$\def\Spec{\mathop{\rm Spec}}
\def\R{{\bf R}}
\def\Ep{{\rm E}^+}
\def\L{{\rm L}}
\def\EpL{\Ep\L}$
One can argue that an object of the right category of spaces in measure theory is not a set equipped with a σ-algebra of measurable sets,
but rather a set $S$ equipped with a σ-algebra $M$ of measurable sets and a σ-ideal $N$ of negligible sets, i.e., 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 enhanced measurable spaces, since they are measurable spaces enhanced with a σ-ideal of negligible sets.
A morphism of enhanced measurable spaces $(S,M,N)→(T,P,Q)$ is a map $S\to T$ 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 in “Equivalences of measure spaces” (see also Kelley's “Decomposition and representation theorems in measure theory”)
that for an enhanced measurable space $(S,M,N)$ that admits a faithful measure (meaning $μ(A)=0$ if and only if $A∈N$) 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 $S$ modulo equality almost everywhere is Dedekind-complete;
- The Radon-Nikodym theorem is true for $(S,M,N)$;
- The Riesz representation theorem is true for $(S,M,N)$ (the dual of $\L^1$ is isomorphic to $\L^∞$);
- Equivalence classes of bounded functions on $S$ form a von Neumann algebra (alias W*-algebra).
An enhanced measurable space that satisfies these conditions (including the existence of a faithful measure) is called localizable.
This theorem tells us that if we want to prove anything nontrivial about measurable spaces, we better restrict ourselves to localizable enhanced 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 nonmeasurable set $G$ and a family of singleton 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 naive version of (1).
But restricting to localizable enhanced measurable spaces does not eliminate all the pathologies:
one must further restrict to the so-called compact and strictly localizable enhanced measurable spaces,
and use a coarser equivalence relation on measurable maps: $f$ and $g$ are weakly equal almost everywhere
if for any measurable subset $B$ of the codomain the symmetric difference $f^*B⊕g^*B$ of preimages of $B$ under $f$ and $g$ is a negligible subset of the domain.
(For codomains like real numbers this equivalence relation coincides with equality almost everywhere.)
An enhanced measurable space is strictly localizable if it splits as a coproduct (disjoint union) of σ-finite (meaning there is a faithful finite measure)
enhanced measurable spaces.
An enhanced measurable space $(X,M,N)$ is (Marczewski) compact if there is a compact class $K⊂M$
such that for any $m∈M∖N$ there is $k∈K∖N$ such that $k⊂m$.
Here a compact class is a collection $K⊂2^X$ of subsets of $X$ such that for any $K'⊂K$ the following finite intersection property holds:
if for any finite $K''⊂K'$ we have $⋂K''≠∅$, then also $⋂K'≠∅$.
The best argument for such restrictions is the following Gelfand-type duality theorem for commutative von Neumann algebras.
Theorem.
The following 5 categories are equivalent.
- The category of compact strictly localizable enhanced measurable spaces with measurable maps modulo weak equality almost everywhere.
- The category of hyperstonean topological spaces and open continuous maps.
- The category of hyperstonean locales and open maps.
- The category of measurable locales (and arbitrary maps of locales).
- The opposite category of commutative von Neumann algebras and normal (alias ultraweakly continuous) unital *-homomorphisms.
I actually prefer to work with the opposite category of the category of commutative von Neumann algebras,
or with the category of measurable locales.
The reason for this is that the point-set definition of a measurable space
exhibits immediate connections only (perhaps) to descriptive set theory, and with additional effort to Boolean algebras,
whereas the description in terms of operator algebras or locales immediately connects measure theory to other areas of the central core of mathematics
(noncommutative geometry, algebraic geometry, complex geometry, differential geometry, topos theory, etc.).
Additionally, note how the fourth category (measurable locales) is a full subcategory of the category of locales.
Roughly, the latter can be seen as a slight enlargement of the usual category of topological spaces,
for which all the usual theorems of general topology continue to hold (e.g., Tychonoff, Urysohn, Tietze, various results about paracompact and uniform spaces, etc.).
In particular, there is a fully faithful functor from sober topological spaces (which includes all Hausdorff spaces) to locales.
This functor is not surjective, i.e., there are nonspatial locales that do not come from topological spaces.
