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A countable algebra of sets generating the intended sigma-algebra. Standard Borel spaces admit such algebras by taking finite Boolean combinations from a countable topological base. Finite linear combinations of its indicators are dense in $L^2$ for every Borel probability measure, by the Monotone class theorem. Countability permits a common full-measure convergence set for all test indicators.

The Conditional Fatou lemma gives $X_n\ge \mathbb{E}[X_{\infty }\mid {\mathcal{F}}_n]$ for the dyadic density martingale. Integration first on dyadic sets, then extension by the Monotone class theorem, makes $\nu =\mu -X_{\infty }\lambda$ a nonnegative measure. Its finite-level density is $Y_n=X_n-\mathbb{E}[X_{\infty }\mid {\mathcal{F}}_n]$, which tends to zero almost everywhere for Lebesgue measure. The density level sets separate a singular measure argument shows that this zero-limit set has $\nu$-measure zero. Thus $\nu$ and $\lambda$ are mutually singular measures, obtaining the decomposition without assuming it to identify the martingale limit.

Assume the uniform estimate obtained in the geometric setting of part (i). The limit $g$ is bounded and continuous on the finite-alphabet symbolic space, so $p(x)=e^{g(E(x))}$ is measurable, positive, and bounded. Possible duplicate itineraries at partition endpoints do not affect its Lebesgue measure density.

For a level-$n$ interval ${\Delta }_w$, the estimate givesFix a level-$m$ cylinder $C_v$ and subdivide its interval into level-$n$ intervals, with $n\ge m$. Integrating these inequalities and summing the subcylinder masses yieldsLetting $n\to \infty$ proves the exact equality ${\int }_{{\Delta }_v}pdx=\nu (C_v)$. The same argument on the entire interval gives $\int p=1$. Shrinking cylinders generate the interval Borel sigma-algebra, so the Monotone class theorem identifies $p(x)dx$ with the pushforward ${\pi }_{\ast }\nu$.

The symbolic Gibbs measure in this setting is shift invariant, and coding satisfies $T\circ \pi =\pi \circ \sigma$ away from the endpoint exceptions. These exceptions form a null set: bounded $p$ and shrinking cylinders give zero mass to individual endpoints. Therefore, for every Borel set $B$,Consequentlyis an absolutely continuous invariant measure. This implication uses the uniform cylinder estimate directly and does not rely on naming an invariance theorem in place of the requested argument.

Put $Z_n=\mathbb{E}[X_{\infty }\mid {\mathcal{F}}_n]$. It is a conditional-expectation martingale. We justify why its limit is $X_{\infty }$, rather than just quoting convergence to an unspecified random variable.

First the family is uniformly integrable. For $A=\{Z_n>K\}\in {\mathcal{F}}_n$, nonnegativity and the conditional expectation identity giveThus its tail expectations are at most twice the integrable tail of $X_{\infty }$, uniformly in $n$. The uniformly integrable martingale convergence theorem gives a limit $Z$ almost surely and in $L^1$. For any $A\in {\mathcal{F}}_m$ and $n\ge m$,Pass to the $L^1$ limit and extend from $\mathcal{A}$ to all Borel sets by the Monotone class theorem. The equality of these integrals forces $Z=X_{\infty }$ almost surely. Combining this with the preceding convergence of $X_n$ proves

Write $\eta (A)={\int }_A{X}_{\infty }d\lambda$. This is a finite positive measure. For $A\in {\mathcal{F}}_n$, the conditional expectation inequality givesThe union $\mathcal{A}={\bigcup }_n{\mathcal{F}}_n$ is an algebra of sets generating $\mathcal{B}([0,1))$, since half-open dyadic intervals generate all open sets and hence all Borel sets. The class $\{A:\eta (A)\le \mu (A)\}$ is closed under increasing and decreasing limits, by continuity of these finite measures. The Monotone class theorem extends the inequality from $\mathcal{A}$ to every Borel set. Approximation by nonnegative simple functions and monotone convergence then giveThe difference $\nu =\mu -\eta$ is therefore a finite nonnegative measure, not merely a signed measure. For $A\in {\mathcal{F}}_n$,Consequently its restricted Radon-Nikodym derivative isChoose the cell-average versions of all the conditional expectations. Then $Y_n$ is constant on each level-$n$ cell and equals $2^n\nu (D_k^n)$ there, so these versions are nonnegative at every point.

