Measurable function

Dependencies:

1. σ-algebra

$$ $$ Let $(X,\mathcal{E})$ and $(Y,\mathcal{F})$ 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 $\mathrm{\forall }T\in \mathcal{F},f^{-1}(T)\in \mathcal{E}$.

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