Set function

Dependencies:

1. σ-algebra

$\newcommand{\Fcal}{\mathcal{F}}$ $\newcommand{\Scal}{\mathcal{S}}$ Let $\Omega$ be a set (called the ground set) and let $\Fcal$ be a subset of the power-set of $\Omega$. Then the function $f: \Fcal \to \mathbb{R}$ is called a set function over $(\Omega, \Fcal)$.

For notational convenience, we denote $f(e) = f(\{e\})$ for $e \in \Omega$.

If $\Omega$ is finite and $\Fcal$ is not specified, then we let $\Fcal$ be $2^{\Omega}$ by default. If $\Omega$ is infinite, we assume $(\Omega, \Fcal)$ to be a sigma-algebra over $\Omega$.

For any sets $A, B \subseteq \Omega$, we define $f(A \mid B) := f(A \cup B) - f(B)$. $f(A \mid B)$ is called $f$'s marginal for $A$ over $B$.

Dependency for:

1. Submodular function
2. Supermodular function
3. Additive set function
4. Subadditive and superadditive set functions
5. Maximin share of a set function
6. Fair division

Info:

• Depth: 2
• Number of transitive dependencies: 2

Transitive dependencies:

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