Write the residual process asFor $s<t<1$, the filtration tells us whether each $U_i$ has arrived by time $s$, and its exact arrival time if it has. On $\{U_i>s\}$, its conditional law is uniform on $(s,1)$. Independence of the original $U_i$ means that the histories of the other arrivals do not alter this law. HenceThis conditional identity can also be checked against products of bounded functions of the censored arrival times and extended to ${\mathcal{F}}_s$ by the Monotone class theorem. Summing the identity gives $\mathbb{E}[M_t\mid {\mathcal{F}}_s]=M_s$.

Each $M_t$ is adapted and integrable, withIts paths are smooth between the finitely many arrival times and have downward jumps of size ${\theta }_i/(1-U_i)$ at arrivals before time one. The indicators use $U_i>t$, so the value at a jump is the post-jump value: paths are right continuous with finite left limits on $[0,1)$. Thus $M$ is a càdlàg martingale, specifically the uniform-arrival residual mass martingale.

The sampling rule in the PDF is with replacement. Thus conditional on the current proportion $y=j/n$, the next count is $W\sim \mathrm{Bin}(n,y)$. Put $\Delta =W/n-y$ and $q=y(1-y)$. Direct binomial moments giveThe fourth centered moment of a binomial distribution implies $\mathbb{E}|\Delta {|}^3\le (\mathbb{E}{\Delta }^2\mathbb{E}{\Delta }^4{)}^{1/2}\le C{n}^{-3/2}$. The accelerated generator of the Wright–Fisher binomial sampling chain isFor $f\in C^3([0,1])$, Taylor's formula uniformly yieldsThe initial condition is $Z_0^n=\lfloor xn\rfloor$, so $X_0^n\to x$. The endpoints are absorbing.

We give the path-space argument, rather than only a formal generator calculation. Regard $X^n$ as a càdlàg process in $D([0,1],[0,1])$ with the Skorokhod $J_1$ topology. The chain proportion is a bounded martingale, with predictable square compensator $V_t^n=n^{-1}{\sum }_{k<\lfloor nt\rfloor }{X}_{k/n}^n(1-X_{k/n}^n)$. For bounded stopping times ${\tau }_n\le {\sigma }_n\le {\tau }_n+{\delta }_n$, optional sampling gives $\mathbb{E}|X_{{\sigma }_n}^n-X_{{\tau }_n}^n{|}^2=\mathbb{E}(V_{{\sigma }_n}^n-V_{{\tau }_n}^n)\le ({\delta }_n+n^{-1})/4$. This bound uses the deterministic number of grid jumps in an interval of length at most ${\delta }_n$. Compact containment is automatic. The Aldous tightness criterion therefore proves tightness.

Furthermore, the fourth-moment bound and a union bound giveEvery subsequential limit is continuous. For each smooth $f$, the discrete compensated processis a martingale. Generator convergence and continuity of any limiting path turn the sum into ${\int }_0^t\frac{1}{2}{X}_s(1-X_s)f^{\prime \prime }(X_s)ds$. These compensated processes are uniformly bounded on $[0,1]$, so their martingale identities pass to the limit against bounded continuous functions of past coordinates. A Monotone class theorem argument extends the identities to the natural filtration. Smooth approximation in the $C^2$ norm extends the test class. Thus every limit solves the martingale problem for $Lf(y)=\frac{1}{2}y(1-y)f^{\prime \prime }(y)$ on $[0,1]$.

