Measureable Function

Last updated Nov 1, 2022

Let $(X, \mathcal{M})$ and $(Y, \mathcal{N})$ be Measure Spaces. Let $f: X \to Y$ be a Function. Then $f$ is a Measureable Function if $$\sigma(f) \subset \mathcal{M}$$

1. We sometimes denote the Set of $(\mathcal{M}, \mathcal{N})$-Measureable Functions as $\mathcal{L}^{0}(\mathcal{M}, \mathcal{N})$. If we are considering Measureable Functions up to null sets, we will refer to the collection of Equivalence Classes as $L^{0}(\mathcal{M}, \mathcal{N})$. We may also use the following abbreviations:
   1. $L^{0}(X, Y)$, when the associated Sigma Algebras are understood to be $\mathcal{M}, \mathcal{N}$.
   2. $L^{0}(_)$ where $_$ is a single argument from the above two forms. A single argument is used when it is well understood what the other argument is and whether it is the source or target.
   3. $L^{0}$ when all arguments are well understood and not ambiguous

• Sigma Algebra induced by Function