All Simple Functions can be Written in Standard Form

Last updated Nov 1, 2022

Let $\phi: X \to \mathbb{C}^{n}$ be a Simple Function. Let $F = \phi(X)$ (which is a Finite Set). Let ${A_{i}} = {\phi^{-1}({a_{i}}) \subset X : a_{i} \in F}$. Then ${A_{i}}$ Partitions $X$ and we can write

$$\phi = \sum\limits_{i=1}^{|F|} a_{i}1_{A_{i}}$$

To establish that ${A_{i}}$ partitions $X$, we show that

1. the collection is Mutually Disjoint.
2. the union of the collection is $X$.

This is all we need to show since ${A_{i}} \subset \mathcal{P}(X)$ by construction. To see (1) observe that for $i, j \in [|F|]$ s.t. $i \neq j$ we have that

$$\begin{align*} A_{i} \cap A_{j} &= \phi^{-1}({a_{i}}) \cap \phi^{-1}({a_{j}})\\ &=\phi^{-1}({a_{i}} \cap {a_{j}})\\ &= \phi^{-1}(\emptyset)\\ &= \emptyset \end{align*}$$ TODO

• Complex Numbers
• Function Image
• Function Preimage
• Indicator Function
• Power Set,
• Set Cardinality

1. Caliu - Deep Learning Architectures - Appendix, unknown page
2. Folland - Real Analysis - Ch 2, unknown page