Unit 4, Part II: Conditional laws given all distal parameters

Part I constructed the selected law $\lambda_Y$ and proved its one-coordinate marginals and invariance. Here we study parts (2) and (3) of B Proposition 3.2 and Lemma 3.3, pp.9–10. Why is the full parameter law unaffected by length selection? Why do the states in every proper subset have a product conditional law even after all parameters are given?

The structural results in B Proposition 2.2 and the pointwise theorem for at most three factors in Lemma 2.1 are explicit hypotheses. We derive the conditional laws under these assumptions. We do not reprove Furstenberg–Zimmer theory, the nonergodic distal pointwise theorem, or all the cited literature.

1. Structural definitions and inputs

1.1 Relative weak mixing

Source: B §2.2, p.4, equation (2.2).

Let $(Y,m,R)$ be a standard invertible probability-preserving system, and let $\mathcal B$ be a factor invariant under both $R$ and $R^{-1}$, viewed as a $\sigma$-algebra. The extension $Y\to\mathcal B$ is relatively weakly mixing if, for bounded functions $f,g$, the condition $\mathbb E(f\mid\mathcal B)=0$ implies

This controls averaged conditional correlations after the part determined by the factor has been removed. The time average remains essential; the statement does not assert decay at each individual time.

1.2 Relative compactness and measure-distality

Source: B p.4, the definitions following (2.2).

For finitely many $F_1,\ldots,F_r\in L^2(m)$, consider the finitely generated $L^\infty(\mathcal B)$-module

It is invariant if composition with both $R$ and $R^{-1}$ preserves this space. The extension is relatively compact if the linear span of all such invariant finitely generated modules is dense in $L^2(m)$. Individual modules need not be closed in $L^2$; closure is taken at the end.

A relatively distal extension is built successively from relatively compact extensions and inverse limits. Without a specified base, distal means distal over the trivial factor. This is a measure-theoretic definition: no metric is chosen on $Y$, and a condition on distances between pairs of points does not replace it. Nonergodic systems are allowed. For example, the identity transformation preserves every one-dimensional subspace and hence is a compact system.

1.3 The precise scope of B Proposition 2.2

Source: B pp.4–6, the five parts of Proposition 2.2 and the subsequent flow version.

We invoke the following structural results, all without an additional ergodicity assumption.

1. Every standard system has a maximal measure-distal factor $W_Y$ containing its invariant factor, and $Y\to W_Y$ is relatively weakly mixing. The maximal distal factor of an extension contains the pullback of the old distal factor. Every measure-preserving automorphism commuting with the transformation preserves $W_Y$.
2. Factors and joinings of distal systems are distal. A distal system is relatively distal over each of its factors. For every nonzero integer $q$, the transformation $R$ is distal if and only if $R^q$ is distal.
3. Relative weak mixing passes to nonzero integer powers. Over any invariant joining of the bases, taking the conditional product of the extension fibers produces a relatively weakly mixing extension.
4. A relatively weakly mixing extension and a relatively distal extension over the same base are relatively disjoint: every invariant joining identifying their base coordinates must make the two systems independent conditional on that base.
5. For a distal transformation $R$, every fixed $d\ge1$, and bounded functions $g_1,\ldots,g_d$,

converges almost everywhere. No general product-of-integrals limit is asserted.

For a probability-preserving flow $S_Y^t$, the maximal distal factor $W_Y$ of the time-one map is preserved by the whole flow, and its action at every nonzero rational time is distal. Statements about the factor action are understood at each fixed time modulo null sets. Time integrals are always computed on the original jointly Borel flow $Y$.

B attributes existence and the relative structure to Furstenberg–Zimmer theory and its cited sources, and discusses the stability properties on pp.5–6. The ergodic case of part (5) comes from Huang–Shao–Ye Theorem C; B p.6 separately discusses the nonergodic decomposition. Citing the ergodic HSY theorem alone does not supply the full nonergodic version needed here.

