Finite measured groupoids and AFD algebras

Written by GPT-6.1 Sol (OpenAI), Ultra, September 2026. Self-checked by the writing AI. New original text: public domain (CC0).

Finite equivalence classes give matrix algebras with a measurable center. They need not give finite-dimensional algebras. To obtain finite-dimensional approximation, one must also discretize that center. We carry out both steps and prove that an ergodic principal measured AF groupoid of type $\mathrm{II}_1$ has an AFD factor of type $\mathrm{II}_1$.

We use standard Borel spaces, countable separating families, measure completion, monotone convergence and Hilbert-space operator theory. The injective-image input is Lusin–Souslin, Theorem 3.6, with Polish topology refinement, Theorem 2.2. Section 2 derives the needed Borel-map version; Section 8 gives the counting and operator-model details. The finite-trace local AFD criterion and separable uniqueness theorem are proved in Hyperfinite finite factors. General modular weights or expectations are not needed for the kernel construction.

The freely readable equivalence-relation construction is Anantharaman–Popa, §1.5. We use the type $\mathrm{II}_1$ hypothesis to choose an equivalent invariant probability measure; we do not infer invariance from quasi-invariance alone. The AF exhaustion and its fixed finite class sizes are given hypotheses.

1. The measured equivalence relation

Principality identifies the groupoid with a Borel equivalence relation $\mathcal R\subset X\times X$, with countable classes, on a standard Borel space $X$. The arrow $(x,y)$ runs from $y$ to $x$; multiplication is

Its source counting measure relative to a probability measure $\mu$ is

The measured-relation datum includes the source-counting measure (2) and its measurable counting functions. The conull AF model constructed below has explicit countably many stage graphs. Section 8 proves counting and saturation measurability on that model directly from those graphs. We use the given measure to pass to the conull model; we do not assume a selection theorem for an arbitrary countable relation.

Invariance means that $\nu$ is unchanged by the flip $(x,y)\mapsto(y,x)$. Equivalently, every Borel partial bijection whose graph lies in $\mathcal R$ preserves the restricted measure. The forward implication follows by comparing its graph and inverse graph over any Borel subset of its domain. The converse follows by decomposing the relation into countably many such graphs; our finite-class labels below provide that decomposition in the AF case.

The source AF hypothesis supplies increasing Borel subrelations

where each $\mathcal R_n$ has a fixed finite class size $k_n$ almost everywhere. This is the source's type $\mathrm I_{k_n}$ condition. Statements and functions are taken modulo null sets.

There is no difficulty in discarding common exceptional classes. If $N\subset X$ is null, invariance gives

The first set has a nonempty source fiber exactly on the saturation of $N$, so that saturation is null as well. Thus countably many null exceptions can be removed without retaining arrows into them.

We can also make the AF model exact on a conull invariant set. Put $\mathcal R_\infty=\bigcup_n\mathcal R_n$ and let $N_0$ be the measurable set of sources with at least one arrow in $\mathcal R\setminus\mathcal R_\infty$. Equation (3) makes $N_0$ null. It is saturated: if $y\in N_0$ and $z\sim y$ had no missing arrows, every point of their common orbit would be $\mathcal R_\infty$-equivalent to $z$; transitivity would contradict the missing arrow at $y$. Remove $N_0$ and the null saturations of the countably many finite-stage class-size exceptions. On the resulting conull invariant model, $\mathcal R=\mathcal R_\infty$ literally and every stage has its stated finite class size. Its labeled stage graphs then describe every arrow.

If the initially given measure is merely equivalent to an invariant probability $\mu$, the associated regular algebras are normally isomorphic. Indeed if $d\mu'/d\mu=h>0$, the source counting measures satisfy $d\nu'/d\nu(x,y)=h(y)$, and

is unitary. Left convolution does not change the source coordinate $y$, so it intertwines under $V$. We may therefore work with the invariant probability furnished by the type $\mathrm{II}_1$ assumption.

2. Borel labels for finite classes

We need actual measurable matrix coordinates, not just a choice of labels on each finite class.

First recall the Borel-map form of the injective-image theorem. If $f:Y\to Z$ is an injective Borel map between standard Borel spaces, refine compatible Polish topologies on $Y$ so that the inverse images of a countable base on $Z$ become open and closed. F4 permits each refinement and their countable join, without changing the Borel sets. Then $f$ is continuous. Lusin–Souslin makes its image, and the image of every Borel subset, Borel. Thus $f$ is a Borel isomorphism onto its image. This uses the declared topology-refinement input; injectivity alone does not make arbitrary projections preserve Borel sets.

