constant function 1(x) = 1, by id the identity function id(x) = x, and by 1 E the indicator function of the set E. When we consider maps into extended real numbers, they are not identically ∞. We denote by E the closure of a set E, and by a union of disjoint sets. O(X ), C (X ), and K (X ) stand, respectively, for the collections of open, closed, and compact subsets of X .
We are interested in repeated quasi-integrals, i.e. repeated quasi-integration with respect to topological measures on locally compact spaces. Since X × Y is locally compact iff X and Y are locally compact, we assume that X and Y are locally compact.
We will use the following definitions and facts.
Definition 2.1 A topological measure on a locally compact space X is a set function μ : C (X ) ∪ O(X ) → [0, ∞] satisfying the following conditions: The following theorem is [8,Theorem 4.9].
Theorem 2.1 Let μ be a topological measure on a locally compact space X . The following are equivalent: For f ∈ C c (X ) we have 0 ∈ f (X ). By a singly generated subalgebra of C c (X ) generated by f we mean the smallest closed subalgebra of C c (X ) containing f ; it has the form When X is compact, by a singly generated subalgebra of C(X ) generated by f we mean the smallest closed subalgebra of C(X ) containing f and 1; it has the form: (Cf. [9,Lemma 1.4]).

Definition 2.2
Let X be locally compact. A quasi-integral (or a quasi-linear functional) on C c (X ) is a map ρ : C c (X ) −→ R such that: When X is compact, we call ρ a quasi-state if ρ(1) = 1.
In this paper we are interested in quasi-integrals on C c (X ) and compact-finite topological measures for the reason given in the next remark.

Remark 2.2
There is an order-preserving isomorphism between compact-finite topological measures on X and quasi-integrals on C c (X ), and μ is a measure iff the corresponding functional is linear. See [9,Theorem 3.9] for this result and [19,Theorem 3.9] for the first version of the representation theorem. We outline the correspondence.
(I) Suppose μ is a compact-finite topological measure on a locally compact space X , f ∈ C c (X ). Then there exists a finite measure m f on R with supp m f ⊆ f (X ) such that thus, If μ is finite then The measure m f is the Stieltjes measure associated with the function Define a quasi-integlral ρ = ρ μ on C c (X ) by: We also write ρ μ ( f ) = X f dμ. If μ is a measure then ρ μ ( f ) = X f dμ in the usual sense. On singly generated subagerbras ρ μ acts as follows: for every where [a, b] is any interval containing f (X ). (See [9,Proposition 2.11, Theorem 2.12].) (II) Let ρ be a quasi-integral on C c (X ). The corresponding compact-finite topological measure μ = μ ρ is given as follows:

Remark 2.3
Suppose X is locally compact.
(1) Let ρ be a quasi-integral. If f , g ≥ 0, f · g = 0 then f , g belong to the same singly generated subalgebra.
(2) It is easy to see that if c ≥ 0 and μ is a topological measure then ρ cμ = cρ μ .
The following is a part of [9,Theorem 4.5].

Theorem 2.2 Suppose X is locally compact and ρ is a quasi-integral on
where μ is the compact-finite topological measure corresponding to ρ. In particular, for any f ∈ C c (X )

Remark 2.4
If we replace condition (TM1) in Definition 2.1 by finite additivity on compact sets only, we obtain the definition of a deficient topological measure. Deficient topological measures correspond to certain nonlinear functionals that generalize quasilinear functionals. See [8,14,20,21], and [6] for more information.
We would like to conclude this section with some examples.