2. Relative weak mixing and $L^2$ replacement

Source: B Lemma 2.3, pp.6–7, equation (2.4). The norm argument below can be proved directly from (RWM) and its stability under powers.

2.1 The full statement

Suppose $Y\to\mathcal B$ is relatively weakly mixing, $a_1,\ldots,a_d$ are pairwise distinct nonzero integers, and $g_i$ are bounded. Then

A telescoping expansion expresses the difference of the products as a finite sum. Each term contains a function with zero $\mathcal B$-conditional mean. After rescaling these finitely many terms, it suffices to prove that, if $|g_i|\le1$ and some $g_s$ satisfies $\mathbb E(g_s\mid\mathcal B)=0$, the original product average tends to zero in $L^2$.

2.2 Fixing the correlation gap reduces the number of factors

Use the complex inner product linear in its first variable. Set

Fix $h\ge1$. By measure preservation, shifting all terms by $-a_sn$ yields

Induct on the number of factors. The difference slopes $a_i-a_s$ are pairwise distinct and nonzero. The induction hypothesis therefore replaces the remaining functions in the average by their $\mathcal B$-conditional expectations, with an error tending to zero in $L^2$. Pairing with the fixed bounded function $q_{s,h}$ makes the integrated error tend to zero as well.

The remaining product after replacement is $\mathcal B$-measurable and bounded in absolute value by $1$. Thus

For fixed $h$, deleting the last $h$ terms costs only $O(h/N)$. When $d=1$ there is no remaining product, so (Corr) holds directly. This starts the induction.

No existence of a Cesàro limit for the correlations is required. A bound for the limit superior suffices.

2.3 Using the limit superior in the finite Hilbert-space inequality

In the finite estimate from Unit 1, fix $H$, let $N\to\infty$, and apply (Corr). We obtain

Since $\mathbb E(\overline{g_s}\mid\mathcal B)=0$, relative weak mixing for $R^{a_s}$ gives

Use $\|u\|_1\le\|u\|_2$ on a probability space and apply Cauchy–Schwarz over $h$. Then

Finally let $H\to\infty$. The centered product average tends to zero, and the finite telescoping expansion proves (Replace).

2.4 Conditional expectation commutes with flow time shifts

Suppose the factor $\mathcal B$ is preserved by every $S_Y^t$. For fixed $t$,

To prove this, test against any bounded $\mathcal B$-measurable function $H$. Write the integral on the left as $\int G(H\circ S_Y^{-t})\,\mathrm{d}m_Y$. Since $H\circ S_Y^{-t}$ remains $\mathcal B$-measurable, the measure-preserving change of variables gives the integral on the right.

Consequently, (Replace) extends to continuous averages of deterministic length. Write $t=n+s$ with $0\le s<1$. For each fixed $s$, apply the result to $G_i\circ S_Y^{a_is}$. For every $s$, the $L^2$ error tends to zero and is uniformly bounded by the function norms. Minkowski's inequality and dominated convergence over $s$ make the integrated $L^2$ error tend to zero. The final incomplete unit interval costs $O(L^{-1})$.

This conclusion concerns deterministic lengths $L$. It cannot be applied directly to the point-dependent lengths $N_a(y)$. The next section first removes the effect of selection using pointwise convergence, and only then applies the norm replacement.

3. From discrete pointwise convergence to all continuous lengths

Source: B p.10, the two paragraphs in the proof of Proposition 3.2 concerning parameter laws and proper-subset laws.

3.1 The order of Fubini in the time decomposition

Suppose the product averages with the fixed slopes are known to converge pointwise on a system $(Z,\nu,R)$. Now let $R=S_Z^1$ come from a flow, and fix bounded functions $G_i$. For each fixed $s\in[0,1)$, apply the pointwise theorem to the shifted functions $G_i\circ S_Z^{is}$. It follows that

converges for almost every $z$.

