Unit 4, Part I: From pointwise failure to a selected law

We turn to §3 of B, beginning with Proposition 3.1 and the first part of Proposition 3.2, pp.7–10. The aim is to encode a possible failure of fourfold pointwise convergence in a countably additive probability measure, retaining a positive correlation, the one-coordinate marginals, and invariance under the action with different coordinate speeds.

Determining the conditional distributions given all parameters also requires distal and relatively weakly mixing structure. That is the subject of Part II. The fourfold main theorem is not treated here as an independently verified conclusion.

1. The two convergence inputs

1.1 Reduction to a standard space

Once four functions are fixed, all their integer translates define a standard Borel factor. See B §2.1, pp.3–4, and the full construction in the standard-factor supplement. Pulling back a Borel full-measure convergence set recovers convergence on the original space, so the final conclusion does not require that space to be standard.

Henceforth let $(X,\mu,T)$ be a standard Borel probability system, with $T$ invertible, bimeasurable, probability-preserving, and mixing. Set $I=\{1,2,3,4\}$.

1.2 The pointwise input for at most three factors

B Lemma 2.1 (p.4) asserts that, for every standard invertible probability-preserving system $(Y,m,R)$, every nonempty proper subset $J\subsetneq I$, and fixed bounded functions $g_i$,

converges almost everywhere. It requires neither ergodicity nor mixing of $R$, and does not identify the general limit as a product of integrals. The proof in Appendix A includes the analytic modifications needed for nonconsecutive triples of coefficients. We take this precise statement as a hypothesis.

On the original mixing system $T$, Unit 1 established that averages with distinct nonzero integer slopes have the product of integrals as their $L^2$ limit. Comparing norm convergence with the pointwise input above identifies the pointwise limit of every proper-subset average on $X$ as the same product.

This step does not use the higher-order mixing assertion in the companion manuscript R. That stronger result enters later in B, when independence of the auxiliary groups is maintained.

2. A real centered witness to fourfold failure

Source: B Proposition 3.1, p.8, the first paragraph of the proof.

Suppose B Theorem 1.1 fails for some fixed quadruple. First write

In the multilinear expansion, the constant term is the proposed limit. Every term containing fewer than four centered functions is a proper-subset average, which already tends pointwise to zero. The product average of all four centered functions must therefore still fail to tend to zero on a set of positive measure.

Split each centered complex function into its real and imaginary parts. The expansion has finitely many terms, each a product of four real centered functions multiplied by a fixed complex constant. If all these real product averages tended to zero almost everywhere, the original product average would also tend to zero on the intersection of finitely many full-measure sets. Thus some real tuple still witnesses failure.

After rescaling, take real functions $f_i$ such that

fails to tend to zero on a positive-measure set. The norm theorem from Unit 1 still gives $A_N\to0$ in $L^2$.

Since $A_N$ is a bounded real sequence, a point where it does not tend to zero has either a positive limit superior or a negative limit inferior. Decompose by countably many positive thresholds and, if necessary, change the sign of one $f_i$. We can choose $\delta>0$ and a measurable set $E$ such that

The set $E$ and threshold $\delta$ are now fixed. Lengths satisfying the inequality may depend on $x$.

3. Selecting from a countable family of lengths

Source: B p.8, the second paragraph of the proof of Proposition 3.1.

3.1 A finite ratio grid preserves a positive deviation

For summands bounded in absolute value by $1$, Unit 1 proved

Choose a sufficiently small mesh bound $\eta$ and a finite grid $F\subseteq[1,2]$ containing $1,2$. The parameter $\eta$ is fixed first; for example, take $\eta<\delta/8$.

If $2^k\le N<2^{k+1}$, choose $c\in F$ such that $|N/2^k-c|\le\eta$. Set $M=\lfloor c2^k\rfloor$. Then

For sufficiently large $k$, the right side is less than $\delta$. Thus the arbitrarily late deviations in (F) can be replaced by deviations at grid lengths $M=\lfloor c2^k\rfloor$ satisfying $A_M(x)>\delta$.

