meaning explanation

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# The meaning explanations of type theory

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## Idea

[[Per Martin-Löf]]'s **meaning explanations of type theory** provide a way to justify the correctness of the rules of [[type theory]].  The meaning of the [[judgements]] of type theory are defined by [[computation]] of closed [[terms]] to [[canonical form]] in some untyped notion of [[computation]], such as a [[lambda calculus]] extended with constants for the [[term introduction|constructors]] and [[term elimination|eliminators]] of type theory.

From a technical/mathematical point of view, the meaning explanations give rise to a [[realizability model]] of type theory, where realizers are terms in some untyped notion of computation, such as an extended lambda calculus.  That is, it can be regarded as a description of how types are assigned to untyped computations.  However, the term "meaning explanations" also refer to the "philosophical" or "pre-mathematical" component, which regards this as an "explanation" of the "constructive validity" of the [[judgements]] of the type theory.

The standard meaning explanations validate not just the rules of [[intensional type theory]] but also the stronger rules of [[extensional type theory]].  Thus, if one regards it as the "intended [[semantics]]" then one may believe in the extensional theory.  However, like any particular semantics for a formal theory, the meaning explanation also validates many other "rules" as well which are not usually added to type theories, so it is not so clear that meaning explanations in their own right argue for extensionality.  They can also be modified so as to no longer satisfy extensionality.

Martin-Löf has been quoted as saying

> Some people feel that when I have presented my meaning explanations I have said nothing.  And this is how it should be. The standard semantics should be just that: standard. It should come as no surprise.

## Definition

Suppose we have given some collection of computational objects which we call "terms", with an *untyped* notion of "reduction" or "[[computation]]".  These could be terms in a [[lambda calculus]], or elements of a [[partial combinatory algebra]], or even programs in a real-world programming language such as C, Java, or Python.
We write $x\Rightarrow y$ to mean that $x$ computes or reduces to $y$ (in some number of steps).
We then define simultaneously the following [[predicates]], where $a,b,A$ are [[terms]]:

* $A$ is a type, often written $A\;type$.
* $a$ has type $A$, written $a:A$.
* $a$ and $b$ are equal elements of type $A$, written $a=b:A$.

The definition is by [[induction-recursion]].  We define the property "$A$ is a type" *inductively*, and simultaneously we define *recursively* two functions on such $A$, one giving the collection of terms of type $A$, and the other giving the collection of pairs of terms considered equal as elements of $A$.  Note that each such collection may itself be a prior inductive definition.

The definition is intended to satisfy some properties:

* Equality:
  * $a:A$ if and only if $a=a:A$. (Reflexivity)
  * If $a=b:A$ then $b=a:A$. (Symmetry)
  * If $a=b:A$ and $b=c:A$ then $a=c:A$. (Transitivity)

* Reverse reduction:
  * If $V\;type$ and $A \Rightarrow V$ then $A\;type$.
  * The other judgment forms similarly respect reverse reduction in all positions.

By reflexivity, the specification of which terms have a type is technically redundant. It is just reiterating the reflexivity instances of equality. So for this meaning explanation, the recursive part of the definition is assigning types their meaning as [[partial equivalence relations]] (PERs) on terms.

We expect $a:A$ *if and only if* $a=a:A$, rather than just the one, usual direction because we want equality to only be defined on elements. By symmetry and transitivity, $a=b:A$ implies $a=a:A$ and $b=b:A$, so then by the unusual direction, $a:A$ and $b:A$.

The reverse reduction properties are saying that the relevant information about a term comes from the value it reduces to. Meaning explanations provide a sense in which type theory is about computational behavior. For this system, the computational behavior of a term is to return a value. Note however that not all terms reduce to a value, for example, the $\Omega$ combinator: $(\lambda x.x\,x) (\lambda x.x\,x)$.

The clauses we should include in the inductive-recursive definition depend on what type constructors we want to put into our type theory; in each case the base notion of untyped computation must include corresponding operators.  Here are three paradigmatic examples.

### Natural numbers

Suppose that we have terms $N$, $0$, and $s(x)$, along with a [[recursion|recursor]] $rec(n,z_0,x\,r. z_s)$ such that
$$\frac{n \Rightarrow 0 \qquad z_0 \Rightarrow v}
{rec(n,z_0,x\,r.z_s) \Rightarrow v}$$
and
$$\frac{n' \Rightarrow s(n) \qquad z_s[n/x,rec(n,z_0,x\,r.z_s)/r] \Rightarrow v}
{rec(n',z_0,x\,r. z_s) \Rightarrow v}.$$
The term $rec(n,z_0,x\,r. z_s)$ is intended to denote the value at $n$ of a recursively defined function specified to be $z_0$ at zero and $z_s$ at successors.  Here $x$ and $r$ are [[variables]] which may occur free in $z_s$ but are bound in $rec(n,z_0,x\,r. z_s)$, representing the predecessor and the result of the last recursive call, respectively.

Then to explain the [[natural numbers type]], we include a clause in the [[inductive definition]] of "$A$ is a type" which says

* if $A\Rightarrow N$, then $A$ is a type.

The corresponding clause for elements is

* if $A\Rightarrow N$, then the collection of terms $a$ such that $a:A$ is defined inductively by the clauses:
  * if $a\Rightarrow 0$, then $a:A$.
  * if $a\Rightarrow s(b)$ and $b:A$, then $a:A$.

Note that the recurrence in this inductive definition doesn't go via the overall recursive definition of $a:A$. In other words, to put the above clause more pedantically, we first define

* an inductive predicate $IsNat$:
  * if $a\Rightarrow 0$, then $IsNat(a)$.
  * if $a\Rightarrow s(b)$ and $IsNat(b)$, then $IsNat(a)$.

Then say

* if $A\Rightarrow N$, then the collection of terms $a$ such that $a:A$ is those such that $IsNat(a)$.

The clause for equality is

* if $A\Rightarrow N$, then the collection of pairs $a,b$ such that $a=b:A$ is defined inductively by the clauses:
  * if $a\Rightarrow 0$ and $b\Rightarrow 0$, then $a=b:A$.
  * if $a\Rightarrow s(a')$ and $b\Rightarrow s(b')$ and $a'=b':A$, then $a=b:A$.

With a similar separation of the inductive definition.

### Function types

Suppose that we have a term constructor yielding "$A\to B$" for terms $A$ and $B$, and similarly $app(f,a)$ and $\lambda x.t$, such that
$$\frac{f \Rightarrow \lambda x.t \qquad t[a/x] \Rightarrow v}
{app(f, a) \Rightarrow v}.$$
Then to explain the [[function type]], we include a clause in the inductive definition of "$A$ is a type" which says

* if $A$ is a type and $B$ is a type and $C\Rightarrow (A\to B)$, then $C$ is a type.

The clause for elements is:

* if $C\Rightarrow (A\to B)$ for types $A$ and $B$, and for any $a=a':A$ we have $app(f,a)=app(f,a'):B$, then $f:C$.  In other words, the collection of elements of type $A\to B$ is the collection of terms $f$ such that for any $a=a':A$ we have $app(f,a)=app(f,a'):B$.

For equality, we have:

* if $C\Rightarrow (A\to B)$ for types $A$ and $B$, and for any $a=a':A$ we have $app(f,a)=app(g,a'):B$, then $f=g:C$.

These are not inductive definitions as we had for the natural numbers; they are merely simple definitions.  On the other hand, we are now using the fact that the function mapping types to their collections of element pairs is recursively defined, since in order to make this definition, we must invoke it at $A$ and $B$. The negative occurrence of $a=a':A$ in this clause (and similar occurrences in other clauses for higher order types) is the reason why the whole explanation cannot just be defined by mutual induction.

This clause for function types is not the one Martin-Löf proposed. Unlike all of Martin-Löf's clauses, this one gives *all* terms of a function type directly; it doesn't just give the canonical terms first, with the rest defined via reduction. This is because it defines the elements in reference to the elimination form, $app$, instead of the introduction form, $\lambda$, as Martin-Löf did. Here's Martin-Löf's clause for elements (adapting for this non-dependent case):

* if $C\Rightarrow (A\to B)$ for types $A$ and $B$, $f \Rightarrow \lambda x.b$ for some (variable $x$ and term) $b$, and for any $a=a':A$ we have $b[a/x]=b[a'/x]:B$, then $f:C$.

The element equality clause gets an analogous change. (Exercise!)

Both styles of function type validate all the usual function rules of extensional type theory.

### Identity types

Suppose we have a term constructor yielding $Id_A(a,b)$ for terms $A,a,b$, and similarly some term $refl$.
Then to explain the [[identity type]], we add to the inductive definition of "$A$ is a type" the clause

* If $A$ is a type, $a:A$ and $b:A$, and $C \Rightarrow Id_A(a,b)$, then $C$ is a type.

Here we are finally using the fact that typehood and elements are defined by induction-recursion: this clause in the *inductive* definition of typehood refers to the value of the *recursive* elementhood function at some smaller type.

The clause for elements is:

* If $A$ is a type, $a:A$, $b:A$, and $C \Rightarrow Id_A(a,b)$, then $p:C$ just if $a=b:A$ and $p \Rightarrow refl$.  So, the collection of elements of $\Id_A(a,b)$ contains something only if $a=b:A$, in which case it contains precisely those terms which reduce to refl.

The clause for equality is:

* If $A$ is a type and $a:A$, $b:A$, and $C \Rightarrow Id_A(a,b)$, then $p=q:C$ just if $a=b:A$, $p\Rightarrow refl$, and $q\Rightarrow refl$.

Because we have deliberately only put $refl$ (and terms reducing to it) into the identity type, this definition of identity types will validate [[axiom UIP|uniqueness of identity proofs]] (UIP). Moreover, because $Id_A(a,b)$ is inhabited if and only if $a=b:A$, it validates the "equality reflection" rule of [[extensional type theory]].

There are simple ways to modify the meaning explanation so as to no longer validate the extensional rules.  For instance, if we add to the semantics additional "free variables", so that there are "neutral terms" which do not compute to any canonical form, and modify the rules of the interpretation so as to say that any normalizable neutral term inhabits every type, then extensionality will no longer hold, since the identity type (like any type) will contain all the variables.  This is maybe not a particularly compelling model of intensionality, but there are proposals for better ways to expand the meaning explanation to deal with [[intensional type theory]].

### Other judgements

#### Type equality

The easy way is to say that

* $A=B$ means that
  * $A\;type$ and $B\;type$, and
  * for all $a$, $a:A$ if and only if $a:B$, and
  * for all $a$ and $b$, $a=b:A$ if and only if $a=b:B$.

This validates the conversion rules. It's extensional in a manner similar to [[material set theory]], which seems appealing, given the rest of the setup. But it turns out to be inconvenient in this case, because for technical reasons, many rules will require more premises in order to be valid. So it's a trade-off. [[Nuprl]] uses a more intensional type equality, inductively defining $A=B$ along with $A\;type$, to simplify the rules.

#### Hypothetical judgements

TODO!

Intuitively, open terms are multi-argument dependent functions. The hypothetical judgements are defined in terms of the basic judgements on closed terms by substituting a lot of appropriate closed terms. Respect for equality is enforced in a manner to agree with the function type constructor.