Joint measurability of the flow makes the set where the Cauchy condition fails measurable in $Z\times[0,1)$: it is expressed by countable unions and intersections involving measurable finite sums. For each fixed $s$, its $z$-section has measure zero. Fubini therefore shows that, for almost every $z$, convergence holds for almost every $s$.

The summands have a common bound, so for each such $z$ we may apply dominated convergence over $s$. Observe that

The continuous averages thus converge first at integer lengths. Comparing a length $L$ with $\lfloor L\rfloor$ introduces normalization and endpoint errors of $O(L^{-1})$. This proves convergence through all real lengths.

3.2 Any selected lengths tending to infinity have the same limit

At each good starting point, the preceding conclusion holds through all real $L\to\infty$. Thus however $N_a(z)$ depends on that point, the condition $N_a(z)\to\infty$ ensures convergence to the same pointwise limit.

Bounded convergence then identifies the limit of the integrated deterministic-length averages with that of the selected-length averages. Pointwise convergence is established before selection is allowed. A qualitative $L^2$ theorem alone does not justify the same inference.

4. Deterministic lengths determine the full parameter law

4.1 Parameters and conditional measures

Let $\omega_Y:Y\to W_Y$ be the maximal distal factor of the time-one map, and let $\nu_Y=(\omega_Y)_*m_Y$. On the standard spaces, choose regular conditional probabilities so that

For a state in $\lambda_Y$, written $(y_i)_{i\in I}$, define

Conditional probabilities need only be defined outside a $\nu_Y$-null set. Since each $P_i$ has marginal $\nu_Y$, changing versions affects finitely many coordinates only on a $\Lambda_Y$-null set.

4.2 Proposition 3.2(2): statement and deduction

For each fixed bounded $H_i:W_Y\to\mathbb C$, the paper's equation (3.5) is

The right side is an ordinary limit through deterministic lengths, rather than an ultrafilter limit.

The reason is that $W_Y$ is distal. To avoid presupposing a factor model strictly equivariant simultaneously for all real times, define directly on the original flow

For each fixed $s$, the factor is preserved at that time, so $G_{i,s}$ is a $W_Y$-measurable function class. Choose a bounded representative on $W_Y$, apply the discrete distal pointwise theorem to the time-one factor transformation, and pull back to $Y$. Then

converges for almost every $y$. Only countably many integers $n$ occur here. After fixing $s$, their exceptional null sets can be removed by a countable union.

The finite sums themselves are jointly measurable on $(s,y)$, since they use the original flow $S_Y$ and Borel versions of $\omega_Y$. Apply Fubini on $Y\times[0,1)$ and then the time decomposition from §3 to obtain pointwise convergence through all continuous lengths. No joint measurability in $s$ is required of the chosen factor representatives, and no separate pointwise factor action for all real times is constructed.

Along $N_a(\pi_Yy)$, the same pointwise limit persists. Apply bounded convergence to the integral and compare with (Selected) from Part I. This proves (Parameter).

Thus length selection may change the joint law of the four complete states, but it cannot change the deterministic-length law of all their distal parameters.

5. First condition on the subset's own parameters

5.1 Removing selection from a proper-subset law

Fix a nonempty proper subset $J\subsetneq I$. B Lemma 2.1 gives pointwise convergence on $Y$ for slopes $i\in J$, without requiring $Y$ to be mixing. The time decomposition in §3 therefore identifies this subset's selected law with its deterministic-length law.

Only now apply (Replace) to the deterministic-length averages, replacing each state function $G_i$ by

The bar denotes conditional averaging on the factor, not complex conjugation.

5.2 Including a parameter weight

For each $i\in J$, take a bounded parameter function $h_i$. Apply the preceding argument to the state functions

whose conditional expectations are $\overline G_i(w)h_i(w)$. It follows that

Finite linear combinations of parameter products on $P_J=(P_i)_{i\in J}$ generate its $\sigma$-algebra. By a bounded monotone-class argument, the parameter weight can be any bounded $H(P_J)$. Thus

Since the spaces are standard, first establish these identities for indicators from a countable algebra generating the Borel $\sigma$-algebra. Determine the conditional probability on one common full-measure parameter set, then extend to all bounded measurable functions. Equation (Subarray) is therefore equivalent to

5.3 Why this is not yet the paper's equation (3.6)

Equation (Subarray) conditions only on the subset's own parameters $P_J$. The paper requires conditioning on all parameters $P$, including those outside $J$. Additional conditioning can destroy independence, so $P_J$ cannot simply be replaced by $P$ in the conditioning field. The next section supplies this step using relative disjointness.

