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On the convergence sets of operator sequences on spaces of homogeneous type G. A. Karagulyanab a Faculty of Mathematics and Mechanics, Yerevan State University, Yerevan, Republic of Armenia b Institute of Mathematics of National Academy of Sciences of RA, Yerevan, Republic of Armenia
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   § 1. Introduction1.1. A historical overviewLet be an infinite sequence of real functions. We denote by $C(f)\subset \mathbb{R}$ the convergence set of the sequence (1.1), that is, the set of points $x\in \mathbb{R}$ such that $\lim f_n(x)$ exists. A classical theorem of Hahn and Sierpinski (see [12] and [35]) asserts that if the functions (1.1) are continuous, then $C(f)$ is an $F_{\sigma\delta}$-set and, conversely, every $F_{\sigma\delta}$-set is the convergence set for a sequence of continuous functions. The first part of this statement is immediate since the convergence set of (1.1) can be written in the form
$$
\begin{equation*}
C(f)=\bigcap_{m=1}^\infty\bigcup_{n=1}^\infty\bigcap_{k=1}^\infty \biggl\{x\colon |f_{n+k}(x)-f_n(x)|\leqslant \frac 1m\biggr\},
\end{equation*}
\notag
$$
and this is an $F_{\sigma\delta}$-set if the functions are continuous (see Definition 1). The second part of the Hahn–Sierpinski theorem requires a construction of a sequence of continuous functions for which a given $F_{\sigma\delta}$-set is a convergence set (see also [14], p. 307). Note that the complement of the convergence set $C(f)$ is the divergence set of (1.1), which we denote by $D(f)$. So $G_{\delta\sigma}$-sets completely characterize the divergence sets of sequences of continuous functions. One can also consider the unbounded divergence set $UD(f)\subset D(f)$ of the sequence (1.1), which consists of the points $x$, satisfying $\limsup |f_n(x)|=\infty$. It is also known that for a set to be the $UB$-set for a sequence of continuous functions it is necessary and sufficient to be a $G_\delta$-set (see [12], [35] and [14]). Hence we have a topological characterization of $C$-, $D$- and $UD$-sets in the class of sequences consisting of continuous functions. Such characterization problems have been considered in many fields of analysis (Fourier series, analytic functions on the unit ball, power series, differentiation theory), and there are many published papers and open problems, some of which are considered in the last section. The reader can find earlier reviews of problems concerning the convergence or divergence sets of Fourier series in trigonometric, Walsh and Haar orthogonal systems in [38]–[40]. We also refer to the monograph [7], related to boundary exceptional sets of analytic functions. Let us state a few remarkable examples of characterization theorems. The first one provides a complete characterization of the sets of nondifferentiability points of continuous functions. Theorem A (Zahorski [41]). A set $E\subset \mathbb{R}$ can be the set of nondifferentiability points of a real continuous function if and only if it is a union of a $G_\delta$-set and a $G_{\delta\sigma}$-nullset. In fact, the main ingredient of this theorem is the construction of a continuous function whose set of points of nondifferentiability coincides with a given set that is a union of a $G_\delta$-set and a $G_{\delta\sigma}$-set of measure zero. To implement this construction subtle technique was used in [41] (see also [32] and [9] for alternative simplified proofs of Theorem A). The next well-known result is an extension of Kolmogorov’s celebrated theorem [25] on an example of an integrable function whose Fourier series is everywhere divergent. Theorem B (Zeller [43]). Let $E$ be an $F_\sigma$-set on the circle $\mathbb{T}=\mathbb{R}/2\pi$. Then there exists a function $f\in L^1(\mathbb{T})$ whose Fourier series converges at each point $x\in E$ and diverges unboundedly whenever $x\in \mathbb{T}\setminus E$. This theorem, in combination with the general results mentioned above, provides a complete characterization of $UD$-sets for the class of Fourier series of functions in $L^1(\mathbb{T})$. Subsequently, an analogous result for Walsh series was proved in [27] and [28]. Note that it is an open problem to characterize $D$ or $UD$ sets of Fourier series of functions in $L^p(\mathbb{T})$, $1<p\leqslant \infty$, or continuous functions (see [38], p. 107). Also note that, in accordance with results due to Carleson and Hunt (see [4] and [15]), the Fourier series of functions from $L^p(\mathbb{T})$, $p>1$ converge almost everywhere, so the divergence sets of these series are nullsets. One can find some partial results concerning the characterization of convergence or divergence sets of Fourier series in [1]–[3], [37], [10], [24], [16], [20]–[22], [29] and [17]. Those papers mostly provide constructions of different examples of Fourier series divergent on a given nullset, and the most general theorem in this context is Kahane–Katznelson’s one [16] on the existence of a continuous function whose trigonometric Fourier series diverges on a given nullset (see also [23], Ch. II, § 3, for a detailed discussion of divergence sets). We also refer to the papers [20] and [29], which provide a complete characterization of the $C$-, $D$- and $UD$-sets of Fourier–Haar series. This author obtained in [19] general characterization theorems for certain operator sequences, which cover many results of the papers mentioned above, providing a complete characterization of the convergence sets of Fourier series in the Haar and Franklin systems, as well as of positive-order Cesàro means of trigonometric and Walsh Fourier series. We observe here that Körner [26] constructed a $G_\delta$-set that is not the convergence set of any trigonometric series. This example says that for ordinary partial sums of trigonometric or Walsh–Fourier series, in some open problems a pure topological characterization of convergence sets can fail to exist (see a discussion in [38] and [39]). The next result provides a complete characterization of radial convergence sets of bounded analytic functions on the unit disc. It was a solution of a longstanding problem stated by Collingwood and Lohwater in [7]. Theorem C (Kolesnikov [25]). Let $D$ be the unit disc with boundary circle $\Gamma$. For a set $E\subset \Gamma$ to be the radial convergence set for a bounded analytic function on $D$ it is necessary and sufficient to be an $F_{\sigma\delta}$ set of full measure. 1.2. The main results of the paperIn the present paper we obtain complete characterization theorems for certain operator sequences living in general spaces of homogeneous type. These generalize the results of [18] and [19], where operators on the interval $[0,1]$ were considered. By a quasi-distance on a set $X$ we mean a nonnegative function $d(x,y)$ defined on $X\times X$ such that
Definition 1. Recall that a set in a topological space is of $G_\delta$-type provided it is a countable intersection of open sets, and a set is $G_{\delta\sigma}$ if it is a countable union of $G_\delta$-sets. $F_\sigma$-sets are countable unions of closed sets, and $F_{\sigma\delta}$-sets are countable intersections of $F_\sigma$-sets. Definition 2. The distance between two sets $A$ and $B$ in a space of homogeneous type $X$ is denoted by The notation $A\Subset B$ is used for the relation $\operatorname{dist}(A,B^{\rm c})>0$. Note that $A\Subset B$ implies $B^{\rm c}\Subset A^{\rm c}$. Let $L^1(X)$ denote the space of Lebesgue integrable functions on a space of homogeneous type $(X,d,\mu)$, and suppose that $M(X)$ is the normed space of bounded measurable functions on $X$ with the norm $\|f\|_M=\sup_{x\in X}|f(x)|$. We consider sequences of linear operators which can have some of the following properties:
