Theory and Applications of Categories, Vol. 23, No. 12, 2010, pp. 243–250.

A CATEGORICAL APPROACH TO INTEGRATION

REINHARD BÖRGER

Abstract. We present a general treatment of measures and integrals in certain (monoidal closed) categories. Under appropriate conditions the integral can be defined by a universal property, and the universal measure is at the same time a universal multiplicative measure. In the multiplicative case this assignment is right adjoint to the formation of the Boolean algebra of idempotents. Now coproduct preservation yields an approach to product measures.

1. Introduction

Integration is a map that assigns to each integrable function its integral. If we exclude improper integrals and admit signed measures (which may be negative), integration is a linear map from some vector space LB of integrable functions to ℝ. It is desirable that every function in this space can be integrated with respect to every bounded measure; thus we consider spaces of bounded measurable functions. Now we observe that mapping each set to its indicator function (sometimes called characteristic function) also yields a charge (i.e. a finitely additive set function) from B to a set of simple functions (i.a. measurable functions attaining only finitely many values) on B. Moreover it is a given by a universal property. Naturally this is not restricted to real-valued measures but also works for vector measures. The integral of a function x with respect to the “indicator function measure” is x itself.

Furthermore, a map m : B → ℝ has a linear extension to the vector space of all simple functions if and only if m is a charge. It is possible to introduce topologies on the spaces such that integration is the unique continuous linear extension of this map. It is convenient to work with abstract Boolean algebras rather than set algebras. Then certain analogues of function spaces can be defined by the universal property, though their elements can no longer be interpreted as functions in the usual way. So integration becomes a morphism in a suitable category rendering certain diagrams commutative. A measure should be a map from some internal Boolean algebra B over a reasonable category X to some object in some additive category A; its objects should carry a vector space structure. Then the set M(B, A) of measures from some (internal) Boolean algebra B to A is in bijection with A(LB, A). Convergence theorems like dominated and monotone convergence should mean continuity in certain topologies; these should be encoded in X. It is convenient to work in a cartesian closed category (e.g. sequential spaces); this yields nice spaces of measurable functions and nice spaces of measures. Then σ-additivity can be interpreted as finite additivity together with continuity in an appropriate topology; this leads to a special choice of the object of A. Moreover, it is often useful to impose certain completeness conditions on the target spaces of the measures. In the classical situation each function this enforces inclusion of bounded measurable functions, because they are uniform limits of step functions and thus limits in the given topology.

Dedicated to Dominique Bourn on the occasion of his sixtieth birthday

Received by the editors 2009-05-30 and, in revised form, 2010-04-06.

Published on 2010-04-28 in the Bourn Festschrift.

2000 Mathematics Subject Classification: 06E05 16A32, 18A15, 18A30, 18A35, 18A40, 18E05, 28A30, 28A33, 28A40, 28A45, 46G10.

Key words and phrases: internal Boolean algebra, universal measure, multiplicative measure, product measure, Boolean algebra of idempotents, symmetric monoidal closed category, cartesian closed category.

\begin{matrix}c \\ ◯ \end{matrix} Reinhard Börger, 2010. Permission to copy for private use granted.

In the preprint [3] we give a tratment in some special categories as in the author’s habilitationsschrift [1] based on Linton’s [5] approach. Here we want to reveal the categorical reasons behind the machinery and point out which conditions are necessary for which conclusions. This more flexible approach allows us also to consider other sorts of convergence and to see what holds in a more general context, e.g. for positive measures, if we consider only semi-additive categories rather than additive ones. The main categorical ideas become clear without convergence; for X the category of sets and A the category of modules over a commutative unital ring the results are presented in [2].

If A is a symmetric monoidal closed category (i.e. has an appropriate tensor product), there is a canonical notion of a commutative monoid in A; this is a triple (A, r, u) of an A ∈ A and A-morphisms r : A ⊗ A → A (“multiplication”) and u : I → A (“unit”), subject to appropriate axioms. Then the universal measure turns out to be also a universal multiplicative measure, i.e. the set M^{×}(B, A) of multiplicative measures from some object B ∈ |B| for the category B of (internal) Boolean algebras in X to some A ∈ |A| is in bijection with B(L̂B, A), where L̂B is LB with some canonical algebra structure corresponding to pointwise multiplication. For valuable comments on monoidal categories the author is indebted to his deceased colleague Harald Lindner. Here a measure is called multiplicative if it maps the top element to the unit element of the algebra and binary intersections to products. In general, multiplicative measures are quite rare; e.g. real-valued ones correspond to ultrafilters.