6. Relative disjointness extends the conditioning to all parameters

Source: B p.10, the last paragraph of the proof of Proposition 3.2. The conclusion is equation (3.6).

6.1 The subset states are relatively weakly mixing over their parameters

Write $\Lambda_J$ for the law of $P_J$. Section 5 identified the measure on the subset states as

It is invariant under the different-speed time-one action $(S_Y^i)_{i\in J}$. Each extension $Y\to W_Y$ is relatively weakly mixing for $S_Y^i$. Structural input §1.3(3) therefore makes $(Y^J,\lambda_J)\to(W_Y^J,\Lambda_J)$ relatively weakly mixing.

This particular conditional-product statement can also be checked directly by computing conditional correlations. Set

with every function bounded, and suppose at least one $f_s$ satisfies $\mathbb E(f_s\mid W_Y)=0$. Conditional on the full base parameter $w=(w_i)_{i\in J}$, the fiber measure is already a product. Hence the conditional correlation is

where $R_J=(S_Y^i)_{i\in J}$ and $R_W=(S_{W_Y}^i)_{i\in J}$.

Bound the other factors by constants. The squared $L^2(\Lambda_J)$ norm is then controlled by the squared norm of factor $s$ in $L^2(\nu_Y)$. This uses only that the marginal of $\Lambda_J$ at coordinate $s$ is $\nu_Y$; the parameters need not be independent. Taking Cesàro averages over $h$ makes the bound tend to zero by relative weak mixing.

A general bounded function of conditional mean zero admits $L^2$ approximation by finite tensor sums with their base conditional expectations subtracted. After centering, each tensor telescopes into terms with at least one conditionally centered factor. The approximation error is uniform in time; for example,

The order of approximation can be fixed explicitly. First choose a centered finite tensor sum $F_0$ so that $\|F-F_0\|_2$ is small. Keep this bounded $F_0$ fixed. Depending on $\|F_0\|_\infty$, choose a bounded finite tensor sum $G_0$ with $\|G-G_0\|_2$ sufficiently small. The identity

together with measure preservation by $R_J$ and the $L^2$ contraction property of conditional expectation bounds the error independently of $h$ by

First take the time-average limit for the finite tensor sums, then send both errors to zero. This proves (RWM) without assuming an unstated common $L^\infty$ bound for every approximating tensor.

6.2 The full parameter system is relatively distal over the same base

The full parameter system is $(W_Y^I,\Lambda_Y)$, with time-one action $(S_{W_Y}^i)_{i\in I}$. Each coordinate system is distal. The measure $\Lambda_Y$ has one-coordinate marginals $\nu_Y$ and is invariant under this product action, so it is a joining of those distal systems.

By structural input §1.3(2), the full parameter system is distal and is relatively distal over its factor $(W_Y^J,\Lambda_J)$.

We now have two extensions over the same base: the relatively weakly mixing extension formed by the $J$ complete states, and the relatively distal extension formed by all parameters.

6.3 The selected law is a joining of these two extensions

Push $\lambda_Y$ forward to the joint distribution of

The two components agree exactly on the $J$ parameters, and their joint distribution is invariant under the corresponding time-one action.

Relative disjointness, structural input §1.3(4), now applies. This joining must make the two components independent conditional on $P_J$.

If $\Lambda_Y(\mathrm{d}P\mid P_J)$ denotes the conditional law of all parameters given the subarray, the joint law must be

Relative disjointness is an explicitly invoked structural theorem. The preceding two subsections check its hypotheses in this application.

