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For $T_n=T\wedge n$, Itô formula and the differential equation show thatup to $T$. After localization this is a martingale, and boundedness of $V$ permits optional stopping. Thus, conditionally on $X_0$,On $\{T<\infty \}$, path continuity gives $X_T\in \{\ell ,r\}$ and hence $V(X_T)=1$. On $\{T=\infty \}$, boundedness of $V$ makes $e^{-\lambda n}V(X_n)\to 0$. The dominated convergence theorem therefore yieldson $\{\ell <X_0<r\}$, with the stated convention when $T=\infty$.

The total variation distance isThe problem is a well-posed Bayesian inverse problem in total variation when every $v\in {\mathbb{R}}^N$ determines a unique posterior ${\mu }^v$ and

The heat solution operator at positive time is bounded from $L^2[0,1]$ to $C[0,1]$, so the finite sensor map $G:X\to {\mathbb{R}}^N$ is bounded and continuous. Therefore $\Phi (u;v)$ is jointly continuous and $0<e^{-\Phi }\le 1$. The normalizer satisfies $0<Z(v)\le 1$. If $v_n\to v$, the dominated convergence theorem gives bothand convergence in $L^1({\mu }_0)$ of the normalized posterior densities. Since total variation is one half of this $L^1$ distance for absolutely continuous measures, $d_{\mathrm{TV}}({\mu }^{v_n},{\mu }^v)\to 0$. Existence, uniqueness, and continuous dependence all follow.

Let $({\sigma }_j,u_j,v_j)$ be a singular system for $A$. The Picard criterion for $f\in \mathrm{dom}(A^{†})$ isafter discarding the component in $(\mathrm{ran}A{)}^{\perp }$. For $0<\tau <\Vert A{\Vert }^{-2}$, the partial series acts diagonally:For each $j$ the multiplier in the numerator tends to one and lies in $[0,1]$. The Picard summability condition therefore supplies an ${\ell }^2$ dominating sequence, so the dominated convergence theorem givesThis is the series form of Landweber iteration.

Change variables $x={\lambda }_{n}y$ and use the profiles from part 5:Since $v_n\to Q{e}^{i\gamma }$ in $L^2$, the difference made by replacing $|v_n{|}^2$ with $|Q{|}^2$ tends to zero against the bounded function $\varphi ({\lambda }_n\cdot )$. For each fixed $y$, continuity gives $\varphi ({\lambda }_{n}y)\to \varphi (0)$, and dominated convergence then givesThus the mass measures $|u(t_n,x){|}^{2}dx$ converge weakly to the Dirac delta function $\Vert Q{\Vert }_2^2{\delta }_0$.

For $u\in C^2(\overline{D})$, Itô formula shows thatis a martingale. Take expectations and let $t\to \infty$. The function $u$ is bounded on the compact set $\overline{D}$, while $\Delta u$ is bounded and ${\mathbb{E}}_{x}T<\infty$ by part (a). The dominated convergence theorem therefore gives Dynkin formula for Brownian motion

Apply part (b) to $u_n$. Its zero boundary value givesThe assumptions say that $\Delta u_n(y)\to 0$ for every $y\in D$ and that $|\Delta u_n|\le C$ uniformly. Thus the integrand converges pointwise to zero and is dominated by $CT$, whose expectation is finite by part (a). The dominated convergence theorem gives $u_n(x)\to 0$ for every $x\in D$.

Use the continuous semimartingale decomposition $X=X_0+M+A$, where $M$ is a continuous local martingale and $A$ is a continuous adapted finite-variation process. Pointwise limits preserve predictability, so $H$ is predictable; it is bounded by the common bound for the $H_n$.

Localize so that $[M{]}_{\infty }$ and the total variation $V(A{)}_{\infty }$ are bounded. The Doob L2 maximal inequality and the Itô isometry giveby the dominated convergence theorem. For the finite-variation part,almost surely, again by dominated convergence, now for each sample path. Hence the two integrals converge uniformly in probability after every localization. Part (b) removes the localization and provesu.c.p.

A measurable-function class is pointwise separable if it has a countable subclass such that every member is the pointwise limit of a sequence from that subclass. With an integrable common envelope, dominated convergence theorem makes both empirical averages and population expectations converge along each such sequence. Thus the supremum of the absolute empirical errors equals the supremum over the countable subclass and is measurable. A finite integrable bracketing cover provides an integrable envelope by taking the maximum absolute value of all its endpoints.