But M^{×}(B, A) is also in bijection with the set of all Boolean morphisms from B to the Boolean algebra ΘA of idempotents of A. This yields an adjunction L̂ ⊣ Θ; thus the left adjoint functor L̂ preserves colimits, in particular binary coproducts. So for the coproduct B_{0} ˇ⊗B_{1} of two Boolean algebras B_{0}, B_{1} we obtain an isomorphism L(B_{0} ˇ⊗B_{1}) ≅ LB_{0} ˇ⊗LB_{1}. This allows us to define product measures, and we obtain a version of Fubini’s Theorem.

2. Measures and bimeasures

In a category X with finite products we can define the category B of internal Boolean algebras, we assume every Boolean algebra to have a top element 1. An object B = (\underline{B}, t_{∨}, t_{∧}, t_{0}, t_{1}, t_{⊥}) ∈ |B| is given by an object \underline{B} ∈ |X | and X -morphisms t_{∧}, t_{∨} : \underline{B} ×\underline{B} → \underline{B}, t_{0}, t_{1} : 1 → \underline{B}, t_{⊥} : \underline{B} → \underline{B}, where the last one corresponds to the complementation x ↦ x^{⊥}. These morphisms should satisfy the usual axioms, i.e. for each X ∈ X the hom-set X (X, \underline{B}) is a Boolean algebra (i.e. a complemented distributive lattice).

Now let A be a semi-additive category, i.e. a category enriched over the symmetric monoidal category of commutative monoids. Assume that A has finite products; note that by semi-additivity they are also finite coproducts. Let U : A → X be a functor that preserves finite products; think of U as a forgetful functor. This induces a canonical structure of a commutative monoid on X (X, UA) for all X ∈ X, A ∈ A. If A also has coequalizers, then it has all finite colimits; for additive A it suffices that A has cokernels.

For B = (\underline{B}, t_{∧}, t_{∨}, t_{0}, t_{1}, t_{⊥}) ∈ |B|, A ∈ |A|, we call an X -morphism m : B → UA an A-valued measure on B if it satisfies mt_{0} = m.Uo and m.U(p + q) = m.t_{∨} + m.t_{∧}, where the addition on X (\underline{B}, UA) is as above, p, q : A ⊕ A → A are the canonical projections, and o : 0 → A is the unique morphism in the semi-additive category A (with zero object, i.e. empty product 0). M(B, A) denotes the set of all A-valued measures on B; then M is a functor from B^{op} × A into the category of sets. Since U preserves finite products, it follows immediately that for all A, A′ ∈ |A| the induced map A(A, A′) → X (UA, UA′) is a homomorphism of commutative monoids. Now for each B ∈ |B| the set-valued functor M(B, −) on A is a finite limit of representable functors.

2.1. Theorem. If the functor U from a semi-additive category A with finite colomits into a category X has a left adjoint F : X → A, then for each B = (\underline{B}, t_{∧}, t_{∨}, t_{0}, t_{1}, t_{⊥}) ∈ B the functor M(B, −) is represented by the codomain LB of the joint coequalizer of the pairs id_{\underline{B}⊕\underline{B}}, (t_{∨} + t_{∧}) : \underline{B} ⊕ \underline{B} → \underline{B} and o, t_{0} : 0 → \underline{B}. In particular, M(B, −) is representable, and L : B → A is a functor.

Proof. By adjunction, for each X ∈ |X | the functor X (X, U(−)) is represented by F X ∈ |A|. Therefore the above colimit in A induces a limit of representable functors, that yields that M(B, −) is represented by LB. The functoriality of F follows from the Yoneda Lemma.

F B can be viewed as a space of integrable functions on B. Representability of M(B, −) can be interpreted as a universal property of the measure χ_{B} ∈ M(B, LB) that corresponds to the identity morphism on B. Then χ_{B}(x) plays the role of the indicator function of x. The induced map LB → A can be viewed as integration with respect to m; usually it is a continuous linear extension; a suitable completeness condition allows to extend the integral to more general measurable functions than just step functions. σ-Additivity means the same as finite additivity plus continuity in a reasonable sequential topology. So it might be appropriate to work in a category X of topological spaces. This approach automatically leads to vector integration, since A = ℝ is not necessary. Observe that integration along χ_{B} is the identity morphism.