3.2 Measurable selection without a general selection theorem

Fix an enumeration of $F\times\mathbb Z_{\ge0}$. For each integer $a\ge1$ and each $x\in E$, choose the first candidate pair in this enumeration satisfying

and set $N_a(x)=c2^k$. The preceding argument ensures that a candidate exists. If needed, the selected $k$ can also be required to exceed a fixed rounding threshold.

The admissible set for each candidate is measurable. The set on which it is the first admissible candidate is obtained by removing the finitely many preceding admissible sets. Hence $N_a$ is measurable. For $x\notin E$, define $N_a(x)=2^a$. We then have everywhere

and, on $E$, we have $A_{\lfloor N_a(x)\rfloor}(x)>\delta$. Using a deterministic length on the complement is essential to controlling its contribution later.

4. The suspension flow and a phase test

Source: B §3.1, equation (3.1) on p.7, and the phase construction and equation (3.3) on p.8.

4.1 Embedding integer time in continuous time

Set

Braces denote the fractional part. The addition identity for integer parts implies $S_C^{s+t}=S_C^sS_C^t$. For fixed $t$, integer powers of $T$ preserve the measure, and the circle rotation $u\mapsto\{u+t\}$ preserves Lebesgue measure. It follows that $S_C^t$ preserves $m_C$. Invertibility gives meaning to negative integer parts.

Write the spatial coordinate as $\xi(x,u)=x$ and the phase as $\upsilon(x,u)=u\pmod1$. Extend the previous selection by $N_a(x,u)=N_a(x)$, independently of the initial phase.

4.2 A phase test that prevents additional integer carries

Choose nonnegative, nonzero continuous functions on the circle, $\alpha,\beta$, with respective supports contained in

Define the continuous real function on $\mathbb T^4$ by

If $v_i=\upsilon(S_C^{it}(x,u))=u+it\pmod1$, then

Write $t=n+s$ with $0\le s<1$. Then $\Psi=\alpha(u)\beta(s)$. When this is nonzero, we have $u\in(1/4,1/2)$ and $s\in(0,1/16)$, so for $i=1,2,3,4$,

Thus $\lfloor u+i(n+s)\rfloor=in$, and the spatial coordinate is exactly

The phase test preserves the spatial readings of the original average by excluding the phase values that could cause an additional carry.

4.3 A uniform comparison of continuous and discrete averages

Set $b_\beta=\int_0^1\beta(s)\,\mathrm{d}s>0$. For any real $L\ge2$, integrating over the first $\lfloor L\rfloor$ complete unit intervals gives

The error is uniform in $x,u,L$. It comes from three bounded endpoint changes: the final incomplete unit interval, the difference between normalizations $1/L$ and $1/\lfloor L\rfloor$, and shifting the summation range from $0,\ldots,\lfloor L\rfloor-1$ to $1,\ldots,\lfloor L\rfloor$. Since all summands are bounded, each error is $O(L^{-1})$, with constants depending only on the fixed $\alpha,\beta$.

5. The selected averages retain a uniform positive correlation

In (Phase), take $L=N_a(x,u)$. Since $N_a\ge2^a$, the integrated uniform error is $O(2^{-a})$. On $E\times[0,1)$, the integral of the main term is at least

On $E^c\times[0,1)$, the selected length is the deterministic value $2^a$. The absolute value of the main-term integral is therefore at most

This is where the zero $L^2$ limit of the centered fourfold average is used. Consequently, there is $\varepsilon_0>0$ such that, for all sufficiently large $a$,

This is equation (3.2) of B Proposition 3.1. Ordinary $L^2$ convergence is not applied to arbitrary point-dependent lengths. Bad lengths are deliberately retained on the positive-measure set, while deterministic lengths are used on its complement.

6. From finite selected laws to a countably additive limit measure

Source: B §3.2, the definition and first part of the proof of Proposition 3.2, p.9.

6.1 Extensions and finite-stage distributions

Let $(Y,m_Y,S_Y^t)$ be a standard Borel probability-preserving flow, and let $\pi_Y:Y\to C$ be a flow factor: it preserves measure and intertwines the action at each fixed time, modulo null sets. The flow is jointly measurable. Pull back the selected length as $N_a(\pi_Yy)$.

