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An $L^2$ observable of an invertible probability measure-preserving system is almost periodic when its Koopman operator orbit is totally bounded in the global $L^2$ norm. A noninvertible system uses its forward orbit. This is the Hilbert-space dynamical definition, distinguished from uniform almost periodicity for continuous functions of a real variable.

A probability measure-preserving system is compact if every $L^2$ observable has a relatively compact Koopman operator orbit in the global $L^2$ norm. For an invertible system one may use all integer times; for a general transformation use nonnegative times. This is a measure-theoretic definition, stronger than compactness of the underlying topological space.

An invertible measure-preserving system is ergodic exactly when the fixed space of its Koopman operator on $L^2$ consists only of almost-everywhere constant functions. One direction applies invariance to level sets; the other applies it to indicator functions of invariant sets.

An invertible measure-preserving system has a measurable inverse modulo null sets and preserves its measure in both time directions. Its Koopman operator on $L^2$ is unitary.

For irrational $\alpha$, the irrational skew shift is the skew product $T_{\alpha }(x,y)=(x+\alpha ,y+x)$ on the torus $(\mathbb{R}/\mathbb{Z}{)}^2$. It preserves normalized Lebesgue measure and has iterates $T_{\alpha }^n(x,y)=(x+n\alpha ,y+nx+\alpha n(n-1)/2)$ modulo one. Its Koopman operator sends the Fourier basis character $e_{r,s}=e^{2\pi i(rx+sy)}$ to $e^{2\pi ir\alpha }{e}_{r+s,s}$. The resulting infinite chains of Fourier coefficients show that it is an ergodic transformation. Moreover, uniform equidistribution of an irrational skew shift proves unique ergodicity.

A weakly mixing system has no nonconstant Koopman operator eigenfunction. If $U_{T}f=\lambda f$, then $f(x)\overline{f(y)}$ is invariant under $T\times T$, contradicting the product characterization of weak mixing unless $f$ is constant.

The discrete Bochner–Herglotz theorem says that a sequence $r:\mathbb{Z}\to \mathbb{C}$ is a positive-definite function, meaningif and only if there is a unique finite positive Borel measure $\sigma$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ such thatThe mass is $\sigma (\mathbb{T})=r(0)$. The necessity of positive definiteness follows by integrating $|{\sum }_j{c}_j{e}^{2\pi ijt}{|}^2$.

For sufficiency, positive definiteness makes the relevant Gram matrices Hermitian positive semidefinite, so $r(-n)=\overline{r(n)}$ and $|r(n)|\le r(0)$. DefineTaking $c_j=e^{-2\pi ijt}$ in the positive-definiteness condition gives $p_N(t)\ge 0$, while ${\int }_{\mathbb{T}}{p}_N(t)dt=r(0)$. Moreover,Thus the integrals against every trigonometric polynomial converge. These polynomials are uniformly dense in continuous functions on the circle: smooth periodic approximation followed by Fourier inversion for smooth functions gives this density. The uniform mass bound $r(0)$ then makes the integrals against every continuous $\phi$ converge as well. Their limit is a positive linear functional $L$ with $|L(\phi )|\le r(0)\Vert \phi {\Vert }_{\infty }$. The standard Riesz-Markov-Kakutani representation theorem gives a finite positive Borel measure $\sigma$ with $L(\phi )=\int \phi d\sigma$. The preceding coefficient computation proves its required Fourier coefficients. Finally, two such measures agree on all trigonometric polynomials, hence on all continuous functions by uniform density, and therefore are identical. This proves the Bochner–Herglotz theorem, including existence and uniqueness; if $r(0)=0$, the estimate $|r(n)|\le r(0)$ gives the zero measure directly.

Now let $U$ be the Koopman operator $Uf=f\circ T$ on the complex L2 space of the probability measure-preserving system. We use $⟨f,g⟩=\int f\overline{g}d\mu$, linear in the first entry. Since $U$ is an isometry, the sequence $r_f(n)=⟨U^{n}f,f⟩$ for $n\ge 0$, extended by $r_f(-n)=\overline{r_f(n)}$, is positive-definite: its matrix is the Gram matrix of $f,Uf,\dots$. The Bochner–Herglotz theorem gives its spectral measure of a Koopman observable ${\sigma }_f$.