Definition 2.3 A set A is bounded if
Then ν is a solid-set function (see [10,Example 15.5]), and ν extends to a unique topological measure on X . Let K i be the closed ball of radius 1 centered at p i for i = 1, 2. Then K 1 , K 2 and C = K 1 ∪ K 2 are compact solid sets, ν(K 1 ) = ν(K 2 ) = π, ν(C) = 4π . Since ν is not subadditive, it can not be a measure. The quasi-linear functional corresponding to ν is not linear. ]. The resulting topological measure is not a measure. For instance, when X is the square and n = 1, it is easy to represent X = A 1 ∪ A 2 ∪ A 3 , where each A i is a compact solid set containing one point from P. Then ν(A i ) = 0 for i = 1, 2, 3, while ν(X ) = 1. Since ν is not subbadditive, it is not a measure, and the quasi-linear functional ρ corresponding to ν is not linear. In [9,Example 4.13] we take n = 2 and show that there are f , g ≥ 0 such that ρ( f + g) = ρ( f ) + ρ(g). An interesting property of quasi-linear functionals is demonstrated by ρ. If X is locally compact, non-compact, n = 1, for the functional ρ we consider a new functional ρ g defined by The new functional ρ g corresponds to a set function (in, fact, a deficient topological measure) obtained by integrating g over closed and open sets with respect to a topological measure ν. We can choose g ≥ 0 so that ρ g is no longer linear on singly generated subalgebras, but only linear on singly generated cones. See [7,Example 35, Theorem 43] for detail.
For more examples of topological measures and quasi-integrals on locally compact spaces see [5] and the last sections of [10] and [9].  [21] and [8].

Remark 2.5
We can say that, in general, the collection of all regular Borel measures and all Radon measures is properly contained in the collection of all topological measures, which, in turn, is properly contained in the collection of all deficient topological measures (see also [8,Remark 4.3].) We can state the same about corresponding functionals.

Almost simple quasi-integrals
A nontrivial topological measure assumes at least two values. Topological measures that assume exactly two values are important for proving results about repeated quasiintegration.
Definition 3.1 Let X be locally compact. A topological measure is called simple if it only assumes values 0 and 1. A topological measure is almost simple if it is a positive scalar multiple of a simple topological measure. A quasi-integral is simple (almost simple) if the corresponding topological measure is simple (almost simple). Lemma 3.1 Suppose X is locally compact and μ is a compact-finite topological measure that assumes more than two values. Let ρ be the corresponding quasi-integral.
Functions f 1 , f 2 belong to the same singly generated subalgebra.
Then f 1 f 2 = 0 and by calibrating f i we may assume that ρ( (This proof is adapted from part of an argument in [13,Theorem 1].) The last statement follows from Remark 2.3.
The next theorem extends results for a simple quasi-state on C(X ) where X is compact, given in [2,Sect. 2]. Theorem 3.1 Let X be locally compact. The following are equivalent for a quasiintegral ρ on C c (X ): we have: ρ(gh) = ρ(g) ρ(h) for g, h in the singly generated subalgebra B( f ).
Proof (i) ⇒ (ii). If ρ is simple, i.e. the corresponding topological measure μ is simple, then the measure m f in part (I) of Remark 2.2 is a point mass. From formula (2) we see that m f is a point mass at y = ρ( f ). To show that y ∈ f (X ), suppose the opposite, and choose an open set W such that y ∈ W , W ∩ f (X ) = ∅. Then m f (W ) = 1, while μ( f −1 (W )) = μ(∅) = 0, which contradicts formula (1).

Remark 3.1
If μ is almost simple but not simple, write μ = cμ , where μ is simple and c > 0. Then ρ = cρ , where quasi-integrals ρ and ρ correspond to μ and μ , and ρ is no longer multiplicative on singly generated subalgebras.
The following theorem follows immediately from Theorem 3.1 and Lemma 3.1.
Theorem 3.2 Let X be locally compact. The following are equivalent for a quasiintegral ρ on C c (X ): When X is compact the equivalence of (ii) and (iv) in Theorem 3.2 with the condition "ρ is simple" is given by [22,Theorem 3.10].

Repeated quasi-integrals
Let μ be a compact-finite topological measure on X with corresponding quasi-integral ρ, and ν be a compact-finite topological measure on Y with corresponding quasiintegral η.

Remark 4.1 If the set A is closed/compact/open then so is the set
where π 1 : X ×Y → X is the canonical projection, contains supp f y for any y. We have f y ∈ C c (X ), and f x ∈ C c (Y ).

