Lebesgue measure

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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
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Created 2025-06-17 ֍ Last Modified 2025-09-18