Let * A* be a fixed set of axioms. Then for a given arbitrary monetary value measure

*can we make a good alternative for it? In other words, can we find a monetary value measure that satisfies*

**Ψ***and is the best approximation of the original*

**A***? For a Grothendieck topology*

**Ψ***on*

**J***, define*

**χ***to be a full sub-category whose objects are all sheaves for*

**Sh(χ, J) ⊂ Set**^{χop}*. Then, it is well known that*

**J***a left adjoint*

**∃***in the following diagram.*

**π**_{J}**Sh(χ, J) → Set ^{χop}**

**Sh(χ, J) ← _{πJ} Set^{χop}**

**π _{J}**

**(Ψ) ← Ψ**

The functor * π_{J}* is known as Sheafification functor, which has the following limit cone:

for sieves * I*,

*and*

**K***This also satisfies the following theorem.*

**U.*** 1.0* If

*is a sheaf for*

**π**_{J}**(Ψ)**

**J*** 1.1* If

*is a sheaf for*

**Ψ****, then for any**

*J**,*

**U ∈ χ**

**π**_{J}**(Ψ)(U) ≅ L(U)**The theorem suggests that for an arbitrary monetary value measure, the sheafification functor provides one of its closest monetary value measures that may satisfy the given set of axioms. To make this certain, we need a following definition.

* 2.0* Let

*be a set of axioms of monetary value measures*

**A****2.1*** M(A)* := the collection of all monetary value measures satisfying

**A****2.2*** M_{O}* := collection of all monetary value measures

* 2.3* A is called complete if

**π _{JM(A)} (M_{O}) ⊂ M(A)**

* 3.0* Let

*be a complete set of axioms. Then, for a monetary value measure*

**A***is the monetary value measure that is the best approximation satisfying*

**Ψ ∈ M**_{O}, π_{JM(A}(Ψ)*.*

**A**Let us investigate if the set of axioms of concave monetary value measures is complete in the case of * Ω = {1, 2, 3}* with a

**σ-field F := 2**^{Ω}We enumerate all possible * sub-σ-fields* of

*, that is, the shape of the category*

**Ω***,*

**χ = χ(Ω)**where,

**U _{∞} := **

**F := 2**

^{Ω}**U _{1} := {Φ, {1}, {2, 3}, Ω}**

**U _{2} := {Φ, {2}, {1, 3}, Ω}**

**U _{3} := {Φ, {3}, {1, 2}, Ω}**

**U _{4} := {Φ, Ω}**

The Banach spaces defined by the elements of * χ* are

**L _{∞} := L := L(U_{∞}) := {a, b, c | a, b, c ∈ ℜ}**

**L _{1} := L(U_{1}) := {a, b, b | a, b ∈ ℜ}**

**L _{2} := L(U_{2}) := {a, b, a | a, b ∈ ℜ}**

**L _{3} := L(U_{3}) := {a, a, c | a, c ∈ ℜ}**

**L _{0} := L(U_{0}) := {a, a, a, | a ∈ ℜ}**

Then a monetary value measure * Ψ : χ^{op} → Set* on

*is determined by the following six functions*

**χ**We will investigate its concrete shape one by one by considering axioms it satisfies.

For **Ψ ^{1}_{∞} :**

*we have by the cash invariance axiom,*

**L**_{∞}→ L_{1},**Ψ ^{1}_{∞} (a, b, c) = Ψ^{1}_{∞} ((0, b – c, 0) + (a, c, c))**

**= Ψ ^{1}_{∞} ((0, b – c, 0)) + (a, c, c)**

**= (f _{12} (b – c), f_{11} (b – c), f_{11} (b – c)) + (a, c, c)**

**= (f _{12} (b – c) + a, f_{11} (b – c) + c, f_{11} (b – c)+ c)**

where * f_{11}, f_{12} : ℜ → ℜ* are defined by

**(f**_{12}(x), f_{11}(x), f_{11}(x)) =

**Ψ**^{1}_{∞}_{ }

**(0, x, 0).**Similarly, if we define nine functions

* f_{11}, f_{12}, f_{21}, f_{22}, f_{31}, f_{32}, g_{1}, g_{2}, g_{3} : ℜ → ℜ *by