### The rules of type theory

The claim is that by inspecting these definitions, one can arrive at the formal rules of [[type theory]] for manipulating judgments such as "$A$ is a type", "$a:A$", and "$a=b:A$", which no longer refer to any given notion of computation.  However, Martin-Löf famously wrote

> ...there are ... certain limits to what verbal explanations can do when it comes to justifying axioms and rules of inference.  In the end, everybody must understand for himself.

Moreover, as mentioned above, the meaning explanation (like any particular semantics) validates many other "rules" which are not usually added to type theories, such as
$$ x : Id_N(0,1)  \vdash  5 : N \to N $$
(However in [[Nuprl]] rules such as these are, in fact, included.)  Thus, some discrimination is required, as well as some choices (e.g. whether to add the extensionality rule for identity types).

## References

* [[Per Martin-Löf]], _Intuitionistic Type Theory_ (Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980). Napoli, Bibliopolis, 1984. ([pdf](http://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf))

* [[Per Martin-Löf]], _Constructive mathematics and computer programming_ ([pdf](http://www.cs.tufts.edu/~nr/cs257/archive/per-martin-lof/constructive-math.pdf))

* [[Per Martin-Löf]]. _On the Meanings of the Logical Constants and the Justifications of the Logical Laws_, Nordic Journal of Philosophical Logic, 1(1): 11–60, 1996 ([pdf](http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf))

* [[UF-IAS-2012]], _[Meaning Explanations](https://ncatlab.org/ufias2012/published/Meaning%20Explanations)_

* [[Jonathan Sterling]], _Type Theory and its Meaning Explanations_, [arxiv](https://arxiv.org/abs/1512.01837)

Proposal for [[homotopy type theory]]:

* {#Tsementzis17} [[Dimitris Tsementzis]], _A Meaning Explanation for HoTT_ ([philsci-archive.pitt.edu/12824](http://philsci-archive.pitt.edu/12824/)))

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Meas

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## Idea

The [[category]] whose [[objects]] are [[measurable spaces]] and whose [[morphisms]] are [[measurable functions]] is often denotes _Meas_, or similar.

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measurable cardinal

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## Definition

A [[cardinal number]] $\kappa$ is **measurable** if some (hence any) set of cardinality $\kappa$ admits a two-valued [[measure]] which is $\kappa$-additive, or equivalently an [[ultrafilter]] which is $\kappa$-complete.

## Properties

Any measurable cardinal is, in [[ZFC]], necessarily [[inaccessible cardinal|inaccessible]], and in fact much larger than the smallest inaccessible.  In fact, if $\kappa$ is measurable, then there is a $\kappa$-complete ultrafilter $\mathcal{U}$ on $\{\lambda | \lambda \lt \kappa\}$ which contains the set $\{\lambda | \lambda \lt \kappa$ and $\lambda$ is inaccessible $\}$.  In particular, there are $\kappa$ inaccessible cardinals smaller than $\kappa$. Note that in [[ZF]] it is consistent that $\omega_1$, a successor cardinal, is measurable.

It follows from this that the existence of any measurable cardinals cannot be proven in [[ZFC]], since the existence of inaccessible cardinals cannot be so proven.  Thus measurable cardinals are a kind of [[large cardinal]].  They play an especially important role in large cardinal theory, since any measurable cardinal gives rise to an [[elementary embedding]] of the universe $V$ into some submodel $M$ (such as an [[ultrapower]] by a countably-complete ultrafilter), while the "critical point" of any such embedding is necessarily measurable.

Measurable cardinals are sometimes said to mark the boundary between "small" large cardinals (such as inaccessibles, [[Mahlo cardinal]]s, and [[weakly compact cardinal]]s) and "large" large cardinals (such as [[strongly compact cardinal]]s, [[supercompact cardinal]]s, and so on).

## In category theory

The existence or nonexistence of measurable cardinals can have noticeable impacts on [[category theory]], notably in terms of the properties of the category [[Set]].

For instance, the existence of a measurable cardinal is equivalent to the existence of an [[exact functor]] $F: Set \to Set$ that is not naturally isomorphic to the identity.  This was essentially proved by V. Trnková, and it was rediscovered by Blass in his paper "Exact functors and measurable cardinals" ([Blass 1976](#Blass_76)).

Furthermore, the category $Set^{op}$ has a [[small category|small]] [[dense subcategory]] if and only if there does *not* exist a [[proper class]] of measurable cardinals.  Specifically, the subcategory of all sets of cardinality $\lt\lambda$ is dense in $Set^{op}$ precisely when there are no measurable cardinals larger than $\lambda$.  In particular, the full subcategory on $\mathbb{N}$ is dense in $Set^{op}$ precisely when there are no measurable cardinals at all.

This is theorem A.5 of [[Locally Presentable and Accessible Categories]].

## Related entries

* [[large cardinal]]

* [[real-valued-measurable cardinal]]

## References

* M. Adelman, [[Andreas Blass|A. Blass]], _Exact functors, local connectedness and measurable cardinals_ , Rend. Sem. Mat. Fis. Milano **54** (1984) pp.9-28.

* {#Blass_76} [[Andreas Blass]], _Exact Functors and Measurable Cardinals_ , Pacific J. Math. **63** (1976) pp.335-346. ([euclid](https://projecteuclid.org/euclid.pjm/1102867389))

* [[Andreas Blass]], _Corrections to: 'Exact Functors and Measurable Cardinals'_ , Pacific J. Math. **73** (1977) p.540. ([euclid](https://projecteuclid.org/euclid.pjm/1102810622))

* {: #MR0175954 } [[John Isbell]], _Adequate subcategories_ , Illinois J. Math. **4** (1960) pp.541-552. [MR0175954](http://www.ams.org/mathscinet-getitem?mr=0175954).
([euclid](https://projecteuclid.org/euclid.ijm/1255456274))

* [[John Isbell]], _Subobjects, adequacy, completeness and categories of algebras_ , Rozprawy Mat. **36** (1964) pp.1-32. ([toc](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-0dbcb276-0b92-49eb-b504-a9963119ea3e))

* [[David P. Blecher]], [[Nik Weaver]], *Quantum measurable cardinals* ([arXiv:1607.08505](https://arxiv.org/abs/1607.08505))

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measurable field of Hilbert spaces

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## Idea

A __measurable field of Hilbert spaces__ is the exact analogue of a [[vector bundle]] over a [[topological space]] in the setting of [[fiber bundles]] of infinite-dimensional [[Hilbert spaces]] over [[measurable spaces]].

## Definition

The original definition is due to [[John von Neumann]] (Definition 1 in [Neumann](#Neumann)).

We present here a slightly modernized version, which can be found in many modern sources, e.g., [Takesaki](#Takesaki).

\begin{definition}
Suppose $(X,\Sigma)$ is a [[measurable space]]
equipped with a [[σ-finite measure]] $\mu$,
or, less specifically, with a [[σ-ideal]] $N$ of negligible subsets
so that $(X,\Sigma,N)$ is an [[enhanced measurable space]].
A __measurable field of Hilbert spaces__ over $(X,\Sigma,N)$
is a family $H_x$ of [[Hilbert spaces]] indexed by points $x\in X$
together with a [[vector subspace]] $M$
of the [[product]] $P$ of [[vector spaces]] $\prod_{x\in X} H_x$.
The elements of $M$ are known as __measurable sections__.
The pair $(\{H_x\}_{x\in X},M)$ must satisfy the following conditions.

* For any $m\in M$ the function $X\to\mathbf{R}$ ($x\mapsto \|m(x)\|$)
is a measurable function on $(X,\Sigma)$.
* If for some $p\in P$, the function $X\to\mathbf{C}$ ($x\mapsto\langle p(x),m(x)\rangle$) is a measurable function on $(X,\Sigma)$ for any $m\in M$,
then $p\in M$.
* There is a countable subset $M'\subset M$ such that for any $x\in X$, the closure of the span of vectors $m(x)$ ($m\in M'$) coincides with $H_x$.

\end{definition}

The last condition restrict us to bundles of separable Hilbert spaces.
One can also define bundles of nonseparable Hilbert spaces,
but this cannot be done simply by dropping the last condition.

## Serre–Swan-type duality

The category of measurable fields of Hilbert spaces on $(X,\Sigma,N)$ (as defined above) is equivalent
to the category of countably-generated [[W*-modules]] over the [[commutative von Neumann algebra]]
$\mathrm{L}^\infty(X,\Sigma,N)$.

(If we work with bundles of general, possibly nonseparable Hilbert spaces,
then the [[W*-modules]] do not need to be countably generated.)

## Related entries

* [[duality between algebra and geometry]]

* [[Serre-Swan theorem]]

* [[W*-module]]

* [[commutative von Neumann algebra]]

## References

* {#Neumann} [[John von Neumann]], _On Rings of Operators. Reduction Theory_, The Annals of Mathematics 50:2 (1949), 401.  [doi](http://dx.doi.org/10.2307/1969463).

* {#Takesaki} [[Masamichi Takesaki]], _Theory of Operator Algebras.  I_, Springer, 1979.

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measurable function

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# Measurable functions
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## Idea

The _measurable functions_ are those functions between [[measurable spaces]] amenable to treatment in [[measure theory]], hence they are the evident _[[homomorphisms]]_ between [[measurable spaces]].

## Definitions

Given [[measurable spaces]] $X$ and $Y$, a __measurable function__ from $X$ to $Y$ is a [[function]] $f\colon X \to Y$ such that the [[preimage]] $f^*(T)$ is [[measurable set|measurable]] in $X$ whenever $T$ is measurable in $Y$.

In classical measure theory, it is usually assumed that $Y$ is the [[real line]] (or a variation such as the extended real line or the complex plane) equipped with the [[Borel sets]].  Then $f$ is measurable if and only if $f^{-1}(I)$ is measurable whenever $I \subseteq Y$ is an interval.  More generally, if $Y$ is any [[topological space]] equipped with the Borel sets, then $f$ is measurable if and only if $f^{-1}(I)$ is measurable whenever $I \subseteq Y$ is [[open set|open]].

In some variations of measure theory based on $\delta$‑ or $\sigma$-rings instead of on $\sigma$-algebras, it is necessary to allow [[partial functions]] whose domain is a [[relatively measurable set]].  Classically (when $Y$ is the real line), one achieves (for purposes of [[integration]]) essentially the same result by requiring only that $f^{-1}(I)$ be measurable whenever $I \subseteq Y$ is an interval that does not contain $0$; in other words, one effectively assumes that $f$ is zero wherever it would otherwise be undefined.

## Modulo null sets

If (as in a [[measure space]], a [[Cheng space]], or a [[localisable measurable space]]), we have a notion of [[null sets]] (or [[full sets]]) in $X$ and $Y$, then we may allow a measurable function to be an __[[almost function]]__: a [[partial function]] whose domain is full.  Specifically, an almost function $f\colon X \to Y$ is __measurable__ if the [[preimage]] of every full set in $Y$ is full in $X$ and the preimage of every measurable set in $Y$ is, if not quite measurable in $X$, at least equal to a measurable set in $X$ up to a full set in $X$.  (To emphasise this last change, we may call such functions __almost measurable__.)  Additionally, we consider two measurable almost functions to be [[equal]] (or [[equivalent]] if one prefers) if they are __[[almost equality|almost equal]]__: their [[equaliser]] is full.

