Our “cake” C is some set. We wish to partition C among n players, whom we shall refer to as Player 1, Player 2, …, Player n. For each i = 1, 2, …, n, Player i uses a measure mi to evaluate the size of pieces of cake (i.e., subsets of C). Unless otherwise noted, we shall always assume that C is non-empty.

Definition 1.1 A σ- algebra on C is a collection of subsets W of C satisfying that

1. a. C ∈ W,

2. b. if A ∈ W then C\A ∈ W, and

3. c. if Ai ∈ W for every i ∈ N, then (∪i∈NAi) ∈ W (where N denotes the set of natural numbers).

Definition 1.2 Assume that some σ-algebra W has been defined on C. A countably additive measure on W is a function µ : W → R (where R denotes the set of real numbers) satisfying that

1. a. µ(A) ≥ 0 for every A ∈ W,

2. b. µ(ø) = 0, and

3. c. if A1, A2, … is a countable collection of elements of W and this collection is pairwise disjoint, then µ(∪i∈NAi) = ∑i∈N µ(Ai).

4. In addition, µ is

5. d. non-atomic if and only if, for any A ∈ W, if µ(A) > 0 then for some B ⊆ A, B ∈ W and 0 < µ(B) < µ(A) and

6. e. a probability measure if and only if µ(C) = 1.