We always suppose that $X$ is a space of homogeneous type unless other assumptions are made. The main results of this paper read as follows. Theorem 1. If an operator sequence (1.4) satisfies (U1)–(U4), then for any $G_{\delta\sigma}$-nullset $E\subset X$ and each $\varepsilon>0$ there exists a function $f\in L^\infty(X)$ such that (1) $\mu(\operatorname{supp} f)<\varepsilon$; (2) $U_n(x,f)$ diverges at any $x\in E$; (3) $U_n(x,f)\to f(x)$ if $x\in X\setminus E$. Theorem 2. Let the operator sequence (1.4) satisfy (U1)–(U4), and for a function $\psi\colon [0,\infty)\to [0,\infty)$ let Then for every $G_{\delta}$-nullset $E\subset X$ and each $\varepsilon>0$ there exists a measurable function $f$ such that (1) $\mu(\operatorname{supp} f)<\varepsilon, \quad \int_X\psi(|f|)<\infty$; (2) $\limsup_{n\to\infty}|U_n(x,f)|=\infty$ for all $x\in E$; (3) $U_n(x,f)\to f(x)$ if $x\in X\setminus E$. Observe that if, apart from properties (U1)–(U4), we also assume the continuity of the $U_n(x,f)$ as functions of $x$, then Theorem 1 gives a complete characterization of the $C$- and $D$-sets of sequences $U_n(x,f)$, where $f\in L^p(X)$ and $1\leqslant p\leqslant \infty$. Similarly, Theorem 2 provides a complete characterization of the $UD$-sets of such sequences when $1\leqslant p<\infty$. One cannot claim for the function $f$ in Theorem 2 to be bounded like in Theorem 1, since in that case condition (U2) would imply the boundedness of $U_n(x,f)$ for all $x\in X$. In the case when $X$ coincides with $[0,1]$, Theorems 1 and 2 were proved in [19], and we essentially use some arguments from [18] and [19] in the present work. The approach provided in the proofs of Theorems 1 and 2 enables us to obtain also the following pure divergence result, where the operator sequences satisfy only conditions (U1) and (U3). This result is a generalization of an analogous theorem in [18] for operators living on $[0,1]$. Theorem 3. If an operator sequence (1.4) satisfies (U1) and (U3), then for every nullset $E\subset X$ there exists a set $G$ such that for the indicator function $f=\mathbf{1}_G$ the sequence of $U_n(x,f)$ diverges at each point $x\in E$. Moreover, Remark 1. With a slight change of the proofs, Theorems 1–3 can be stated and proved for operators $U_r$ where the parameter $r$ ranges over an infinite partially ordered set $R$. Namely, we can consider a partially ordered set $R$ with a relation $<$ satisfying the following: (1) there is a unique maximal element $r_\infty\in R$; (2) for any $r<r_\infty$ there exist infinitely many elements $s$ satisfying $s>r$. Then, given a process (or a sequence) $x_r$ in $X$, $r\in R$, the convergence $\lim_{r\to r_\infty} x_r=a$ means the following: for any open set $G\supset a$ there exists $r\in R$ such that for each $s$ such that $r<s<r_\infty$ we have $x_r\in G$. We note that Theorem 3, as well as the results in [18], generalize the results of [24], [1], [10], [27], [28] and [33], where the authors considered various Fourier partial sums (namely, trigonometric, Walsh and Haar) instead of the operators $U_n$. Examples of operators satisfying (U1)–(U4) and corollaries to Theorems 1 and 2 are considered in the next section. § 2. Examples and applications2.1. Approximation of the identity on metric measure spacesLet $X$ be a space of homogeneous type. A sequence of kernel functions $K_n(x,y)\in L^\infty(X\times X)\cap L^1(X\times X)$ is said to be an approximation of the identity if it satisfies the conditions
$$
\begin{equation}
\int_X K_n(x,y)\,dy\rightrightarrows1 \quad\text{as } n\to\infty,
\end{equation}
\tag{2.1}
$$
$$
\begin{equation}
\int_{\{y\colon d(x,y)>\delta\}} K_n^*(x,y)\,dy\rightrightarrows0 \quad\text{as } n\to\infty \quad\text{for every } \delta>0
\end{equation}
\tag{2.2}
$$
and where the convergence in (2.1) and (2.2) holds uniformly with respect to $x$ over $X$, and
$$
\begin{equation*}
K_n^*(x,y)=\sup_{t\colon d(x,t)\geqslant d(x,y)} |K_n(x,t)|.
