One can argue that an object of the right category of spaces in measure theory is not a set equipped
with a $sigma$-algebra of measurable sets, but rather a set $S$ equipped with a $sigma$-algebra $M$ of measurable
sets and a $sigma$-ideal $N$ of $M$ consisting of sets of measure $0$.
The reason for this is that you can hardly state any theorem of measure theory or probability
theory without referring to sets of measure $0$.
However, objects of this category contain less data than the usual measured spaces,
because they are not equipped with a measure.
Therefore I prefer to call them measurable spaces.
A morphism of measurable spaces $(S,M,N)to(T,P,Q)$ is a map $Sto T$ such that
the preimage of every element of $P$ is a union of an element of $M$ and a subset of an element of $N$
and the preimage of every element of $Q$ is a subset of an element of $N$.
Irving Segal proved that for a measurable space the following properties are equivalent:
- The Boolean algebra $M/N$ of equivalence classes of measurable sets is complete;
- The space of equivalence classes of all bounded (or unbounded) real-valued functions on $S$ is Dedekind-complete;
- Radon-Nikodym theorem is true for $(S,M,N)$;
- Riesz theorem is true for $(S,M,N)$;
- Equivalence classes of bounded functions on $S$ form a von Neumann algebra (aka $W^*$-algebra).
- $(S,M,N)$ is a coproduct (disjoint union) of points and real lines.
A measurable space that satisfies these conditions is called localizable.
This theorem tells us that if we want to prove anything nontrivial about measurable
spaces, we better restrict ourselves to localizable measurable spaces.
We also have a nice illustration of the claim I made in the first paragraph:
None of these statements would be true without identifying objects that differ
on a set of measure $0$.
For example, take a non-measurable set $G$
and a family of one-element subsets of $G$ indexed by themselves.
This family of measurable sets does not have a supremum in the Boolean algebra
of measurable sets, thus disproving a naïve version of (1).
Another argument for restricting to localizable spaces is the following version of
Gelfand-Neumark theorem:
The category of localizable measurable spaces is equivalent to the category of commutative von Neumann algebras (aka $W^*$-algebras) and their morphisms (normal unital homomorphisms of $^*$-algebras).
I actually prefer to define the category of localizable measurable spaces
as the opposite category of the category of commutative $W^*$-algebras.
The reason for this is that the classical definition of measurable space
exhibits immediate connections only to descriptive set theory (and with additional
effort to Boolean algebras), which are mostly
irrelevant for the central core of mathematics,
whereas the description in terms of operator algebras immediately connects
measure theory to other areas of the central core (noncommutative geometry,
algebraic geometry, complex geometry, differential geometry etc.).
Also it is easier to use in practice. Let me illustrate this statement
with just one example: When we try to define measurable bundles of Hilbert spaces
on a localizable measurable space set-theoretically, we run into all sorts of problems
if the fibers can be non-separable, and I do not know how to fix this problem in the set-theoretic framework.
On the other hand, in the algebraic framework we can simply say that a bundle
of Hilbert spaces is a Hilbert module over the corresponding $W^*$-algebra.
Categorical properties of $W^*$-algebras (hence of localizable measurable spaces)
were investigated by Guichardet.
Electronic version of this paper is available here.
Let me mention some of his results.
The category of localizable measurable spaces admits equalizers and coequalizers,
arbitrary coproducts and hence arbitrary colimits.
It also admits products, although they are quite different from what one might think.
For example, the product of two real lines is not $Bbb R^2$ with the two obvious projections.
The product contains $Bbb R^2$, but it also has a lot of other stuff, for example, the diagonal of $Bbb R^2$,
which is needed to satisfy the universal property for the two identity maps on $Bbb R$.
The more intuitive product of measurable spaces ($Bbb Rtimes Bbb R=Bbb R^2$) corresponds to the spatial
tensor product of von Neumann algebras and forms a part of a symmetric monoidal
structure on the category of measurable spaces.
See Guichardet's paper for other categorical properties (monoidal structures
on measurable spaces, flatness, existence of filtered limits, etc.).
Finally let me mention pushforward and pullback properties of measures on measurable spaces.