As it turns out, all measurable locales (excluding discrete ones) are nonspatial.
Thus, measure theory is part of (pointfree) general topology, in the strictest sense possible.
The non-point-set languages (2–5) are also 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 compact strictly localizable enhanced measurable space in a point-set way, we run into all sorts of problems
if the fibers can be nonseparable, and I do not know how to fix this problem in the point-set framework.
On the other hand, in the algebraic framework we can simply say that a bundle of Hilbert spaces is a Hilbert W*-module over the corresponding von Neumann algebra.
Categorical properties of von Neumann algebras (hence of compact strictly localizable enhanced measurable spaces)
were investigated by Guichardet in “Sur la catégorie des algèbres de von Neumann”.
Let me mention some of his results, translated in the language of enhanced measurable spaces.
The category of compact strictly localizable enhanced measurable spaces admits equalizers and coequalizers, arbitrary coproducts, hence also arbitrary colimits.
It also admits products (and hence arbitrary limits), although they are quite different from what one might think.
For example, the product of two real lines is not $\R^2$ with the two obvious projections.
The product contains $\R^2$, but it also has a lot of other stuff, for example, the diagonal of $\R^2$,
which is needed to satisfy the universal property for the two identity maps on $\R$.
The more intuitive product of measurable spaces ($\R\times\R=\R^2$) 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.).
Another property worthy of mentioning is that the category of commutative von Neumann algebras
is a locally presentable category, which immediately allows one to use the adjoint functor theorem to construct commutative
von Neumann algebras (hence enhanced measurable spaces) via their representable functors.
Finally let me mention pushforward and pullback properties of measures on enhanced 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, denote $\L_p(M)=\L^{1/p}(M)$, where $M$ is an enhanced measurable space.
Here $p$ can be an arbitrary complex number with a nonnegative real part.
We do not need a measure on $M$ to define $\L_p(M)$.
For instance, $\L_0$ is the space of all bounded functions (i.e., the commutative von Neumann algebra corresponding to $M$),
$\L_1$ is the space of finite complex-valued measures (the dual of $\L_0$ in the ultraweak topology),
and $\L_{1/2}$ is the Hilbert space of half-densities.
I will also talk about extended positive part $\EpL_p$ of $\L_p$ for real $p$.
In particular, $\EpL_1$ is the space of all (not necessarily finite) positive measures on $M$.
Pushforward for $\L_p$-spaces.
Suppose we have a morphism of enhanced measurable spaces $M\to N$.
If $p=1$, then we have a canonical map $\L_1(M)\to\L_1(N)$, which just the dual of $\L_0(N)→\L_0(M)$ in the ultraweak topology.
Geometrically, this is the fiberwise integration map.
If $p≠1$, then we only have a pushforward map of the extended positive parts, namely, $\EpL_p(M)→\EpL_p(N)$, which is nonadditive unless $p=1$.
Geometrically, this is the fiberwise $\L_p$-norm.
Thus $\L_1$ is a functor from the category of enhanced measurable spaces to the category of Banach spaces
and $\EpL_p$ 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 pullbacks of $\L_p$-spaces (in particular, $\L_1$-spaces) one needs to pass to a different category of enhanced measurable spaces.
In the algebraic language, if we have two commutative von Neumann algebras $A$ and $B$,
then a morphism from $A$ to $B$ is a usual morphism of commutative von Neumann algebras $f\colon A\to B$
together with an operator valued weight $T\colon\Ep(B)\to\Ep(A)$ associated to $f$.
Here $\Ep(A)$ denotes the extended positive part of $A$.
(Think of positive functions on $\Spec A$ that can take infinite values.)
Geometrically, this is a morphism $\Spec f\colon\Spec B\to\Spec A$
between the corresponding enhanced measurable spaces and a choice of measure on each fiber of $\Spec f$.
Now we have a canonical additive map $\EpL_p(\Spec A)\to\EpL_p(\Spec B)$,
which makes $\EpL_p$ into a contravariant functor from the category of enhanced measurable spaces
and measurable maps 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 $T\colon B\to A$ (think of a fiberwise finite complex-valued measure).
Then $\L_p$ becomes a functor from the category of enhanced 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 $f\colon A\to B$ and in this (and only this) case we do not need $T$.
