Title: Bisymmetric functionals revisited or a converse of the Fubini theorem

We observe that bisymmetry is in fact the assertion of the Fubini theorem and we describe the form of general bisymmetric operations on some function spaces. Mathematics Subject Classification. 39B52, 43A07, 47A67.


Introduction
Let us observe that the well known arithmetic mean may be treated as a functional on the space of all functions from Z 2 to R. In fact, if x : Z 2 −→ R is an arbitrary function then the arithmetic mean M (x) is given by It is rather easy to observe that M satisfies the following conditions: (a) M is linear, for every c ∈ R, here 1 : G −→ R is defined by 1(t) = 1, t ∈ G; (f) monotonic, i. e.
x ≥ 0 =⇒ M (x) ≥ 0. (1.2) Now, from (a) we infer that Moreover, because of (f) ((1.2), which is a consequence of (b)) we obtain k∈{0,...,n−1} Further, because of (c) we get Hence, if M satisfies (a), (f) and (c) then it has to be the arithmetic mean. Therefore, we have to develop some other instruments to investigate means on finite groups. It turns out that linearity is in a sense too much -it implies the form (1.3). We know several means that are not linear, but bisymmetric, i.e. satisfy the functional equation (B)

History
In the 1940's Aczél [1, p. 281] proved the following theorem on the form of bisymmetric means. (Without symmetry M is given by where q ∈ (0, 1).)

Remark 1.2.
It is noteworthy that the formula expressing bisymmetry is actually a form of the equality appearing in the celebrated Fubini theorem on iterated integrals.
A more general result was obtained by Münnich et al. [11] (Theorem 2). and (1.5) Then there are strictly monotone increasing, continuous function ϕ : I → R and constants α 0 , . . . , α n−1 ∈ (0, 1) such that There is also another result in this spirit. Let A, B ∈ R, A < B and denote by D (A, B) the family of all distribution functions F : R −→ [0, 1] such that • F is constant in stretches and has only a finite number of discontinuities; Note that for every ξ ∈ [A, B] the function E ξ given by E ξ (x) = 1 2 (1 + sgn(x − ξ)) belongs to D (A, B). The following was proved by de Finetti [5] (cf. Hardy et al. [7, p. 158]).

Then there is a function φ, continuous and strictly increasing in [A, B], for which
Conversely, if M is defined by ( ), for a φ with the properties stated, then it satisfies (a), (b) and (c), so that these conditions are necessary and sufficient for the representation of M in the form ( ).

Consistent aggregation
One of the problems in which bisymmetric operations naturally arise is the so called consistent aggregation, appearing in the procedure of aggregating outputs of different producers who, with different production functions, make several goods. In this connection the following equation comes up (cf. e.g. [2]) x(0, ·)) , . . . , G (n, x(n − 1, ·))) .
Then for every x : Now, it is clearly seen that "consistent aggregation" is in fact the "Fubini theorem".
Finally, let us mention the quite recent papers by Leonetti et al. [8] and G lazowska with the former authors [6] where a question of the commutativity of integral means was considered. The results are also in the spirit of the Fubini theorem. We shall use one of their theorems to prove our main result.
In the sequel we will formulate a proposition which characterizes means satisfying bisymmetry or, as we have tried to convince the reader, the assertion of the Fubini theorem. We prove Proposition 3.1 for families of finite valued functions and then extend its result to the case of some function spaces, in which finite valued functions form a dense subset (cf. Corollaries 3.1, 3.2 and 3.3). In order to do this however, we need to introduce some notations and definitions. Introduce a relation ≤ in M (Ω) (M + (Ω)) in the following way. If f, g ∈ M (Ω) (M + (Ω)) then

