Measure

Dependencies:

1. σ-algebra

$\newcommand{\Fcal}{\mathcal{F}}$ $\newcommand{\Scal}{\mathcal{S}}$ A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

Let $\Fcal$ be a $\sigma$-algebra over $X$. A function $\mu: \Fcal \mapsto \mathbb{R} \cup \{\infty, -\infty\}$ is called a measure over $(X, \Fcal)$ iff it satisfies all of these properties:

• non-negativity: $\forall A \in \Fcal, \mu(A) \ge 0$.
• $\mu(\{\}) = 0$.
• $\sigma$-additivity: Let $\Scal = \{A_1, A_2, \ldots\}$ be a countable subset of $\Fcal$, such that all sets in $\Scal$ are pairwise-disjoint. Then \[ \mu\left( \bigcup_{A \in \Scal} A \right) = \sum_{A \in \Scal} \mu(A) \]

The triple $(X, \Fcal, \mu)$ is called a measure space.

Dependency for:

1. Random variable
2. Probability
3. Conditional probability (incomplete)

Info:

• Depth: 2
• Number of transitive dependencies: 2

Transitive dependencies:

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