## Related concepts

* [[pushforward measure]]

## Literature

* Dave Applebaum, *Measurable Functions* ([pdf](http://www.applebaum.staff.shef.ac.uk/Ch2MeasFn.pdf))

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measurable locale

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# Measurable locales
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## Idea

Measurable locales are certain [[locales]], which may serve as a basis for [[measure theory]] much as general locales serve as a basis for [[topology]] (especially in [[constructive mathematics]]).  Specifically, a measurable locale is equivalent to a compact strictly [[localizable measurable space]] (or [[dual equivalence|dual]] to a [[commutative von Neumann algebra]]).

Ironically, in [[constructive mathematics]], measurable locales are *not* locales (except for the [[empty space]]), on pain of [[excluded middle]].

The concept is due to [[Dmitri Pavlov]].

## Definitions

At present, there is no purely [[order theory|order-theoretic]] definition of measurable locales.  However, there are a few other ways of defining them.

### In classical mathematics

First we give definitions appropriate for [[classical mathematics]].

As a preliminary step, consider the [[category]] $Comp Bool Alg_{sup}$ of [[complete boolean algebras]] and [[supremum]]-preserving [[homomorphisms]] of boolean algebras.  This is a [[full subcategory]] of the category $Frm$ of [[frames]] and so its [[opposite category]] $Comp Bool Alg_{sup}^{op}$ is a full subcategory of the category [[Loc]] of locales.  The category $Meas Loc$ of measurable locales is yet further a full subcategory of $Comp Bool Alg_{sup}^{op}$, so our job is simply to specify *which* complete boolean algebras are the [[objects]] of this category.  There are at least three equivalent ways of doing so.

1. By one definition, a complete boolean algebra $L$ is a __measurable locale__ if there is a (complete) compact strictly [[localizable measure space]] $X$ such that $L$ is [[isomorphic]] (as a boolean algebra) to the boolean algebra $\mathcal{M}/\mathcal{N}$ of [[measurable subsets]] of $X$ modulo [[null subsets]].  Note that the measure on $X$ is irrelevant except to specify the null subsets.  In this way, $Meas Loc$ becomes [[equivalence of categories|equivalent]] to the category $CSLEMS$ of (complete) compact strictly localizable [[enhanced measurable spaces]].  (Not every measure space has a *complete* boolean algebra as $\mathcal{M}/\mathcal{N}$; those which do are called *localizable*, and an enhanced measurable space is a structure with both $\mathcal{M}$ and $\mathcal{N}$ specified.)

2. By another definition, a complete boolean algebra $L$ is a __measurable locale__ if for every element $x \neq 0$ of $L$ there is a [[continuous valuation]] (see [below](#normalmeasures)) $\mu$ on $L$ valued in $[0,\infty]$ such that $\mu(x) = 1$.  (It would be enough to demand that $0 \lt \mu(x) \lt \infty$, since we can rescale the valuation.)

3. By yet another definition, a complete boolean algebra $L$ is a __measurable locale__ if there is a commutative $W^*$-[[W-star-algebra|algebra]] $A$ such that $L$ is [[isomorphic]] (as a boolean algebra) to the boolean algebra $Proj(A)$ of [[projection operators]] in $A$; (see [[idempotent operator]] for a construction of $Proj(A)$).  In this way, $Meas Loc$ becomes [[dual equivalence|dual]] to the category $Comm W^* Alg$ of commutative $W^*$-algebras.  For this definition, we need not require that $L \cong Proj(A)$ be complete (or even a boolean algebra); this can be proved.

Whichever of these equivalent definitions is adopted, a __measurable function__ between measurable locales is simply a [[continuous map]] between them as locales; these are the [[morphisms]] of $Meas Loc$.  In more elementary terms, a measurable function from $L$ to $M$ (thought of as measurable locales) is a function from $M$ to $L$ (thought of as Boolean algebras) that preserves all [[joins]] (and then it also preserves finite [[meets]] and so is a [[frame]] [[homomorphism]]).  Or we can think of this function's [[right adjoint]], a function from $L$ to $M$ that preserves all [[meets]].

### In constructive mathematics

(This section is not due to Pavlov.)

For purposes of [[constructive mathematics]], [[Toby Bartels]] suspects that it is appropriate to use the definition from $W^\star$-algebras, so long as we allow norms to be [[lower real numbers]].  (If they are all [[located real number|located]], then $Proj(A)$ has [[decidable equality]], which we don\'t want to require.)  The other definitions seem more difficult to work with (primarily because they require completeness explicitly).

For this definition (as was remarked above), we need not require that $L \cong Proj(A)$ be complete.  Constructively, we *cannot* require this; $Proj(A)$ need not be complete (although it is still a boolean algebra).  Indeed, consider the point (see [the examples](#examples)), based on $A \coloneqq \mathbb{C}$, which is *not* the complete [[Heyting algebra]] of *all* [[truth values]] but only the (possibly incomplete) boolean algebra $\{\bot, \top\}$ of *classical* truth values (corresponding to the self-adjoint idempotent complex numbers $0$ and $1$).

It seems that there is some notion of completeness that applies here; $L$ should in some sense be 'measurably complete'.  In the point, for example, the [[subsingleton]] $\{* \;|\; P\}$, where $P$ is a truth value, is measurable iff $P$ is true or false, so the supremum of $\{\top \;|\; P\}$ exists in $L$ under the same circumstances.  Figuring this out could allow the definitions through measurable spaces or continuous valuations to work.

Simpson's theory of [[sigma-locale]]s, see below, is developed in classical framework, but could be a good starting point for a constructive theory, as it avoids the focus on complements.  (But the main problem is completeness, not complements.)

## Examples {#examples}

The [[empty space]], which is [[initial object|initial]] in $Meas Loc$, is the [[terminal object|terminal]] boolean algebra with one element.

The [[point]], which is [[terminal object|terminal]] in $Meas Loc$, is the [[initial object|initial]] boolean algebra $\{\bot, \top\}$ of (classical) [[truth values]].

The [[real line]] is the boolean algebra of Lebesgue-[[measurable sets]] of [[real numbers]] modulo the [[null sets]].  This is complete as a boolean algebra because ... \[argument needed\] (and even constructively, it is still a boolean algebra).

Applying the classification of $W^*$-algebras, we find that (up to [[isomorphism]]), every measurable locale is a [[disjoint union]] (dually a [[direct product]] of boolean algebras) of $\{\bot, \top\}^I$ for a set $I$ of arbitrary cardinality.  If $I$ is finite, we get discrete spaces; if $I$ is countable, we get the real line.  Note that a single real line is already isomorphic to the union of countably infinitely many real lines.  (Of course, we can\'t expect this [[classification theorem]] to hold constructively.)

### Borel measurable spaces

To do [[measure theory]], it\'s important to know how to interpret the [[real line]] with the [[Borel sets]] (rather than Lebesgue sets) as a measurable space.  This is actually a little tricky, because [[Lebesgue measure]] is not complete on the Borel sets, and passing to the Lebesgue sets gives a different measurable space.  We might simply take the Borel sets as they are (so that only the empty set is null), but then this is not complete as a boolean algebra.

One might suspect that there is no Borel real line in $Meas Loc$, which would cast serious doubts on that as a category for measure theory.  But we can argue by [[abstract nonsense]] that it must exist, and more generally that any [[locale]] (including the [[locale of opens]] of any [[topological space]]) has a __Borel measurable locale__, as follows:

... <http://mathoverflow.net/questions/31603/why-do-probabilists-take-random-variables-to-be-borel-and-not-lebesgue-measurab/31724#31724> ... [[representable functor theorem]] ...

In the case of the real line, the resulting measurable locale is actually rather complicated: something like an uncountable disjoint union of points and lines.

This process is a [[functor]] $B\colon Loc \to Meas Loc$, which we can also extend to $Top \to Loc \to Meas Loc$.

### Lebesgue measurable spaces

Similarly, [[smooth manifolds]] give rise to (much simpler!) measurable locales by taking the Lebesgue sets (possible since we know which sets are null sets on a smooth manifold).  This process is a [[functor]] $L\colon SmoothMan_subm \to MeasLoc$, where the [[morphisms]] in $SmoothMan_subm$ are the [[smooth map|smooth]] [[submersions]].

In the case of the [[real line]], we get the usual line described under the basic examples above.  In fact, *any* [[second-countable space|second-countable]] smooth manifold with positive dimension gives rise to this measurable locale; the really interesting thing about $L$ is what it does to the morphisms.

## Measures {#normalmeasures}

On *any* complete boolean algebra $L$, given any [[abelian monoid]] $R$ equipped with a [[convergence space|convergence structure]] (such as $[0,\infty]$), a __[[normal measure]]__ on $L$ valued in $R$ is a [[function]] $\mu\colon L \to R$ such that:

*  _[[inclusion-exclusion|additivity]]_: $\mu(\bot) = 0$ and $\mu(x \wedge y) + \mu(x \vee y) = \mu(x) + \mu(y)$;

*  _[[Scott continuity|continuity]]_: if $S$ is a downward-[[codirected set|directed]] [[subset]] of $L$ whose [[infimum]] is the [[bottom element]], then the [[net]] $(\mu(x))_{x\colon S}$ converges to $0$.

One could, of course, define a garden-variety [[measure]] by requiring continuity only when $S$ is the image of a decreasing [[sequence]], but apparently normal measures are what we want for measurable locales.  Normal measures are also known as [[continuous valuations]].

Absolutely continuous measures on, say, Lebesgue space correspond to normal measures on the real line as a measurable locale.  ([[Lebesgue measure]] is not normal on the $\sigma$-algebra of measurable sets, but it is normal on the boolean algebra of measurable sets modulo null sets, which is what we want.)

This should work in constructive mathematics, with $[0,\infty]$ (for a positive measure) being be the space of nonnegative [[lower reals]].

## Related work

This is related to $\sigma$-[[sigma-locale|locales]].

* Alex Simpson, _Measure, randomness and sublocales_ [link](http://www.sciencedirect.com/science/article/pii/S0168007211001874)

which is part of his work to develop [[synthetic probability theory]].

## References

The basic definition and some elementary properties are given in

* [[Dmitri Pavlov]], _Gelfand-type duality for commutative von Neumann algebras_,
[arXiv:2005.05284](https://arxiv.org/abs/2005.05284).

This paper also establishes  [[Gelfand-type duality for commutative von Neumann algebras]].

The theory of measurable locales originated on [[MathOverflow]], in various questions and answers by [[Dmitri Pavlov]].  Here is an index:

* [[Dmitri Pavlov]], [answer](http://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th/49542#49542) to _Is there a category structure one can place on measure spaces so that category-theoretic products exist?_ (version: 2011-01-08)

[[!redirects measurable locale]]
[[!redirects measurable locales]]
[[!redirects MeasLoc]]
[[!redirects Meas Loc]]


measurable space

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# Measurable spaces
* table of contents
{: toc}

## Idea

Measurable spaces are the traditional prelude to the general theory of [[measure]] and [[integration]].  Basically, a measure is a recipe for computing the size --- e.g., length, area, volume --- of [[subsets]] of a given [[set]] $X$.  The structure of a 'measurable space' picks out those subsets of $X$ for which the size is well-defined; these subsets are called 'measurable'.  The measure on $X$ is then an operation that assigns a number to each measurable subset saying how big it is.