Notations and definitions
Actually, ≤ is a preorder in M (Ω) (M + (Ω)), since f ≤ g and g ≤ f imply that f = g μ-almost everywhere. Also, we say that for f, g ∈ M (Ω) (M + (Ω)) in other words For each n ∈ N we call a sequence of sets (A 0 , . . . , A n−1 ) ⊂ A n satisfying a partition of Ω. The family of all partitions of Ω will be denoted by P(Ω) in the sequel. Observe that x k 1 A k : n ∈ N, We will also consider the following families of functions x : Ω × Ω −→ R : x ij 1 Ai×Bj : m, n ∈ N, (A 0 , . . . , A n−1 ), Our aim is to study functionals defined on families of measurable functions, containing the finite valued ones. First, let us define the properties of functionals in question. Let . Take two functions from F (Ω) (F + (Ω)) corresponding to the partition, say Assume that x < y. In view of (2.2) we get that x k ≤ y k , for each k ∈ {0, . . . , n − 1}, unless μ(A k ) = 0. For some k 0 ∈ {0, . . . , n − 1} we have to get x k0 < y k0 , and the latter has to occur together with μ(A k0 ) > 0. Remark 2.2. As usual, we will not distinguish functions that are equal μalmost everywhere, cf. the assumption (2.6). Therefore in the definition of F (Ω) and F + (Ω) , we shall restrict ourselves to the case of such partitions More exactly, from the equivalence class of functions equal μ-a.e. we shall pick those which correspond to partitions not including nullsets. It is always possible, just by joining the nullsets to the sets of positive measure and taking the corresponding value. In the sequel we will deal also with some partitions with sets of measure zero as components, this will be the case for instance if we take a partition which results as the intersection of two partitions. More exactly, if Π = (A 0 , . . . , A n−1 ) and Π = (B 0 , . . . , B m−1 ) are two partitions of Ω, then Then the function y corresponds to a partition containing sets with positive measure only. We will use the partition Π ∧ Π in the proof of Proposition 3.1.
Let us introduce a new definition.
and if x(s, ·) ∈ F for every s ∈ Ω then we put We say that a function x : Ω × Ω → R is admissible if x(·, t) ∈ F for every t ∈ Ω and x(s, ·) ∈ F for every s ∈ Ω. We show the following.
. Then x is admissible and and We can write it in the form which proves our Lemma. Similarly we prove that (cf. (2.10)) any function which is reflexive and strictly increasing, for instance suppose that (2.14) Let us choose arbitrary , defined by (2.11). We get by (2.12)) and (2.13 In view of (2.15) we get Moreover, the functions F, G are strictly increasing, which follows from the assumption (2.14) (cf. Remark 2.1). We could now use results on generalized bisymmetry from [2] or [9] but we will apply a slightly different method.
We will use in the sequel the following. Let us conclude the present section with the following remark.
Remark 2.4. We are going to give a representation of functionals M satisfying some additional conditions. We will start with determining M (y) for finite valued functions y : Ω −→ R(R + ). To determine M (y) we will use the Theorem 1.2. However, this theorem works "separately" in every dimension n ∈ N. The dimension is strictly connected with the partition generating y, i.e. such partition Π = (A 0 , . . . , A n−1 ) ∈ P(Ω) that y = n−1 i=0 x i 1 Ai . However, any given finite function may be generated by different partitions. For instance the function y = (x 0 , x 1 , x 1 ) is generated by partitions (A 0 , A 1 , A 2 ) and (A 0 , Ω \ A 0 ). Therefore in the sequel we have to prove that the definition of M (y) is independent of the partitions generating y, in particular that we may use the same generating function ϕ for every partition. We will do it in the proof of Proposition 3.1 below.