Proposition 4.1 Suppose X × Y is locally compact and f
Here π 1 : X × Y → X and π 2 : X × Y → Y are canonical projections.
Proof Let y ∈ Y . We shall show that T ρ ( f ) is a continuous function at y. Compact C = π 1 (supp f ) contains supp f y . Let > 0. For each x ∈ C let U x be a neighborhood of x and V x,y be a neighborhood of y such that | f (x, y) − f (x , y )| < whenever (x , y ) ∈ U x ×V x,y . Open sets U x cover C, so let U x 1 , . . . , U x n be a finite subcover, and . Therefore, f y − f y < (for any y ∈ V y ). Since supp f y , supp f y ⊆ C, by Theorem 2.2 for any y ∈ V y we have: and the continuity of T ρ ( f ) at y follows. By Since X and Y are locally compact, for x ∈ X let U (x) be a neighborhood of x such that U (x) is compact in X , and let V (y) be a neighborhood of y ∈ Y such that V (y) is compact in Y . Open sets U (x) × V (y) cover supp f , so let U 1 × V 1 , . . . , U n × V n be a finite subcover of supp f . Let G = V 1 ∪ . . . ∪ V n , a compact in Y . For each x ∈ X and each y / ∈ G we have(x, y) / ∈ supp f , so f (x, y) = 0. This means that f y = 0 for each y / ∈ G. Then T ρ ( f )(y) = ρ( f y ) = 0 for each y / ∈ G. Hence, T ρ ( f ) ∈ C c (Y ).
We would like to know whether η × ρ = ρ × η. We shall see later that, unlike the case of linear functionals, this is not usually the case. For

Remark 4.3 (1)
If μ is the compact-finite topological measure corresponding to quasiintegral ρ, ν is the compact-finite topological measure corresponding to η, let ν ×μ and μ×ν be the compact-finite topological measure corresponding to quasi-integrals η×ρ and ρ × η, respectively. Using part (2) of Proposition 4.2 and part (II) of Remark 2.2 it is easy to see that By Definition 4.3, Theorem 2.2, and Proposition 4.1 the opposite inequalities also hold, so (2) It is easy to see that if η and ρ are homogeneous, then so are η × ρ and ρ × η; and that if η and ρ are positive, then so are η × ρ and ρ × η. Yet, ρ × η and η × ρ are not always quasi-integrals. The criterion for η × ρ to be a quasi-integral is given in Theorem 4.1.

Theorem 4.1 (1) If the compact-finite topological measure corresponding to ρ assumes more than two values and η × ρ is a quasi-integral, then η is linear. (2) If η is linear or ρ is simple then η × ρ is a quasi-integral. (3) Suppose η and ρ are quasi-integrals. η × ρ is a quasi-integral iff η is linear or ρ is almost simple.
Proof 1. Our proof follows part of [13,Theorem 1]. It is given for completeness and because intermediate results from the proof are needed elsewhere in the paper. Suppose that η × ρ is a quasi-integral and the compact-finite topological measure μ corresponding to ρ assumes more than two values. By Lemma 3.1 choose For any y ∈ Y , (g(y) f 1 )(h(y) f 2 ) = 0, so using Remark 2.3 we have: i.e.
Since η × ρ is a quasi-integral and ( f 1 ⊗ g)( f 2 ⊗ h) = 0, using Proposition 4.2 and Remark 2.3 we see that Thus, η is linear on C c (Y ).  If μ is a compact-finite topological measure on X (with corresponding quasi-integral ρ) and ν is a compact-finite topological measure on Y (with corresponding quasiintegral η) we may define a compact-finite product topological measure ν × μ on X × Y (corresponding to quasi-integral η × ρ) if either ν is a measure or μ is an almost simple topological measure. Similar results hold for topological measure μ × ν corresponding to quasi-integral ρ × η.

Theorem 4.2 If ρ and η are simple quasi-integrals, then so is η × ρ.
Proof Assume that ρ and η are simple.
contains the ranges of f and T ρ ( f ). Since ρ is simple, by part (iii) of Theorem 3.1 we have Proof If ρ and η are almost simple, write ρ = cρ , η = kη where c, k > 0 and ρ , η are simple quasi-integrals. By Lemma 4.1 and Theorem 4.2 η × ρ = ck(η × ρ ) is almost simple. Now assume that η × ρ is almost simple. Write η × ρ = cξ where c > 0 and ξ is a simple quasi-integral. Suppose that neither of ρ, η is almost simple. By Lemma Since (g 1 ⊗ h 1 ) · (g 2 ⊗ h 2 ) = 0, by Remark 2.3 functions g 1 ⊗ h 1 , g 2 ⊗ h 2 belong to the same singly generated subalgebra. Since η × ρ is simple, it is multiplicative on this subalgebra and then using Proposition 4.2 we have: Then The contradiction shows that at least one of ρ, η must be almost simple. If ρ is almost simple and η is not almost simple, we pick g ∈ C + c (X ) such that ρ(g) = 1. With h 1 , h 2 as above, and g instead of g 1 , g 2 , the argument above shows that we again obtain ξ(g 1 g 2 ⊗ h 1 h 2 ) = 1/c and ξ(g 1 g 2 ⊗ h 1 h 2 ) = 0. Thus, the case "ρ is almost simple, and η is not almost simple" is impossible. Similarly, the case "η is almost simple and ρ is not almost simple" is impossible. Therefore, both ρ and η are almost simple.
We extend Theorem 4.2 to: Theorem 4.4 Suppose ρ, η, and η × ρ are quasi-integrals. If any two of them are simple, then so is the third one.