In short: you get a [[measure space]] by placing a measure on a measurable space.

Ideally, all subsets would be measurable, but this contradicts the [[axiom of choice]] for the basic example of [[Lebesgue measure]] on the [[real line]].  Although it is possible to use nonstandard [[foundations]] of mathematics in which all subsets of the real line are Lebesgue measurable, any general theory that includes that example and is more general than those foundations requires some explicit notion of measurable space (or an alternative such as a [[measurable locale]]).

In any case, measurable spaces are of some interest in their own right, even without a measure on them.

## Definitions

We give first the usual notion, assuming the validity of [[excluded middle]] and [[power sets]]; see below for alternative versions, including the [[constructive mathematics|constructive]] and [[predicative mathematics|predicative]] theories.

Given a [[set]] $X$, a __$\sigma$-[[sigma-algebra|algebra]]__ is a collection of [[subsets]] of $X$ that is closed under [[complement|complementation]], [[countable set|countable]] [[unions]], and countable [[intersections]].  A __measurable space__, by the usual modern definition, is a set $X$ equipped with a $\sigma$-algebra $\Sigma$.  The elements of $\Sigma$ are called the __[[measurable sets]]__ of $X$ (or more properly, the measurable subsets of $(X,\Sigma)$).

Given measurable spaces $X$ and $Y$, a __[[measurable function]]__ from $X$ to $Y$ is a [[function]] $f\colon X \to Y$ such that the [[preimage]] $f^*(T)$ is measurable in $X$ whenever $T$ is measurable in $Y$.  Measurable spaces and measurable functions form a [[category]] [[Meas]], which is [[topological concrete category|topological]] over [[Set]].

In classical measure theory, it is usually assumed that $Y$ is the [[real line]] (or a variation) equipped with the [[Borel sets]] (see the examples below).  Then $f$ is measurable if and only if $f^{-1}(I)$ is measurable whenever $I \subseteq Y$ is an interval.

## Variations

We will briefly examine variations of the notion of measurable space, from those most like the standard to those most unlike it.  Most of these are discussed at articles dedicated to them.

Historically, people have used more general notions that $\sigma$-algebras, such as algebras, $\delta$-rings, and similar concepts whose names you can probably now guess; these are all discussed at [[sigma-algebra]].  These are all more general than $\sigma$-algebras, being possibly not closed under some operations.  When using some of these more general rings of measurable sets, it is necessary to allow [[partial functions]] whose domain is a [[relatively measurable set]] as measurable functions; for details, see [[measurable function]].

An __enhanced measurable space__ has, in addition to the $\sigma$-algebra of measurable sets, a $\sigma$-[[sigma-ideal|ideal]] of __measurable [[null sets]]__.  That is, besides the set $X$ and the $\sigma$-alebra $\Sigma$, we have a collection $N \subseteq \Sigma$ that is closed under [[countable set|countable]] unions and taking subsets (within $\Sigma$).  (The elements of $N$ are the *measurable* null sets; a __[[null set]]__ is *any* subset of a measurable null set.)  One can equivalently specify a $\delta$-[[delta-filter|filter]] of __measurable [[full sets]]__; the full sets are the [[complements]] of the null sets.  Either way, this allows us to use almost measurable [[almost functions]] up to [[almost equality]], as described at [[measurable function]].

In [[constructive mathematics]], because [[complement|complementation]] doesn\'t behave nicely, the concept of $\sigma$-algebra is not so useful.  It\'s also essential to use [[almost functions]] to avoid a paucity of measurable functions.  One solution, due to [[Henry Cheng]], may be found at [[Cheng measurable space]]; briefly, we use [[disjoint pairs]] $(A,B)$ of sets instead of individual measurable sets and use formal complements in the algebra, as well as a notion of [[full sets]].  Assuming [[excluded middle]], a Cheng measurable space is actually equivalent to a measurable space equipped with null (or full) sets, as in the previous paragraph.

In order to have the most important theorems of [[measure theory]], it is necessary and sufficient to restrict to [[localizable measures]].  Since localizability refers only to the null (or full) sets, we can actually speak of a __[[localizable measurable space]]__: a measurable space equipped with null (or full) sets as above, with the property that the [[boolean algebra]] of measurable sets [[quotient algebra|modulo]] the null sets is [[complete lattice|complete]].

Another approach to measure theory, more abstract, is to ignore the set $X$ and use *only* the $\sigma$-algebra $\Sigma$, as an abstract [[boolean algebra]] equipped with [[countable set|countable]] [[suprema]]; this is called a __[[measurable algebra]]__ (or a [[measure algebra]] when equipped with a [[measure]]).  A measurable algebra might also can be equipped with a $\sigma$-ideal of null sets (or a $\delta$-filter of full sets), but really it is simpler to take the [[quotient algebra]], which is itself a perfectly good measurable algebra.  Even if a measurable algebra is a [[complete lattice]], it can still be pathological; but if it has enough [[normal measure]]s, then we have a __[[measurable locale]]__; the [[category]] of measurable locales is [[equivalence of categories|equivalent]] to that of [[localizable measurable spaces]] (from the previous paragraph).

Yet another category equivalent to localizable measurable spaces and measurable locales is the [[opposite category]] of the category of [[commutative ring|commutative]] [[von Neumann algebras]]; this is really a version of the [[Gelfand–Neumark theorem]].  Then a *[[noncommutative geometry|noncommutative]]* (localizable) measurable space is (the formal dual of) *any* von Neumann algebra.  In this way, measure theory may be seen as a branch of [[operator algebra]] theory (at least if one assumes that only localizable measurable spaces are well enough behaved to be worthy of study).

## Examples

*  Of course, the [[power set]] of $X$ is closed under all operations, so it is a $\sigma$-algebra.  Thus every set becomes a __[[discrete space|discrete]] measurable space__.

*  If $X$ is a [[topological space]], then the $\sigma$-algebra generated by the open sets (or equivalently, by the closed sets) in $X$ is the __Borel $\sigma$-algebra__; its elements are called the __[[Borel sets]]__ of $X$. In particular, the Borel sets of [[real number]]s are the Borel sets in the [[real line]] with its usual topology.

*  If a measurable space $(X,\Sigma)$ is equipped with a measure $\mu$, making it into a [[measure space]], then the sets of measure zero form a $\sigma$-[[sigma-ideal|ideal]] of $\Sigma$ (that is an [[ideal]] that is also a sub-$\sigma$-ring).  Let a __[[null subset|null set]]__ be *any* set (measurable or not) contained in a set of measure zero; then the null sets form a $\sigma$-ideal in the [[power set]] of $X$.  Call a set __$\mu$-measurable__ if it is the union of a measurable set and a null set; then the $\mu$-measurable sets form a $\sigma$-algebra $\Sigma_\mu$ called the __completion__ of $\Sigma$ under $\mu$, and the measurable space $(X,\Sigma_\mu)$ is the __completion__ of $(X,\Sigma)$.

*  In particular, the __Lebesgue-measurable__ sets in the real line are the elements of the completion of the Borel $\sigma$-algebra under [[Lebesgue measure]].

## Properties

### Relation to von Neumann algebras
 {#RelationToVonNeumannAlgebras}

One version of the [[Gelfand-Naimark theorem|Gel'fand–Naimark theorem]] states that the category of [[commutative algebra|commutative]] $W^*$-[[W*-algebra|algebras]] is [[dual equivalence|dual]] to the category of *localizable* measurable spaces.  (As such, arbitrary $W^*$-algebras may be interpreted as 'noncommutative' measurable spaces in a sense analogous to [[noncommutative geometry]].)  See the [references below](#ReferencesRelationToVonNeumannAlgebras).

To make this work correctly, we cannot simply define localizability as a [[property]] of measurable spaces; instead, a __[[localizable measurable space]]__ is a measurable space (a set $X$ with a $\sigma$-algebra $\Sigma$) with a $\sigma$-[[sigma-ideal|ideal]] $\mathcal{N}$ of $\Sigma$ and $\mathcal{P}X$ simultaneously (called the ideal of __[[null sets]]__) such that $\Sigma/\mathcal{N}$ is a [[complete lattice]]; and a [[morphism]] of localizable measurable spaces is a measurable function, with the property that the [[preimage]] of any null set is null, up to an [[equivalence relation]] where $f \cong g$ if $\{ x \;|\; f(x) \neq g(x) \}$ is a null set.

The requirement that $\Sigma/\mathcal{N}$ be complete is the real localizability condition here; the trick of equipping a measurable space with a $\sigma$-ideal of null sets (or equivalently a $\sigma$-[[filter]] of [[full sets]]) and taking measurable functions only up to equivalence is a common one in other situations.

Localizable measurable spaces can also be studied via the lattice $\Sigma/\mathcal{N}$, which is a [[frame]]; the morphisms correspond to certain [[continuous maps]] between [[locales]], and thus we are studying locales with [[extra structure]], called __[[measurable locales]]__.

### Relation to Boolean toposes

In terms of [[topos theory]], measurable spaces are closely related to [[Boolean toposes]] (e.g. [Jackson 06](#Jackson06), [Henry 14](#Henry14)).

## In alternative foundations

While Lebesgue measure on $\mathbb{R}^n$ can be done in very weak [[foundations]], a general theory of measure and measurable spaces seems to require powerful [[set theory|set-theoretic]] machinery.  Indeed, not much seems to be possible in [[predicative mathematics|predicative]] contexts, and the (nonpredicative) [[constructive mathematics|constructive]] theory is noticeably more complicated than the classical theory.  On the other hand, the classical theory has its own complications, with nonmeasurable sets and functions that can be proved to exist but which seem to never arise in practice.  Instead, there are classically false but apparently consistent foundations in which measure theory is extremely simple.

### Predicative theory
{#Predicative}

The main problem for measure theory in [[predicative mathematics]] is getting your hands on a $\sigma$-algebra.  Once you\'ve got that, you\'ve got a measurable space (obviously) and go on to [[measure space]], where there are no new difficulties.  However, what is (say) a Borel set in the real line?  This is difficult, if not impossible, to explain predicatively.  (In the case of [[Lebesgue measure]], there *are* ways to describe the Lebesgue-measurable sets predicatively, but these do not seem to generalise to a broader theory.)

Note that there is no real problem in describing what, say, an open set is.  Not only can this be done for the real line in the usual $\epsilon$-$\delta$ way, but it is easy to take *any* collection of subsets of any set $X$, call that collection a [[subbase]], and describe which sets are the open sets in the [[topological space|topology]] generated by that subbase.  The reason is that there are only two steps in moving from a subbase to a topology, and while the latter step is too impredicative to allow one to speak of the *set* of all open sets, it\'s OK if you only want to talk about *individual* open sets.  (To be explicit: given a collection $B$ to be used as subbase, a set $G$ is _open_ if, for every point $x$, if $x \in G$, then there exist a [[natural number]] $n$ and elements $A_1, \ldots, A_n$ of $B$ such that $x \in A_i$ for each $i$ and, for every point $y$, if $y \in A_i$ for each $i$, then $y \in G$.  Since we quantify only over points and natural numbers, not over sets or functions, this is a predicative definition, and it\'s easy to prove that the open sets satisfy the axioms of a topology.)

