measure of union of nested sets converges to measure of limiting set

Evaluation Error: SyntaxError: Unexpected token '>'

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:1:1214378)
at DataviewApi.executeJs (plugin:dataview:19607:18)
at DataviewCompiler.eval (plugin:digitalgarden:10763:23)
at Generator.next (<anonymous>)
at fulfilled (plugin:digitalgarden:77:24)

Theorem (continuity of measure)

Suppose ${E_{k}}$ is a countable collection of measurable sets such that $E_{1} \subset E_{2} \subset \dots$ Then

m (⋃_{k = 1}^{\infty} E_{k}) = lim_{n \to \infty} m (⋃_{k = 1}^{n}) = lim_{n \to \infty} m (E_{n})

Proof

The second equality is because $E_{n} = ⋃_{k = 1}^{n} E_{k}$ . So it is enough to just show $m (⋃_{k} E_{k}) = lim_{n \to \infty} m (E_{n})$ .

We can do this by showing the countable union as the countable disjoint union (recall that algebras have closure under finite disjoint countable unions).

So define $F_{1} = E_{1}$ and $F_{k} = E_{k} ∖ E_{k - 1}$ . Each $F_{k}$ is measurable since $F_{k} = E_{k} \cap E_{k - 1}^{c}$ and the collection ${F_{k}}$ is disjoint. Then for all $n \in N$ , we have

\begin{aligned} ⋃_{k = 1}^{n} F_{k} = E_{n} & ⋃_{k = 1}^{\infty} F_{k} = ⋃_{k = 1}^{\infty} E_{k} \\ ⟹ m (⋃_{k = 1}^{\infty} E_{k}) & = \sum_{k = 1}^{\infty} m (F_{k}) \\ = lim_{n \to \infty} \sum_{k = 1}^{n} m (F_{k}) \\ = lim_{n \to \infty} m (⋃_{k = 1}^{n} F_{k}) \\ = lim_{n \to \infty} m (E_{n}) \end{aligned}

Since measure of finite disjoint measurable sets is the sum of the measures. Thus we have shown the desired equality.

\begin{matrix} ◼ \end{matrix}

File	Last Modified
Functional Analysis Lecture 11	2025-07-15
Functional Analysis Lecture 8	2025-07-14
integral is 0 if and only if the function is 0 almost everywhere	2025-07-14

File

Last Modified

Functional Analysis Lecture 11

2025-07-15

Functional Analysis Lecture 8

2025-07-14

integral is 0 if and only if the function is 0 almost everywhere

2025-07-14

measure of union of nested sets converges to measure of limiting set

References

See Also

Mentions