Measurable function

Dependencies:

1. σ-algebra

$\newcommand{\Fcal}{\mathcal{F}}$ $\newcommand{\Ecal}{\mathcal{E}}$ Let $(X, \Ecal)$ and $(Y, \Fcal)$ be $\sigma$-algebras. Let $f: X \mapsto Y$ be a function.

Let $T \subseteq Y$. Define $f^{-1}(T) = \{x \in X: f(x) \in T\}$.

$f$ is said to be measurable iff $\forall T \in \Fcal, f^{-1}(T) \in \Ecal$.

Dependency for:

1. X and Y are independent implies X and f(Y) are independent
2. Random variable

Info:

• Depth: 2
• Number of transitive dependencies: 2

Transitive dependencies:

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