We first prove exactly how an atom of a measure detects an eigenfunction. Fix $\zeta =e^{2\pi i{t}_0}$ and setThe mean ergodic theorem, applied to the isometry $V={\zeta }^{-1}U$, says these averages converge in L2 space to the orthogonal projection $P_{\zeta }f$ onto $\ker (U-\zeta I)$. Its usual Hilbert-space proof works without invertibility: the orthogonal complement of $\overline{\mathrm{ran}(I-V)}$ is $\ker (I-V^{\ast })=\ker (I-V)$, and the averages of $(I-V)h$ telescope to $(h-V^{N}h)/N\to 0$. The equality of the two kernels follows from $V$ being an isometry, by expanding $\Vert Vh-h{\Vert }^2$ when $V^{\ast }h=h$.

Expanding the averaged norm through the spectral measure of a Koopman observable givesThe integrand is at most one and converges to $1_{\{t_0\}}$. By the dominated convergence theorem,Thus a nonzero atom corresponds to a nonzero Koopman operator eigenfunction, with eigenvalue $\zeta$, in the cyclic spectral data of $f$.

We also need the Wiener Fourier coefficient lemma, which here has a short proof. For any finite positive circle measure $\sigma$ with coefficients $r(n)$,The bracket is bounded in absolute value by one and tends to the indicator of the diagonal $t=s$. Another application of the dominated convergence theorem givesIn particular the limit is zero for an atomless measure.

Suppose there are no nonconstant Koopman operator eigenfunctions. For every $f$ in the mean-zero L2 space, all its projections $P_{\zeta }f$ vanish: a nonzero projection would be a nonconstant eigenfunction, since it has mean zero. Hence ${\sigma }_f$ is atomless, and the Wiener Fourier coefficient lemma givesThe Cauchy-Schwarz inequality also gives vanishing averaged absolute autocorrelations. For two mean-zero observables $f,g$, the sesquilinear polarization identitythen gives vanishing averaged absolute cross-correlations. Taking $f=1_A-\mu (A)$ and $g=1_B-\mu (B)$ proveswhich is weak mixing.

Conversely, this set formulation of weak mixing extends to all pairs of mean-zero L2 space observables by finite-simple-function approximation and the uniform estimate $|⟨U^{n}f,g⟩|\le \Vert f{\Vert }_2\Vert g{\Vert }_2$. If $Uh=\zeta h$ and $h\ne 0$, the isometry property gives $|\zeta |=1$. If $\zeta \ne 1$, invariance of the integral gives $\int h=0$. If $\zeta =1$ and $h$ is nonconstant, replace it by $h-\int h$, a nonzero mean-zero eigenfunction. But in either casefor every $n$, contradicting weak mixing. We have proved weak mixing is equivalent to absence of nonconstant eigenfunctions, including for noninvertible transformations.

Finally, the doubling map $Tx=2x(\mathrm{mod}1)$ preserves normalized Lebesgue measure, as substitution on its two inverse branches shows. Let $f\in L^2(\mathbb{T})$ be an eigenfunction, $f(2x)=\zeta f(x)$, with Fourier coefficients $c_k$. The Fourier basis calculation givesSince $\zeta \ne 0$, all odd coefficients vanish. Every nonzero integer is $2^{r}m$ with $m$ odd, and repeated use of $\zeta c_{2^{r}m}=c_{2^{r-1}m}$ makes all nonzero coefficients vanish. Completeness of the Fourier basis then says $f$ is constant. Therefore the doubling map has no nonconstant eigenfunctions, and

Use the convention that level $k$ counts the $k$ nonzero shifts, hence an arithmetic progression of length $k+1$. The SZ property at level $k$ means that every measurable set $A$ of positive measure satisfiesThis is the averaged multiple-recurrence property, stronger than merely finding one positive intersection. Shifting the indexing convention from $k+1$ terms to $k$ terms does not change the all-levels assertion. We work with probability measure-preserving systems.