Lemma 2.1. A Borel equivalence relation with all classes of size $k<\infty$ has a Borel partition

such that each $A_i$ meets every class once. With $Y=A_1$, there are Borel isomorphisms $\phi_i:Y\to A_i$, with $\phi_1=\operatorname{id}_Y$, which list each class.

Proof. Choose a countable Borel family $(C_n)$ separating the points of $X$. Give two distinct points the order $x\prec y$ when, at the first index where their membership bits differ, $x\notin C_n$ and $y\in C_n$. This is a strict linear order. Its graph is Borel: it is the countable union, over that first index, of the Borel conditions that all earlier bits agree and the two indicated bits differ. Every nonempty finite subset has a unique least element in this order. If $X$ is empty, take all the sections and coordinate maps to be empty.

We first produce one Borel section on a Borel subset $Z\subseteq X$ on which the restricted relation has classes of one fixed size $m\geq1$. Use a compatible Polish topology on $X$. The complete analytic-image and Borel-set facts and separation theorem, Proposition 2.2, Corollary 2.6 and Theorem 2.4, have the following consequences: a coordinate projection of a Borel subset of a finite power of $X$ is analytic, and disjoint analytic subsets of $X$ can be separated by a Borel set. The first assertion follows by viewing the Borel set as a continuous Polish image and composing with the coordinate projection. These facts apply to Borel subsets $Z$ of the ambient $X$ without requiring their relative topology to be Polish.

Let

This is analytic, as the projection of its displayed Borel witness relation. For the opposite case, consider the Borel set of witnesses

Write $V_Z$ for its projection onto the first coordinate. When $m=1$, this definition means $T_{Z,1}=Z$ and $V_Z=Z$. For any $m$, the set $V_Z$ is analytic. A witness lists exactly the other $m-1$ points of the restricted class, so it exists precisely when $x$ is its least point. Thus

Separation gives a Borel set $B\subseteq X$ containing $V_Z$ and disjoint from $U_Z$. Hence $B\cap Z=V_Z$, so $V_Z$ itself is Borel. It meets every restricted class exactly once. The exact class size is used in the exhaustive-witness implication; an upper bound alone would not justify it.

Start with $Z_1=X$. Take $A_1=V_{Z_1}$, remove it, and apply the same construction to $Z_2=X\setminus A_1$, whose restricted classes have size $k-1$. Continue for $k$ steps. At step $i$, the Borel set $Z_i=X\setminus\bigcup_{j<i}A_j$ has exactly $k-i+1$ members of each original class, so $A_i=V_{Z_i}$ is Borel and selects one of them. After the last step nothing remains. The sets $A_i$ therefore form the required partition and each meets every original class once.

Put $Y=A_1$. For each $i$, the Borel graph

has both coordinate maps bijective onto their indicated ranges. The injective Borel-map theorem proved above makes each coordinate map a Borel isomorphism. Composing the inverse of the first coordinate map with the second gives a Borel isomorphism $\phi_i:Y\to A_i$. For $i=1$, equivalence and the one-point section property give $H_1=\{(t,t):t\in Y\}$, so $\phi_1=\operatorname{id}_Y$. These maps list exactly each class. $\square$

For an invariant measure, the maps $\phi_i$ preserve its restriction on $Y$, so

is a probability measure. More precisely $\mu(\phi_i(E))=\mu(E)$ for every Borel $E\subset Y$.

Apply this lemma to every $\mathcal R_n$. Its finitely many pair-coordinate graphs, as $n$ varies, cover $\mathcal R$ modulo the null set in (3). In particular all counting functions, kernel sums and null-set saturations used here can be evaluated through countably many Borel graphs.

3. The regular algebra and its trace

Put $H=L^2(\mathcal R,\nu)$. Let $\mathcal C_n$ be the bounded measurable kernels supported on $\mathcal R_n$, extended by zero outside it. The union $\mathcal C=\bigcup_n\mathcal C_n$ is a unital convolution *-algebra:

For $f,g$ from two stages, all nonzero terms lie in the larger subrelation, whose finite class size bounds the sums. The diagonal kernel $1_\Delta$ is the convolution identity.