This cannot be done with $\sigma$-algebras, since we need uncountably many sets.  To be sure, each individual step is predicative, and we can freely talk about $G_\delta$ sets and the like, but to define a Borel set we need to quantify over all countable ordinals.  While it is possible to hypothesise the existence of an uncountable ordinal $\omega_1$ and be predicative 'over' $\omega_1$ (and after all, everything else in this section is only predicative over the first infinite ordinal $\omega_0$, which we only have if we accept an [[axiom of infinity]]), this cannot be constructed predicatively.  (The immediate definition of $\omega_1$ as the [[Hartog's number]] of $\omega_0$ uses [[power set]]s; while the construction of an uncountable ordinal by applying the [[well-ordering theorem]] to the [[function set]] $\mathbf{N}^{\mathbf{N}}$ doesn\'t seem to use reasoning that requires the existence of power sets as long as you don\'t also throw in [[excluded middle]], it does use reasoning that is not accepted by any predicative school that I know.)

So as far as I ([[Toby Bartels]]) can tell, there is no general predicative theory of measurable spaces, only an ad hoc theory of Lebsegue measurability.  I would be delighted to learn otherwise!

### Constructive theory

From a constructive perspective, there are a couple of related problems with the classical theory.  One is that the notion of $\sigma$-algebra is highly suspicious, because it relies on an operation, [[complement]]ation, that behaves very differently in the [[intuitionistic logic]] that [[constructive mathematics]] uses.  The other is that, even you acept the definition of $\sigma$-algebra anyway (after all, the Lebesgue-measurable sets on the real line do still form one), there may be very few measurable functions.

Indeed, if we set aside the general theory of measurable spaces and simply do Lebesgue measure ad hoc in a constructive (even predicative) way, we find that instead of measurable [[functions]] we really want measurable [[partial functions]] whose domain of definition is a [[full set]].  This suggests that if we want to define the concept of measurable function, then we have to know what the full sets are.

There is a way out, due to [[Henry Cheng]], for both of these problems at once.  Instead of dealing with individual sets, we will deal with pairs of [[disjoint sets]].  The intuition is that we use disjoint pairs $(A,B)$ such that $A \cup B$ is full ---with $(A,\neg{A})$ being the motivating example in the classical theory---, but we let the $\sigma$-algebra itself tell us which pairs those are.  Once we fix a particular measure, we may find additional pairs whose union is full, somewhat like finding additional measurable sets when taking the completion in the classical theory (although taking the completion is a separate phenomenon here), but that\'s all right; the important thing is that each pair chosen really is full in any measure used (much as each set in a classical $\sigma$-algebra must actually be measurable by any measure used).

See details at [[Cheng space]].

### In dream mathematics

While measure theory only gets more complicated in constructive mathematics, it becomes much easier in [[dream mathematics]].

... more coming ...

## Related concepts

* [[measurable locale]]

* [[noncommutative measurable space]]

## References

### General

For Cheng\'s theory of measure spaces, see the 1985 edition of Bishop & Bridges, _Constructive Analysis_.  (And the references therein, obviously, but I haven\'t read those.)

### Relation to von Neumann algebra
 {#ReferencesRelationToVonNeumannAlgebras}

A discussion of the abstract properties of the category of localizable measurable spaces and its relation to [[von Neumann algebras]] is in

* Alain Guichardet, _Sur la catégorie des algèbres de von Neumann_  (French) Bull. Sci. Math. (2) 90 1966 41–64. ([MSN](http://ams.org/mathscinet-getitem?mr=201989)) ([scan](http://dmitripavlov.org/scans/guichardet.pdf))

A modern treatment of this discussion can be found in

* [[Dmitri Pavlov]], *Gelfand-type duality for commutative von Neumann algebras*, Journal of Pure and Applied Algebra **226** 4 (2022) 106884 [[arXiv:2005.05284](https://arxiv.org/abs/2005.05284), [doi:10.1016/j.jpaa.2021.106884](https://doi.org/10.1016/j.jpaa.2021.106884)]

A useful series of expositions along these lines is in

* [[Dmitri Pavlov]], _On measurable spaces_

  * [MO comment 1](http://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20820#20820)

  * [MO comment 2](http://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th/49542#49542)

See also

* Ryszard Paweł Kostecki, _$W^\ast$-algebras and noncommutative integration_ ([pdf](http://www.fuw.edu.pl/~kostecki/wstarint.pdf))

### Relation to Boolean toposes

* {#Jackson06} Matthew Jackson, _A sheaf-theoretic approach to measure theory_, 2006 ([pdf](http://www.andrew.cmu.edu/~awodey/students/jackson.pdf))

* {#Henry14} [[Simon Henry]], _From toposes to non-commutative geometry through the study of internal Hilbert spaces_, 2014  ([pdf](http://www.normalesup.org/~henry/Thesis.pdf))

category: probability

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measurable subset

# Measurable subsets
* table of contents
{: toc}

## Idea

The measurable subsets of a [[measure space]] $(X,\mu)$ are those [[subsets]] $A$ of the [[underlying set]] $X$ for which the measure $\mu(A)$ is defined (at all, even possibly as infinite).  Intuitively, one might expect every subset of $X$ to be measurable, and this is the case in some examples, but in the standard example of [[Lebesgue measure]] on the [[real line]], this is incompatible with the [[axiom of choice]].  (On the other hand, in [[dream mathematics]], where the full axiom of choice fails, every subset of the real line *is* Lebesgue measurable.)  Regardless, every subset $A$ of $X$ has both an [[outer measure]] $\mu^*(A)$ and an [[inner measure]] $\mu_*(A)$.

The concept actually makes sense in any [[measurable space]] as well as in related contexts such as [[Cheng spaces]] and [[measurable locales]].

## Definitions

Typically, the notion of measurable subset of $X$ is given axiomatically by a [[structure]] on the [[set]] $X$, usually a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{M}$.  The a subset of $X$ is __measurable__ if it belongs to $\mathcal{M}$.

Sometimes $\mathcal{M}$ is a weaker structure, such as a $\delta$-ring; see other variants at [[sigma-algebra]].  Then we require some subsidiary notions: $S$ is __relatively measurable__ if $S \cap T$ belongs to $\mathcal{M}$ whenever $T$ does, and $S$ is __$\sigma$-measurable__ if it is a [[union]] of a [[countable set|countable]] [[family of subsets|family]] of elements of $\mathcal{M}$.

(The terms 'relatively measurable' and '$\sigma$-measurable' are [[Toby Bartels|my own]]; I cannot find them in the literature.  In the case of $\sigma$-measurable sets, the terminology follows a standard pattern.  Halmos uses relatively measurable sets, but he doesn\'t seem to give them a name.)

## Modulo null sets

Besides the $\sigma$-algebra of measurable subsets, we may place another structure on $X$, a $\sigma$-[[sigma-ideal|ideal]] $\mathcal{N}$ in $\mathcal{M}$.  (This structure also exists, for example, for any [[measure space]], and already for a [[Cheng space]] or a [[localisable measurable space]].)  Then a __[[null set]]__ is any subset of $X$ (measurable or not) contained in an element of $\mathcal{N}$.  The null sets form a $\sigma$-ideal $\bar{\mathcal{N}}$ of the [[power set]] $\mathcal{P}X$, and we may equivalently begin with the $\sigma$-ideal of null sets as long as every null set is contained in a measurable null set.

This allows two complementary modifications to the notion of measurable set:

1. We may accept the [[union]] of any measurable set and any null set as measurable.  Since this changes the meaning of 'measurable', we may speak of $\mathcal{M}$-measurable and $(\mathcal{M},\mathcal{N})$-measurable sets.  The collection of such sets may be denoted $\mathcal{M} \cup \bar{\mathcal{N}}$ (applying $\cup$ pointwise).

2. We may regard two measurable sets as equivalent if their [[symmetric difference]] is a null set (and hence an element of $\mathcal{N}$).  This defines an [[equivalence relation]] on $\mathcal{M}$; the collection of [[equivalence classes]] is denoted $\mathcal{M}/\mathcal{N}$.

Note that $(\mathcal{M} \cup \bar{\mathcal{N}})/\mathcal{N} \cong \mathcal{M}/\mathcal{N}$; indeed, this diagram commutes:
$$ \array {
   \mathcal{M}             & \hookrightarrow & \mathcal{M} \cup \bar{\mathcal{N}} \\
   \downarrow              &                 & \downarrow \\
   \mathcal{M}/\mathcal{N} & \cong           & (\mathcal{M} \cup \bar{\mathcal{N}})/\mathcal{N}
} $$
Accordingly, one may skip the former modification if one intends to also perform the latter.  Nevertheless, even when using $\mathcal{M}/\mathcal{N}$ as the lattice of measurable 'sets', if one considers a subset $A$ of $X$ and asks whether $A$ is 'measurable', one usually means whether $A \in \mathcal{M} \cup \bar{\mathcal{N}}$.

One could equally well begin with a $\delta$-[[delta-filter|filter]] $\mathcal{F}$, although a $\sigma$-ideal is more traditional.  Then a __[[full set]]__ is any subset of $X$ that contains in an element of $\mathcal{F}$.  (If we start with the $\delta$-filter $\bar{\mathcal{F}}$ in $\mathcal{P}X$, then every full set must contain a measurable full set.)  In [[constructive mathematics]], full sets are more fundamental for such examples as [[Lebesgue measure]].  In any case, the modifications are as follows:

1. We may accept the [[intersection]] of any measurable set and any full set as measurable; the collection of such sets may be denoted $\mathcal{M} \cap \bar{\mathcal{F}}$.

2. We may regard two measurable sets as equivalent if their [[biconditional]] is a full set; the collection of equivalence classes is denoted $\mathcal{M}/\mathcal{F}$.

Then we have this commuting diagram:
$$ \array {
   \mathcal{M}             & \hookrightarrow & \mathcal{M} \cap \bar{\mathcal{F}} \\
   \downarrow              &                 & \downarrow \\
   \mathcal{M}/\mathcal{F} & \cong           & (\mathcal{M} \cap \bar{\mathcal{F}})/\mathcal{F}
} $$

## Abstract measurable sets

Already we have seen that we may be more interested in [[equivalence classes]] of measurable sets than in the sets themselves.  We may well start with any appropriate algebra in the place of $\mathcal{M}/\mathcal{F}$ above and regard its elements are 'measurable sets'.  This may be done in the theory of [[measurable locales]] and other 'pointless' approaches to measure theory.

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measure space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Integration theory
+-- {: .hide}
[[!include integration theory - contents]]
=--
=--
=--

# Measure spaces
* table of contents
{: toc}

## Idea

In [[measure theory]], measure spaces are used in the general theory of measure and [[integration]], somewhat analogous to the role played by [[topological spaces]] in the study of continuity.

For the general theory of measure spaces, we first need a _[[measurable space]]_ $(X, \Sigma)$, that is a [[set]] equipped with a collection $\Sigma$ of __measurable sets__ complete under certain operations.  Then this becomes a measure space $(X, \Sigma, \mu)$ by throwing in a [[function]] $\mu$ from $\Sigma$ to a space of values (such as the [[real line]]) that gets along with the set-theoretic operations that $\Sigma$ has.  If $E$ is a measurable set, then $\mu(E)$ is called the __measure__ of $E$ with respect to $\mu$.