Here is the Furstenberg correspondence principle construction proving the Szemerédi theorem. Let $D\subseteq \mathbb{N}$ have positive upper density $\delta ={\mathrm{limsup}}_N|D\cap \{1,\dots ,N\}|/N$. Put $z\in \{0,1{\}}^{\mathbb{Z}}$ equal to the indicator of $D$ on the positive coordinates and zero elsewhere, and let $S$ be the left shift. The binary full shift $\Omega =\{0,1{\}}^{\mathbb{Z}}$ is a compact metric space in the product topology, and $S$ is a homeomorphism. Choose $N_i\to \infty$ attaining the upper density and define the empirical measuresWe use the standard weak compactness fact that every sequence of Borel probability measures on a compact metric space has a weakly convergent subsequence, where weak convergence means convergence of integrals of continuous functions. Pass to a subsequence with limit $\mu$. For every continuous $F$, telescoping givesPassing to the limit proves $S_{\ast }\mu =\mu$. The cylinder set $A=\{w:w(0)=1\}$ is clopen, so its indicator is continuous andIf every measure-preserving system has the SZ property at level $k$, there is some $n\ge 1$ for which $\mu (C_n)>0$, where $C_n={\bigcap }_{j=0}^k{S}^{-jn}A$. This is also a clopen cylinder set, so ${\mu }_i(C_n)\to \mu (C_n)>0$. For some sufficiently large $i$, at least one term of the empirical average is positive: $S^{m}z\in C_n$ for some $1\le m\le N_i$. Unwinding the coordinates givesTherefore $D$ contains an arithmetic progression of length $k+1$ with positive common difference. Having the SZ property at every level gives the infinite-density form of the Szemerédi theorem.

The finite uniform form follows from the same argument. If, for fixed $\delta >0$ and $k$, arbitrarily large $N_i$ admitted sets $D_i\subseteq \{1,\dots ,N_i\}$ with $|D_i|\ge \delta N_i$ and no such arithmetic progression, replace $z$ above by the indicator $z_i$ of $D_i$, extended by zero outside that interval. Their empirical measures still have an invariant weak limit with $\mu (A)\ge \delta$. For every fixed positive $n$, however, ${\mu }_i(C_n)=0$, since a point contributing to that cylinder would be an actual progression in $D_i$. Thus $\mu (C_n)=0$ for all positive $n$, contradicting the SZ property. Consequently a sufficiently large $N$ depending only on $\delta ,k$ forces the required progression.

A compact measure-preserving system is one for which every observable $f\in L^2(\mu )$ has relatively compact forward orbit $\{U^{n}f:n\ge 0\}$ under the Koopman operator $Uf=f\circ T$. Equivalently, each of these orbits is totally bounded in the global $L^2$ norm; compactness refers to observables, not just to the underlying topological space. For an invertible system it is usual to use all integer times. One may also define compactness by density of almost periodic observables: the two definitions coincide because $\Vert U^n(f-g){\Vert }_2=\Vert f-g{\Vert }_2$, so a finite net for an approximating observable is a finite net, with slightly larger radius, for $f$.

To prove the SZ property, fix $A$ with $a=\mu (A)>0$, put $f=1_A$, and chooseCompactness gives a finite $\epsilon$-net for its orbit with centers $U^{r_1}f,\dots ,U^{r_s}f$, where $r_i\ge 0$. Let $M={\max }_i{r}_i$. For every integer $n\ge M+1$, some center satisfies $\Vert U^{n}f-U^{r_i}f{\Vert }_2<\epsilon$. Since $U$ is an isometry, cancellation givesThus the return set $R=\{r\ge 1:\Vert U^{r}f-f{\Vert }_2<\epsilon \}$ meets every interval $\{n-M,\dots ,n\}$ for $n\ge M+1$. In particular, $R$ is a syndetic set with lower density at least $1/(M+1)$. This proves the needed syndetic near returns of an almost periodic observable without any assumption of ergodicity.

For $r\in R$, telescoping and the isometry property giveAs $f$ is an indicator, $\Vert U^{jr}f-f{\Vert }_2^2=\mu (A△T^{-jr}A)$. The union bound therefore givesAll other multiple-intersection measures are nonnegative. Averaging over the bounded-gap set $R$ yieldsThus every compact measure-preserving system has the SZ property at every level, including nonergodic systems and, under the forward-orbit definition, noninvertible transformations.

Use probability measure-preserving systems: $T:X\to X$ is measurable and $\mu (T^{-1}A)=\mu (A)$. Its invariant sigma-algebra isErgodicity means that every invariant measurable set has measure zero or one:Equivalently, every invariant measurable function is constant almost everywhere. Indeed all rational sublevel sets of a real invariant function belong to $\mathcal{I}$, so the zero-or-one alternative forces its distribution to concentrate at one value. Apply this to real and imaginary parts for complex functions. Conversely, an invariant set has an invariant indicator, so the function criterion implies the set criterion. This is the invariant-function characterization of ergodicity.

