nLab
null set

Null and full sets

Idea

In measure theory, a null set is a subset of a measure space (or measurable space) that is so small that it can be neglected: it might as well be the empty subset; its measure is zero. Similarly, a full set is a subset that is so large that it might as well be the improper subset (the entire space). One also says that a null set has null measure and a full set has full measure.

Traditionally, full sets are not usually referred to explicitly; in classical mathematics, they are simply the complements of null sets. However, they are often referred to implicitly through such terminology as ‘almost everywhere’. Also, in constructive mathematics, full sets are more fundamental than null sets; they are not simply the complements of the latter.

Definitions

The definitions depend on the context.

In a measure space

In a traditional measure space, we have an abstract set XX, a σ\sigma-algebra (or similar structure) \mathcal{M} consisting of the measurable subsets of XX, and a measure μ\mu mapping each measurable set AA to a real number (or similar quantity) μ(A)\mu(A), the measure of AA.

A measurable subset BB of XX is full if, given any measurable set AA, μ(AB)=μ(A)\mu(A \cap B) = \mu(A); an arbitrary subset of XX is full if it's a superset of a full measurable set. Dually, a measurable set BB is null if, given any measurable set AA, μ(AB)=μ(A)\mu(A \cup B) = \mu(A); an arbitrary subset of XX is null if it's a subset of a null measurable set.

Some equivalent characterisations (constructively valid for measures on Cheng spaces except as stated):

  • A measurable set BB is null iff μ(C)=0\mu(C) = 0 for every measurable subset of BB.
  • If μ\mu is a positive measure, then a measurable set BB is null iff μ(B)=0\mu(B) = 0.
  • If μ\mu is a finite measure with total measure II, then a measurable set BB is full iff μ(C)=I\mu(C) = I for every measurable superset of BB.
  • If μ\mu is both positive and finite (so a probability measure up to rescaling), then a measurable set BB is full iff μ(B)=I\mu(B) = I.
  • Using excluded middle, a set is null iff its complement is full, and a set is full iff its complement is null. (Even constructively, if a set is null, then its complement is full.)
  • Even constructively, a measurable set is null iff its measurable complement (the complement in the algebraic structure of complemented pairs in a Cheng measurable space) is full, and a measurable set is full iff its measurable complement is null.

In untraditional measurable spaces

In topological manifolds

Logic of full/null sets

A property of elements of XX (given by a subset SS of XX) can be considered modulo null sets. We say that the property ϕ\phi is true almost everywhere if it is true on some full set, that is if \{X | \phi} is full. Dually, we say that ϕ\{X | \phi}\phi is true almost nowhere if {Xϕ}\{X | \phi\} is null. It is better to use the negation of ‘almost nowhere’, although the terminology for this is not really standard; say that ϕ\phi is true somewhere significant if {Xϕ}\{X | \phi\} is non-null.

Note that being true almost everywhere is a weakening of being true everywhere (given by the universal quantifier \forall), while being true somewhere significant is a strengthening of being true somewhere (given by the particular quantifier \exists). Indeed we can build a logic out of these. Use essi,ϕ[i]\ess\forall i, \phi[i] or essϕ\ess\forall \phi to mean that a predicate ϕ\phi on XX is true almost everywhere, and use essi,ϕ[i]\ess\exists i, \phi[i] or essϕ\ess\exists \phi to mean that ϕ\phi is true somewhere significant. Then we have:

ϕessϕ\forall \phi \;\Rightarrow\; \ess\forall \phi
essϕϕ\ess\exists \phi \;\Rightarrow\; \exists \phi
ess(ϕψ)essϕessψ\ess\forall (\phi \wedge \psi) \;\Leftrightarrow\; \ess\forall \phi \wedge \ess\forall \psi
ess(ϕψ)essϕessψ\ess\exists (\phi \wedge \psi) \;\Rightarrow\; \ess\exists \phi \wedge \ess\exists \psi
ess(ϕψ)essϕessψ\ess\forall (\phi \vee \psi) \;\Leftarrow\; \ess\forall \phi \wedge \ess\forall \psi
ess(ϕψ)essϕessψ\ess\exists (\phi \vee \psi) \;\Leftrightarrow\; \ess\exists \phi \vee \ess\exists \psi
ess¬ϕ¬essϕ\ess\forall \neg{\phi} \;\Leftrightarrow\; \neg\ess\exists \phi

and other analogues of theorems from predicate logic. Note that the last item listed requires excluded middle even though its analogue from ordinary predicate logic does not.

A similar logic is satisfied by ‘eventually’ and its dual (‘frequently’) in filters and nets.

Revised on February 19, 2013 13:06:52 by Toby Bartels (98.23.143.39)