Let \((\Omega', \mathcal{A}')\) be a measurable space, \(\Omega\neq \emptyset\) and \(f\colon \Omega\to \Omega'\) a mapping. A map \(g\colon \Omega\to \bar{\mathbb{R}}\) is \(\sigma(f)\)-\(\mathcal{B}(\bar{\mathbb{R}})\)-measurable if and only if a measurable map \(\varphi\colon (\Omega',\mathcal{A}')\to (\bar{\mathbb{R}},\mathcal{B}(\bar{\mathbb{R}}))\) exists with \(g=\varphi \circ f\). [1, Korollar 1.97]
Proof
Follows almost immediately from (0x667d03c3)
.
See also Link to heading
- Factorization Lemma for “Borel Set”-valued functions
- Factorization Lemma for “Standard Borel Space”-valued functions
References Link to heading
- A. Klenke, Wahrscheinlichkeitstheorie. Berlin, Heidelberg: Springer Berlin Heidelberg, 2020. doi:10.1007/978-3-662-62089-2