Define left and right convolution by

These are bounded operators. For left convolution, if both absolute row and column sums of $f$ are bounded by $C$, weighted Cauchy–Schwarz gives on each orbit

Integrating over the source coordinate proves $\|L_f\|\le C$. For $f\in\mathcal C_n$ we may take $C=k_n\|f\|_\infty$. Flip invariance makes

antiunitary, and $Q_f=J L_{f^*}J$, proving the right bound as well. Finite-sum multiplication gives

Define the regular von Neumann algebra

This is the associated groupoid algebra in the usual bounded-Schur-kernel definition, not a smaller algebra caused by (3). In fact let $f$ have bounded absolute row and column sums. Then $|f|\le C$ and $f\in L^2(\nu)$, since $\int\sum_x|f(x,y)|^2\,d\mu(y)\le C^2$. The same bound (8) defines its left operator. It commutes with each $Q_g$, $g\in\mathcal C$: first check on kernel vectors $\xi\in\mathcal C$. Both composite sums are finite because $g$ and $\xi$ lie in finite-class stages. Interchanging their intermediate indices gives equality, which extends by density and boundedness. Its cutoffs $f_n=f1_{\mathcal R_n}$ belong to $\mathcal C_n$, converge to $f$ in $L^2(\nu)$, and have the same uniform operator bound. The commuting right algebra used below has a dense orbit on the diagonal vector. Convergence on that vector therefore extends to strong convergence on all of $H$, giving $L_f\in M$. The reverse inclusion is immediate because every finite-stage kernel is a Schur kernel. This also matches the source's left regular convolution construction.

Lemma 3.1. The diagonal vector $\Omega=1_\Delta$ is cyclic and separating for $M$, and

is a faithful normal tracial state. For $f\in\mathcal C$,

Proof. We have $L_f\Omega=Q_f\Omega=f$. The bounded functions on $\mathcal R_n$ are dense in $L^2(\mathcal R_n,\nu)$, and (3) makes their union dense in $H$. Here $\nu(\mathcal R_n)=k_n<\infty$. Thus both convolution algebras have a dense orbit on $\Omega$. The right algebra commutes with $M$, so $x\Omega=0$, $x\in M$, forces $xQ_f\Omega=0$ on a dense set and hence $x=0$. This proves separation and faithfulness; $\|\Omega\|=1$ and vector functionals are normal.

The first identity in (12) follows from $L_f\Omega=f$. Invariance under the flip gives

These are $\tau(L_fL_g)$ and $\tau(L_gL_f)$. Thus $\tau$ is tracial on the unital generating -algebra. Bounded strong Kaplansky approximation extends this identity first in one variable and then in the other to $M$. Finally $\|L_f\|_{2,\tau}=\|L_f\Omega\|=\|f\|_{L^2(\nu)}$. $\square$

The dense right orbit just proved also justifies the strong-cutoff assertion preceding the lemma: for $g\in\mathcal C$,

Uniform boundedness and density complete that assertion.

Bounded 2-norm convergence here implies sigma-strong* convergence, also when $M$ is not a factor. Suppose $z_a\in M$, $\|z_a\|\le C$, and $\|z_a\|_{2,\tau}\to0$. Commutation with the right algebra gives

The right orbit is dense, so the common bound extends convergence to every vector. Traciality gives $\|z_a^*\|_{2,\tau}=\|z_a\|_{2,\tau}$, and the same argument applies to the adjoints. Thus $z_a\to0$ strongly*. Both $z_a^*z_a$ and $z_az_a^*$ are bounded and weakly null. Finite-rank approximation of a trace-class functional, with their common bound controlling its trace-norm tail, makes both products ultraweakly null. The concrete predual of $M$ is a quotient of the trace-class predual of $B(H)$, so every normal positive functional on $M$ tends to zero on these products. These are precisely the sigma-strong* tests. The concrete-predual foundation remains explicit; this argument makes no factoriality assumption.

4. Finite stages are matrix fields

Fix a stage, and write $k=k_n$, $Y=A_1$ and $\phi_i$ for its labels. Set $v_{ij}=L_{1_{\operatorname{Gr}(\phi_i\phi_j^{-1})}}$. Direct convolution gives

For $d\in L^\infty(X,\mu)$, its diagonal kernel gives the multiplication operator $D_d\xi(x,y)=d(x)\xi(x,y)$. This diagonal representation is normal and faithful.

Every kernel in $\mathcal C_n$ is determined by

Convolution is matrix multiplication of these fields, and involution is matrix adjoint. Its operator is $\sum_{i,j}v_{i1}D_{F_{ij}1_Y}v_{1j}$. Consequently the stage algebra is

This image is already a von Neumann algebra. To verify closedness, an operator $T$ belongs to it exactly when all $v_{1i}Tv_{j1}$ lie in the weakly closed diagonal corner $D_{L^\infty(Y)}$. The reverse implication follows by reconstructing $T$ with the finite matrix-unit sum (13). Each corner condition is ultraweakly closed. The entry maps and their inverse corner extraction are normal; faithfulness also follows from $L_f\Omega=f$.

