> [!tldr] > The **standard machine** (informal name) refers to a boilerplate strategy for proving results in measure theory. > > Suppose we wish to prove that all measurable functions $f \in m\mathcal{F}$ have some "linear" property $p$. The standard machine proves the result by: > - Showing that the result holds for all indicator functions $\mathbf{1}_{A}$ for $\forall A \in \mathcal{F}$. > - Use linearity to extend the result to all simple functions. > - For a generic $f$, approximate it by a sequence of simple functions $(f_{n}) \to f$, then pass the property to $f$ by: > - Either assume $f\in m\mathcal{F}$ and use the DCT, > - Or assume $f \in m\mathcal{F}^{+}$ and use the MCT with $(f_{n})\nearrow f$.