|analytic integration||cohomological integration|
|measurable space||Poincaré duality|
|measure||orientation in generalized cohomology|
|volume form||(virtual) fundamental class|
|Riemann/Lebesgue integration of differential forms||push-forward in generalized cohomology/in differential cohomology|
For the general theory of measure spaces, we first need a measurable space , that is a set equipped with a collection of measurable sets complete under certain operations. Then this becomes a measure space by throwing in a function from to a space of values (such as the real line) that gets along with the set-theoretic operations that has. If is a measurable set, then is called the measure of with respect to .
The original notation for an integral (going back to Gottfried Leibniz) was
(where would be replaced by some formula in the variable ). In modern measure theory, we can now understand this as the integral of the measurable function on the interval relative to Lebesgue measure. If we wish to generalise from Lebesgue measure to an arbitrary measure and generalise from to an arbitrary measurable set , then we can write
instead. Now, if is not given by a formula, then there is no need for the dummy variable , which gives
However, it has been more common to keep the symbol ‘’ and write
(Note that ‘’ can be read as ‘with respect to’ in both (1) and (4), 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 (2) with
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 on with respect to is simply the measure of with respect to , as can be seen in (3). Compare also notation for Radon–Nikodym derivatives.
It is also possible to take the entire expression ‘’ as the name of the measure, writing even where the common notation is . In that case, the common expression (4) is literally the same as (what would otherwise be) (3), although (5) is not quite the same as (what would otherwise be) (2).
We will use (3) below (although other forms may well be found on other pages).
There is also some variation in notation as to whether to use a roman ‘’ or an italic ‘’; roman is more common in England and italic in America. But of course, that variation should not cause any difficulties!
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.
(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:
The first of these conditions will follow for all of the generalised notions of measure below, but the others usually will not. Related query discussion is archived here.
From now on, we drop (2); the next step is to generalise the target of , as follows: * Use for a finite measure. * Use (instead of ) for a positive measure. * Use for a signed measure. * Use for a complex-valued measure. * Use an arbitrary topological vector space for a vector-valued measure. * In principle, one could go further yet; 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 centipedes.
Some futher terms: * We can combine conditions; for example a finite positive measure takes values in . * A measure is bounded if, for some (finite) real number , for every measurable set . * A measure is -finite if every measurable set is a union of countably many sets with finite measure.
Remarks: * The property that is increasing holds for all positive measures in the name but may fail for others. * A positive measure that satisfies (2) must be a probability measure as defined earlier; that is, it satisfies for all . * When is allowed as a value of , 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 is not allowed as a value for a signed measure. It works just as well to allow and forbid . It is even possible to allow both, but this is a little trickier, so we deal with it later.
Another possibility is to generalise the source of ; instead of using a -algebra on , we could use a -ring or even a -ring. These versions are mostly more about changing the definition of measurable space, so refer there for details of the definition; however, we note that (3), when is a -ring, should state that the left-hand side exists (that is, the union is measurable) if the right-hand side converges. Generalising in this way is complementary to generalising the target above; in particular it may allow one to avoid dealing with . For example, while Lebesgue measure is only a positive measure on a -algebra, it is a finite positive measure on the -ring of bounded measurable sets. Indeed, every signed measure gives rise to finite measure on its -ring of finitely measurable sets (defined below); conversely, every -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 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 of extended real numbers. Here we have the problem that, even when considering finite additivity, we might have to add and . While we might simply require that this never happens (so that at least one of and must be finite if they have opposite signs and ), this does not include some examples that we want. To deal with this, we define an extended measure to be a formal difference of positive measures; whenever this is not of the form and is otherwise undefined. Note that the set of extended measures on is a quotient set of the set of pairs of positive measures; we say that if whenever either side is defined, that is if they are the same as partial functions from to .
In Henry Cheng's constructive theory of measure, the definition of measurable space becomes more complicated; the main point is that a single measurable set is replaced by a complemented pair . Once that is understood, very little needs to be changed to define a measure space.