Now we enrich the structure of M(B, A). First observe that M(B, A) is isomorphic to A(LB, A) and a commutative submonoid of X (\underline{B}, UA) ≅ A(F B, A). If A is an additive category, M(B, A) is even always an abelian group. If X is cartesian closed, we can define an object of measures as a subobject of the cartesian power (UA)\underline{B} ∈ |X |; this subobject is mapped to M(B, A) ⊂ X (\underline{B}, UA) by U if U is represented by the terminal object. Now assume that we have a symmetric monoidal structure on A with a functor ⊗ : A×A → A, a unit object I ∈ |A|, natural associativity morphisms α(A_{0}, A_{1}, A_{2}) : (A_{0} ⊗ A_{1}) ⊗ A_{2} → A_{0}⊗(A_{1}⊗A_{2}) for all A_{0}, A_{1}, A_{2} ∈ |A|, unit isomorphisms ρA : A⊗I → A, λA : I⊗A → A and a symmetry σ(A_{0}, A_{1}) : A_{0} ⊗ A_{1} → A_{1} ⊗ A_{0} for A_{0}, A_{1} ∈ |A|. Then we can use the above isomorphism and obtain a function space object [L̂B, A] in A.

By a symmetric monoidal functor we mean a functor U : A → X together with a natural transformation β that assigns to each (A_{0}, A_{1}) ∈ |A × A| an X -morphism β(A_{0}, A_{1}) : UA_{0} × UA_{1} → U(A_{0} ⊗ A_{1}) and also with an X -morphism j : T → UI, where T ∈ |X | is the terminal object (i.e. the unit object in the cartesian symmetric monoidal (closed) structure), subject to the following conditions: Uα(A_{0}, A_{1}, A_{2}).β(A_{0} ⊗ A_{1}, A_{2}).β(A_{0}, A_{1}) × id_{UA_{2}} = β(A_{0}, A_{1} ⊗ A_{2}).id_{UA_{0}} ⊗ β(A_{1}, A_{2}).t for A_{0}, A_{1}, A_{2} ∈ |A| and the canonical associativity isomorphism t : (UA_{0} × UA_{1}) × UA_{2} → UA_{0} × (UA_{1} × UA_{2}). Moreover, UλA.β(I, A).j × id_{UA} and UρA.β(A, I).id_{UA} × j are the canonical projection isomorphisms T × UA → UA and UA × T → UA for all A ∈ |A|; moreover, we assume Uσ(A_{0}, A_{1}).β(A_{0}, A_{1}) = β(A_{1}, A_{0}).s for A_{0}, A_{1} ∈ |A| and the canonical interchange isomorphism s : A_{0}×A_{1} → A_{1}×A_{0}. This can be done analogously for an arbitrary symmetric monoidal structure on X (not only for the cartesian one). A symmetric monoidal functor always induces a functor between the corresponding categories of commutative monoids; in our case a commutative monoid (A, r, u) in A (with A ∈ |A|, r : A⊗A → A, u : I → A) is mapped to the commutative monoid (UA, Ur.β(A, A), Uu.j).

For a left adjoint F ⊣ U we obtain A-morphisms F (X_{0} ×X_{1}) → F (UF X_{0} ×UF X_{1}) → F U(X_{0} ⊗ X_{1}) → X_{0} ⊗ X_{1}, where the first arrow is given by the units X_{i} → UF X_{i}; the second one is F β(F X_{0}, F X_{1}), and the third one is the counit. Moreover, the A-morphism F j : F T → F UI followed by the counit F UI → I yields an A-morphism F T → I. In this way, F becomes a symmetric comonoidal functor, i.e. the analogues of the above equations hold in A^{op}. If the above natural morphisms F (X_{0} × F X_{1}) → F X_{0} ⊗ F X_{1} and F T → I are isomorphisms, their inverses yield a symmetric monoidal structure on F, hence F and U are both symmetric monoidal functors. In this case we call F ⊣ U a symmetric monoidal adjunction; it induces an adjunction between the corresponding categories of monoids (cf. [4]). Note that β need not be a natural isomorphism, and j need not be invertible either.

For B_{0}, B_{1} ∈ B and A ∈ A an element of M(B_{0}, [LB_{1}, A]) corresponds to an Amorphism LB_{0} → [LB_{1}, A] by adjunction; the monoidal structure on A yields a corresponding A-morphism LB_{0}⊗LB_{1} → A. If the above joint coequalizers c_{i} : F B_{i} → LB_{i} exist for i ∈ {0, 1}, c_{i} corresponds to χ_{B_{i}} : B_{i} → ULB_{i}. Composing U[L(c_{0}⊗c_{1}), A] with our above A-morphism B_{0} → U[LB_{0} ⊗ LB_{1}, A] yields another A-morphism LB_{0} ⊗ LB_{1} → A.