The trace is

Indeed each $\phi_i$ preserves $\mu|_Y$, and (5) gives the normalization. The algebras $M_n$ are increasing and generate $M$.

Lemma 4.1. For a finite set of elements of $M_n$ and $\varepsilon>0$, one unital finite-dimensional subalgebra of $M_n$ approximates every element within $\varepsilon$ in $\|\cdot\|_{2,\tau}$.

Proof. The finite collection of bounded entries in (14) can be approximated simultaneously by one finite measurable partition $\mathcal P$ of $Y$: discretize their real and imaginary ranges and take the common refinement. Average each matrix field over each positive-measure cell of $\mathcal P$, entry by entry. The resulting field $E_{\mathcal P}F$ lies in

a unital finite-dimensional algebra.

If each real and imaginary entry varies by at most $\delta$ on a cell, its difference from its cell average has magnitude at most $\sqrt2\delta$. Formula (16) gives

Choose $\delta$ for the prescribed $\varepsilon$. Zero-measure cells are irrelevant. Matrix averaging is also contractive in operator norm, because the norm of an average is at most the average of the norms. No abstract conditional expectation theorem is needed for this finite-partition operation. $\square$

Theorem 4.2. The regular algebra $M$ is finite and AFD. This conclusion does not require ergodicity.

Proof. Finiteness is supplied by the faithful normal trace. The increasing union of $M_n$ is a unital *-algebra generating $M$, so Kaplansky density approximates any finite family $x_1,\ldots,x_r\in M$ strongly*, by elements $y_j$ in one common stage with $\|y_j\|\le\|x_j\|$. In particular $\|x_j-y_j\|_{2,\tau}$ can be made arbitrarily small. Lemma 4.1 then approximates the $y_j$ by $b_j=E_{\mathcal P}y_j$ in one finite-dimensional algebra, with $\|b_j\|\le\|y_j\|$. The triangle inequality makes every $\|x_j-b_j\|_{2,\tau}$ arbitrarily small while $\|x_j-b_j\|\le2\|x_j\|$. The bounded 2-norm argument in Section 3 therefore gives approximation in every prescribed sigma-strong* neighborhood. This proves AFD for the finite algebra, including its nonfactor cases. $\square$

5. Ergodicity and type

Lemma 5.1. The center of $M$ consists exactly of the diagonal functions invariant under $\mathcal R$.

Proof. Let $z\in Z(M)$ and put $\eta=z\Omega$. For each bounded diagonal function $d$, its left and right multiplications agree on $\Omega$. Centrality of $z$ and commutation with the right algebra give

A countable separating family on $X$ forces $\eta$ to vanish off the diagonal. Write $\eta=h1_\Delta$, $h\in L^2(X,\mu)$. For every measurable $E\subset X$,

Hence $h\in L^\infty(X,\mu)$, with $\|h\|_\infty\le\|z\|$. The separating vector gives $z=D_h$.

Commutation with every matrix unit $v_{ij}$ at every stage says $h(\phi_i(t))=h(\phi_j(t))$ almost everywhere. By (3) this is precisely $h(x)=h(y)$ for $\nu$-almost every arrow. Conversely that equality makes $D_h$ commute with every convolution generator.

The countably many stage graphs preserve null sets. Thus an almost invariant level set of $h$ has an exactly saturated representative modulo a null set: take its saturation under those graphs; every newly added piece comes from a null violation of invariance. This justifies using the usual measured ergodicity definition. $\square$

Theorem 5.2. An ergodic principal orbitally countable measured AF groupoid of type $\mathrm{II}_1$ has an AFD factor of type $\mathrm{II}_1$. Its predual is separable, and it is isomorphic to the tracial infinite tensor product of $M_2$.

Proof. Ergodicity makes every invariant bounded function constant, so Lemma 5.1 makes $M$ a factor. Its trace is finite and faithful.

The invariant probability $\mu$ has no atoms. On a standard Borel space every probability atom contains a point of positive mass: use a countable separating family, take on each set the side carrying the atom's full measure, and intersect those sides. Their intersection has the atom's measure and contains at most one point. If such a point $x$ had mass $c>0$, invariance applied to singleton arrows would give mass $c$ to every point in its orbit. That orbit would be finite. Its positive-measure saturation would be conull by ergodicity, putting the relation in type I, contrary to its specified type $\mathrm{II}_1$. Thus $L^\infty(X,\mu)\subset M$ is diffuse and infinite dimensional. A finite factor containing it cannot be a finite matrix algebra; it is of type $\mathrm{II}_1$.