For a bounded measurable function on $Y\times\mathbb R$, denoted $H$, define the probability integral

Measurability of the length and joint measurability of the flow justify the integral. The inner normalization makes its total mass $1$. Push $Q_a$ forward under

to $Y^4$. This defines a countably additive probability measure $\lambda_{Y,a}$. For product test functions,

The one-coordinate marginals at a finite stage need not already equal $m_Y$. We must first show that they converge to $m_Y$.

6.2 Convergence of one-coordinate marginals through all lengths

Fix a bounded measurable function $G$ and an index $i\in I$. The ordinary one-function pointwise ergodic theorem for the flow implies that

converges almost everywhere through all real lengths $L\to\infty$. This also follows directly from discrete Birkhoff. Set

For an integer length $L=N$, the average above equals

Birkhoff supplies an almost-everywhere limit $L_i(y)$, with $\int L_i\,\mathrm{d}m_Y=\int\widetilde G_i\,\mathrm{d}m_Y=\int G\,\mathrm{d}m_Y$. Comparing a noninteger length with its integer part gives an error $O(\|G\|_\infty/L)$, so convergence holds through all real lengths.

Since $N_a(\pi_Yy)\ge2^a$, the averages along these point-dependent lengths still converge to the same $L_i(y)$. They remain bounded by $\|G\|_\infty$, and bounded convergence yields

No ergodicity of $Y$ or flow-invariance of the selector is assumed. A separate full-measure convergence set for each fixed $G$ suffices; no common null set for every bounded function is needed.

6.3 One fixed ultrafilter for every system

At the end of p.8, B fixes a single free ultrafilter $\mathcal U$, used for all subsequent extensions. For a bounded complex sequence $(z_a)$, the statement $z=\lim_{a\to\mathcal U}z_a$ means that, for every $\varepsilon>0$, the set $\{a:|z_a-z|<\varepsilon\}$ belongs to $\mathcal U$.

A free ultrafilter contains no finite set and contains every cofinite set. Compactness ensures that a bounded sequence has a unique ultrafilter limit. This limit respects finite linear combinations and continuous operations and agrees with an ordinary limit when one exists. Only these properties are used; it need not be interpreted as an actually selected common subsequence.

Choosing convergent subsequences independently for different systems could destroy compatibility with factor projections. A common $\mathcal U$ ensures that the same limiting operation is used after test functions are pulled back.

6.4 Establishing countable additivity

Realize the standard Borel space $Y$ as a Borel subset of a compact metric space $K$. Regard $\lambda_{Y,a}$ as probability measures on $K^4$. For $F\in C(K^4)$, define

The functional $\Lambda$ is positive and linear, with $\Lambda(1)=1$ and $|\Lambda(F)|\le\|F\|_\infty$. The Riesz representation theorem produces a countably additive Borel probability measure $\lambda$ on $K^4$. Countable additivity comes from this representation step; taking ultrafilter limits separately on arbitrary sets would not establish it.

Equation (Marginal) holds for every continuous one-coordinate function. Hence each one-coordinate marginal of $\lambda$ equals $m_Y$, where $m_Y$ is regarded as a measure on $K$ concentrated on $Y$. Therefore

Restrict to the Borel subset $Y^4$ to obtain the required probability measure $\lambda_Y$, with every marginal equal to $m_Y$.

6.5 Extension to all bounded measurable product tests

Source: B p.9, the $L^1$ approximation following the positive-functional construction on a compact space.

First assume $|G_i|\le1$. Continuous functions on the compact metric space are dense in $L^1(m_Y)$. Choose $g_i\in C(K)$ with $|g_i|\le1$ and $\|G_i-g_i|_Y\|_1<\varepsilon$. The common bound is retained by truncating to the complex unit disk.

A telescoping expansion gives

Under $\lambda_Y$, the right side has integral at most $4\varepsilon$, since the marginals are exactly $m_Y$. Under $\lambda_{Y,a}$, apply (Marginal) to each fixed bounded function $|G_i-g_i|$. The ultrafilter limit of the integral is again at most $4\varepsilon$.