On the other hand, if X is cartesian closed, U[LB_{1}, A] is canonically equivalent to a subobject of (UA)^{\underline{B}^{1}} the X -morphism ŵ : \underline{B}_{0} → U[LB_{1}, A] yields a morphism \underline{B}_{0} → (UA)^{\underline{B}^{1}}; by cartesian closedness this corresponds to some w : \underline{B}_{0} × \underline{B}_{1} → UA. Since ŵ is a measure, we get w.(t_{∨,0} + t_{∧,0}) × id_{\underline{B}_{1}} = w.(p_{0} + q_{0}) × id_{\underline{B}_{1}} and wt_{0,0} = o; we might say roughly that b is a measure in the first component. Since the codomain of ŵ is the object [LB̂_{1}, A] of measures, we also obtain b.(id_{\underline{B}_{0}} ×(t_{∨,1}+t_{∧,1})) = w.(id_{B_{0}} ×(p_{1}+q_{1})) and bt_{0,1} = o, i.e. w is also a measure in the second component (for B_{i} = (\underline{B}_{i}, t_{∧,i}, t_{∨,i}, t_{0,i}, t_{1,i}, t_{⊥,i}). So we call w a bimeasure in this case. Conversely, if w is a bimeasure, i.e. satisfies the above equations, we get the morphisms ŵ as above and then the morphism LB_{0} ⊗ LB_{1} → A. A bimeasure w : B_{0} ×B_{1} → ℝ with values in [0, 1] may indicate the joint probability of some event from B_{0} and some event from B_{1}; it describes their dependence. For independent Boolean algebras B_{0}, B_{1} and probability measures m_{0} : B_{0} → ℝ, m_{1} : B_{1} → ℝ we have w(x_{0}, x_{1}) = m_{0}(x_{0})m_{1}(x_{1}) for all x_{0} ∈ X_{0}, x_{1} ∈ X_{1}.

2.2. Lemma. Assume that F ⊣ U : A → X is a monoidal adjunction for the cartesian structure on a cartesian closed category X with finite limits and that A is semi-additive and has finite colimits. Then the set-valued functor that assigns to each pair of Boolean algebras B_{0}, B_{1} in B the set of A-valued bimeasures for each A ∈ A is represented by LB_{0} × LB_{1}.

3. Multiplicative measures and idempotents

If we have two Boolean algebras B_{0}, B_{1} ∈ |B| (of events), whose dependence is described by a bimeasure w, it may be useful to have common Boolean algebra and a probability measure on it. One important case is independence, (i.e. w(x, y) = m_{0}(x)m_{1}(y) for some m_{i} ∈ M(B_{i}, ℝ)); the other extremal case is B_{0} = B_{1} =: B and w(x, y) := m(x ∧ y) for some probability measure m : B → ℝ. It turns out that mt_{∧} : B × B → UA is an A-valued bimeasure on B for all B = (\underline{B}, t_{∧}, t_{∨}, t_{0}, t, 1, t_{⊥}) ∈ |B|. In particular, for the universal measure χ_{B} : B → LB, it corresponds to some r : LB ⊗ LB → LB by 2.2. Moreover, for the terminal object T of X the X -morphism t_{1} : T → \underline{B} corresponds to some A-morphism I ≅ F T → F B by adjunction; composition with the canonical morphism u : F B → LB. By a direct check we see that L̂B := (LB, r, u) is even a monoid in the symmetric monoidal category A; this follows from associativity and the unit law of ∧ in Boolean algebras.

Now assume that A is even a symmetric monoidal category and that F ⊣ U is a symmetric monoidal adjunction; then L̂ is a functor from B into the category R of commutative monoids in A. For B ∈ |B| and Â = (A, r, u) ∈ |R| we call an m ∈ X (\underline{B}, U\underline{A}) a multiplicative Â-valued measure on B if it is an A-valued measure on B and satisfies mt_{∧} = r.(m ⊗ m) and mt_{1} = uj for the canonical isomorphism j : LT → I. In the classical case of set-functions these equations mean m(x ∧ y) = m(x)m(y) for all x, y ∈ \underline{B} and m(1) = 1. Then we denote the space of all Â_{0}-valued multiplicative measures on B by M^{×}(B, Â). Multiplicative measures do not play an important role in practice; e.g. multiplicative real-valued measures correspond to ultrafilters. But the universal measure χ_{B} : B → LB is always multiplicative if the space LB is equipped with pointwise multiplication: The indicator function of the intersection of two sets is the product of the indicator functions, and the constant function with value is a unit element.