## Notation {#notation}

The original notation for an [[integral]] (going back to [[Gottfried Leibniz]]) was
\[ \label{Leibniz} \int_a^b f(x) \,\mathrm{d}x \]
(where $f(x)$ would be replaced by some formula in the variable $x$).  In modern measure theory, we can now understand this as the integral of the [[measurable function]] $f$ on the [[interval]] $[a,b]$ relative to [[Lebesgue measure]].  If we wish to generalise from Lebesgue measure to an arbitrary measure $\mu$ and generalise from $[a,b]$ to an arbitrary measurable set $S$, then we can write
\[ \label{full} \int_{x\in{S}} f(x) \,\mu(\mathrm{d}x) \]
instead.  Now, if $f$ is not given by a formula but rather explicitly named, then there is no need for the dummy variable $x$, so we should write
\[ \label{simple} \int_S f \,\mu .\]
However, it has been more common to keep the symbol '$\mathrm{d}$' and write
\[ \label{excessive} \int_S f \,\mathrm{d}\mu .\]
(Note that '$\mathrm{d}$' can be read as 'with respect to' in both (eq:Leibniz) and (eq:excessive), although meaning different things; in the former case, it indicates the dummy variable, while in the latter case, it indicates the measure.)  This notation then leads to replacing (eq:full) with
\[ \label{switched} \int_{x\in{S}} f(x) \,\mathrm{d}\mu(x) .\]
This last notation, however, hides the fact that integrating a function with respect to a measure is a way of multiplying a function by a measure to get a new measure; the integral of $f$ on $S$ with respect to $\mu$ is simply the measure of $S$ with respect to $f \mu$, as can be seen in (eq:simple).  Compare also notation for [[Radon-Nikodym derivatives]].

It is also possible to take the entire expression '$\mathrm{d}\mu$' as the name of the measure, writing $\mathrm{d}\mu(A)$ even where the common notation is $\mu(A)$.  In that case, the common expression (eq:excessive) is literally the same as (what would otherwise be) (eq:simple), although (eq:switched) is not quite the same as (what would otherwise be) (eq:full).

We will use (eq:simple) below (although other forms may well be found on other pages).

See [Usenet discussion](http://groups.google.com/group/sci.math.research/browse_thread/thread/e28593bfd6b83aac/67a61d19e8f4d57f), and contrast (eq:switched) with the [Stieltjes integral](https://en.wikipedia.org/wiki/Stieltjes_integral).  (The point is that it is $\mathrm{d}x$, not just $x$, that gives us the relevant measure in (eq:Leibniz).)  The notation (eq:simple) has also been used in an introductory graduate-level course by [[John Baez]].

There is also some variation in notation as to whether to use a roman '$\mathrm{d}$' or an italic '$\mathit{d}$'; roman is more common in England and italic in America.  But of course, that variation should not cause any difficulties!

## Definitions

A __measure space__ is a [[measurable space]] equipped with a measure.  There are many different kinds of measures; we start with the most specific and then consider generalisations.  The motivating example is [[Lebesgue measure]] on the [[unit interval]].

### Probability measures

Let $(X, \Sigma)$ be a measurable space.  A __[[probability measure]]__ on $X$ (due to [[Andrey Kolmogorov]]) is a [[function]] $\mu$ from the collection $\Sigma$ of measurable sets to the [[unit interval]] $[0,1]$ such that:

1.  The measure of the entire space is [[one]]:  $\mu(X) = 1$;
2.  Countable additivity:  $\mu(\bigcup_{i = 1}^{\infty} S_i) = \sum_{i=1}^{\infty} \mu(S_i)$ whenever the $S_i$ are mutually [[disjoint sets|disjoint]].

(Part of the latter condition is the requirement that the sum on the right-hand side must converge.)

It is sometimes stated (but in fact follows from the above) that:

*  Finitary additivity:  $\mu(S \cup T) = \mu(S) + \mu(T)$ whenever $S$ and $T$ are disjoint.
*  $\mu$ is increasing:  $\mu(A) \leq \mu(B)$ if $A \subseteq B$.
*  The measure of the [[empty set]] is [[zero]]:  $\mu(\emptyset) = 0$;

The first of these conditions will follow for all of the generalised notions of measure below, but the second usually will not. Related query discussion is archived [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2829&Focus=28141#Comment_28141).

### Generalizations

From now on, we drop the condition $\mu(X)=1$; the next step is to generalize the [[target]] of $\mu$, as follows:

*  Use $[0,\infty)$ (instead of $[0,1]$) for a (finite) __positive measure__.
*  Use $\mathbf{R} = {]{-\infty,\infty}[}$ for a (finite) __signed measure__ (alias __charge__).
*  Use $\mathbf{C}$ for a (finite) __complex-valued measure__.
*  Use an arbitrary [[topological vector space]] $V$ for a __vector-valued measure__.
*  In principle, one could go further yet; $V$ just needs an analogue of addition with a notion of infinitary sum (such as a [[topological abelian group]] has).  But until someone suggests a useful example, we will leave this to the [[centipede mathematics|centipedes]].

We define an __infinite measure__ by replacing the domain of $\mu$ by
an ideal $\Sigma'$ of $\Sigma$ such that the following
saturation condition is satisfied: if $\{S_i\}_{i\in I}$ is a disjoint
family of elements of $\Sigma'$ and $\sum_{i\in I}|\mu|(S_i)$ exists (and is finite),
then $\bigcup_{i\in I}S_i\in\Sigma'$.
The countable additivity condition should now be modified to require
$\bigcup_{i\in I}S_i\in\Sigma'$.

An infinite measure $\mu$ is __semifinite__ if for any $S\in\Sigma\setminus\Sigma'$
there is $T\in\Sigma'$ such that $T\subset S$ and $\mu(T)\gt0$.
The [[Radon-Nikodym theorem]] shows that semifinite complex-valued
measures that are absolutely continuous with respect to some fixed
[[localizable measure]] form a [[free module]] over the [[algebra]]
of complex-valued measurable functions (not necessarily bounded).

Some further terms:

*  We can combine conditions; for example a __finite positive measure__ takes values in ${[{0,\infty}[}$.
*  A measure is __bounded__ if, for some (finite) real number $M$, $|\mu(S)| \leq M$ for every measurable set $S$.  (This requires that the target of $\mu$ have a real-valued notion of absolute value or norm, so a vector-valued measure should be valued in a [[Banach space]] or something similar.)
*  A measure is __$\sigma$-finite__ if every measurable set is a union of countably many sets with finite measure.

Remarks:

*  The property that $\mu$ is increasing holds for all positive measures but may fail for others.
*  A positive measure $\mu$ that satisfies $\mu(X)=1$ must be a probability measure as defined earlier; that is, it satisfies $\mu(S) \leq 1$ for all $S$.
*  When $\infty$ is allowed as a value of $\mu$, then the requirement in (3) that the sum converges should be interpreted in this light; that is, the sum may diverge to infinity.  (For a positive measure, therefore, the convergence criterion is vacuous in [[classical mathematics]].)
*  Notice that $-\infty$ is not allowed as a value for a signed measure.  It would work just as well to allow $-\infty$ and forbid $\infty$.  It is even possible to allow both, but this is a little trickier (because of $-\infty + \infty$), so we deal with it later (at the end of this subsection).

Another possibility is to generalize the [[source]] of $\mu$; instead of using a $\sigma$-algebra on $X$, we could use a $\sigma$-ring or even a $\delta$-ring.  These versions are mostly more about changing the definition of [[measurable space]], so refer there for details of the definitions; however, we note that (3), when $\Sigma$ is a $\delta$-ring, should state that the left-hand side exists (that is, the union is measurable) if the right-hand side converges.  Generalizing $\Sigma$ in this way is complementary to generalizing the target above; in particular it may allow one to avoid dealing with $\infty$.  For example, while Lebesgue measure is only a positive measure on a $\sigma$-algebra, it is a *finite* positive measure on the $\delta$-ring of bounded measurable sets.  Indeed, every signed measure gives rise to finite measure on its $\delta$-ring of finitely measurable sets (as defined below); conversely, every $\sigma$-finite measure can be recovered from this by imposing (3) in all cases.

Yet another possibility is to drop countable additivity, replacing it with finite additivity.  The result is a __finitely additive measure__, sometimes called a __charge__ to avoid the [[red herring principle]]; in contrast, the usual sort of measure may be called __countably additive__.  For a charge, one could replace $\Sigma$ with an algebra (or even a ring) of sets; again see [[measurable space]] for these definitions.

Finally, an __extended measure__ takes values in the set $[-\infty,\infty]$ of [[extended real numbers]].  Here we have the problem that, even when considering finite additivity, we might have to add $\infty$ and $-\infty$.  While we might simply require that this never happens (so that at least one of $\mu(S)$ and $\mu(T)$ must be finite if they have opposite signs and $S \cap T = \empty$), this does not include some examples that we want (and in fact it would follow that $\infty$ and $-\infty$ cannot both be values of $\mu$ after all).  To deal with this, we define an extended measure to be a formal difference $\mu^+ - \mu^-$ of positive measures; $\mu(S) = \mu^+(S) - \mu^-(S)$ whenever this is not of the form $\infty - \infty$ and is otherwise undefined.  Note that the set of extended measures on $X$ is a [[quotient set]] of the set of pairs of positive measures; we say that $\mu = \nu$ if $\mu(S) = \nu(S)$ whenever either side is defined, that is if $\mu$ and $\nu$ are the same as [[partial functions]] from $\Sigma$ to $[-\infty,\infty]$.

Any extended measure restricts to an infinite signed measure, taking $\Sigma'\subset\Sigma$
to be the set of elements of $\Sigma$ with a finite measure.
Vice versa, an infinite signed measure $\mu$
canonically extends to an extended measure:
we can define $\mu_+$ and $\mu_-$ as usual and then take
the formal difference $\mu_+-\mu_-$.

### Constructive theory

In Henry Cheng\'s [[constructive mathematics|constructive]] theory of measure, the definition of [[measurable space]] becomes more complicated; the main point is that a single measurable set $S$ is replaced by a _complemented pair_ $(S,T)$.  Once that is understood, very little needs to be changed to define a measure space.

In the requirements (1--3), the constants $\empty$ and $X$ and the operation $\union$ are interpreted by formal [[de Morgan duality]], as explained at [[Cheng measurable space]].  The convergence requirement in (3) should be interpreted in the strong sense of located convergence and is no longer trivial for positive measures.  We must add a further requirement to enforce the idea that $\mu(S,T)$ is the measure of $S$ alone, as follows:

*  $\mu(S,T) = \mu(S,U)$ whenever $(S,T)$ and $(S,U)$ are both complemented pairs.

In general, a _measurable set_ is any set $S$ such that $(S,T)$ is a complemented pair for some set $T$; the term 'measurable set' in the classical theory should be interpreted as either 'mesurable set' or 'complemented pair' in the constructive theory, depending on context.  Usually both interpretations will actually work, but often only the first set of the pair will matter, thanks to the axiom above.

We will mention other occasional fine points in the constructive theory when they occur; the main outline does not change.