To identify this problem with the stated SDE, tests equal to $y$ and $y^2$ on $[0,1]$ (using bounded smooth extensions) give a continuous martingale with bracket ${\int }_0^t{X}_s(1-X_s)ds$. On an enlargement carrying independent Brownian motion $W$, setIts bracket is $t$, and the Lévy characterization of Brownian motion makes it Brownian. The zero-coefficient portion of $dX$ has zero bracket, so $d{X}_t=\sqrt{X_t(1-X_t)}d{B}_t$. Conversely Itô's formula shows that any such SDE solution solves the limiting martingale problem. The allowed uniqueness of the SDE, understood at least as uniqueness in law, therefore makes every subsequential limit have the same law. ConsequentlyThe limit is the Wright–Fisher diffusion. Linear interpolation has the same weak limit in the uniform topology because the maximum jump tends to zero. The results applied are Aldous tightness, the vanishing-jump continuous-limit criterion, passage of uniformly bounded martingale identities under weak convergence, Lévy characterization, and uniqueness in law; the estimates above verify their hypotheses.

Use a standard Borel probability space so that conditional measures can be realized by measures. The disintegration theorem for a probability measure states the following. For a measurable map $\pi :X\to Y$ between standard Borel spaces and $\nu ={\pi }_{\ast }\mu$, there is a measurable probability kernel $y\mapsto {\mu }_y$, unique for $\nu$-almost every $y$, such thatMeasurability means $y\mapsto {\mu }_y(A)$ is measurable for every Borel set $A$. More generallyfor integrable $f$, with equality of the last expression almost everywhere. The ${\mu }_y$ are the conditional measures of a factor. This also defines conditional measures for a countably generated sub-sigma-algebra, or a measurable partition represented by such a map. Sigma-algebras are considered modulo null sets where necessary.

For an invariant probability measure, disintegrate over the invariant sigma-algebra $\mathcal{I}$. On a standard probability space it can be represented, modulo null sets, by a countably generated measurable factor $\pi$. One justification for this countability is the separability of $L^2$: a countable dense family of $\mathcal{I}$-measurable functions generates $\mathcal{I}$ modulo null sets. The resulting conditional measures are the ergodic components:Their invariance is one of the permitted facts. It remains to prove that almost every component is ergodic, rather than simply invariant.

Choose a countable generating algebra $\mathcal{A}$ generating the Borel sigma-algebra of $X$. Apply the Birkhoff ergodic theorem simultaneously to its indicators. For $\mu$-almost every $z$ and all $A\in \mathcal{A}$,Disintegration transfers this common full-measure set to ${\mu }_y$-almost every $z$, for $\nu$-almost every $y$. On that fiber the conditional expectation is constant and equals ${\mu }_y(A)$. Thus, simultaneously for every $A\in \mathcal{A}$,Fix such a $y$, for which ${\mu }_y$ is also invariant. Applying the Birkhoff ergodic theorem to the system with measure ${\mu }_y$ identifies these limits withFinite linear combinations of the algebra's indicators are dense in $L^2({\mu }_y)$, by the Monotone class theorem. Since conditional expectation is an $L^2$ contraction, its projection onto invariant functions consequently sends every $L^2({\mu }_y)$ function to its constant integral. In particular, for any ${\mu }_y$-invariant set $D$,and so ${\mu }_y(D)\in \{0,1\}$. This proves almost every component is ergodic. The countable-test proof of ergodicity of conditional components avoids taking an invalid intersection of uncountably many full-measure sets.

For rotation by $1/5$, take the quotient map $\pi (x)=5x(\mathrm{mod}1)$. Its fibers are exactly the five-point orbits. The ergodic components of a rational circle rotation are thereforeEquivalently the component through $x$ is $\frac{1}{5}{\sum }_{j=0}^4{\delta }_{x+j/5}$. To check the disintegration, for bounded measurable $f$ use the change of variables on the five consecutive fifth-intervals:The rotation cyclically permutes the five atoms, preserving their equal masses. A set invariant modulo ${\mu }_y$ must contain all or none of this finite orbit, since every atom has positive mass. Thus each ${\mu }_y$ is ergodic. Finally, the invariant sigma-algebra is the pullback under $\pi$: a function invariant under translation by $1/5$ is constant on each orbit and is a measurable function of $5x(\mathrm{mod}1)$. This confirms that these are the conditional measures over $\mathcal{I}$, not merely some decomposition into invariant measures.

