null set

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.

The definitions depend on the context.

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 $\mu(B) = I$.
- 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.

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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** if it is true on some full set, that is if `\{X | \phi}`

is full. Dually, we say that $\{X | \phi}\phi$ is true **almost nowhere** 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.

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