+-- {: .query}
I need to check Bishop & Bridges to see if there are any other changes, but I don\'t think so; that is, I went through the following, and it all seems correct as it is.  ---Toby
=--

### Measures on a $\sigma$-frame

In measure theory, a __measure__ on a [[sigma-frame|$\sigma$-frame]] or more generally a [[sigma-complete lattice|$\sigma$-complete]] [[distributive lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a [[valuation (measure theory)|valuation]] $\mu:L \to [0, \infty]$ such that the elements are mutually disjoint and the probability valuation is denumerably/countably additive
$$\forall s\in L^\mathbb{N}. \forall m \in \mathbb{N}. \forall n \in \mathbb{N}. (m \neq n) \wedge (s(m) \wedge s(n) = \bot)$$
$$\forall s\in L^\mathbb{N}. \mu(\Vee_{n:\mathbb{N}} s(n)) = \sum_{n:\mathbb{N}} \mu(s(n))$$

### Subsidiary definitions

Given a measure space $(X,\Sigma,\mu)$, a __$\mu$-null $\Sigma$-measurable set__ is a measurable set $N$ such that $\mu(S) = 0$ whenever $S \subseteq N$ is measurable; a __$\mu$-[[null set]]__ is any subset of a null measurable set.  In a positive measure space, we don\'t have to bother with $S$; $N$ will be a null measurable set as long as $\mu(N) = 0$.

A __$\mu$-full $\Sigma$-measurable set__ is a measurable set $F$ such that $\mu(S) = \mu(S \cap F)$ for every measurable set $S$; a __$\mu$-[[full set]]__ is any superset of a full measurable set.  In a probability measure space, we don\'t have to bother with $S$; $F$ will be a full measurable set as long as $\mu(F) = 1$.  Classically, a full set is precisely the [[complement]] of a null set, but this doesn\'t hold in the constructive theory.

A property of elements of $X$ holds __$\mu$-almost everywhere__ if the set of values where it holds is full.

A measure is __complete__ if every full set is measurable.  We may form the __completion__ of a measure space by accepting as a measurable set the intersection of any set and a full set; these __$\mu$-measurable sets__ will automatically form a $\sigma$-algebra (or whatever $\Sigma$ originally was).  Classically, a measure is complete if and only if every null set is measurable and a set is $\mu$-measurable if and only if it is the [[symmetric difference]] between a measurable set and a null set.

A __finitely $\mu$-measurable set__ is a measurable set $M$ such that $\mu(S)$ is finite whenever $S \subseteq M$ is measurable; a __$\sigma$-finitely $\mu$-measurable set__ is any union of countably many finitely measurable sets.  Again, we don\'t have to bother with $S$ in a positive measure space.  Note that a measure space is ($\sigma$)-finite if and only if every measurable set is ($\sigma$)-finitely measurable.  The finitely measurable sets form a $\delta$-ring, and the $\sigma$-finitely measurable sets form a $\sigma$-ring.

Recall that a $\Sigma$-[[measurable function]] from $(X,\Sigma)$ to some other measurable space is any function $f$ such that the [[preimage]] under $f$ of a measurable set is always measurable (or something more complicated in the constructive theory).  Now that we have a measure space, let a __$\mu$-measurable function__ be a [[partial function]] $f$ from $X$ to some other measurable space such that the domain of $f$ is full and the preimage under $f$ of a measurable set is always $\mu$-measurable (that is measurable in the completion of $\mu$), and let two such functions be __$\mu$-equivalent__ if their [[equaliser]] is a full set.  We are really interested in the [[quotient set]] under this equivalence and so identify equivalent $\mu$-measurable functions.  Classically, every $\mu$-measurable function is equivalent to some (total) $\Sigma$-measurable function, so the definition is simpler in that case; however, partial functions still come up naturally in the classical theory, so it can be convenient to allow them rather than (as is usually done in a rigorous treatment) systematically replacing them with total functions.

A __$\mu$-integrable function__ is a $\mu$-measurable function $f$ such that the integral $\int_S f \,\mu$ (as defined below) exists for every measurable set $S$; it is enough to check $S = X$.  Equivalently, we may say that it is a $\mu$-measurable function $f$ such that the extended measure $f \mu$ (also defined below) is actually a finite measure.  (In any case, we get a finite measure $f \mu$ if $f$ is integrable.)

### Integration

In the following, 'measurable' will mean $\mu$-measurable.  That is, we assume that $\mu$ is complete and identify $\mu$-equivalent functions.  We will also assume that $\mu$ is a positive measure until I make sure of what must be done to generalise.

Given a measure $\mu$, a measurable set $S$, and a measurable function $f$, we will define the integral
$$ \int_S f \,\mu $$
(see [above](#notation) for variations in notation) in stages, from the simplest form of $f$ to the most arbitrary.

Each measurable subset $S \subseteq X$ induces a measurable [[characteristic function]] $\chi_S\colon X \to \mathbb{R}_+$ where $\chi_S(x) = 1$ if $x \in S$, $\chi_S(x) = 0$ if $x \in \neg{S}$.  (In the constructive theory, where $S$ is a complemented pair, $\chi_S$ is a partial function with a full domain, so $\chi_S f$ is still a measurable function as defined above.)  In general, we have
$$ \int_S f \,\mu = \int_X \chi_S f \,\mu ,$$
so from now on we will assume that we are integrating over all of $X$ (and drop the subscript).

A positive __[[simple function]]__ is a finite $\mathbb{R}_+$-linear combination of measurable characteristic functions; the first form of integral that we define is
$$ \int \sum_{1 \leq i \leq n} a_i \chi_{S_i} \,\mu = \sum_{1 \leq i \leq n} a_i \mu(S_i) .$$

The integral is extended to all measurable functions $f\colon X \to [0, \infty]$ by the rule
$$ \int f \,\mu = \sup \{ \int s \,\mu \;|\; 0 \leq s \leq f, s\; simple \} $$
if this [[supremum]] converges.  Classically, the integral either converges or diverges to infinity, so $\int f \,\mu$ exists in some sense in any case; the possibilities are more complicated constructively.

For any measurable function $f\colon X \to [-\infty, \infty]$, define $f_+$ and $f_{-}$ by
$$ f_+(x) = \max\{f(x), 0\}, \qquad f_{-}(x) = \max\{-f(x), 0\} $$
so that $f = f_+ - f_{-}$, ${|f|} = f_+ + f_{-}$.  Then the final definition is
$$ \int f \,\mu = \int f_{+} \,\mu - \int f_{-} \,\mu $$
if both integrals on the right converge.  Classically, the other possibilities are $\infty$, $-\infty$, and $\infty - \infty$; not much can be done with the latter.

A measurable function $f$ is **integrable** with respect to $\mu$ if this integral converges.  It can be proved that all of the definitions above are consistent; that is, if the final definition is applied to a simple function, then it agrees with the original definition.

If $f$ takes values in the field $\mathbb{C}$ of [[complex numbers]] or in some more general [[Banach space]] $V$, then we can still ask whether ${|f|}$ is integrable.  If it is, then we say that $f$ is __absolutely integrable__.  We can then define the integral of $f$; we always have
$$ {\|\int f \,\mu\|} \leq \int {\|f\|} \,\mu .$$
This integral is easy to define if $V$ has a basis; for example, a measurable complex-valued function $f\colon X \to \mathbb{C}$ is integrable iff both its real and imaginary parts are integrable, and we have
$$ \int f \,\mu = \int \Re{f} \,\mu + \mathrm{i} \int \Im{f} \,\mu .$$

+-- {: .query}
I need to check [[HAF]] for more details here in the general case.  In particular, something can be integrable without being absolutely integrable (although not if it\'s complex-valued, of course) or indeed even without being valued in a (pseudo)normed space.
=--

The vector space of $V$-valued integrable functions is itself a Banach space, using the norm
$$ {\|f\|_1} \coloneqq \int {\|f\|} \,\mu .$$
Note that we must use the notion of measurable function as an equivalence class of functions to get a Banach space here; otherwise we have only a pre-Banach space (that is, a complete pseudonormed vector space).

This Banach space is called a __[[Lebesgue space]]__ and is denoted $L^1(\mu,V)$, $L^1(X,V)$, or just $L^1$, depending on context.  The default value of $V$ is usually either $\mathbb{R}$ or $\mathbb{C}$, depending on the author.  More general Lebesgue spaces of the form $L^p$ also exist; $f$ is in $L^p$ precisely when ${|f|^p}$ is integrable, and we use
$$ {\|f\|_p} \coloneqq \root p {\int {\|f\|^p} \,\mu} $$
as the norm.  (At least, this is so for $p \in {]{0,\infty}[}$; see [[Lebesgue space]] for generalizations to other values of $p$.)

## The algebra of measures

Note that
$$ (f \mu) (S) = \int \chi_S f \,\mu $$
makes $f \mu$ into a $V$-valued measure whenever $f$ is an integrable $V$-valued function.  When $f$ is $[-\infty,\infty]$-valued and $\mu$ is a signed measure, then $f$ is an extended measure which is finite iff $f$ is integrable.  We have
$$ (f g) \mu = f (g \mu) .$$
Thus integration can be seen as a way of multiplying a function by a measure to get another measure.

The [[Radon-Nikodym derivative]] is about reversing this (dividing two measures to get a function).

Other topics: absolute continuity, etc.  (Refer to <http://tobybartels.name/notes/#Radon>.)

## Noncommutative measure theory

Every commutative [[von Neumann algebra]] is isomorphic to the [[Lebesgue space]] $L^\infty(X,\mu)$ where $\mu$ is some measure (which is irrelevant) on a [[localisable measurable space]], and this extends to a [[dual category|duality]] between localisable measurable spaces and commutative von Neumann algebras.  This is similar to the correspondence between commutative $C^*$-[[C-star algebra|algebras]] and [[locally compact Hausdorff space]]s, which is the central approach to [[noncommutative geometry]].  It is useful to exploit the intuition that the theory of (noncommutative) von Neumann algebras is a noncommutative analogue of classical measure theory.

## Examples

* [[counting measure]]

* [[Haar measure]]

* [[Borel measure]]

* [[Radon measure]]

* [[Gaussian measure]]

* [[spectral measure]]

* [[Wiener measure]]

## Related concepts

* The pointless version of the notion of measurable space is the notion of _[[measurable locale]]_.

* In the context of [[fiber integration]] in [[generalized cohomology]], the analog of a measure is an [[orientation in generalized cohomology]].

* The [[density of a subset]] can be considered as taking the measure of sets normally considered to be not well behaved in measure theory, such as infinite subsets of the natural numbers.

## References

See the references at _[[measure theory]]_.

For measures on [[sigma-frame|$\sigma$-frames]], see

* Alex Simpson, [Measure, randomness and sublocales](https://www.sciencedirect.com/science/article/pii/S0168007211001874).

Discussion via [[Boolean toposes]] is in

* {#Jackson06} Matthew Jackson, _A sheaf-theoretic approach to measure theory_, 2006. ([pdf](http://www.andrew.cmu.edu/~awodey/students/jackson.pdf))

* {#Henry14} [[Simon Henry]], _Measure theory over boolean toposes_, Mathematical Proceedings of the Cambirdge Philosophical Society, 2016 ([arXiv:1411.1605](https://arxiv.org/abs/1411.1605))

category: analysis

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Measure Theory

A book on [[measure theory]] by [[David H. Fremlin]].

