Prof. Hans Wilhelm Alt: Script Analysis III -- Properties μ -almost everywhere

Outer measure and null sets

LEBESGUE Integral

Equivalence relation

Index

Properties μ -almost everywhere

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Definition

Let S and μ as above, and let A(x) be an expression for x ∈S . We say

A(x) holds for μ-almost all (μ-a.a.) x ∈S,

A holde μ-almost everywhere (μ-a.e.) ,

if there is a μ -null set N ⊂S , such that A(x) is true for all x ∈S∖N . Using quantors this means that

∃ N ⊂S :

(

μ^*(N)=0 , ∀ x ∈S∖N : A(x)

)

For example, if f, g : S → R , then the statement f=g μ -almost everywhere means, that

∃ N ⊂S : μ^*(N)=0 and ∀ x ∈S∖N : f(x)=g(x) ,

in words: There exists a subset N ⊂S with μ^*(N)=0 , such that for all x ∈S∖N the equality f(x)=g(x) is satisfied.

Note: One might think, that for N ∈B the property μ^*(N)=0 implies that μ(N)=0 , which in general is false. Also it is not true, that the outer measure on B is greater or equal to the underlying measure. The contrary holds: From the definition of μ^* it follows that μ^*≦μ on B .

Version 1.7
H.W. Alt - 02.01.2007

Outer measure and null sets

LEBESGUE Integral

Equivalence relation

Index