Many of the important theorems of measure theory fail to hold in full generality. (See below under Theorems for which theorems we're talking about.) Often these theorems are stated for $\sigma$-finite measures, but they do hold a bit more generally than that. In fact, they hold for localizable measures, and this fact characterizes the localizable measures.
The following definition is in elementary terms; but we will see that there are many other characterizations.
Let $\mu$ be a positive measure on an abstract set $X$. (That is, certain subsets of $X$, forming a $\sigma$-algebra, are measurable by $\mu$, and $\mu$ maps these sets to the space $[0,\infty]$ of lower real numbers in a monotone and countably additive way.)
Given two measurable subsets $E$ and $F$, $E$ essentially contains $F$ if the set
is full; or equivalently (using excluded middle) if $F \setminus E$ is null. (This is a preorder on the measurable sets.)
Then $\mu$ is localizable if the following conditions both apply:
Semifiniteness: Every measurable set $E$ with positive measure essentially contains a measurable set with finite positive measure. (We may strengthen ‘essentially contains’ to ‘contains’ in this clause.)
Essential suprema: Given any collection $\mathcal{C}$ of measurable sets, there is a measurable set $E$ such that:
(This set $E$ is essentially unique, in that it essentially contains and is essentially contained in any other set with the same property; we call $E$ the essential union $\ess \bigcup \mathcal{C}$ of $\mathcal{C}$.)
We generalize to measures taking place in some space other than $[0,\infty]$: a $\mu$ is localizable if ${|\mu|}$ is, as long as $\mu$ has an absolute value (or total variation?) ${|\mu|}$ that takes values in $[0,\infty]$.
Of course, a set equipped with a localizable measure is a localizable measure space.
Every $\sigma$-finite measure is localizable. (Since this includes so many examples, the theorems below are often stated for $\sigma$-finite measures.)
Any counting measure is localizable (but $\sigma$-finite only on a countable set).
The following sheaf condition is fundamental: Given any cover $\mathcal{U}$ of $X$ (a localizable measure space) by measurable sets and a $\mathcal{U}$-indexed family $f$ of partial measurable functions (with $\dom f_A = A$), if always $f_A = f_B$ almost everywhere on $A \cap B$ (meaning that there is a full subset $E$ of $X$ such that $f_A = f_B$ on $A \cap B \cap E$), then there exists a (necessarily unique up to almost equality) measurable (and total) function $\ess \bigcup f$ such that always $\ess \bigcup f = f_A$ almost everywhere on $A$. (This is Theorem 213N in Fremlin. I don't know if it characterizes localizable measures.)
Slightly more generally, start with any family of partial measurable functions on $X$ and treat it as a cover of (any representative of) the essential union of its domains.
… other results such as the Radon–Nikodym theorem …
These ideas are elaborated at measurable space Boolean toposes and Boolean valued model?s and forcing.
Whether a semifinite measure $\mu$ is localizable depends only on which measurable sets are full (or null). Sometimes it is convenient to equip a measurable space with a $\delta$-filter of full sets (or a $\sigma$-ideal of null sets), without equipping it with the measure of any other set. A localizable measurable space is a measurable space so equipped, such that every collection of measurable sets has an essential union. (It is a theorem, I believe, that every localizable measurable space is capable of supporting a semifinite, hence localizable, measure with the same full/null sets.)
Knowing the full/null sets is also sufficient to define almost equality of measurable functions between measurable spaces; this gives us a category $Loc Meas$ of localizable measurable spaces. This category is dual to the category of commutative von Neumann algebras; hence an arbitrary von Neumann algebra may be viewed as a noncommutative localizable measurable space (in the sense of noncommutative geometry).
If $\mathcal{M}$ is the $\sigma$-algebra of measurable sets and $\mathcal{F}$ is the $\delta$-filter of full sets (or $\mathcal{N}$ is the $\sigma$-ideal of null sets), then we may form the quotient boolean algebra $\mathcal{M}/\mathcal{F}$ (or $\mathcal{M}/\mathcal{N}$) by identifying each full set with the entire space (or identifying each null set with the empty set). Then a measurable space is localizable iff this quotient is complete; this is the real point of the notion of localizability.
This boolean algebra is all the structure needed to specify a measure on the original space (at least one that is absolutely continuous in that it has at least the requisite full/null sets). We can now abstract away the underlying set and simply look at the complete boolean algebra. However, it is not true that every complete boolean algebra is capable of arising in this way from a measurable space. Those which do so arise may be called measurable locales, since a complete boolean algebra is a frame and measurable functions (up to almost equality) between the measurable spaces correspond to continuous maps between the frames, thought of as locales. (That is, $Loc Meas$ is equivalent to $Meas Loc$, a sort of pun.)
Without requiring localizability, the boolean algebra $\mathcal{M}/\mathcal{F}$ (or $\mathcal{M}/\mathcal{N}$) is called a measurable algebra?; equipped with a measure, we have a measure algebra?. Thus a measurable locale is the same thing as a localizable measurable algebra, and one may also speak of localizable measure algebras (with the category of these equivalent to the category localizable measure spaces, so long as morphisms in the latter are again only defined up to almost equality).
The axiom of choice is indispensable for the development above, as stated. (The reason is that one constantly makes choices among essentially equivalent measurable sets, or among almost equal measurable functions.) However, the category of localizable measurable spaces (and measurable functions up to almost equality) is (assuming choice) equivalent to the category of measurable locales, which may prove to be more tractable without choice, even in constructive mathematics. That said, nobody has worked out a constructive development of this yet. (In particular, identifying which boolean algebras we want is more difficult; perhaps surprisingly, they are still boolean, but they are no longer necessarily complete!)
This giant treatise on all of measure theory is free (in both senses) online:
Last revised on May 3, 2019 at 20:21:53. See the history of this page for a list of all contributions to it.