AFD follows from Theorem 4.2. Each $\mathcal R_n$ is a standard Borel space with finite counting measure. Its $L^2$ space is separable, using simple functions from a countable Borel generator. Their increasing dense union makes $H$ separable. The concrete predual of $M$ is a quotient of the separable trace-class predual of $B(H)$, so it is separable. The local finite AFD uniqueness theorem now identifies $M$ with the tracial $M_2$ product. $\square$

This proof uses the given AF exhaustion. It does not need the theorem converting amenability of measured relations to approximate finiteness.

6. Exercises with complete solutions

Exercise 1. Why does a relation with classes of size two on $X=Y\times\{1,2\}$, with a nonatomic probability space $Y$, fail to have a finite-dimensional regular algebra?

Solution. Relate $(t,1)$ only to $(t,2)$, and give the two coordinates equal weight. Formula (15) gives $M_2(L^\infty(Y))$. Its center contains all scalar fields $a(t)1_2$, an infinite-dimensional algebra because $Y$ is nonatomic. Class finiteness controls matrix size, not the dimension of the coefficient algebra. Finite partitions of $Y$ give the finite-dimensional approximants in (17).

Exercise 2. Verify the measure normalization in (16) for a single matrix-unit field.

Solution. For $F=1_E e_{ij}$, $E\subset Y$, its $L^2$ norm squared is $\mu(E)$, since the kernel has one nonzero arrow per source point in $\phi_j(E)$, and $\mu(\phi_j(E))=\mu(E)$. Formula (16) gives $\beta(E)/k=\mu(E)$. Its trace is zero if $i\ne j$, and $\mu(E)$ if $i=j$. In particular the identity has trace $k\mu(Y)=1$, whereas a constant diagonal matrix unit has trace $1/k$.

Exercise 3. On a finite class of size three, check the orientation of convolution using the arrows $(x_1,x_2)$ and $(x_2,x_3)$.

Solution. Their graph kernels are matrix units $e_{12}$ and $e_{23}$. In (6) the only surviving intermediate point is $x_2$, giving $e_{12}*e_{23}=e_{13}$, the arrow $(x_1,x_3)$. Reversing their order gives zero. The involution reverses an arrow and complex conjugates its coefficient, so $e_{12}^*=e_{21}$.

Exercise 4. Prove the strong convergence claim for Schur-kernel cutoffs without claiming operator-norm convergence.

Solution. Let $f_n=f1_{\mathcal R_n}$. Formula (3) and dominated convergence give $\|f_n-f\|_{L^2(\nu)}\to0$; the Schur bounds give a common operator bound. For $g\in\mathcal C$, commutation gives

whose norm tends to zero. The vectors $Q_g\Omega$ span a dense subspace. Approximate an arbitrary vector by this subspace and use the common bound to control the error. This proves strong convergence. It supplies no estimate forcing $\|L_{f_n}-L_f\|\to0$.

Exercise 5. Show that the finite-partition matrix average in Lemma 4.1 preserves positivity and the identity.

Solution. If $F(t)\ge0$ almost everywhere, then for every vector $\zeta\in\mathbb C^k$ its cell average satisfies

The same computation applies to matrices of such fields, giving complete positivity. The constant identity averages to the identity. This elementary finite integral explains the norm bound, without importing general von Neumann expectation theory.

Exercise 6. Why is a bounded invariant diagonal function central even before ergodicity is assumed?

Solution. For $h(x)=h(z)$ on almost every arrow, the products $D_hL_f$ and $L_fD_h$ have kernels $h(x)f(x,z)$ and $f(x,z)h(z)$. They agree for every generating kernel $f$. Hence $D_h$ commutes with their von Neumann algebra. Ergodicity is used only to make these central functions scalar.

Exercise 7. Identify exactly what fails if one tries to deduce a factor from a finite-class exhaustion without ergodicity.

Solution. The construction still gives a faithful finite trace and the finite-dimensional approximation proof still works. However a nontrivial invariant subset $E\subset X$ gives a central projection $D_{1_E}$, by Lemma 5.1. Its trace is $\mu(E)\in(0,1)$, so it is neither zero nor one. Thus the algebra is not a factor. The example in Exercise 1 has precisely this obstruction.

Exercise 8. If a probability measure assigns an atom of mass $c>0$ to one point in an invariant measured relation, show that its orbit has at most $1/c$ points.

Solution. For each equivalent point $y$, the singleton partial bijection $x\mapsto y$ preserves measure, so $\mu(\{y\})=c$. Distinct orbit points are disjoint atoms. Any $N$ of them have total measure $Nc\le1$, so $N\le1/c$. The orbit is therefore finite. Ergodicity then concentrates the measure on that orbit, the type I case excluded in Theorem 5.2.

