Lebesgue measure

[[concept]]

[!themes] Topics

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)

Definition

Lebesgue Measure

The Lebesgue measure of a measurable set E⊂M is given by

m(E)=m∗(E)

Where m∗(⋅) is the outer measure (see below)

Outer measure

Outer Measure

For A⊂R, we define the outer measure of A

m∗(A)=inf{∑nℓ(In):{In} countable and open s.t. A⊂⋃nIn}

We can see that m∗(A)≥0 for all A.

Example

Consider the set A={0} containing a single point. m∗({0})=0.

Let ϵ>0. Then

{0}⊂(−ϵ2,ϵ2)⟹m∗({0})≤ℓ(−ϵ2,ϵ2)=ϵ⟹m∗({0})=0

Thus the (outer) measure is 0!

^example

References

References

See Also

Mentions

Mentions

File Last Modified
almost everywhere 2025-07-14
changing measurable functions on a measure zero set preserves measurability 2025-07-14
Functional Analysis Lecture 8 2025-07-14
Functional Analysis Lecture 9 2025-07-14
lebesgue measurable 2025-07-14
countable sets have outer measure zero 2025-07-08
desirable properties for measure 2025-07-08
Functional Analysis Lecture 6 2025-07-08
Functional Analysis Lecture 7 2025-07-08
outer measure has countable subadditivity 2025-07-08
outer measure of subsets are bounded by their supersets 2025-07-08
unions of measurable sets are measurable 2025-07-08

Created 2025-06-17 Last Modified 2025-07-08