lebesgue integral of sum is sum of the integral

[[concept]]

Topics

const fieldName = "theme"; // Your field with links
const oldPrefix = "Thoughts/01 Themes/";
const newPrefix = "Digital Garden/Topics/";
 
const relatedLinks = dv.current()[fieldName];
 
if (Array.isArray(relatedLinks)) {
    // Map over the links, replace the path, and output only clickable links
    dv.el("span",
        relatedLinks
            .map(link => {
                if (link && link.path) {
                    let newPath = link.path.startsWith(oldPrefix)
                        ? link.path.replace(oldPrefix, newPrefix)
                        : link.path;
                    return dv.fileLink(newPath);
                }
            })
            .filter(Boolean).join(", ") 
	// Remove any undefined/null items
    );
} else {
    dv.el(dv.current().theme);
}
 
 

Theorem

If $f,g\in L^{+}(E)$ then ${\int }_E(f+g)={\int }_{E}f+{\int }_{E}g$

Proof

Let $\{{\phi }_n{\}}_n$ and $\{{\psi }_n{\}}_n$ be two sequences of simple functions such that $0\le {\phi }_1\le {\phi }_2\le \cdots \le f$ with ${\phi }_n\to f$ and $o\le {\psi }_1\le {\psi }_2\le \cdots \le g$ with ${\psi }_n\to g$ pointwise then $0\le {\phi }_1+{\psi }_1\le {\phi }_2+{\psi }_2\le \cdots \le f+g$ Where ${\phi }_n+{\psi }_n\to f+g$ pointwise and each ${\phi }_i+{\psi }_i$ are simple (since they are the sum of simple functions). So by the Monotone Convergence Theorem and linearity for simple functions we have

\int _{E} (f+g) &= \lim_{ n \to \infty } \int _{E} (\varphi_{n} + \psi_{n}) \\ \text{(linearity)}&= \lim_{ n \to \infty } \int _{E} \varphi_{n} + \int _{E} \psi_{n} \\ \text{(MTC)}&=\int _{E} f+\int _{E} g \end{align}$$

References

References

See Also

Mentions

Mentions

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>
}