Exercise 9. Explain why changing to an equivalent invariant measure in (4) does not require multiplying left-convolution kernels by a density ratio.

Solution. Source-coordinate densities multiply a vector by $h(y)^{-1/2}$. Left convolution sums over the first coordinate while keeping $y$ fixed:

Thus $VL_f=L_fV$ between the two $L^2$ spaces. A right-convolution formula changes the source coordinate and would behave differently for a noninvariant measure. Only the invariant-probability model is used for the simple right formula (7).

Exercise 10. The algebras $B_{\mathcal P}$ chosen for different finite tests need not form an increasing sequence. Why is this enough here, and how does one obtain the increasing matrix sequence for the final factor?

Solution. The definition of local AFD asks for one finite-dimensional algebra for each finite set and neighborhood; it does not demand that choices for different tests be nested. Theorem 4.2 proves exactly that property. In the separable $\mathrm{II}_1$ factor, the earlier finite AFD lesson's dyadic replacement and exact containment arguments convert successive local approximations into an increasing sequence of matrix subfactors. Its uniqueness proof then identifies the closure with the tracial $M_2$ product. A claim of nesting directly from unrelated partitions at unrelated relation stages would need a separate argument.

7. Elementary completion of finite-partition averaging

This section supplies the scalar integration, measurable partition, and matrix averaging details in Lemma 4.1 and Exercise 5. Its starting point is the unital matrix-field identification (15), with the usual field norm and order, and the normalized trace formula (16). In particular, $\beta(Y)=1$,

The argument applies to any probability measure space, including one with atoms and one whose sigma-algebra is completed. Sections 2, 3 and 8 establish the Borel labels, the faithful normal operator model and the trace construction in (15)-(16).

Bounded scalar integration

For a complex simple function on a finite disjoint measurable partition, define

Two representations have a common refinement formed from the intersections of their cells. Finite additivity on that refinement shows that $I(s)$ is independent of the representation and complex linear. The finite sums also give positivity for real nonnegative simple functions and

Every bounded measurable complex function $h$ is a uniform limit of simple functions: divide its bounded real and imaginary ranges into finitely many half-open intervals, use their measurable inverse images, and let the interval widths tend to zero. If $s_n\to h$ uniformly, the preceding bound makes $I(s_n)$ Cauchy. Define $I(h)=\lim_n I(s_n)$. For another uniform simple approximation $u_n\to h$, the same bound on $s_n-u_n$ shows that the two limits agree. Approximating sums proves linearity. A nonnegative real function has nonnegative simple approximants, so positivity passes to the limit and real order bounds pass to integrals. Since $|s_n|\to|h|$ uniformly,

The construction on a measurable subset $C$ gives $I_C$, with the bound $|I_C(h)|\leq\beta(C)\|h\|_\infty$. A bounded function supported on a null set therefore has integral zero, and the integral depends only on the almost-everywhere class.

These integrals agree with the usual Lebesgue integrals of bounded functions. For a nonnegative bounded $h$, choose grid approximants with $s_n\leq h\leq s_n+\eta_n$, where $\eta_n\downarrow0$. If $u$ is any nonnegative simple function below $h$, then $u\leq s_n+\eta_n$. Thus the supremum of the simple integrals below $h$ lies between $I(s_n)$ and $I(s_n)+\beta(Y)\eta_n$. These bounds have the same limit. Positive and negative parts extend the agreement to real functions; real and imaginary parts extend it to complex functions. All integrals used below can consequently be obtained from finite sums and uniform limits.

One common partition and a strict error bound

Let $F^{(1)},\ldots,F^{(r)}\in\mathcal A$ and $\varepsilon>0$. Choose measurable representatives for their finitely many entries. A single measurable null set contains every violation of chosen finite essential bounds on those entries. Redefine all entries as zero on that set. They are now bounded everywhere and represent the original fields. If the family is empty, use $\mathcal P=\{Y\}$.

For a nonempty family set

For each real and imaginary entry $h$, take the inverse images of

Only finitely many bins meet the bounded range. The half-open convention assigns every value to one bin, including a value on a boundary. Take all intersections of the $2rk^2$ resulting partitions and discard empty intersections. This gives one finite measurable partition $\mathcal P$ of $Y$, on whose cells every real and imaginary entry has oscillation at most $\delta$.

On a cell with $\beta(C)>0$, define the matrix average entry by entry:

Boundedness and finite measure justify every scalar integral. Almost-everywhere equal representatives give the same averages. The formula makes no division on a null cell; assigning zero there gives the same $L^\infty$ field as any other assignment.