Products of topological measures and Fubini's theorem
The next two theorems describe how ν × μ acts on sets.
If μ is finite, for a compact set K in X × Y we also have (y). The argument follows that in [13,Theorem 2].

Proof We shall show that for any open
Observe that the function y → μ(U y ) is lower semicontinuous, hence, νmeasurable. [Indeed, suppose μ(U y ) > α, and choose compact K ⊆ μ(U y ) such that μ(K ) > α. Since K × {y} ⊆ U , there is a neighborhood V of y such that K × V ⊆ U . Then for y ∈ V we have K ⊆ U y , so μ(U y ) > α.] Let f ∈ C c (X × Y ), supp f ⊆ U . Then f y ∈ C c (X ) and supp f y ⊆ U y for each y ∈ Y . Thus ρ( f y ) ≤ μ(U y ) and (η×ρ) For each y choose a compact K (y) ⊆ U y such that μ(K (y)) > g(y) − . Since K (y) × {y} ⊆ U , there is a neighborhood (with compact closure) V (y) of y such that K × V (y) ⊆ U and |g(y) − g(z)| < for z ∈ V (y). Choose V (y 1 ), . . . , V (y n ) that cover supp g, and let E = n i=1 K (y i ) × V (y i ). Then compact E ⊆ U and we choose f ∈ C c (X × Y ) such that 1 E ≤ f , supp f ⊆ U . If y ∈ supp g, say, y ∈ V (y i ), then K (y i ) ⊆ E y and 1 K (y i ) ≤ f y . Then by part (II) of Remark 2.2 If ν is finite, for K compact in X × Y we also have: Then for y ∈ K , say, y ∈ W (y i ), we have f y = 1 on C(y i ), so 1 = μ(C(y i )) ≤ ρ( f y ) = T ρ ( f )(y). Thus T ρ ( f ) = 1 on K . Then by part (II) of Remark 2.2 Taking the supremum over K ⊆ B(U ) shows that ν(B(U )) ≤ (ν × μ)(U ).
for any open set U ⊆ X × Y . When ν is finite, the formula for compact K can be proved as in Theorem 5.1. Proof We first prove (as in [13,Corollary 3]) that if μ and ν are simple but not measures then η × ρ = ρ × η. If μ and ν are not measures, by Theorem 2.1 they are not subadditive, and we may find open sets U , If μ and ν are almost simple but not measures, write μ = cμ , ν = kν , where c, k > 0 and μ , ν are simple, but not measures. With simple quasi-integrals ρ , η corresponding to μ , ν we have η × ρ = ck(η × ρ ) = ck(ρ × η ) = ρ × η.
Both η × ρ and ρ × η are quasi-integrals. From Theorem 4.1 we see that this happens only when (a) both μ and ν are measures, or (b) at least one of μ or ν is a positive scalar multiple of a point mass, or (c) both μ and ν are almost simple, but not measures. The first two cases produce η × ρ = ρ × η, by Fubini's Theorem or Lemma 5.2. In the last case (c), by Lemma 5.1 η × ρ = ρ × η. This finishes the proof.

Remark 5.1
As in the compact case (see [13,p. 2166]), we have the following interesting phenomenon: if μ and ν are almost simple, but not measures, then ν × μ and μ × ν are different, even though they agree on rectangles. This holds even when X = Y and μ = ν. Linear combinations d(μ × ν) + (m − d)(ν × μ), where m = μ(X ) 2 and 0 ≤ d ≤ m, give uncountably many topological measures that agree on rectangles, but are distinct. This is impossible for measures on product spaces, as they are determined by values on rectangles.