In set theory, a null set N ⊂ R is a set that can be covered by an countable
union of intervals of arbitrarily small total length. The notion of null set in
set theory anticipates the development of Lebesgue measure since a null set
necessarily has measure zero. Definition
Suppose A is a subset of the real line R such that
where the Un are intervals and |U| is the length of U, then A is a null set.
In terminology of mathematical analysis, this definition requires that there be a
sequence of open covers of A for which the limit of the lengths of the covers
is zero. Null sets include all finite sets, all
countable sets, and even some uncountable sets such as the Cantor set.
Properties The empty set is always a null set. More
generally, any countable union of null sets is null. Any measurable subset of a
null set is itself a null set. Together, these facts show that the m-null sets of
X form a sigma-ideal on X. Similarly, the measurable m-null sets form a
sigma-ideal of the sigma-algebra of measurable sets. Thus, null sets may be
interpreted as negligible sets, defining a notion of almost everywhere.
Lebesgue measure The Lebesgue measure is the standard way
of assigning a length, area or volume to subsets of Euclidean space.
A subset N of R has null Lebesgue measure and is considered to be a null
set in R if and only if: Given any positive number ε, there is a
sequence {In} of intervals in R such that N is contained in the union of the
{In} and the total length of the union is less than ε.
This condition can be generalised to Rn, using n-cubes instead of intervals. In
fact, the idea can be made to make sense on any differentiable manifold, even if
there is no Lebesgue measure there. For instance:
With respect to Rn, all 1-point sets are null, and therefore all countable sets
are null. In particular, the set Q of rational numbers is a null set, despite
being dense in R. The standard construction of the Cantor
set is an example of a null uncountable set in R; however other constructions
are possible which assign the Cantor set any measure whatsoever.
All the subsets of Rn whose dimension is smaller than n have null Lebesgue
measure in Rn. For instance straight lines or circles are null sets in R2.
Sard's lemma: the set of critical values of a smooth function has measure zero.
if λ is Lebesgue measure for R and π is Lebesgue measure for R2, then the
product measure λ x λ = π . In terms of null sets, the following equivalence has
been styled a Fubini's theorem: For A⊂ R2 and
Uses Null sets play a key role in the
definition of the Lebesgue integral: if functions f and g are equal except on a
null set, then f is integrable if and only if g is, and their integrals are
equal. A measure in which all subsets of null
sets are measurable is complete. Any non-complete measure can be completed to
form a complete measure by asserting that subsets of null sets have measure
zero. Lebesgue measure is an example of a complete measure; in some
constructions, it's defined as the completion of a non-complete Borel
measure. = A subset of the Cantor set which is
not Borel measurable = The Borel measure is not complete. One
simple construction is to start with the standard Cantor set K, which is closed
hence Borel measurable, and which has measure zero, and to find a subset F of
K which is not Borel measurable. First, we have to know that every set of
positive measure contains a nonmeasurable subset. Let f be the
Cantor function, a continuous function which is locally constant on Kc, and
monotonically increasing on [0, 1], with f(0) = 0 and f(1) = 1. Obviously, f(Kc)
is countable, since it contains one point per component of Kc. Hence f(Kc)
has measure zero, so f(K) has measure one. We need a strictly monotonic
function, so consider g(x) = f(x) + x. Since g(x) is strictly monotonic and
continuous, it is a homeomorphism. Furthermore, g(K) has measure one. Let E
⊂ g(K) be non-measurable, and let F = g−1(E). Because g is injective, we have
that F ⊂ K, and so F is a null set. However, if it were Borel measurable,
then g(F) would also be Borel measurable =−1(F) is the preimage of F through the
continuous function h = g−1.) Therefore, F is a null, but non-Borel measurable
set. Haar null
In a separable Banach space, the group operation moves any subset A ⊂ X to the
translates A + x for any x ∈ X. When there is a probability measure μ on the
σ-algebra of Borel subsets of X, such that for all x, μ(A + x) = 0, then A is
a Haar null set. The term refers to the null invariance
of the measures of translates, associating it with the complete
invariance found with Haar measure. Some algebraic properties of topological
groups have been related to the size of subsets and Haar null sets. Haar null
sets have been used to in Polish groups to show that when A is not a meager set
then A–1A contains the identity element. This property is named for Hugo
Steinhaus since it is the conclusion of the Steinhaus theorem.
See also Cantor function
Measure Empty set
References Marek Capinski & Ekkehard Kopp Measure,
Integral and Probability, page 16, Springer ISBN 1-85233-781-8 .
Frank Jones Lebesgue Integration on Euclidean Spaces, page 107, Jones &
Bartlett ISBN 0-86720-203-3 . John C. Oxtoby Measure and Category,
page 3, Springer-Verlag ISBN 0-387-05349-2 .