The average of a real coordinate whose values lie in $[m\delta,(m+1)\delta)$ lies in the closed interval $[m\delta,(m+1)\delta]$, by positivity and real order bounds for the integral. Each real and imaginary difference from its average therefore has absolute value at most $\delta$, and each complex entry difference has squared magnitude at most $2\delta^2$. Formula (16) gives, for each original field,

Hence every error is at most $\varepsilon/2<\varepsilon$. This proves the constant in (18) and the strict tolerance in Lemma 4.1.

Positive cells and representatives before completion

If there are $s$ positive-measure cells, their indicators are nonzero orthogonal central projections summing to $1$ in $L^\infty(Y,\beta)$. Null cells are zero projections. The map

is a unital $*$-homomorphism onto the algebra in (17). It is injective because a nonzero constant matrix cannot vanish almost everywhere on a positive-measure cell. Thus $B_{\mathcal P}$ is unital and has dimension $sk^2$. There is a positive cell since $\beta(Y)=1$. All null cells may be adjoined to one positive cell without changing its projection, its averages, or the estimates almost everywhere.

Suppose the given sigma-algebra is the completion of $\Sigma_0$. Each of the finitely many cells has a $\Sigma_0$-representative differing from it inside a $\Sigma_0$-measurable null set. Successively subtract earlier representatives to make them disjoint, then adjoin their uncovered complement to the first representative. Every change lies inside the finite union of those null sets. The resulting sets form an actual $\Sigma_0$-partition and determine the same projections and averages. In the standard Borel setting $\Sigma_0$ can be the Borel sigma-algebra. This finite replacement uses the definition of completion and finite set operations.

Contractivity and complete positivity

Let $M=\|F\|$, so $\|F(t)\|_{\mathrm{op}}\leq M$ almost everywhere. For $u,v\in\mathbb C^k$, linearity and the scalar integral bound give

Taking the supremum over unit vectors proves $\|\overline F_C\|_{\mathrm{op}}\leq M$ and $\|E_{\mathcal P}F\|\leq\|F\|$. This establishes the operator bound used in Theorem 4.2.

At matrix level $q$, a matrix of fields is a field of $qk$-by-$qk$ matrices, and the amplified map still averages each entry over the same cells. If that field $G$ is positive almost everywhere, then for every $w\in\mathbb C^{qk}$,

Each cell average is positive. This works for every $q$, proving complete positivity and supplying the amplification step in Exercise 5. The same bilinear estimate proves contractivity at every matrix level.

The identity and all cell-constant fields are fixed, proving unitality and idempotence onto $B_{\mathcal P}$. Constant matrices may be brought through finite scalar integrals, so for $b,d\in B_{\mathcal P}$,

Summing the cell integrals in (16) proves $\tau(E_{\mathcal P}F)=\tau(F)$. Moreover, the integral of every entry of $F-\overline F_C$ on its cell is zero. Hence for cell-constant $b$,

Expanding the trace square gives

so this averaging is also contractive in the tracial $2$-norm. These conclusions concern the finite-partition map constructed here.

The scalar finite-bin method can be compared with Sheldon Axler, Measure, Integration & Real Analysis, author revision of 12 June 2026, Theorem 2.89, printed p.65 / PDF p.80. The complete proof on that page constructs measurable simple approximants and proves uniform convergence for bounded real functions. Section 7 gives the scalar integration and matrix arguments, including the complex entries and every matrix amplification. Sections 2, 3 and 8 and the linked operator foundations supply the Borel, representation and topology inputs.

8. Counting, diagonal closure and the measure model

The measured groupoid includes the source-counting measure (2). The passage to the exact conull AF model in Section 1 uses this given measure and its flip invariance. Once that model is chosen, the finite labels give the following direct formulas for every counting operation in the proof.

Counting through disjoint stage graphs

List all the pair-coordinate graphs of all the finite stages as $G_1,G_2,\ldots$, including the diagonal. They cover the relation. Replace $G_j$ by

These sets are disjoint Borel graphs of partial Borel bijections. Indeed, a Borel subset of the graph of a Borel bijection is obtained by restricting its domain to the inverse image of that subset under its Borel graph map. Its domain and range are Borel, since the bijection and its inverse are Borel. Write the restricted bijection as $\psi_j:U_j\to V_j$, with graph $P_j=\{(\psi_j(y),y):y\in U_j\}$.