6.4 Conditional probabilities given all parameters

In (Join), the conditional law of the $J$ states depends only on $P_J$ and not on the other parameters. Thus, for any bounded $H(P)$ and bounded state functions $G_i$,

By the definition of conditional expectation, this is equivalent to

Again use a countable generating algebra on the standard spaces to identify conditional probabilities. This gives the paper's equation (3.6):

For empty $J$, the assertion is merely that the empty product equals $1$. The conclusion does not include $J=I$; the full four-state correlation may remain nonzero.

7. The normalized parameters still lie in the maximal distal factor

Source: B Lemma 3.3, p.10, immediately after (3.7).

Return to the normalized flow from Part I:

On parameter $i$, the time-one action is $S_{W_Y}^{i/b}$. This rational-time action is distal. Indeed, for $r=p/q\ne0$,

The right side is an integer power of the distal time-one action, so it is distal. A transformation is distal if and only if a nonzero integer power is distal. Therefore the underlying transformation $S_{W_Y}^{p/q}$ on the left is also distal.

The full parameter array is an invariant joining of these rational-speed distal systems, hence is itself a distal factor. By maximality, it lies in the maximal distal factor of the time-one map of $M_b(Y)$.

Under further iteration, the pullback of an old distal factor to an extension remains contained in the new maximal distal factor. Repeated application proves the conclusion of Lemma 3.3 for the original parameter arrays at every finite stage.

Phase parameters also lie in these factors. The time-one map of the unit-roof suspension preserves the phase $u$, so the phase factor has the identity action and is compact. The phase weight that retained the positive correlation in Part I can therefore be used as a function of the full distal parameter.

8. Order of the deduction and remaining dependencies

The order of the argument matters.

1. Use pointwise convergence for at most three factors to identify the proper-subset selected-length law with the deterministic-length law.
2. Apply relative $L^2$ replacement to deterministic-length averages, obtaining the conditional product given the subset's own parameters.
3. Show that the state system is relatively weakly mixing over the parameter subarray, while the full parameter system is relatively distal over that same subarray.
4. Use relative disjointness to extend the conditioning from the subarray to the full parameter array.

The conditional product in the second step alone does not imply the fourth. Nor may relative $L^2$ replacement be applied directly at lengths depending on the starting point.

The structural material still requiring independent verification is the full scope of B Proposition 2.2, especially the nonergodic distal pointwise theorem and relative disjointness, together with the analytic proof used for B Lemma 2.1. This part does not use the assertion that ordinary mixing implies mixing of every finite order. That companion result enters separately when the auxiliary-group structure is maintained later.

Original text and prerequisite sources

• B p.4, equation (2.2) and the following definitions: relative weak mixing, finitely generated modules, relative compactness, and relative distality.
• B pp.4–6, Proposition 2.2: the structural statements invoked individually here.
• B pp.6–7, Lemma 2.3, equation (2.4): relative $L^2$ replacement and its continuous-time version.
• B pp.9–10, Proposition 3.2(2)(3), equations (3.5)–(3.6): the full parameter law and conditioning on all parameters.
• B p.10, Lemma 3.3: distality of the normalized parameters and finite iteration.

Huang–Shao–Ye, Pointwise convergence of multiple ergodic averages and strictly ergodic models, arXiv:1406.5930v2, Theorem C, p.2, assumes an invertible, ergodic, measure-distal system on a Lebesgue probability space. It concludes almost-everywhere convergence for every fixed finite order, without requiring a product limit. The nonergodic extension is a separate point to justify in B.

Jamneshan, An uncountable Furstenberg–Zimmer structure theory, arXiv:2103.17167v4, Theorem 4.1(ii)', p.24, characterizes relatively compact extensions through invariant finitely generated $L^\infty$-modules. Its statement takes the $L^2$ closure after the union of these modules; individual modules need not already be closed. This formulation has been checked here, without treating that paper or all structural consequences used in B as independently proved in these notes.