For an irrational rotation of the circle, $T(x)=x+\alpha (\mathrm{mod}1)$ with $\alpha \notin \mathbb{Q}$, an invariant $g\in L^2(m)$ has Fourier coefficients satisfyingFor $k\ne 0$ the multiplier is nonzero. Completeness of the Fourier series basis therefore makes $g$ constant. Apply this to an invariant indicator to prove the irrational rotation is ergodic. A second example is the cyclic permutation of $q$ points with equal masses: a nonempty invariant set contains the whole cycle and hence has measure one.

The identity on $[0,1]$ with Lebesgue measure is not ergodic, since $[0,1/2]$ is invariant and has measure $1/2$. Nor is rotation by $1/5$ on the circle: the setis invariant and has measure $1/2$. These examples also show why preservation of measure does not imply ergodicity.

Let $U=U_T$ be the Koopman operator, $Uf=f\circ T$, and setThe Von Neumann mean ergodic theorem says that, for $f\in L^2(\mu )$, these Cesàro averages converge in $L^2$ to the orthogonal projection onto the invariant functions:For an invertible system, $U$ is a unitary operator. The Hilbert-space theorem states more generally that the Cesàro averages of any unitary operator converge strongly to its fixed-space projection.

Here is the requested invertible-case proof. Since $U^{\ast }=U^{-1}$,Thus the orthogonal decomposition for unitary ergodic averages isOn the first summand, $A_{N}f=f$. On a vector $(I-U)g$, telescoping givesBecause $\Vert A_N\Vert \le 1$, approximation extends this convergence to the closure of the range. The orthogonal decomposition now proves the theorem for every $f$. An invariant $L^2$ function is $\mathcal{I}$-measurable, and an $\mathcal{I}$-measurable function is invariant by approximation with indicators; hence the projection is the stated conditional expectation.

In an ergodic system, invariant functions are constants by the criterion already proved. Their orthogonal projection is obtained by taking the inner product with $1$, whose norm is one. Therefore the ergodic limit is the space average:

The Birkhoff ergodic theorem states that, for $f\in L^1(\mu )$,The limit is invariant, integrable, and has the same integral as $f$. On a probability space convergence also holds in $L^1$. In an ergodic system the limit is $\int fd\mu$ almost everywhere.

For the topological sharpening, take a continuous map $T$ on a compact metric space. Unique ergodicity means there is exactly one invariant Borel probability measure $\mu$. The uniform ergodic convergence for uniquely ergodic systems says that every continuous real or complex $f$ satisfiesThus convergence holds at every point and uniformly in that point. The continuity, compactness and continuous-observable hypotheses belong to this theorem; it is not a claim of uniform convergence for all measurable $L^1$ functions.

For an invertible probability measure-preserving system, an almost periodic observable is an $f\in L^2(\mu )$ whose Koopman operator orbit is totally bounded:Equivalently, for every $\epsilon >0$ it has a finite $\epsilon$-net in $L^2$. For a noninvertible system use the forward orbit $n\ge 0$.

A factor map $\phi :X\to Y$ satisfies ${\phi }_{\ast }\mu =\nu$ and $\phi T=S\phi$ almost everywhere. Disintegrate $\mu ={\int }_Y{\mu }_{y}d\nu (y)$ over this map and writefor the conditional L2 norm. A relatively almost periodic observable is an $f\in L^2(\mu )$ such that, for every $\epsilon >0$, there are finitely many $g_1,\dots ,g_r\in L^2(\mu )$ withThe center may depend on both $n$ and $y$; the finite list itself does not. Since $\mathbb{Z}$ is countable, the exceptional null sets can be combined. A compact extension of a measure-preserving system is a factor map for which these relatively almost periodic observables are dense in $L^2(\mu )$. This density definition should not be replaced by a claim that every $L^2$ observable already satisfies the uniform fiberwise finite-net condition.

For a proper factor map example, let $Y=\{-1,1{\}}^{\mathbb{Z}}$ carry independent fair coordinates and the two-sided Bernoulli shift $S$, with $(Sy{)}_j=y_{j+1}$. Setusing the product of the Bernoulli measure and the equal two-point measure. These are invertible measure-preserving systems, and the projection preserves measure and intertwines the transformations, so it is a two-to-one factor map.