A bounded continuous factor $k_1$ and an integrable factor $k_2$ give a bounded velocity-integral gain on bounded continuous functions, with bound $\Vert Kf{\Vert }_{\infty }\le \Vert k_1{\Vert }_{\infty }\Vert k_2{\Vert }_1\Vert f{\Vert }_{\infty }$. The dominated convergence theorem proves joint continuity of the velocity integral. Combined with damped transport, the factorial bound for a Volterra iterate proves existence on arbitrary finite time slabs.

If a negative differentiable concave utility function satisfies $\mathbb{E}U(\theta X)>-\infty$ on an open interval, outer secants at four points bound the sample derivative on a smaller compact interval. Negativity makes the endpoint values absolutely integrable. The dominated convergence theorem then gives $d\mathbb{E}U(\theta X)/d\theta =\mathbb{E}[X{U}^{\prime }(\theta X)]$ and continuous derivative when $U^{\prime }$ is continuous. This is a justification of differentiation, rather than an assumption of integrability of marginal utility.

If every eigenvalue of a positive definite symmetric matrix $A$ tends to zero, then the anisotropic theta functional equation and the dominated convergence theorem show that $(\det A{)}^{1/2}{\theta }_{\Lambda }(A)\to \mathrm{covol}(\Lambda {)}^{-1}$. Indeed all eigenvalues of $A^{-1}$ tend to infinity, the zero term of the dual Gaussian theta sum is one, and its remaining terms tend to zero while being dominated by a fixed summable Gaussian. Merely requiring $\det A\to 0$ is insufficient.

For a time-independent strictly positive Dirichlet realization of an elliptic operator and $\psi \in L^2(U)$, the eigenfunction expansionsolves $u_t+Lu=0$ with homogeneous Dirichlet boundary conditions. For $t\ge \epsilon >0$, every power ${\lambda }_m^N$ times the exponential is bounded, so the series and all its time derivatives converge in all the Sobolev domains of powers of an elliptic Dirichlet operator. Elliptic regularity and the Sobolev embedding theorem give smoothness up to the spatial boundary for positive time. Parseval identity and the dominated convergence theorem give $u(t)\to \psi$ in $L^2$ as $t\downarrow 0$.

For nonzero compact operator $A$, step $0<\tau <2/\Vert A{\Vert }^2$, and data admitting a minimum-norm least-squares solution $x^{†}$, zero-initialized Landweber iteration satisfiesThe Picard criterion makes the majorant summable, so the dominated convergence theorem proves this limit. No mesh-independent or singular-value-independent strict contraction factor is required. At $\tau =2/\Vert A{\Vert }^2$, the top singular mode has multiplier $-1$ and may not converge. Without solvable projected data, the iterates need not have a limit in the solution Hilbert space, even when their residuals approach the least-squares infimum.

For an L-diffusion with bounded coefficients and $f\in C_b^{1,2}$, $M^f$ is a continuous martingale. Freeze the time argument along a deterministic partition, apply the spatial martingale problem on each interval, and pass to the limit by the dominated convergence theorem. The same proof works if the absolute drift coefficients and the trace of the diffusivity have integrable time integrals along the path on every finite horizon. Local hypotheses alone give a local martingale conclusion.

If $X\ge 0$ and $\mathbb{E}{e}^{tX}<\infty$ for some finite $t\ge 0$, the estimator ${\hat{m}}_n(s)=n^{-1}{\sum }_i{e}^{s{X}_i}$ satisfies ${\sup }_{s\in [0,t]}|{\hat{m}}_n(s)-\mathbb{E}{e}^{sX}|\to 0$ almost surely. The dominated convergence theorem makes the population moment-generating function uniformly continuous on this interval. Parameter subintervals give function brackets whose endpoints are ordered exponentials and whose widths are the corresponding increments of the population mean. Apply the uniform strong law from finite L1 bracketing. Continuity in the parameter makes the supremum measurable.

If $\Theta$ is compact, $q(\theta ,x)$ is continuous in $\theta$ and measurable in $x$, and $\mathbb{E}{\sup }_{\Theta }|q(\theta ,X)|<\infty$, then an independent and identically distributed sample satisfies ${\sup }_{\Theta }|P_n{q}_{\theta }-P{q}_{\theta }|\to 0$ almost surely. Uniform continuity for each $x$, the dominated convergence theorem, and finite parameter nets produce arbitrarily narrow function brackets.

Weak convergence to zero gives pointwise convergence by the continuous evaluation functionals and uniform boundedness in the supremum norm by the weakly bounded set criterion. The dominated convergence theorem for Lebesgue measure then proves the displayed implication. The finite measure of the interval and uniform norm bound are the relevant hypotheses.