The Lebesgue Measure Function

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Desirable properties of the measure function

Defined for all subsets of

(Defined for all subsets of ):

is defined on all subsets of and produces a non-negative length (mathematically: )

Related to the notion of length

(Related to the notion of length):

Length of an interval is given by

Translation Invariance

If a set is shifted left or right by a certain number : ), - (this defines the operation ) then its measure should not change: - (here we use the operation )

Sigma Additivity

(Sigma Additivity):

If a set is a union of disjoint subsets then its measure equal to the sum of the individual measure - this is the sigma-additivity part

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Appendix

• Tags:

• Reference:
  • ; Measure Theory Lecture 01
• Related:
  • A non-measurable set

Anki Cards

Translation Invariance

TARGET-DECK: Measure Theory

What is Translation Invariance for the Lebesgue Measure Function? ::

If a set is shifted left or right by a certain number : ), - (this defines the operation )

then its measure should not change: - (here we use the operation )

Sigma Additivity

What is Sigma Additivity for the Lebesgue Measure Function? ::

If a set is a union of disjoint subsets then its measure equal to the sum of the individual measure - this is the sigma-additivity part