The integral of each continuous product is already determined by $\Lambda$. Letting $\varepsilon\downarrow0$ proves B (3.4) for all bounded measurable $G_i$:

Rescaling handles general bounded functions. This extension also shows that the resulting measure is independent of the compact model: indicators of measurable rectangles are product tests of this kind and uniquely determine a probability measure on $Y^4$.

7. Invariance without an invariant selector

Source: B p.9, the last paragraph in the proof of Proposition 3.2.

Fix a real $s$. In (Selected), replace function $i$ by $G_i\circ S_Y^{is}$. The finite-stage inner integral becomes

Substitute $r=t+s$. The only change from the original integral is the shift of its interval from $[0,N_a]$ to $[s,N_a+s]$. If $|G_i|\le1$, the intervals have symmetric difference of length at most $2|s|$, so the pointwise error is at most

This bound is uniform in $y$ and tends to zero for fixed $s$. Taking ultrafilter limits gives

Product tests determine the measure, so $\lambda_Y$ is invariant under the different-speed product action

as claimed.

The comparison always uses the same starting point $y$ and the same length $N_a(\pi_Yy)$. It does not replace $y$ by $S_Y^sy$, and never requires $N_a$ to be flow-invariant. This is why point-dependent selection is compatible with invariance of the limiting law.

8. Compatibility with extensions and inverse limits

Source: B p.10, the first paragraph in the proof of Proposition 3.2.

8.1 Compatibility with factor projections

Let $\theta:Y'\to Y$ be a measure-preserving flow factor over $C$. Thus

modulo null sets for the respective measures. On $Y'$, test (Selected) with $G_i\circ\theta$. The selected length is the same $N_a\circ\pi_{Y'}$, equal by the factor relation to $N_a\circ\pi_Y\circ\theta$.

Push the functions and integral down to $Y$. At each finite stage, the product-test integral equals the corresponding integral on $Y$. If equivariance holds only modulo null sets at each fixed time, joint measurability and Fubini justify its use inside the time integral. No single set with pointwise equivariance simultaneously for all real times is required.

Every system uses the same ultrafilter, so

Here $\theta^4$ applies $\theta$ coordinatewise. Equation (Factor) is the precise compatibility with projections claimed in the paper.

8.2 Realizing the inverse limit on the full countable product

The treatment of countable inverse limits in B p.10 is brief. We use the full-product realization in A Proposition 3.2 (p.8), which accommodates factor relations holding modulo null sets at each fixed time.

Let $Y_1\leftarrow Y_2\leftarrow\cdots$ be a countable chain of extensions over $C$, with bonding maps $\theta_n:Y_{n+1}\to Y_n$. Take the full standard Borel product

For the first $r$ coordinates, use $m_{Y_r}$ and finitely many bonding maps to define the joint law. These finite-dimensional distributions are consistent. The extension theorem for consistent cylinder measures on standard Borel spaces therefore gives a probability measure $m_\infty$. Its coordinate marginals are $m_{Y_n}$, and it is concentrated on sequences satisfying all compatibility relations

simultaneously. There are countably many such relations, each holding with probability one, so their common compatibility set is measurable and has full measure.

On all of $\Omega$, define the coordinatewise flow

This is a jointly Borel action. Fix $t$. The probability of a cylinder involving only the first $r$ coordinates can be computed on $Y_r$. The bonding maps are equivariant modulo null sets at this fixed time, and $S_{Y_r}^t$ preserves measure, so the cylinder's probability is unchanged. Cylinders determine the probability measure. Thus each fixed $t$ preserves $m_\infty$.

Keeping the full product as the underlying space is important. The full-measure compatibility set is not required to be pointwise invariant simultaneously for all real times. The coordinate projections $\theta_m^\infty:\Omega\to Y_m$ are measure-preserving flow factors, with compatibility modulo null sets. The first coordinate supplies the factor onto $C$. This realizes the inverse limit in the category being used.