Finally, if we restrict ourselves to an even smaller subcategory defined by the additional condition $T(1)=1$
(i.e., $T$ is a conditional expectation; think of a 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 enhanced 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,
the integration map on 1-densities is defined via Poincaré duality (to avoid any dependence on measure theory) etc.
There is a forgetful functor that sends a smooth manifold to its underlying enhanced measurable space.
Of course, the story does not end here, there are many
other interesting topics to consider: products of measurable spaces,
the difference between Borel and Lebesgue measurability, conditional expectations, etc.
An index of my writings on this topic is available.
Best Answer
I like the simple slogan: homotopical algebra is the nonlinear generalization of homological algebra. Let me assume that you value and appreciate homological algebra in the broadest sense as a fundamental, successful and highly applicable tool in many areas of math (otherwise I can't conceive of an argument that would be convincing for this question). At the coarsest level homological algebra is based on the idea of resolutions, i.e. that to perform algebraic operations on objects we should describe them in terms of objects that behave well for the given operations.
Now let's observe that homological algebra is a linear theory, in the sense that it deals with things like vector spaces, modules over a ring, and more generally objects of abelian categories. What if your interests involve more complicated objects that are not linear? for example, rings, algebras, varieties, manifolds, categories etc? philosophically it still makes sense that we have much to gain by resolving in some appropriate sense. Homotopical algebra is the language and toolkit built for this explicit purpose, and with many explicit applications. The $\infty$-language in my mind is just a very convenient and relatively friendly apparatus to understand, navigate and apply this theory.
Some key examples:
$\bullet$ Hodge theory. For me (and I assume many other algebraic geometers) the first instance of homotopical algebraic thinking I encountered was Deligne's construction of mixed Hodge structures on the cohomology of complex algebraic varieties, one of the most powerful tools in modern algebraic geometry. The idea is that the functor "de Rham cohomology" is very wonderfully behaved on smooth complex projective varieties, and most importantly carries a rich extra structure, a pure Hodge structure. We can take advantage of this for say any singular projective variety if we use the idea of resolution, in the form of a simplicial object (a convenient nonlinear version of a chain complex) --- we replace the variety by a simplicial smooth projective variety which is equivalent in the appropriate sense, in particular will produce the same measurement (cohomology). The existence of such is deep geometry (resolution of singularities) but its explicit applications don't require explicit knowledge of this geometry. It now follows that the singular variety's cohomology carries the appropriate derived version of a pure Hodge structure, namely a mixed Hodge structure.
$\bullet$ The tangent complex. Another seminal circa 1970 application is the Quillen-Illusie theory of the tangent complex. Again we want to do basic geometry - this time calculus - on a singular variety, or perhaps let's say a commutative ring, so we resolve it in the sense that befits the problem. We like affine spaces for taking derivatives etc, so if we want to calculate derivatives (tangent spaces) on a singular variety we should resolve it by such --- replace a ring by an appropriate free resolution (this time a COsimplicial variety). This gives us a way to extend the basic tools of calculus to singular varieties, with many corresponding applications.
$\bullet$ The virtual fundamental class. This is an elaboration on the previous point which is much more recent. We would like now to integrate on a class of singular varieties, so need a version of the fundamental class. The varieties in question arise as moduli spaces (say in Gromov-Witten or Donaldson-Thomas theories), which means they are relatively easy to resolve in a natural way (express as a derived moduli problem). As ordinary varieties they are very badly behaved (eg are not even equidimensional) but the derived moduli problem naturally carries a fundamental class.
$\bullet$ In representation theory the key objects of study are again nonlinear --- associative algebras (or equivalently their categories of modules). Thus to perform algebraic operations on these algebras we gain much by allowing ourselves to resolve them. As mentioned above the geometric Langlands program is one place where homotopical language is extremely useful, but one can find the same issues in studying say modular representations of finite groups (eg the theory of support varieties and stable module categories). More generally Hochschild/cyclic theory, the "calculus" of associative algebras/the fundamental invariants of noncommutative geometry, are natural applications of homotopical algebra. There are many spectacular achievements in this area, one famous one being the Deligne conjecture/Kontsevich formality/deformation quantization circle of ideas. The cobordism hypothesis, in my view one of the pinnacles of homotopical algebra, has among its many facets a vast generalization of Hochschild theory.