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Let C denote the Lie group (ℝ/ℤ, +), and let ℤ_n denote the subgroup of C generated by [1/n].

Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of images {g+X | g ∊ C} of X by the action of the group ℤ_n forms a partition of C.

Call such an X an "1/n equal part" of C.

We now try to create a finitely additive measure 𝜇 "from the top down" on C by assigning to each 1/n equal part X of C the value 𝜇(X) = 1/n.

Assume that, for all positive integers n, this assignment is made for all (X,n) such that X is a 1/n equal part of C.

Questions:

1. Does 𝜇 extend to a finitely additive measure on an algebra of sets containing all such X ?

2. What about a 𝜎-additive measure on a 𝜎-algebra of sets containing all such X ?

3. And if this is possible, can such a 𝜇 extend ordinary Lebesgue measure?

Let C denote the Lie group (ℝ/ℤ, +), and let ℤ_n denote the subgroup of C generated by [1/n].

Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of images {g+X | g ∊ ℤ_n} of X by the action of the group ℤ_n forms a partition of C.

Call such an X an "1/n equal part" of C.

We now try to create a finitely additive measure 𝜇 "from the top down" on C by assigning to each 1/n equal part X of C the value 𝜇(X) = 1/n.

Assume that, for all positive integers n, the assignment 𝜇(X) = 1/n is made for all (X,n) such that X is a 1/n equal part of C.

Questions:

1. Does 𝜇 extend to a finitely additive measure on an algebra of sets containing all such X ?

2. What about a 𝜎-additive measure on a 𝜎-algebra of sets containing all such X ?

3. And if this is possible, can such a 𝜇 extend ordinary Lebesgue measure?

Let C denote the Lie group (ℝ/ℤ, +), and let ℤ_n denote the subgroup of C generated by [1/n].

Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of images {g+X | g ∊ C} of X by the action of the group ℤ_n forms a partition of C.

Call such an X an "1/n equal part" of C.

We now try to create a finitely additive measure 𝜇 "from the top down" on C by assigning to each 1/n equal part X of C the value 𝜇(X) = 1/n.

Assume this assignment is made for all (X,n) such that X is a 1/n equal part of C, for all positive integers n.

Questions:

1. Does 𝜇 extend to a finitely additive measure on an algebra of sets containing all such X ?

2. What about a 𝜎-additive measure on a 𝜎-algebra of sets containing all such X ?

3. And if this is possible, can such a 𝜇 extend ordinary Lebesgue measure?

Let C denote the Lie group (ℝ/ℤ, +), and let ℤ_n denote the subgroup of C generated by [1/n].

Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of images {g+X | g ∊ C} of X by the action of the group ℤ_n forms a partition of C.

Call such an X an "1/n equal part" of C.

We now try to create a finitely additive measure 𝜇 "from the top down" on C by assigning to each 1/n equal part X of C the value 𝜇(X) = 1/n.

Assume that, for all positive integers n, this assignment is made for all (X,n) such that X is a 1/n equal part of C.

Questions:

1. Does 𝜇 extend to a finitely additive measure on an algebra of sets containing all such X ?

2. What about a 𝜎-additive measure on a 𝜎-algebra of sets containing all such X ?

3. And if this is possible, can such a 𝜇 extend ordinary Lebesgue measure?

Let C denote the Lie group (ℝ/ℤ, +), and let ℤ_n denote the subgroup of C generated by [1/n].

Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of images {g+X | g ∊ C} of X by the action of the group ℤ_n forms a partition of C.

Call such an X an "1/n equal part" of C.

We now try to create a finitely additive measure 𝜇 "from the top down" on C by assigning to each 1/n equal part X of C the value 𝜇(X) = 1/n.

Assume this assignment is made for all (X,n) such that X is a 1/n equal part of C, for all positive integers n.

Questions: 1. Can 𝜇 be extended to a finitely additive measure on an algebra of sets containing all such X ? 2. What about a 𝜎-additive measure on a 𝜎-algebra of sets containing all such X? 3. And if this is possible, can such a 𝜇 extend ordinary Lebesgue measure?

Let C denote the Lie group (ℝ/ℤ, +), and let ℤ_n denote the subgroup of C generated by [1/n].

Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of images {g+X | g ∊ C} of X by the action of the group ℤ_n forms a partition of C.

Call such an X an "1/n equal part" of C.

We now try to create a finitely additive measure 𝜇 "from the top down" on C by assigning to each 1/n equal part X of C the value 𝜇(X) = 1/n.

Assume this assignment is made for all (X,n) such that X is a 1/n equal part of C, for all positive integers n.

Questions:

1. Does 𝜇 extend to a finitely additive measure on an algebra of sets containing all such X ?

2. What about a 𝜎-additive measure on a 𝜎-algebra of sets containing all such X ?

3. And if this is possible, can such a 𝜇 extend ordinary Lebesgue measure?