The variance at zero is zero, so $X_0=0$ almost surely. For $s<t$, the increment is Gaussian with mean zero and varianceFor every $u\le s$,The increment and any finite vector of past values are jointly Gaussian. Zero cross-covariances imply their independence, by the factorization of the joint Gaussian characteristic function. The Monotone class theorem extends this independence to the sigma-field generated by all past values, ${\mathcal{F}}_s^X=\sigma (X_u:u\le s)$.

Together with the assumed continuity, these proveThis is the Gaussian-process characterization of Brownian motion. Completion of the natural filtration preserves the independence assertion.

Use the standard convention that the finite-variation process $X$ has càdlàg paths and that the stochastic integral starts from zero. Write $V(X)$ for its total-variation process. The necessary admissibility condition is ${\int }_{(0,t]}|H_s|d{V}_s(X)<\infty$ almost surely for each finite $t$; it holds, for example, for locally bounded $H$. Define the pathwise Lebesgue-Stieltjes integralAn atom at $s>0$ contributes $H_s\Delta X_s$, so this convention also fixes the treatment of jumps.

Fix $t$. For a generator $H_s=1_A{1}_{(a,b]}(s)$ of the predictable sigma-algebra, with $A\in {\mathcal{F}}_a$, the Lebesgue-Stieltjes integral isIt vanishes when $t\le a$ and is ${\mathcal{F}}_t$-measurable otherwise. The total-variation process is adapted: for a càdlàg path its value at $t$ is the supremum over a countable collection of partitions of an interval using rational interior points and endpoint $t$. Thus the positive and negative Lebesgue-Stieltjes measures $d{X}^{\pm }=(dV(X)\pm dX)/2$ have ${\mathcal{F}}_t$-measurable masses on the generator intervals.

The Monotone class theorem, applied separately to these positive random measures on $[0,t]$, extends this measurability from the generators of the predictable sigma-algebra to all nonnegative predictable processes. Truncation and positive/negative decomposition then cover every admissible $H$. Consequently $H\cdot X$ is adapted.

The integrability condition is essential. Taken literally, an arbitrary previsible process cannot always be integrated against a finite-variation process: with $X_t=t$ and deterministic $H_s=s^{-1}$ for $s>0$, $H_0=0$, the proposed value at every $t>0$ diverges. The usual càdlàg convention is also what identifies $dX$ by $dX((a,b])=X_b-X_a$.

A simple predictable process has the formwhere $0=t_0<\cdots <t_m<\infty$, $H_0$ is bounded and ${\mathcal{F}}_0$-measurable, and each $H_k$ is bounded and ${\mathcal{F}}_{t_k}$-measurable. Finite linear combinations of indicators of predictable rectangles are of this form after refining the finitely many time partitions.

Those rectangles form a semiring of sets generating $\mathcal{P}$. The collection of sets whose indicators can be approximated in $L^2(\nu )$ by simple predictable processes is a monotone class: for an increasing sequence, truncate the union and use the finiteness of $\nu$; complements and finite disjoint unions are handled by linearity. The Monotone class theorem therefore puts every $\mathcal{P}$-measurable indicator in the closure. Ordinary measurable simple functions are dense in $L^2(\mathcal{P},\nu )$, so simple predictable processes are dense there as well.

Fix $0\le r\le t$. Since $M$ is a martingale,Every finite vector consisting of $M_{t+s}-M_t$ and past values $M_{r_1},\dots ,M_{r_n}$ has a multivariate normal distribution. Therefore uncorrelated jointly normal variables are independent, so the increment is independent of every finite vector of past values. A Monotone class theorem then extends this to independence from ${\mathcal{F}}_t=\sigma (M_r:0\le r\le t)$. This is the independent increments of a Gaussian martingale.