**(f _{12}(x), f_{11}(x), f_{11}(x)) = Ψ^{1}_{∞}(0, x, 0)**

**(f _{21}(x), f_{22}(x), f_{21}(x)) = Ψ^{2}_{∞}(0, 0, x)**

**(f _{31}(x), f_{31}(x), f_{32}(x)) = Ψ^{3}_{∞}(x, 0, 0)**

**(g _{1}(x), g_{1}(x), g_{1}(x)) = Ψ^{0}_{1}(x, 0, 0)**

**(g _{2}(x), g_{2}(x), g_{2}(x)) = Ψ^{0}_{2}(0, x, 0)**

**(g _{3}(x), g_{3}(x), g_{3}(x)) = Ψ^{0}_{3}(0, 0, x)**

We can represent the original six functions by nine functions

* Ψ^{1}_{∞}(a, b, c) = (f_{12}(b – c) + a, f_{11}(b – c) + c, f_{11}(b – c) + c)*,

* Ψ^{2}_{∞}(a, b, c) = (f_{21}(c – a) + a, f_{22}(c – a) + a, f_{21}(c – a) + a)*,

* Ψ^{3}_{∞}(a, b, c) = (f_{31}(a – b) + b, f_{31}(a – b) + b, f_{32}(a – b) + c)*,

* Ψ^{0}_{1}(a, b, b) = (g_{1}(a – b) + b, g_{1}(a – b) + b, g_{1}(a – b) + b)*,

* Ψ^{0}_{2}(a, b, a) = (g_{2}(b – a) + a, g_{2}(b – a) + a, g_{2}(b – a) + a)*,

**Ψ ^{0}_{3}**

**(a, a, c) = (g**

_{3}(c – a) + a, g_{3}(c – a) + a, g_{3}(c – a) + a)Next by the normahzation axiom, we have

**f _{11}(0) = f_{12}(0) = f_{21}(0) = f_{22}(0) = f_{31}(0) = f_{32}(0) = g_{1}(0) = g_{2}(0) = g_{3}(0) = 0**

partially differentiating the function in * Ψ^{1}_{∞}(a, b, c), *we have

**∂****Ψ ^{1}_{∞}**

**(a, b, c)**

**/∂**

*a = (1, 0, 0)***∂Ψ ^{1}_{∞}**

**(a, b, c)/∂b = (f’**

_{12}(b – c), f’_{11}(b – c), f’_{11}(b – c))**∂****Ψ ^{1}_{∞}**

**(a, b, c)**

**/∂**

*c =**(- f’*_{12}(b – c), 1 – f’_{11}(b – c), 1 – f’_{11}(b – c))Therefore, by the monotonicity, we have * f’_{12}*(x) = 0 and 0 ≤

*≤ 1. Then by the result of the normalization axiom, we have*

**f’**_{11}* x ∈ ℜ*,

*. Hence,*

**f**_{12}**(x) = 0***,*

**∀ x ∈ ℜ****f _{12}(x) = f_{22}(x) = f_{32}(x) = 0**

With this knowledge, let us redefine the three functions **f _{1}, f_{2}, f_{3} :**

*by*

**ℜ → ℜ****(0, f _{1}(x), f_{1}(x)) = Ψ^{1}_{∞}(0, x, 0)**

**(f _{2}(x), 0, f_{2}(x)) = Ψ^{2}_{∞}(0, 0, x)**

**(f _{3}(x), f_{3}(x), 0) = Ψ^{3}_{∞}(x, 0, 0)**

Then, we have a new representation of the original six functions

**Ψ ^{1}_{∞}(a, b, c) = (a, f_{1}(b – c) + c, f_{1}(b – c) + c)**

**Ψ ^{2}_{∞}**

**(a, b, c) = (f**

_{2}(c – a) + a, b, f_{2}(c – a) + a)**Ψ ^{3}_{∞}**

**(a, b, c) = (f**

_{3}(a – b) + b, f_{3}(a – b) + b, c)**Ψ ^{0}_{1}**

**(a, b, b) = (g**

_{1}(a – b) + b, g_{1}(a – b) + b, g_{1}(a – b) + b)**Ψ ^{0}_{2}**

**(a, b, a) = (g**

_{2}(b – a) + a, g_{2}(b – a) + a, g_{2}(b – a) + a)**Ψ ^{0}_{3}**

**(a, a, c) = (g**

_{3}(c – a) + a, g_{3}(c – a) + a, g_{3}(c – a) + a)Thinking about the composition rule, we have

**Ψ ^{0}_{∞} = Ψ^{0}_{1} o Ψ^{1}_{∞} = Ψ^{0}_{2} o Ψ^{2}_{∞} = Ψ^{0}_{3} o Ψ^{3}_{∞}**

**g _{1}(a – f_{1}(b – c) – c) + f_{1}(b – c) + c**

**= g _{2}(b – f_{2}(c – a) – a) + f_{2}(c – a) + a**

**=g _{3}(c – f_{3}(a – b) – b) + f_{3}(a – b) + b**

………..

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