Additive set function

Dependencies:

1. Set function

A set function $f: 2^Ω → ℝ$ is additive iff $f(A ∪ B) = f(A) + f(B)$ for all disjoint sets $A$ and $B$.

If $f$ is additive, then $f(∅) = 0$, since $f(∅) = f(∅ ∪ ∅) = f(∅) + f(∅)$.

If $Ω$ is discrete, then $f(S) = \sum_{e ∈ S} f(e)$ for all $S ⊆ Ω$.

Dependency for:

1. A set function is additive iff it is submodular and supermodular iff it is subadditive and superadditive.
2. EEF doesn't imply EF1
3. GMMS doesn't imply APS
4. MMS+APS doesn't imply PROP1 for chores
5. M1S doesn't imply PROP1
6. APS can be > MMS
7. MMS is largest bundle value at most PROP when n=2
8. Additive goods and binary marginals
9. Additive chores and binary marginals
10. Cake cutting: PROP implies EEF for additive valuations
11. EF1 implies PROP1
12. EFX implies EF1
13. EFX implies PROPm
14. MMS implies MXS
15. PROP implies APS

Info:

• Depth: 3
• Number of transitive dependencies: 3

Transitive dependencies:

1. /sets-and-relations/countable-set
2. σ-algebra
3. Set function