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

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

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

1. the collection is Mutually Disjoint

   Mutually Disjoint

   Definition Let $\{A{i}\}{i \in I}$ be a collectio of s indexe by $I$. We say $\{A{i}\}{i \in I}$ is if...

   11/7/2022

   .
2. the union

   Set Union

   Definition 1 Let $A$, $B$ be s. Then $A \cup B := \{x : x \in A \vee x \in B\}$. Proof...

   11/7/2022

   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

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