Finite valued functions
First, let us prove the following.
Proof. I. Our aim is to apply Theorem 1.2 to prove our result, with I = R (R + .) Let us start with a partition Π = (A 0 , . . . , for all (x 0 , . . . , x n−1 ) ∈ R n (R n + ). Let us note that (3.4) holds for all functions y ∈ E Π ⊂ F(Ω)(F + (Ω)), where E Π is the linear space of functions generated by the partition Π in the sense that In particular, if μ(A 0 ) ∈ (0, 1) then formula (3.4) holds for any function w (s,t) ∈ E Π defined as follows.
From (3.4) we infer that for every s, t ∈ I the equality holds.
We are going to show that in (3.4) we may consider a function ϕ instead of ϕ Π , in other words we are going to show that the ϕ Π in the value of M (y) is independent of the partition generating y (cf. Remark 2.4). Indeed, suppose that there is a partition Π = (B 0 , . . . , Similarly as above we can prove that there are a strictly increasing and continuous function ϕ Π : R −→ R (ϕ Π : R + −→ R) and positive numbers β Π ,j , j ∈ {0, . . . , m − 1} such that for all (x 0 , . . . , x m−1 ) ∈ I m , or for all y ∈ E Π . Since where δ 0 : = j∈J γ Π∧Π ,0j . Combining (3.10) with (3.6) (cf. (3.8)) and denoting ω = ϕ • ϕ −1 Π we obtain the equation for all s, t ∈ I or, after substituting u: = ϕ Π (s) and v: = ϕ Π (t) (3.11) for all u, v ∈ ϕ Π (I) = :K, where K is a real non-degenerate interval. Applying a trick used by Daróczy and Páles [4] we see that for all u, v ∈ K. This yields for some c > 0 and d ∈ R and every u ∈ K. Therefore, recalling the definition of ω we get (3.12) for every ξ ∈ R(R + ). In an analogous way we obtain (3.13) for some constants c > 0 and d ∈ R, and all ξ ∈ R(R + ). It easily follows from (3.12) and (3.13) that (cf. (3.8)) for every y ∈ F(Ω), independently of its representation. In other words, we get that the same ϕ is enough to represent M (y), for every y ∈ F(Ω)(F + (Ω)). It remains to deal with the case μ(A 0 ) = 1 but then y = c1 a.e. and thus M (y) = c in view of (2.7), and it is a simple exercise to show that this value is independent of the partition generating y (and moreover M (y) = ϕ −1 (1 · ϕ(c)) for any strictly increasing and continuous function ϕ : I → R).

II.
Suppose now that y, z ∈ F(Ω)(F + (Ω)) are two different functions. Up to now we have checked that M (y) and M (z) are generated by some monotone functions ϕ and ψ, respectively. We are going to show that actually ϕ and ψ are conjugated. To prove it we shall use the assumed equality (2.15) to the function x ∈ F(Ω × Ω)(F + (Ω × Ω)) given by In view of (2.12) and (2.13) we obtain Let us take y = y 0 1 A0 + y 1 1 Ω\A0 and z = z 0 1 B0 + z 1 1 Ω\B0 (it is rather obvious that it is enough to restrict our attention to this special case). Then (2.15) becomes y 1 z 1 )) .
Substituting s = y 0 z 0 , t = y 1 z 0 , u = y 0 z 1 , and v = y 1 z 1 above we get However, this is an equation that was treated by Leonetti et al. [8] in a slightly more general situation (cf. also Theorem 2 in [6]). They showed (cf. Lemma 3 in [8]) that there exist real constants a, b with a = 0 such that A straightforward calculation shows that both ϕ and ψ generate the same M, more exactly we have M (y) = ϕ −1

Ad 2.
As to the additivity of P , consider the following two partitions of Ω: for some disjoint sets A 0 , A 1 ∈ A. We may assume w.l.o.g. that μ(A 0 )μ(A 1 )μ (Ω \ (A 0 ∪ A 1 )) > 0. Then for every s, t ∈ R (R + ) we have (cf. (2.16) or (2.17)) which yields (cf. (3.14) applied to the left-hand side) for all s, t ∈ R (R + ). On the other hand we get for some β 0 , β 1 and all s, t ∈ R(R + ). From (3.17), (3.17) and (3.18), and the properties of ϕ we get which means in particular that and yields the required additivity of P .
It follows that P (A) > 0. To see that the converse also holds, suppose that for some A ∈ A we have P (A) > 0 and μ(A) = 0. Then μ(Ω\A) > 0 and, as it follows from the first part of the present argument, P (Ω \ A) > 0. Let us take s < t, u, v ∈ R (R + ) and define two functions f, g : Ω −→ R by f = u1 A + s1 Ω\A and g = v1 A + t1 Ω\A . Then f < g in the sense of (2.1). Thus M (f ) < M(g) by monotonicity which implies whence it follows that Taking v < u and letting s −→ t in (3.19) we get

