nLab null subset

(Null set redirects here; for the notion in set theory, see empty set.)

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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 $X$, a $\sigma$-algebra (or similar structure) $\mathcal{M}$ consisting of the measurable subsets of $X$, and a measure $\mu$ mapping each measurable set $A$ to a real number (or similar quantity) $\mu(A)$, the measure of $A$.

A measurable subset $B$ of $X$ is full if, given any measurable set $A$, $\mu(A \cap B) = \mu(A)$; an arbitrary subset of $X$ is full if it's a superset of a full measurable set. Dually, a measurable set $B$ is null if, given any measurable set $A$, $\mu(A \cup B) = \mu(A)$; an arbitrary subset of $X$ 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 $B$ is null iff $\mu(C) = 0$ for every measurable subset of $B$.
• If $\mu$ is a positive measure, then a measurable set $B$ is null iff $\mu(B) = 0$.
• If $\mu$ is a finite measure with total measure $I$, then a measurable set $B$ is full iff $\mu(C) = I$ for every measurable superset of $B$.
• If $\mu$ is both positive and finite (so a probability measure up to rescaling), then a measurable set $B$ is full iff $\mu(B) = I$.
• If $\mu$ is complete, then every null set is measurable and every full set is measurable (which is basically the definition of ‘complete’) and consequently the preceding properties continue to hold when the adjective ‘measurable’ is removed.
• 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

Traditionally, a measurable space is simply an abstract set $X$ and a $\sigma$-algebra (or similar structure) $\mathcal{M}$ consisting of the measurable subsets of $X$. There is no notion of null or full subsets of such a space. However, there are two (essentially equivalent) variations of this concept in which null and full subsets do make sense.

One variation is to simply equip the space with a $\delta$-filter of measurable subsets, which are declared to be the full measurable subsets. Then a general full subset is a superset of a measurable full subset, and a null subset is any set whose complement is full. (Alternatively, equip the space with a $\sigma$-ideal of measurable subsets, which are declared to be the null measurable subsets.) In particular, a localizable measurable space is a measurable space so equipped such that the Boolean algebra of measurable sets modulo null sets (or modulo full sets if this is done by identifying the full sets with $X$) is complete.

Another variation, used especially in constructive mathematics, is a Cheng measurable space. This consists of a set $X$ equipped with a $\sigma$-semialgebra of disjoint pairs of subsets of $X$, declared to be the complemented pairs. A set is measurable iff it appears as one component of a complemented pair. A measurable subset is full if it appears as one component of a complemented pair whose other component is empty, or equivalently (given the structure of the algebra of complemented pairs) if it is the union of the two components of any complemented pair. Then a general full subset is a superset of a measurable full subset, and a null subset is any set whose complement is full.

These are actually equivalent concepts. Given a measurable space equipped with a $\delta$-filter of measurable full subsets, define a complemented pair to be a pair of disjoint measurable subsets whose union is full. Conversely, given a Cheng measurable space, the measurable subsets and measurable full subsets as defined above comprise a $\sigma$-algebra and a $\delta$-filter in it. (But constructively, the algebra of measurable subsets, while closed under the appropriate operations, will generally not be a boolean algebra.)

In smooth manifolds

A subset $A$ of an $n$-dimensional smooth manifold $X$ is null or full (respectively) if its preimage under every coordinate chart is a null or full subset (respectively) of the chart's domain (which is an open subset of the Cartesian space $\mathbb{R}^n$) under Lebesgue measure.

This is actually better behaved than it may at first seem. If $A$ is covered by an atlas $(\phi_i\colon U_i \to X)_i$, then $A$ is null or full as soon as $\phi_i^*(A)$ is null/full in $U_i$ for every index $i$. In particular, if $A$ is contained in a single coordinate chart (which is not very likely for a full set but fairly common for null sets), then it is sufficient to check its preimage under that one. This fact depends on the smoothness and fails for topological manifolds.

As we can define a measurable subset of a smooth manifold similarly, this means that every smooth manifold gives rise to a measurable space equipped with a $\delta$-filter of full subsets (and hence to a Cheng measurable space); this space is always localizable.

(Details? Is $C^1$ sufficient? Conversely, is paracompactness necessary to keep the covers manageable?)

Logic of full/null sets

A property of elements of $X$ (given by a subset $S$ of $X$) can be considered modulo null sets. We say that the property $\phi$ is true almost everywhere or almost always if it is true on some full set, that is if $\{X | \phi\}$ is full. Dually, we say that $\phi$ is true almost nowhere or almost never if $\{X | \phi\}$ is null. It is better to use the negation of ‘almost nowhere’, although the terminology for this is not really standard; say that $\phi$ is true somewhere significant if $\{X | \phi\}$ 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 $\ess\forall i, \phi[i]$ or $\ess\forall \phi$ to mean that a predicate $\phi$ on $X$ is true almost everywhere, and use $\ess\exists i, \phi[i]$ or $\ess\exists \phi$ to mean that $\phi$ is true somewhere significant. Then we have:

$$\forall \phi \;\Rightarrow\; \ess\forall \phi$$

$$\ess\exists \phi \;\Rightarrow\; \exists \phi$$

$$\ess\forall (\phi \wedge \psi) \;\Leftrightarrow\; \ess\forall \phi \wedge \ess\forall \psi$$

$$\ess\exists (\phi \wedge \psi) \;\Rightarrow\; \ess\exists \phi \wedge \ess\exists \psi$$

$$\ess\forall (\phi \vee \psi) \;\Leftarrow\; \ess\forall \phi \wedge \ess\forall \psi$$

$$\ess\exists (\phi \vee \psi) \;\Leftrightarrow\; \ess\exists \phi \vee \ess\exists \psi$$

$$\ess\forall \neg{\phi} \;\Leftrightarrow\; \neg\ess\exists \phi$$

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.

Related concepts

• almost surely

References

See also

• Wikipedia, Null set