\end{equation*}
\notag
$$
Consider the operator sequence where the kernels $K_n$ satisfy the above conditions. It is known that if a space of homogeneous type has the property that the space of functions
$$
\begin{equation}
C_{K}(X)=\Bigl\{f\in C(X)\colon \operatorname{supp}(f)\text{ is bounded and } \sup_{x\in X}|f(x)|<\infty \Bigr\}
\end{equation}
\tag{2.5}
$$
is dense in $L^1(X)$, then the operators (2.4) enjoy the following properties (see, for example, [6] and [5]): (1) if a function $f\in L^1(X)$ is uniformly continuous (for instance, is constant) on an open set $G\subset X$, then $U_n(x,f)\rightrightarrows f(x)$ uniformly on any subset $E\Subset G$; (2) if $f\in L^1(X)$, then $U_n(x,f)\to f(x)$ almost everywhere. Such operators obviously satisfy conditions (U1)–(U4). Properties (U1) and (U2) immediately follow from the definition of an approximation of the identity, while properties (U3) and (U4) are weaker versions of the above properties (1) and (2), respectively. So we can state the following result. Proposition 1. If $X$ is a space of homogeneous type such that $C_K(X)$ is dense in $L^1(X)$, then the operators (2.4) obey the assumptions of Theorems 1–3. 2.2. Walsh functionsWe consider the Walsh orthonormal system, defined on the set of all sequences
$$
\begin{equation}
(x_0,x_1,x_2,\dots), \quad\text{where } x_j=0 \text{ or } 1, \qquad j=0,1,2,\dots
\end{equation}
\tag{2.6}
$$
(see [11], § 1). Every such a sequence generates the series which is the dyadic decomposition of some number in $[0,1]$. We note that this correspondence is surjective, but it is not injective since dyadic rationals in $[0,1]$ have two decompositions (2.7), an infinite and a finite one. For a geometric understanding of the set of sequences (2.6) we introduce the extended interval $[0,1]^*$, where each dyadic rational number $x$ is doubled, giving rise to a left-hand point $x-$ corresponding to the finite dyadic decomposition and a right-hand point $x+$ corresponding to the infinite decomposition of $x$. By writing $x+$ or $x-$ for a dyadic irrational number $x$ we mean just the point $x\in [0,1]^*$. We define a generalized dyadic interval by
$$
\begin{equation}
[a,b]=\{x+, x-\in [0,1]^*\colon a<x<b\}\cup\{a+,b-\},
\end{equation}
\tag{2.8}
$$
where
$$
\begin{equation}
a=\frac{j-1}{2^k}, \quad b=\frac{j}{2^k}, \qquad 1\leqslant j\leqslant 2^k, \quad k=0,1,\dots\,.
\end{equation}
\tag{2.9}
$$
One can easily check that the dyadic distance between two points $x,y\in [0,1]^*$,
$$
\begin{equation}
d(x,y)=\inf\bigl\{b-a\colon [a,b] \text{ is a generalized dyadic interval and}\ x,y\in [a,b]\bigr\},
\end{equation}
\tag{2.10}
$$
defines a quasi-distance on $[0,1]^*$. Then each set $E^*\subset [0,1]^*$ has its counterpart $E\subset [0,1]$, obtained by identifying the points in each pair $x+,x-\in [0,1]^*$. So we define the measure of $E^*$ to be the Lebesgue measure of the set $E$. Hence $[0,1]^*$ equipped with such a measure becomes a space of homogeneous type. Moreover, one can easily check that $[0,1]^*$ is a compact space (we do not need this). To define Walsh functions, recall the group operation on $[0,1]^*$ defining the sum of two sequences $\{x_j\}_{j=0}^\infty$ and $\{y_j\}_{j=0}^\infty$ of type (2.6) to be the sequence $\{z_j\}_{j=0}^\infty$, where
$$
\begin{equation*}
z_j= \begin{cases} 1&\text{if } x_j+y_j=1, \\ 0&\text{if } x_j+y_j=0\text{ or }2. \end{cases}
\end{equation*}
\notag
$$
Now we can define the Walsh system of functions $\{w_n(x)\}_{n=0}^\infty$ on $[0,1]^*$. We set $w_0(x)\equiv 1$. For $n\geqslant 1$ we write its dyadic representation $n=\sum_{j=0}^k\varepsilon_j2^j$, then set
$$
\begin{equation}
w_n(x)=(-1)^{\sum_{j=1}^k \varepsilon_jx_j} \quad\text{for } x=(x_0,x_1,\dots)\in [0,1]^*.
