1 Measurable Functions
Measurable functions are morphisms in measurable spaces. We recall that any mapping \(f:X \to Y\) between two sets induces a mapping \(f^{-1}:2^Y\to 2^X\) defined by \[ f^{-1}(E)=\{ x \in X:f(x)\in E \}, \] which preserves unions, intersections and complements. This, if \(\mathcal{G}\) is a \(\sigma\)-algebra on \(Y\), \(\{ f^{-1}(E):E\in\mathcal{G} \}\) is a \(\sigma\)-algebra on \(X\).
For measurable spaces \((X,\mathcal{F})\) and \((Y,\mathcal{G})\), a mapping \(f:X \to Y\) is said to be \((\mathcal{F}, \mathcal{G})\)-measurable if \[ \forall E \in\mathcal{G},~f^{-1}(E):=\{ x \in X:f(x) \in E \}\in\mathcal{F}. \]
Intuitively, we can pull back measurable sets from \(Y\) to \(X\). This is very closely related to the definition of continuity of [[topological-functions]].
Example (Measurable Functions): Consider the mapping \(f(x)=x^2\) mapping within \((\mathbb{R},\mathcal{B}_{\mathbb{R}})\)
We say that a real function \(f:\mathbb{R}\to \mathbb{R}\) is Lebesgue measurable if it is \((\mathcal{F}_{\lambda^*},\mathcal{B}_{\mathbb{R}})\)-measurable. Also, given topological spaces \(X,Y\) we say that \(f:X \to Y\) is Borel measurable if it is \((\mathcal{B}_{X},\mathcal{B}_{Y})\)-measurable. We will see that for functions \(f:\mathbb{R} \to \mathbb{R}\) we have that Borel measurability implies Lebesgue measurability since \(\mathcal{B}_{\mathbb{R}}\subsetneq \mathcal{F}_{\lambda^*}\).
Given measurable spaces \((X,\mathcal{F})\) and \((Y,\mathcal{G})\) where \(\mathcal{G}\) is generated by \(\mathcal{E}\), then \(f:X \to Y\) is \((\mathcal{F},\mathcal{G})\)-measurable if and only if \(\forall E\in \mathcal{E}\), \(f^{-1}(E)\in\mathcal{F}\).
PROOF: The only if implication is trivial. For the converse we observe that \[ \{ E \subset Y:f^{-1}(E)\in \mathcal{F} \}, \] is a \(\sigma\)-algebra that contains \(\mathcal{E}\); it therefore contains \(\mathcal{G}\). \(\square\)
2 Measurable Function Results
For measurable space \((X,\mathcal{F})\) and function \(f:X \to \mathbb{R}\) the following are equivalent: 1. \(f\) is \(\mathcal{F}\)-measurable; 2. \(f^{-1}((a,\infty))\in \mathcal{F}\) for all \(a\in \mathbb{R}\); 3. \(f^{-1}([a,\infty))\in \mathcal{F}\) for all \(a \in \mathbb{R}\); 4. \(f^{-1}((-\infty,a))\in \mathcal{F}\) for all \(a \in \mathbb{R}\); 5. \(f^{-1}((-\infty,a])\in \mathcal{F}\) for all \(a \in \mathbb{R}\).
PROOF: In our results on Borel σ-Algebra we proved that the open and closed rays generate the Borel σ-Algebra. Thus from previous results we have the result. \(\square\)