Construct the selected law on $\Omega$. The factor compatibility already proved in this part gives

Under $\lambda_\Omega$, each state coordinate has marginal $m_\infty$, so the compatibility relations still hold almost everywhere in each state coordinate. The pulled-back stage $\sigma$-algebras are therefore increasing modulo null sets under $\lambda_\Omega$. All stage coordinates generate the full product $\sigma$-algebra, and the union of the pulled-back stage function spaces is dense in $L^2(\lambda_\Omega)$.

More explicitly, a function of finitely many coordinates can be lifted, using compatibility, to the largest stage it involves. Approximation by simple functions from the generating algebra then handles an arbitrary $L^2$ function. This density does not require independence of the four state coordinates.

8.3 Continuity of the flow for the new measure

The paper also establishes strong continuity of the different-speed product flow on the new $L^2$ space. We use the standard fact that a measurable unitary representation on a separable Hilbert space is automatically strongly continuous. The space $L^2$ of a standard probability space is separable, and the original jointly measurable probability-preserving flow induces such a representation. The original flow therefore has the required strong continuity. For a normalized product function $F((y_i))=\prod_iG_i(y_i)$, telescoping and the fixed marginals give

The right side tends to zero as $t\to0$. Finite linear combinations of product functions are dense in the new space, and the action is isometric for the invariant measure. Strong continuity therefore extends to all of $L^2(\lambda_Y)$.

9. Retaining the positive correlation and iterating the construction

9.1 Allowing a joint phase weight in the test

Equation (Selected) was first proved for product functions. The weight $\Psi$ in (Positive) is a joint function of four phases and need not itself be one product.

However, $\Psi\in C(\mathbb T^4)$ admits uniform approximation by trigonometric polynomials. Each trigonometric monomial factors into four one-coordinate phase functions. Apply (Selected) to the finite sums, then pass to the limit using the uniform error and $|f_i|\le1$. This proves B (3.8):

The finite-stage inequality holds for all sufficiently large $a$. Every cofinite set belongs to the free ultrafilter, so the same positive lower bound survives in the limit.

9.2 Normalizing the speed of one coordinate

For any $b\in I$, define the new flow

It preserves probability because different-speed invariance applies at every real time $t/b$. Projection onto coordinate $b$ leaves that coordinate with speed $1$. This projection is therefore an equivariant factor onto the original flow $Y$, and the new system is still an extension over $C$.

The same selected lengths and ultrafilter can consequently be applied again to $M_b(Y)$. This iteration produces the square array used later in the paper. Before iterating, we must determine which conditional laws have been preserved. That is the subject of Part II.

Scope of this part

Conditional on the pointwise theorem for at most three factors in B Lemma 2.1, and using the $L^2$ theorem under ordinary mixing from Unit 1, we have fully derived the positive-correlation selection in Proposition 3.1.

For every standard probability-preserving flow extension, we have also expanded the proof of the existence of the probability measure in Proposition 3.2, its fixed one-coordinate marginals, different-speed invariance, and compatibility with factors and countable inverse limits. The ultrafilter limit was not simply declared countably additive. That property came from representing a positive functional on a compact space and proving concentration of the marginals.

The full distal parameter law in Proposition 3.2 and proper-subset independence conditional on all parameters remain to be proved here. Nor has independence of all four coordinates been asserted: (Retained) preserves a nonzero fourfold correlation.

Guide to the original text

• B pp.3–4, §2.1: the standard-model reduction for fixed functions.
• B p.4, Lemma 2.1: the precise proper-subset pointwise input on general systems.
• B p.7, equation (3.1): the unit-roof suspension flow.
• B p.8, Proposition 3.1, equations (3.2)–(3.3): centered failure, grid selection, and the phase weight.
• B p.9, equation (3.4) and the first part of the proof of Proposition 3.2: a countably additive selected law.
• B p.9, final paragraph, through p.10, first paragraph: invariance and compatibility with projections and inverse limits.
• B p.10, equations (3.7)–(3.8): the normalized flow and retained positive correlation.
• A p.8, the end of the proof of Proposition 3.2: the inverse limit on the full countable product and density of stage functions for the selected law.