3.1. Lemma. Let A be a semi-additive symmetric monoidal closed category with finite colimits; let X be a cartesian closed category with finite limits, and let F ⊣ U : A → X be a symmetric monoidal adjunction. Then for each Boolean algebra in B the functor that assigns to each monoid Â ∈ |R| the set M^{×}(B, Â) of multiplicative Â-valued measures on B is represented by L̂B.

Proof. For m ∈ M^{×}(B, A), Â = (A, r, u) ∈ R, and the induced A-morphism l : LB → A with lχ_{B} = m we obtain lr̂ = r.(l ⊗ l) and lû = u for L̂B = (LB, r̂, û) by multiplicativity of m. Hence l is even (induced by) a monoid homomorphism, i.e. some R-morphism L̂B → A.

The functor M^{×} (a opposed to M) is also representable in the first argument if A is even an additive category (i.e. admits subtraction). Since our adjunction F ⊣ U is a symmetric monoidal adjunction, U transforms an object of R, i.e. a commutative monoid in A, into a commutative monoid in the cartesian closed category X, hence a commutative monoid in X in the usual Linton-Lawvere universally algebraic sense. Moreover the rightadjoint functor U preserves limits, in particular finite products; since A is an additive category, this leads to an addition on the corresponding X -object. This addition and the above monoid multiplication together yield an internal commutative unital ring in X.

In a commutative unital ring R, the idempotent elements (i.e. elements x ∈ R with x^{2} = x) form a Boolean algebra with 0 and 1 as in R, the meet as multiplication; the join given by x ∨ y := x + y − xy and the complement given by x^{⊥} := 1 − x. For A ∈ |R| as above, these operations also yield X -morphisms for the internal ring structure on UA. Moreover, we can restrict them to the object of idempotents; this can be constructed as the equalizer of the identity morphism of UA and r.Uβ(A, A).d where D : UA → UA×UA is the diagonal. This yields an internal Boolean algebra in X, i.e. an object ΘA ∈ |B|. Then Θ : R → B is a functor.

3.2. Theorem. For an additive symmetric monoidal closed category A with finite colimits, a cartesian closed category X with finite limits and a symmetric monoidal adjunction F ⊣ U : A → X and for each commutative unital ring A in (the given symmetric monoidal structure of) A the set-valued functor that maps each internal Boolean algebra B in X to M^{×}(B, A) is representable (by the internal Boolean algebra ΘA of idempotents).

Proof. It has to be shown that every multiplicative A-valued measure M on B induces a B-morphism (i.e. an internal Boolean homomorphism) B → ΘA. This is shown as in ordinary rings; all elements are idempotent, because multiplicativity of the measure implies m(x)^{2} = m(x ∧ x) = m(x) for all x in the Boolean algebra.

If 2 = 1+1 is invertible in a commutative unital ring R, then an R-valued charge m (i.e. an additive set function) with m(1) = 1 on a Boolean algebra B is already multiplicative if it attains only idempotent values. Indeed, for x, y ∈ \underline{B} with x ∧ y = 0 we obtain m(x) + m(y) = m(x ∨ y) = m(x ∨ y)^{2} = (m(x) + m(y))^{2} = m(x)^{2} + 2m(x)m(y) + m(y)^{2} = m(x) + m(y) + 2m(x)m(y), hence 2m(x)m(y) = 0 an thus m(x)m(y) = 0 by invertibility of 2. For arbitrary x, y ∈ \underline{B} this leads to m(x)m(y) = (m(x ∧ y) + m(x ∧ y^{⊥}))(m(x ∧ y) + m(x^{⊥} ∧y)) = m(x∧y)^{2} +m(x∧y)m(x∧y^{⊥})+m(x^{⊥} ∧y)m(x∧y)+m(x^{⊥} ∧y)m(x∧y^{⊥}) = m(x∧y)+0+0+0 = m(x∧y) because x∧y∧x∧y^{⊥} = x^{⊥}∧y∧x∧y = x^{⊥}∧y∧x∧y^{⊥} = 0. By internalization we see that an A-valued measure with mt_{1} = uj as above is multiplicative if id_{A} + id_{A} is invertible in A(A, A).

