Lebesgue null sets - College of Arts and Sciences

Real Analysis Grinshpan

Lebesgue null sets

A set E R is said to be of Lebesgue measure zero or a null set if it can be covered by countably many intervals of arbitrarily small total length:

> 0 (ak, bk) E (ak, bk), (bk - ak) < .

k=1

k=1

Every singleton is a null set.

Every finite set is a null set.

Every subset of a null set is a null set.

A translation of a null set is a null set.

The image of a null set under a Lipschitz transformation is a null set.

Every set of outer content zero is a null set.

A compact (closed and bounded) set is null if and only if it has outer content zero.

A null set necessarily has empty interior.

A countable union of sets of outer content zero is a null set.

A countable union of null sets is a null set. Indeed, suppose that En is a sequence of null sets. Given > 0, choose, for each n, a covering of En by intervals Ik,n of total length less than /2n. Then the union E1 E2 . . . is covered by the doubly indexed sequence of intervals Ik,n of total length

Ik,n

k,n=1

=

Ik,n

n=1 k=1

<

n=1

2n

=

.

Every countable set is a null set.

A nondegenerate interval is not a null set. So R is uncountable.

The set of rational numbers is a null set.

The Cantor set (which is uncountable) is a null set. To prove this, recall that the Cantor set has the form C = n=1 Cn, where Cn consists on 2n closed intervals of length 1/3n each. Thus C Cn and then total length of Cn is (2/3)n 0.

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