lebesgue measurable

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Definition

Lebesgue Measurable

A set E⊂R is Lebesgue measurable if for all A⊂R we have

m∗(A)=m∗(A∩E)+m∗(A∩Ec)

Note

Since for all A,E we have A=(A∩E)∪(A∩Ec), we have by subadditivity that m∗(A)≤m∗(A∩E)+m∗(A∩Ec).

Thus E is measurable if m∗(A∩E)+m∗(A∩Ec)≤m∗(A)

( we only need to show one side of the inequality )

Facts

Theorem

The empty set ∅ and the set of real numbers R are measurable. And a set E is measurable if and only if Ec is measurable.

Proof

These are readily verifiable from the definition of measurability - which is symmetric in E and Ec.

References

References

See Also

Mentions

Mentions

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Created 2025-06-17 Last Modified 2025-07-14