[Website](https://www1.essex.ac.uk/maths/people/fremlin/mt.htm)

Publisher: Torres Fremlin, Colchester.
Later republished by Lulu.com.

  1.  2004\. 108+5 pp.  ISBN: 0-9538129-0-1
  2.  2003\. 563+12 pp.  ISBN: 0-9538129-2-8
  3.  2004\. 693+13 pp.  ISBN: 0-9538129-3-6
  4.  2006\. I: 528 pp.; II: 439+19 pp. ISBN: 0-9538129-4-4
  5.  2015\. I: 329 pp. ISBN: 978-0-9538129-5-0; II: 411 pp. ISBN: 978-0-9538129-6-7

Volume 1. The Irreducible Minimum

11. [[Measure spaces]]
12. [[Integration]]
13. Complements

Appendix

Volume 2. Further topics in the general theory

21. Taxonomy of [[measure spaces]]
22. The fundamental theorem of calculus
23. The [[Radon-Nikodým theorem]]
24. Function spaces
25. Product measures
26. Change of variable in the integral
27. Probability theory
28. [[Fourier analysis]]

Appendix

Volume 3. Measure Algebras

31. [[Boolean algebras]]
32. Measure algebras
33. Maharam's theorem
34. Liftings
35. [[Riesz spaces]]
36. Function spaces
37. Linear operators between function spaces
38. Automorphism groups
39. Measurable algebras

Appendix

Volume 4. Topological Measure Spaces

41. Topologies and measures I
42. Descriptive set theory
43. Topologies and measures II
44. [[Topological groups]]
45. Perfect measures, [[disintegrations]] and processes
46. Pointwise compact sets of measurable functions
47. [[Geometric measure theory]]
48. [[Gauge integrals]]
49. Further topics

Appendix

Volume 5. Set-theoretic Measure Theory

51. Cardinal functions
52. Cardinal functions of measure theory
53. Topologies and measures III
54. [[Real-valued measurable cardinals]]
55. Possible worlds
56. Choice and Determinacy

Appendix

Volume 6. Stochastic Calculus

61. The Riemann-sum integral
62. Martingales
63. Back to work
64. The fundamental theorem of martingales and the S-integral
65. Applications


measure theory

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# Contents
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## Idea

Measure theory studies [[measurable spaces]] and [[measure spaces]].

## History

Measure theory is the field of mathematics that grew out of the [[Lebesgue measure|Lebesgue integral]] and Kolmogorov\'s axioms for [[probability]].

## Some subfields and applications

The general measure theory studies general notions and constructions in measure theory, like the connection to integration, the measure spaces, derivation by measure, [[Caratheodory construction]] and so on.

[[probability theory|Probability theory]] studies special class of measures, so called [[probability measure]]s which are normalized to unity.

Measure theory is very much having a central role in studying so called [[ergodic theory]] of dynamical system.

[[geometric measure theory|Geometric measure theory]] is the geometric study of measures of subsets of Euclidean space and the measure theoretic aspects of various geometric objects, like the integration of classes of currents and their extremization properties.

There is a generalization, the noncommutative measure theory, which is more or less the study of [[von Neumann algebra]], see [Connes (1995)](#Connes1995).

## Related concepts

* [[expectation value]]

* [[valuation (measure theory)]]

* [[noncommutative measure theory]]

## References

### General

A comprehensive five-volume treatise (with a sixth volume forthcoming) is

* [[David H. Fremlin]], _Measure Theory_, Volumes 1–5, Torres Fremlin, Colchester.  Volume 1, 2000; Volume 2, 2001; Volume 3, 2002; Volume 4, 2003; Volume 5, 2008.  [Website](https://www1.essex.ac.uk/maths/people/fremlin/mt.htm).

A more concise two-volume treatise is

* [[Vladimir Bogachev]], _[[Measure theory]]_, Volumes I, II.  Springer, 2007.  ISBN: 978-3-540-34513-8, 3-540-34513-2.

A classical (slightly dated) concise treatise is

* [[Paul Halmos]], _Measure Theory_, D. Van Nostrand Company, 1950.

* Donald L. Cohn, _Measure Theory_, Birkhäuser, 1980.  ISBN: 3-7643-3003-1

Other texts include

* [[Ernst-Erich Doberkat]], _Measures and all that --- A Tutorial_ ([arxiv/1409.2662](http://arxiv.org/abs/1409.2662))

* {#Connes1995} [[Alain Connes]] (1995); _[[Noncommutative Geometry]]_.

* D. H. Fremlin (2001); _Measure Theory_; 5 volumes, [web](http://www.essex.ac.uk/maths/people/fremlin/mt.htm).

### Via topos theory
 {#ReferencesViaToposTheory}

Discussion via [[topos theory]]

* {#Jackson06} Matthew Jackson, _A sheaf-theoretic approach to measure theory_, 2006. ([pdf](http://www.andrew.cmu.edu/~awodey/students/jackson.pdf), [d-scholarship:7348](http://d-scholarship.pitt.edu/7348))

and particularly via [[Boolean toposes]]:

* {#Henry14} [[Simon Henry]], _Measure theory over boolean toposes_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 163 Issue 1, 2016 ([arXiv:1411.1605](https://arxiv.org/abs/1411.1605), [doi:10.1017/S0305004116000700](https://doi.org/10.1017/S0305004116000700))

* [[Asgar Jamneshan]], [[Terence Tao]], _Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration_ ([arXiv:2010.00681](arXiv:2010.00681))


measure theory - contents

**[[measure theory]]**

**[[probability theory]]**

([[quantum probability]])

## Measure theory

* [[measurable space]], [[measurable locale]]

* [[measure]], [[measure space]]

* [[von Neumann algebra]]

* [[geometric measure theory]]

## Probability theory

* [[probability space]]

* [[probability distribution]]

* [[state]]

  * [[states in AQFT and operator algebra|in AQFT and operator algebra]]

  * [[GNS construction]]

  * [[Fell's theorem]]

* [[entropy]], [[relative entropy]]

## Information geometry

* [[information geometry]]

* [[information metric]]

* [[Wasserstein metric]]

## Thermodynamics

* [[thermodynamics]]

* [[second law of thermodynamics]], [[generalized second law of theormodynamics|generalized second law]]

* [[ergodic theory]]

## Theorems

* [[Riesz representation theorem]]

## Applications

* [[machine learning]], [[neural networks]]


measurement

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# Measurement
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## Idea

In principle, any [[interaction]] between two [[physical systems]] is a __measurement__, whereby each system measures the [[state]] of the other.  In practice, we break the symmetry by treating one system as the real physical system $S$ of study and the other system as merely a __measuring apparatus__ $A$ with the sole purpose of measuring $S$.  If the properties of $A$ are more directly [[observable]] than those of $S$, then this is useful.

## Related concepts

* [[physical unit]]

* [[quantum measurement]]

* [[experiment]]

* [[empiricism]]

* [[computable physics]]

## References

* Wikipedia, _[Measurement](http://en.wikipedia.org/wiki/Measurement)_

[[!redirects measurement]]
[[!redirects measurements]]


measurement-based quantum computation

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* table of contents
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## Idea

What has come to be known as *measurement based quantum computation* is a scheme for [[quantum computation]] in which [[quantum gates]] are implemented by partial [[quantum measurement]] on [[entanglement|entangle states]] ([[Bell states]]).

Due to [[wavefunction collapse]] upon [[quantum measurement]], such processes are not [[reversible computation|reversible]] on the total [[quantum state space]] --- conversely, each such measurement-based gate operation "uses up the entanglement-resource" in order to implement ([[reversible computation|reversible]]) [[quantum gates]] on a computational subspace. Therefore one also speaks of "one-way quantum computation" [[Raussendorf &Briegel (2001)](#RaussendorfBriegel01)].

## Related concepts

* [[quantum measurement]]

* [[quantum teleportation]]

* [[quantum computation]]

  * [[adiabatic quantum computation]]

  * [[topological quantum computation]]

## References

The original articles:

* [[Daniel Gottesman]], [[Isaac L. Chuang]], *Quantum Teleportation is a Universal Computational Primitive*, Nature **402** (1999) 390-393 [[doi:10.1038/46503](https://doi.org/10.1038/46503), [arXiv:quant-ph/9908010](https://arxiv.org/abs/quant-ph/9908010)]

* [[Emanuel Knill]], [[Raymond Laflamme]], [[Gerard J. Milburn]], *A scheme for efficient quantum computation with linear optics*, Nature **409** (2001) 46–52 [[doi:10.1038/35051009](https://doi.org/10.1038/35051009)]

* {#RaussendorfBriegel01} [[Robert Raussendorf]], [[Hans J. Briegel]], *A One-Way Quantum Computer*, Phys. Rev. Lett. **86** (2001) 5188 [[doi:10.1103/PhysRevLett.86.5188](https://doi.org/10.1103/PhysRevLett.86.5188)]

* [[Michael Nielsen]], *Quantum computation by measurement and quantum memory*, Physics Letters A **308** (2003) 96–100 [doi:10.1016/S0375-9601(02)01803-0]

* [[Debbie W. Leung]], *Quantum computation by measurements*,  International Journal of Quantum Information **02** 01 (2004) 33-43 [[arXiv:quant-ph/0310189](https://arxiv.org/abs/quant-ph/0310189), [doi:10.1142/S0219749904000055](https://doi.org/10.1142/S0219749904000055)]

Review:

* [[Hans J. Briegel]], [[Dan E. Browne]], [[Wolfgang Dür]], [[Robert Raussendorf]], [[Maarten Van den Nest]], *Measurement-based quantum computation*, Nature Physics **5** 1 (2009) 19-26 [[arXiv:0910.1116](https://arxiv.org/abs/0910.1116), [doi:10.1038/nphys1157](https://doi.org/10.1038/nphys1157)]

* [[Tzu-Chieh Wei]], *Measurement-Based Quantum Computation*, Oxford Research Encyclopedia of Physics (2021) [[arXiv:2109.10111](https://arxiv.org/abs/2109.10111), [doi:10.1093/acrefore/9780190871994.013.31](https://doi.org/10.1093/acrefore/9780190871994.013.31)]

Towards formalizing measurement-based quantum protocols:

* {#DanosKahsefiPanangaden07} [[Vincent Danos]], [[Elham Kashefi]], [[Prakash Panangaden]], *The Measurement Calculus*, Journal of the ACM, **54** 2 (2007) [[arXiv:0704.1263](https://arxiv.org/abs/0704.1263), [doi:10.1145/1219092.1219096](https://doi.org/10.1145/1219092.1219096)]

Using (motivating) the [[ZX-calculus]] for formalizing measurement-based quantum protocols:

* [[Ross Duncan]], [[Simon Perdrix]], *Rewriting Measurement-Based Quantum Computations with Generalised Flow*, in: *Automata, Languages and Programming. ICALP 2010*, Lecture Notes in Computer Science **6199**, Springer (2010) [[doi:10.1007/978-3-642-14162-1_24](https://doi.org/10.1007/978-3-642-14162-1_24)]

* [[Ross Duncan]], *A graphical approach to measurement-based quantum computing* [[arXiv:1203.6242](https://arxiv.org/abs/1203.6242) video exposition:[YT](https://youtu.be/E_zH9C0ds8c?t=648)]