Arbitrary case
We have proved that a continuous, bisymmetric and reflexive functional M which satisfies (2.15) is given by (3.3) for any y ∈ F(Ω) (F + (Ω)). If the set F(Ω) (F + (Ω)) is dense in a function family F (or at least any element of F can be approximated by a sequence from F(Ω) (F + (Ω))) then sometimes, due to the continuity of M, ϕ and ϕ −1 we may pass with lim under the sign of , and we get formula (3.3) for any y ∈ X. Below we present two such cases: the space of bounded functions or the space of integrable functions. Proof. In the space F(Ω) we have defined the functional M by the formula Now, this functional can be uniquely extended to a continuous functional onto the whole B(Ω, R) because F(Ω) is a dense subspace of B(Ω, R). Indeed, for every x ∈ B(Ω, R) there exists a sequence (x n ) n∈N ∈ F(Ω) N , which converges uniformly to x. The sequence (ϕ•x n ) n∈N is then uniformly convergent to ϕ•x, due to the uniform continuity of ϕ in any compact subset of R. Thus we get (considering the finiteness of P (Ω) and continuity of ϕ −1 ) which was to be proved. It remains to show that (3.2) holds (with A = 2 Ω ) but we have already proved it while showing Proposition 3.1. If, additionally, we assume that ϕ is bounded then M is given by (3.3) for every y ∈ L 1 (Ω, R), and P is a probability measure equivalent to μ.
Proof. An additional assumption on ϕ is needed because we want the composition ϕ • y to be in L 1 (Ω, R).
We know that M (y) is given by (3.3) for every y ∈ F(Ω) : it follows from Proposition 3.1. But F(Ω) is a dense subset of L 1 (Ω, R) and therefore for each y ∈ L 1 (Ω, R) there exists a sequence (y n ) n∈N ∈ F(Ω), converging to y in the norm of L 1 . Using the continuity of M and the fact that ϕ is a homeomorphism, we get henceforth If μ is a measure, then we obtain the σ-additivity of P in the following way. Suppose that A = ∞ n=1 A n where A n ∈ A are pairwise disjoint. Fix an N ∈ N and consider the following finite valued function z N = s1 N n=1 An +t1 Ω\ N n=1 An . Then Since lim N →∞ z N = s1 A + t1 Ω\A pointwise (but also in L 1 -norm) we have by the continuity of ϕ and M that  Without any assumption on ϕ we get the following result.

Final remarks
At the 53rd International Symposium on Functional Equations J. Chudziak (cf. [3]) called our attention to the fact that there is a connection between the above results and utility theory. Namely, if u is an arbitrary utility function (i.e. continuous and strictly increasing), we say that λ ∈ R is a certainty equivalent of a gamble X ∈ χ + iff u(λ) = E(u(X)), is called equivalence functional.
Thus we can conclude our present considerations by the following.  21) for every X ∈ χ + , where u : R + → R is a utility function and E is the expectation taken with respect to some probability measure P, equivalent to μ.
We also have Corollary 3.5. Let (Ω, A, μ) be a probability space. Let M : L 1 (Ω, R + ) −→ R be reflexive, μ-strictly increasing, continuous and bisymmetric. Then, according to Proposition 3.1, there exist a probability measure P equivalent to μ and a continuous and strictly increasing function u : R + → R such that M is given by (3.21) for all X ∈ F + (Ω). If, additionally, u is concave and non-negative then M is given by (3.21) for all X ∈ L 1 (Ω, R + ).