Take $f(y,i)=y_0$. Then $U_T^{n}f(y,i)=y_n$. Distinct coordinates are independent and have mean zero and variance one, henceIts orbit has no finite sufficiently small net. Thus the Bernoulli coordinate observable is not almost periodic in global $L^2$. In contrast, the conditional measure over $y$ is the equal measure on $(y,0),(y,1)$, and $U_T^{n}f$ is constant on that fiber with value $y_n\in \{-1,1\}$. The two centers $g_1=1,g_2=-1$ give zero error on every fiber for every $n$. Therefore $f$ is a relatively almost periodic observable for this map.

The example is also a finite-fiber compact extension. For a bounded observable on $X$, each orbit value on a fiber is a pair in a fixed bounded subset of ${\mathbb{C}}^2$, which has a finite net. Its centers can be chosen as functions constant in $y$, with prescribed values on the two labels. Bounded observables are dense in $L^2$, so the extension is compact. The Bernoulli shift is mixing: cylinder events involving disjoint coordinate sets are independent for all sufficiently large shifts, and approximation by cylinder events extends this to arbitrary events. Thus $S$ and $S^2$ are ergodic. If an invariant function on $X$ is written $a(y)+(-1{)}^{i}b(y)$, invariance gives $a\circ S=a$ and $b\circ S=-b$. The first function is constant; the second is $S^2$-invariant and hence constant, and then its sign equation forces zero. This proves that the example's source is also ergodic.

For the positive-subset assertion, interpret the printed $X^{\prime }$ as $X$, or as an invariant full-measure domain of the factor map; the prime is otherwise undefined and does not change the measure-theoretic argument.

First establish conditional norm covariance under a factor map. Invariance of the measures and uniqueness in the disintegration theorem for a probability measure giveon common full-measure sets. For the first equality, $y\mapsto T_{\ast }{\mu }_{S^{-1}y}$ is another disintegration over the same fibers: it is supported on ${\phi }^{-1}\{y\}$ and integrates to $\mu$. Uniqueness identifies it with ${\mu }_y$, and iteration gives the norm formula.

Two elementary facts explain localization of relative almost periodicity to base sets. If $f$ is a relatively almost periodic observable and $C\subseteq Y$ is measurable, then $f{1}_C\circ \phi$ is also relatively almost periodic. Indeed, on a fiber,When that indicator is one, use a center for $U_T^{n}f$; when it is zero, use the additional center $0$. Secondly, relative almost periodicity is closed under convergence in the uniform conditional L2 normIf this norm of $f-f_j$ tends to zero, covariance bounds the corresponding error between every pair of orbit values by the same number. A finite $\epsilon /2$-net for one sufficiently close $f_j$ is consequently an $\epsilon$-net for $f$.

Now take $f=1_A$. The compact extension supplies relatively almost periodic observables $f_j$ withPut $e_j(y)=\Vert f_j-f{\Vert }_y$. The Tonelli theorem givesHence $e_j(y)\to 0$ for $\nu$-almost every $y$. Since ${\int }_Y{\mu }_y(A)d\nu =\mu (A)>0$, choose $\eta >0$ for whichhas positive measure. The Egorov theorem gives a measurable $C\subseteq E$ of positive measure on which $e_j\to 0$ uniformly. DefineThenFor $h_j=f_j{1}_C\circ \phi$, the localization fact makes every $h_j$ relatively almost periodic, whileThe conditional-norm closure fact therefore proves $1_B$ is a relatively almost periodic observable, with $B\subseteq A$ and positive measure. This is the positive-measure almost periodic indicator in a compact extension. No ergodic component or uncountable intersection of exceptional sets is needed in this construction; it in fact works under the same disintegration and compactness hypotheses without using the assumed ergodicity.

A continuous map $F$ of a compact metric space is uniquely ergodic if it has exactly one invariant measure that is a Borel probability measure. The uniqueness requirement ranges over all invariant Borel probability measures.

For an irrational rotation of the circle, let $R_{\alpha }x=x+\alpha$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, and let $\nu$ be any invariant Borel probability measure. For each integer $r$, set ${\chi }_r(x)=e^{2\pi irx}$. Invariance givesFor $r\ne 0$, irrationality forces $e^{2\pi ir\alpha }\ne 1$, so the corresponding Fourier coefficient is zero. For $r=0$ it is one. These are exactly the Fourier coefficients of normalized Lebesgue measure $m$. By the Stone-Weierstrass theorem, trigonometric polynomials are uniformly dense in the continuous functions on the circle group; hence $\nu$ and $m$ integrate every continuous function equally and are the same Borel probability measure. Since $m$ is invariant, the irrational rotation of the circle is uniquely ergodic, with unique measure $m$.

