simple functions can be written as a finite complex linear combination of indicator functions

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at DataviewInlineApi.eval (plugin:dataview:19027:21)
at evalInContext (plugin:dataview:19028:7)
at asyncEvalInContext (plugin:dataview:19038:32)
at DataviewJSRenderer.render (plugin:dataview:19064:19)
at DataviewJSRenderer.onload (plugin:dataview:18606:14)
at DataviewJSRenderer.load (app://obsidian.md/app.js:1:1182416)
at DataviewApi.executeJs (plugin:dataview:19607:18)
at DataviewCompiler.eval (plugin:digitalgarden:10763:23)
at Generator.next (<anonymous>)
at eval (plugin:digitalgarden:90:61)

Proof

Suppose $ϕ : E \to C$ is a simple function where the range $ϕ (E) = {a_{1}, a_{2}, \dots, a_{n}}$ . Define the sets

A_{i} = ϕ^{- 1} ({a_{i}})

Note that each $A_{i}$ is measurable because they are intersections of the measurable sets where $Re (ϕ) = Re (a_{i})$ and $Im (ϕ) = Im (a_{i})$ .

For all $i \neq j$ we have $A_{i} \cap A_{j} = \emptyset$ and $⋃_{i = 1}^{n} A_{i} = E$ . ie, the $A_{i}$ form a finite partition of the domain $E$ . Thus, for all $x \in E$ , we can write

ϕ (x) = \sum_{i = 1}^{n} a_{i} \cdot 1_{A_{i}} (x)

Which is a finite complex linear combination of indicator functions.

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const modules = await cJS()

const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
return function View() {  
	const current = dc.useCurrentFile();
// Selecting `#game` pages, for example. 
	let queryString = `@page and linksto(${current.$link})`;
	let pages = dc.useQuery(queryString);
	
	// check types
	pages = pages.filter( (p) => !modules.typeCheck.checkAll(p, current) ).sort()
	
	
	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}

const { dateTime } = await cJS() return function View() { const file = dc.useCurrentFile(); return <p class="dv-modified">Created {dateTime.getCreated(file)} ֍ Last Modified {dateTime.getLastMod(file)}</p> }

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