\end{equation}
\tag{2.11}
$$
One can see that Walsh functions are continuous in the topology of $[0,1]^*$. Moreover, it is well known that $(C,\alpha)$-means $\sigma_n^\alpha(x,f)$ $\alpha>0$ of the partial sums of Walsh–Fourier series have properties (U1)–(U4) (see [11], Ch. 4). So we can state the following results. Corollary 1. For a set $E\subset [0,1]^*$ to be the divergence (unbounded divergence) set of $\sigma_n^\alpha(x,f)$ for a function $L^p([0,1]^*)$, $1\leqslant p\leqslant \infty$, it is necessary and sufficient to be a $G_{\delta\sigma}$-set ($G_\delta$-set) in the topology of $[0,1]^*$. Remark 2. $(C,\alpha)$-means $\sigma_n^\alpha$ of Walsh–Fourier series can be considered as kernel operators. Nevertheless, it is known that the kernels of $\sigma_n^\alpha$ do not form an approximation of the identity (see [11]). 2.3. SplinesRecall the definition of splines on an interval $[a,b]$. For a knot collection $\Delta = \{t_j\}_{j=1}^{n+k}\subset [a,b]$ such that let $\mathcal{S}_k(\Delta)$ denote the space of $k$-order splines with knots in $\Delta$. These are functions that are polynomials of degree $\leqslant k-1$ on each interval $[t_j,t_{j+1}]$ and have ${k-1-m_j}$ continuous derivatives at each knot $t_j\in \Delta$ of multiplicity $m_i$. Let $P_\Delta$ be the orthoprojection operator onto $\mathcal{S}_k(\Delta)$. Set $|\Delta|=\max_j(t_{j+1}-t_j)$. Given a sequence of fixed $k$-order knot collections $\Delta_n$ such that $|\Delta_n|\to 0$, consider the operator sequence $P_{\Delta_n}$. It is known that such an operator sequence has properties (U1)–(U4). Moreover, in solving de Boor’s conjecture Shadrin [34] proved the uniformly boundedness of these operators on $C[a,b]$, which implies that $P_{\Delta_n}(f)\rightrightarrows f$ whenever $f\in C[a,b]$. Then Passenbrunner and Shadrin [31] proved that $P_{\Delta_n}(f)\to f$ almost everywhere for any $f\in L^1[a,b]$, where they also proved that the kernels of the operators $P_{\Delta_n}$ form an approximation of the identity. Thus we can state the following result.Corollary 2. Let $\Delta_n$ be a sequence of knot collections in $[a,b]$ such that $|\Delta_n|\to0$. Then for a set $E\subset [a,b]$ to be the divergence (unbounded divergence) set of $P_{\Delta_n}(f)$ for a function $L^p[a,b]$, $1\leqslant p\leqslant \infty$, it is necessary and sufficient to be a $G_{\delta\sigma}$-set ($G_\delta$-set) in the standard topology of $[a,b]$. 2.4. More specific examplesLet us also discuss some more specific examples of operator sequences that obey the assumptions of Theorems 1 and 2 and can be deduced from one of the general examples considered above. 1. Partial sums of Fourier series in Haar and Franklin systems with general nodes are spline operator sequences, so they obey the assumptions of Corollary 2. For Haar series characterizations of $C$-, $D$- and $UD$-sets were first given in [29] (for $UD$-sets) and [20] (for $D$-sets). 2. A characterization of convergence and divergence sets of $(C,\alpha)$-means of trigonometric Fourier series with a parameter $\alpha>0$ was given by Zahorski in [42], where the same problem was considered also for Poisson integrals on the unit disc. We just note that both operators are approximations of the identity on the unit circle. 3. Let $(X,d,\mu)$ be a space of homogeneous type such that $C_K(X)$ is dense in $L^1(X)$. Consider a sequence of measurable sets $G_n\subset X$ equivalent to balls $B(0,r_n)$, that is,
$$
\begin{equation}
G_n\subset B(0,r_n)\quad\text{and} \quad \mu(G_n)>c\mu(B(0,r_n)),
\end{equation}
\tag{2.14}
$$