I will talk about more general case of $L^p$ spaces instead of just measures (i.e., $L^1$ spaces).
For the sake of convenience let $L_p(M) := L^{1/p}(M)$, where $M$ is a measurable space.
Here $p$ can be an arbitrary complex number with a nonnegative real part.
Note that you don't need a measure on $M$ to define $L_p(M)$.
In particular, $L_0$ is the space of all bounded functions (i.e., the $W^*$-algebra itself),
$L_1$ is the space of finite complex-valued measures (the dual of
$L_0$ in the $sigma$-weak topology), and $L_{1/2}$ is the Hilbert space of half-densities.
I will also talk about extended positive part $E^+L_p$ of $L_p$ for real $p$.
In particular, $E^+L_1$ is the space of all (not necessarily finite) positive measures.
Pushforward for $L_p$ spaces: Suppose we have a morphism of measurable spaces $Mto N$.
If $p=1$, then we have a canonical map $L_{1}(M) to L_{1}(N)$, which just the dual of $L_{0}(N) to L_{0}(M)$
in the $sigma$-weak topology. Geometrically, this is the fiberwise integration map.
If $pneq 1$, then we only have a pushforward map of the extended positive parts:
$E^+L_p(M) to E^+L_p(N)$, which is non-additive unless $p=1$.
Geometrically, this is the fiberwise $L_p$ norm.
Thus $L_1$ is a functor from the category of measurable spaces to the category of Banach spaces
and $E^+L_p$ is a functor to the category of "positive homogeneous $p$-cones".
The pushforward map preserves the trace on $L_1$ and hence
sends a probability measure to a probability measure.
To define pullback of $L_p$ spaces (in particular, $L_1$ spaces) one needs to pass to a different
category of measurable spaces.
In algebraic language, if we have two $W^*$-algebras $A$ and $B$, then
a morphism from $A$ to $B$ is a usual morphism of $W^*$-algebras $f: Ato B$
together with an operator valued weight $T: E^+(B)→E^+(A)$ associated to $f$.
Here $E^+(A)$ denotes the extended positive part of A (think of positive functions on $mathrm{Spec} A$
that can take infinite values).
Geometrically, this is a morphism $mathrm{Spec} f: mathrm{Spec} B to mathrm{Spec} A $ between the corresponding
measurable spaces and a choice of measure on each fiber of $mathrm{Spec} f$.
Now we have a canonical additive map $E^+L_p(mathrm{Spec} A) to E^+L_p(mathrm{Spec} B)$,
which makes $E^+L_p$ into a contravariant functor from the category of measurable spaces
equipped with a fiberwise measure to the category of “positive homogeneous additive cones”.
If we want to have a pullback of $L_p$ spaces themselves and not just their extended positive parts,
we need to replace operator valued weights in the above definition
by finite complex-valued operator valued weights $T: B to A$ (think of fiberwise complex-valued measure).
Then $L_p$ becomes a functor from the category of measurable spaces to the category
of Banach spaces (if the real part of $p$ is at most $1$) or quasi-Banach spaces (if the real part of $p$ is greater than $1$).
Here $p$ is an arbitrary complex number with a nonnegative real part.
Notice that for $p=0$ we get the original map $f: Ato B$ and in this (and only this) case we don't need $T$.
Finally, if we restrict ourselves to an even smaller subcategory of measurable
spaces equipped with a finite operator valued weight T such that $T(1)=1$
(i.e., T is a conditional expectation; think of fiberwise probability measure), then the pullback map preserves
the trace on $L_1$ and in this case the pullback of a probability measure is a probability measure.
There is also a smooth analog of the theory described above: The category of measurable spaces
and their morphisms is replaced by the category of smooth manifolds and submersions,
$L_p$ spaces are replaced by bundles of $p$-densities, operator valued weights are
replaced by sections of the bundle of relative $1$-densities,
integration map on $1$-densities is defined via Poincaré duality (to avoid circular dependence
on measure theory) etc.
The forgetful functor that sends a smooth manifold to its underlying measurable space
commutes with everything and preserves everything.