Combining the above natural isomorphisms B(B, ΘA) ≅ M^{×}(B, A) ≅ A(L̂B, A), we immediately see:

3.3. Theorem. For an additive symmetric monoidal closed category A with finite colimits, a cartesian closed category X with finite limits and a symmetric monoidal adjunction F ⊣ U : A → X, the functor Θ : R → B that assigns to every internal commutative unital ring A in A the internal Boolean algebra ΘA of idempotents is right adjoint to the functor L̂ : B → R.

So the left adjoint functor L̂ : B → R preserves colimits, in particular binary coproducts. But it is well-known that binary coproducts in R can be constructed as tensor products. The tensor product A_{0} ⊗ A_{1} of two R-objects (i.e. monoids in A) Â_{0} = (A_{0}, r_{0}, u_{0}) and Â_{1} = (A_{1}, r_{1}, u_{1}) is defined by Â_{0}⊗Â_{1} := (A_{0}⊗A_{1}, (r_{0}⊗r_{1}).s, (u_{0}⊗u_{1}).(λI)^{−1}), where s : (A_{0}⊗A_{1})⊗(A_{0}⊗A_{1}) → (A_{0}⊗A_{0})⊗(A_{1}⊗A_{1}) is the canonical interchange isomorphism from the symmetric monoidal structure that interchanges the two middle arguments; remember λI = ρI : I ⊗ I → I. The underlying A-morphisms A_{0}, A_{1} → A_{0} ⊗ A_{1} of the coproduct injections are the composites A_{0} → A_{0}⊗I → A_{0}⊗A_{1} and A_{1} → I⊗A_{1} → A_{0}⊗A_{1}, where the left-hand morphisms are the coherence isomorphisms from the monoidal structure and the right-hand ones are id_{A_{0}} ⊗ u_{1} and u_{0} ⊗ id_{A_{1}}. Combination with 2.2 yields the following

3.4. Theorem. For an additive symmetric monoidal closed category A with finite colimits, a cartesian closed category X with finite limits and a symmetric monoidal adjunction F ⊣ U : A → X and for a coproduct B_{0} ⊗ B_{1} of internal Boolean algebras in X there is a monoid isomorphism L̂(B_{0} ⊗ B_{1}) ≅ L̂B_{0} ⊗ L̂B_{1} (in particular an isomorphism L(B_{0} ⊗ B_{1}) ≅ LB_{0} ⊗ LB_{1} in A), natural in both arguments. For each A ∈ |A|, the set of A-valued bimeasures on B_{0}, B_{1} is in natural bijection with M(B_{0} ⊗ B_{1}, A).

By induction this can be extended to finitely many internal Boolean algebras. If X and A have filtered colimits, they even yield product measures of more than two measures; observe that ⊗ preserves the involved colimits in A by monoidal closedness. If we consider two measures m_{0} ∈ M(B_{0}, A), m_{1} ∈ M(B_{1}, A) for some (A, r, u) ∈ R, the bimeasure r.(m_{0} × m_{1}) : B_{0} × B_{1} → A induces a product measure on B_{0} ⊗ B_{1}; this can be extended to finitely many measures as above, if A has filtered colimits (which are preserved by all − ⊗ A because A is monoidal closed).

References

[1] R. Börger, Ein kategorieller Zugang zur Integrationstheorie, Habilitationsschrift, FernUniversität in Hagen, 1989.

[2] R. Börger, Fubini’s Theorem from a categorical viewpoint, Category Theory at Work, Heldermann, Berlin 1991, 367-375.

[3] R. Börger, What is an integral?, Hagen 2010 (preprint).

[4] G.M. Kelly, Doctrinal adjunction, Proceedings Sydney Category Sminar, 1972/1973, Lecture Notes in Mathematic, Springer, Berlin 420 (1974), 257-280.

[5] F.E.J. Linton, Functorial measure theory, Proceedings of the Confrence on Functional Analysis, Irvine 1966, Thompson, Washington D.C. (1968), 26-49.

Fakultät für Mathematik und Informatik, FernUniversität, 58084 Hagen, Germany

Email: Reinhard.Boerger@FernUni-Hagen.de

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Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Cuill Inc.: valeria@cuill.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Tom Leinster, University of Glasgow, T.Leinster@maths.gla.ac.uk Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca