Measurable function

• categories: Measure Theory, Definition

A function $f:X\to \mathbb{R}$ (or $\mathbb{C}$) defined on a measurable space $(X,\mathcal{A})$ is called measurable if for every Borel Set $B\subset \mathbb{R}$, the preimage of $B$ under $f$ is in $\mathcal{A}$, i.e., $f^{-1}(B)=\{x\in X\mid f(x)\in B\}\in \mathcal{A}.$

Intuition:

A measurable function is one that “preserves” the structure of the $\sigma$-algebra on its domain. It ensures that applying the function to measurable sets gives measurable results.

Key Properties:

1. Preservation of Measure: For any Borel Set $B$, the preimage $f^{-1}(B)$ is measurable.
2. Composition with Measurable Functions: If $f$ and $g$ are measurable, then $f+g$, $fg$, and $cf$ (for any constant $c$) are also measurable.
3. Examples:
   • Continuous functions on a measurable space are measurable.
   • Indicator functions of measurable sets are measurable.