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

Info:

• Depth: 3
• Number of transitive dependencies: 3

Transitive dependencies:

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