For a Borel arrow set $E$, put

Each set on the right is Borel, and disjointness gives the exact, possibly infinite, identity

Thus the counting function is measurable, and monotone convergence defines its integral. For a Borel set $A\subset X$, its saturation is

a Borel set. Images here are images under the stated partial Borel isomorphisms, rather than arbitrary projections. Invariance makes the saturation of a Borel null set null, by the calculation in Section 1. Every set in the completed sigma-algebra differs from a Borel set inside a Borel null set. Consequently saturation preserves measurability after completion, and the same formulas determine completed counting and kernel operations independently of representatives. In particular a completed arrow-null set has empty source fiber outside a null set, since its nonnegative counting function has integral zero. This justifies the almost-everywhere conventions for all the sums above.

For invariance in the reverse direction in Section 1, apply preservation by $\psi_j$ to the restriction $U_j(E)$. The source integral on $E\cap P_j$ is $\mu(U_j(E))$, and the source integral on its flip is $\mu(\psi_j(U_j(E)))$; these are equal. Sum over the disjoint pieces. The forward direction follows by the same calculation on one graph. No uniformization of an arbitrary countable relation enters this AF argument.

The diagonal is a normally represented von Neumann algebra

For bounded measurable $d$, multiplication by $d(x)$ on $H$ has norm at most $\|d\|_\infty$. Its restriction to the diagonal vectors has the usual multiplication norm on $L^2(X,\mu)$, so the norm is exactly $\|d\|_\infty$, including when that norm is zero. For the lower bound, use $1_A\Omega$ on a positive-measure set where $|d|>\|d\|_\infty-\varepsilon$. Pullback along the first coordinate preserves null sets by flip invariance. Thus $D_d$ is well-defined on almost-everywhere classes and faithful.

We can verify weak closedness directly. If $T$ lies in the weak operator closure of the diagonal operators, it commutes both with all $D_d$ and with all right convolutions $Q_f$. Put $\eta=T\Omega$. Since $D_d\Omega=Q_d\Omega$, these commutations give

A countable separating family makes $\eta=h1_\Delta$. Commutation with $D_{1_A}$, for every measurable $A$, gives

Testing the sets where $|h|>\|T\|+1/m$ proves $h\in L^\infty$ and $\|h\|_\infty\leq\|T\|$. Both $T$ and $D_h$ commute with the right algebra and agree on $\Omega$. Its dense right orbit therefore gives $T=D_h$. This proves weak closedness, without an assumption that an arbitrary convergent net is norm bounded. Compressing by $D_{1_Y}$ proves the diagonal-corner closedness used in (15). The finite matrix-unit reconstruction then proves the whole stage closedness.

Here is also an all-net normality argument. Let $0\leq d_a\uparrow d$ in $L^\infty(X,\mu)$. It suffices to consider a common finite upper bound. Put $s=\sup_a\int d_a\,d\mu$, choose indices with integrals tending to $s$, and choose increasing upper bounds $a_n$ for them. The pointwise limit $d_0=\lim_n d_{a_n}$ has integral $s$. For any $a$, directedness implies $\int\max(d_a,d_{a_n})\leq s$. Monotone convergence shows $\int(d_a-d_0)_+=0$. Thus $d_0$ is an essential upper bound for the net, and it equals its essential supremum $d$. On $H$, dominated convergence now gives $D_{d_{a_n}}\to D_d$ strongly: the exceptional $\mu$-null set pulls back to a $\nu$-null set, and the square of the common bound times $|\xi|^2$ is integrable. For $a\geq a_n$, the pointwise inequality $0\leq d-d_a\leq d-d_{a_n}$ gives the same vector estimate. Hence the entire net converges strongly to $D_d$. The representation preserves every bounded increasing positive supremum and is normal. The finite corner extraction and matrix-unit reconstruction in (15) preserve this normality in both directions.

Equivalent probabilities and separability

The density used in (4) can be constructed using only Hilbert space and finite integration. Given equivalent probabilities $\mu,\mu'$, let $\lambda=\mu+\mu'$. On the simple functions,

This functional extends to $L^2(\lambda)$. Hilbert representation gives an integrable function $r$ with $\mu(A)=\int_A r\,d\lambda$ for every measurable $A$. These identities, tested on real and imaginary level sets, give $0\leq r\leq1$; also $\mu'(A)=\int_A(1-r)\,d\lambda$. Equivalence implies $0<r<1$ almost everywhere: $\{r=0\}$ is $\mu$-null and hence $\lambda$-null, and similarly for $\{r=1\}$. Therefore

The integral identity follows first for nonnegative simple functions and then by monotone convergence; it applies to $h1_A$ by truncation. Applying the counting formula to each disjoint graph gives $d\nu'/d\nu(x,y)=h(y)$. The map in (4) is consequently an onto isometry, with inverse multiplication by $h(y)^{1/2}$, and its stated intertwining holds on a dense subspace and then everywhere.