In the requirements (1–3), the constants and and the operation 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 is the measure of alone, as follows:
In general, a measurable set is any set such that is a complemented pair for some set ; 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.
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
Given a measure space , a -null -measurable set is a measurable set such that whenever is measurable; a -null set is any subset of a null measurable set. In a positive measure space, we don't have to bother with ; will be a null measurable set as long as .
A -full -measurable set is a measurable set such that for every measurable set ; a -full set is any superset of a full measurable set. In a probability measure space, we don't have to bother with ; will be a full measurable set as long as . 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 holds -almost everywhere if the set of values where it holds is a full set.
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 -measurable sets will automatically form a -algebra (or whatever originally was). Classically, a measure is complete if and only if every null set is measurable and a set is -measurable if and only if it is the symmetric difference between a measurable set and a null set.
A -finitely measurable set is a measurable set such that is finite whenever is measurable; a -finitely measurable set is any union of countably many finitely measurable sets. Again, we don't have to bother with in a positive measure space. Note that a measure space is ()-finite if and only if every measurable set is ()-finitely measurable. The finitely measurable sets form a -ring, and the -finitely measurable sets form a -ring.
Recall that a -measurable function from to some other measurable space is any function such that the preimage under of a measurable set is always measurable (or something more complicated in the constructive theory). Now that we have a measure space, let a -measurable function be a partial function from to some other measurable space such that the domain of is full and the preimage under of a measurable set is always -measurable (that is measurable in the completion of ), and let two such functions be -equivalent if their equaliser is a full set. We are really interested in the quotient set under this equivalence and so identify equivalent -measurable functions. Classically, every -measurable function is equivalent to some (total) -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 -integrable function is a -measurable function such that the integral (as defined below) exists for every measurable set ; it is enough to check . Equivalently, we may say that it is a -measurable function such that the extended measure (also defined below) is actually a finite measure. (In any case, we get a finite measure if is integrable.)
In the following, ‘measurable’ will mean -measurable. That is, we assume that is complete and identify -equivalent functions. We will also assume that is a positive measure until I make sure of what must be done to generalise.
Given a measure , a measurable set , and a measurable function , we will define the integral
(see above for variations in notation) in stages, from the simplest form of to the most arbitrary.
Each measurable subset induces a measurable characteristic function where if , if . In general, we have
so from now on we will assume that we are integrating over all of (and drop the subscript).
A positive simple function is a finite -linear combination of measurable characteristic functions; the first form of integral that we define is
The integral is extended to all measurable functions by the rule
if this supremum converges. Classically, the integral either converges or diverges to infinity, so exists in some sense in any case; the possibilities are more complicated constructively.
For any measurable function , define and by
so that , . Then the final definition is
if both integrals on the right converge. Classically, the other possibilities are , , and ; not much can be done with the latter.
A measurable function is integrable with respect to 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 takes values in the field of complex numbers or in some more general Banach space , then we can still ask whether is integrable. If it is, then we say that is absolutely integrable. We can then define the integral of ; we always have
This integral is easy to define if has a basis; for example, a measurable complex-valued function is integrable iff both its real and imaginary parts are integrable, and we have
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 -valued integrable functions is itself a Banach space, using the norm
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 , , or just , depending on context. The default value of is usually either or , depending on the author. More general Lebesgue spaces of the form also exist; is in precisely when is integrable, and we use
as the norm.
makes into a -valued measure whenever is an integrable -valued function. When is -valued and is a signed measure, then is an extended measure which is finite iff is integrable. We have
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.
Other topics: absolute continuity, etc.
Every commutative von Neumann algebra is isomorphic to the Lebesgue space where is some measure (which is irrelevant) on a localisable measurable space, and this extends to a duality between localisable measurable spaces and commutative von Neumann algebras. This is similar to the correspondence between commutative -algebras and locally compact Hausdorff spaces, 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.
The pointless version of the notion of measurable space is the notion of measurable locale.