For the irrational skew shift on the two-dimensional torus, writeThe map is invertible, with $T^{-1}(x,y)=(x-\alpha ,y-x+\alpha )$ modulo one, and it preserves $m_2$ as allowed in the question. We first prove the ergodic transformation property by the invariant-function characterization of ergodicity.

Let $F\in L^2(m_2)$ satisfy $F\circ T=F$. Its expansion in the Fourier basis is $F={\sum }_{r,s}{c}_{r,s}{e}_{r,s}$ in $L^2$. Direct calculation of the Koopman operator givesUniqueness of the Fourier coefficients therefore impliesFor $s\ne 0$, the magnitudes of the Fourier coefficients along all distinct indices $(r+js,s)$, $j\in \mathbb{Z}$, are equal. By the Bessel inequality they are square summable, so every such coefficient must be zero. When $s=0$, the relation becomes $c_{r,0}=e^{2\pi ir\alpha }{c}_{r,0}$, which forces $c_{r,0}=0$ for $r\ne 0$. Only $c_{0,0}$ remains. Thus every invariant $L^2$ function is constant, and

To prove unique ergodicity, we will establish uniform averages for every continuous function directly. No theorem on unique ergodicity of skew products is needed. Induction on $n$, using the old first coordinate in the second coordinate of $T$, yieldsIn particular, for a Fourier basis element,If $s=0$ and $r\ne 0$, this is a geometric series in $n$, anduniformly in $(x,y)$.

For $s\ne 0$, we give the finite Van der Corput inequality for finite scalar sequences and its proof. For $|u_n|\le 1$, extend $u_n$ by zero outside $0\le n<N$. Fix an integer $1\le H\le N$ and set $C_h={\sum }_{n=0}^{N-h-1}{u}_{n+h}\overline{u_n}$. Every original summand appears $H$ times in the identityThe Cauchy-Schwarz inequality, followed by expansion of the squared window sums, givesConsequently the Van der Corput inequality for finite scalar sequences isThe finite prefactor is important; the order of limits will be $N\to \infty$ with $H$ fixed, followed by $H\to \infty$.

For the irrational skew shift character sequence above, differencing cancels the quadratic term:For every fixed $h\ge 1$, $s\alpha h$ is irrational, so another geometric series estimate givesuniformly in $(x,y)$. With fixed $H$, the Van der Corput inequality for finite scalar sequences therefore impliesLetting $H\to \infty$ proves that the averages of every nonconstant Fourier basis element converge uniformly to zero. The constant character has average one. This establishes uniform equidistribution of an irrational skew shift on all trigonometric polynomials.

The Stone-Weierstrass theorem makes these trigonometric polynomials uniformly dense in $C({\mathbb{T}}^2)$. If $p$ approximates a continuous $g$ with $\Vert g-p{\Vert }_{\infty }<\epsilon$, thenTaking $N\to \infty$ and then $\epsilon \downarrow 0$ proves uniform convergence to $\int gd{m}_2$ for every continuous $g$.

Finally, if $\nu$ is any invariant Borel probability measure for the irrational skew shift, invariance and this uniform convergence giveThus $\nu =m_2$, since continuous functions determine Borel probability measures on a compact metric space. We concludeThe uniform-average proof also shows that every starting point has the same limiting continuous-function averages, a stronger conclusion than the almost-everywhere assertion provided by the pointwise ergodic theorem.

For a probability measure-preserving system, strong mixing means that for every pair of measurable sets $A,B$,A sequence $a_n$ has convergence in density of a sequence to $a$ when for each $\epsilon >0$ the exceptional set $\{n:|a_n-a|\ge \epsilon \}$ has natural density zero. A weakly mixing measure-preserving transformation has convergence in density of $\mu (T^{-n}A\cap B)$ to $\mu (A)\mu (B)$ for every $A,B$. Equivalently, because these correlations are bounded,Indeed the mean of the absolute discrepancy is at least $\epsilon$ times the exceptional frequency, while it is at most $\epsilon$ plus a uniform bound times that frequency. A signed Cesaro convergence of a sequence without absolute values is insufficient to define weak mixing.

The standard three-cut Chacon map is obtained by cutting and stacking. Start on $[0,1)$ with normalized Lebesgue measure, an initial tower consisting of $[0,2/3)$, and a reservoir $[2/3,1)$ for spacers. At stage $j$, cut every level of the current tower into three equal subintervals, producing three subcolumns. Add one new interval of the same width above the middle subcolumn. Stack the first subcolumn at the bottom, then the middle subcolumn with its spacer, then the third at the top. Define the partial transformation by translation from each level to the next, leaving the current top unmapped. These assignments extend the earlier partial transformation. Repeat indefinitely.

If $h_j$ is the number of levels and $w_j$ their width, thenHence $h_j=(3^{j+1}-1)/2$, $w_j=2/3^{j+1}$, and the stage-$j$ tower has measure $1-3^{-j-1}$. The spacers consume total measure ${\sum }_{j\ge 0}{w}_{j+1}=1/3$, precisely the reservoir. The increasing partial maps give the Chacon map modulo null sets. The tower tops and unused reservoir have measures tending to zero, and the construction yields an invertible measure-preserving transformation. With $0$ marking an original level and $1$ a spacer, the tower words satisfy $W_0=0$ and $W_{j+1}=W_j{W}_{j}1W_j$, so the first new word is $0010$. This describes the spacer placement without needing a proof of well-definedness. The three-cut convention agrees with the classical constant spacer vector $(0,1,0)$ described in Ryzhikov's construction.

For the correlation assertion, take the L2 inner product to be $⟨u,v⟩=\int u\overline{v}d\mu$, linear in its first argument, and put $U=U_T$. The Koopman operator is an isometry on $L^2$, even when $T$ is not invertible. For $n\ge k$ the hint givesTo extend rigorously to all test functions, centre the observable: $v=f-a1$. Invariance of the integral gives $⟨U^{n}v,v⟩=⟨U^{n}f,f⟩-|a{|}^2\to 0$. LetFor each fixed $k$, $⟨U^{n}v,U^{k}v⟩\to 0$ by the same isometry identity. Therefore the limit is zero for every finite linear combination of these orbit vectors. By the Cauchy-Schwarz inequality and $\Vert U^{n}v{\Vert }_2=\Vert v{\Vert }_2$, approximation extends this conclusion to every $g\in M$: the approximation error in the correlation is bounded by $\Vert v{\Vert }_2\Vert g-g_0{\Vert }_2$ uniformly in $n$. For $g\in M^{\perp }$, the correlation is identically zero because $U^{n}v\in M$. The orthogonal decomposition by a closed subspace now gives the result for all $g\in L^2$. ThusThis is decay of autocorrelation implies weak convergence of an observable; no assumption that $T$ is an ergodic transformation, and no invertibility hypothesis was used, and $a=0$ is included.

Finally, strong mixing immediately implies the stated diagonal limit by taking $B=A$. Conversely, suppose that limit holds for every $A$. Apply the correlation result with $f=1_A$ and $g=1_B$. Its hypothesis is exactly $⟨U^n{1}_A,1_A⟩\to \mu (A{)}^2$, and its conclusion isHence strong mixing is equivalent to diagonal set-correlation criterion for mixing. Using the orbit-span proof avoids an invalid polarization argument that would recover only the sum of the two directed cross-correlations from diagonal correlations.

The statement $D-lim{a}_n=a$ means convergence in density of a sequence: for every $\epsilon >0$, the proportion of $n\le N$ with $|a_n-a|\ge \epsilon$ tends to zero. The statement $C-lim{a}_n=a$ means Cesaro convergence of a sequence, $N^{-1}{\sum }_{n=1}^N{a}_n\to a$.

Suppose $|a_n-a|\le M$. Splitting into the indices where $|a_n-a|<\epsilon$ and its complement givesThus density convergence implies Cesaro convergence of the absolute deviations to zero. Conversely, the Markov inequality givesproving the reverse implication.

The product characterization of weak mixing says that if $T$ is weakly mixing, then $T\times S$ is ergodic exactly when $S$ is ergodic; in particular, ergodicity of $T\times T$ characterizes weak mixing.

Suppose $U_{T}f=\lambda f$. Unitarity of the Koopman operator gives $|\lambda |=1$. Weak mixing implies ergodicity, so $|f|$ is almost everywhere constant. On the product,is invariant under $T\times T$. Since the square is ergodic, $F$ is constant, which forces $f$ to be constant almost everywhere. Thus there are no nonconstant Koopman eigenfunctions.