where $c>0$ is a constant. Clearly the operator sequence is a special case of operators (2.4). This sequence is particularly interesting when $X$ coincides with $\mathbb{R}^n$. 4. Dyachkov [8] considered the problem of the characterization of $C$-$D$-sets for the operators
$$
\begin{equation*}
U_\varepsilon(x,f)=\int_{\mathbb R^d}f(x-t)\phi_\varepsilon(t)\,dt \quad\text{as } \varepsilon\searrow 0,
\end{equation*}
\notag
$$
for the kernels $\phi_\varepsilon(t)=\varepsilon^{-d}\phi(t/\varepsilon)$ approximating the identity, where $\phi\in L^1(\mathbb{R}^d)$. 5. The main part of Zahorski’s theorem in [41] (Theorem A) is a construction of a continuous function whose nondifferentiability set is a given set $G=G_1\cup G_2$, where $G_1$ is $G_\delta$-set and $G_2$ is a $G_{\delta\sigma}$-nullset. Without loss of generality one can suppose that $G_1\cap G_2=\varnothing$. So the construction can be split into the constructions of two different functions $f_1$ and $f_2$ whose nondifferentiability sets are $G_1$ and $G_2$, respectively. Then the required function is $f=f_1+f_2$. The function $f_2$ can be sought in the form where $g\in L^\infty(\mathbb{R})$. Then the set of differentiability points of $f_2$ coincides with the set of points $x\in \mathbb{R}$ such that the limit
$$
\begin{equation}
\lim_{|I|\to 0,\,I\ni x}\frac{1}{|I|}\int_Ig(t)\,dt
\end{equation}
\tag{2.17}
$$
exists, where the $I$ are closed intervals containing the point $x$. The existence of a function $g\in L^\infty(\mathbb{R})$ for which the limit (2.17) exists only at the points in a given $G_{\delta\sigma}$-nullset is a part of the general result of Proposition 1.
§ 3. Auxiliary lemmasIt is known from [30] that for any quasi-distance $d$ there exists an alternative quasi-distance $d'$ that is equivalent to $d$ (that is, $C_1d(x,y)\leqslant d'(x,y)\leqslant C_2d(x,y)$) and satisfies a Lipschitz-type condition
$$
\begin{equation}
|d'(x,z)-d'(y,z)|\leqslant L(d'(x,y))^\alpha\bigl(\max\{d'(x,z),d'(y,z)\}\bigr)^{1-\alpha},
\end{equation}
\tag{3.1}
$$
where $L>0$ and $0<\alpha<1$ are some constants. One can see that $d'$ induces the same topology as $d$ does and in view of (3.1) $d'$-balls become open sets in it. Using this we can prove the following result. Lemma 1. Any open set $G$ in a space of homogeneous type admits a representation $G=\bigcup_{k=1}^\infty F_k$, where each $F_k$ is a closed set and $F_k\Subset G$. Proof. Let $\operatorname{dist}'$ denote the distance corresponding to the quasi-distance $d'$ in (3.1). Set
$$
\begin{equation*}
F_k=\biggl\{x\in G\colon \operatorname{dist}'(x,G^{\mathrm c})\geqslant \frac1k\biggr\}, \qquad k=1,2,\dots\,.
\end{equation*}
\notag
$$
It is clear that $F_k\Subset G$. It remains to show that each $F_k$ is closed or, equivalently, $F_k^{\rm c}$ is open. For any $x\in F_k^{\rm c}$ we have $\operatorname{dist}'(x,G^{\rm c})<1/k$. Then for sufficiently small $r>0$ the condition $d'(y,x)<r$ implies that $\operatorname{dist}'(y,G^{\rm c})<1/k$ as follows from (3.1). Thus we obtain $B'(x,r)\subset F_k^{\rm c}$, that is, $F_k^{\rm c}$ is open.
The lemma is proved. The following lemma is a standard property of a topology induced by a quasi-distance. Lemma 2. If two sets $A$ and $B$ in a quasi-metric space $(X,d)$ satisfy ${\operatorname{dist}(\mkern-1mu A,B)\!>\!0}$, then there are disjoint open sets $U$ and $V$ such that $A\Subset U$ and $B\Subset V$. Proof. Let $d'$ be a Lipschitz distance satisfying (3.1). The equivalence of the distances $d$ and $d'$ implies that $\operatorname{dist}'(A,B)>0$, and we also know that each $d'$-ball is an open set. We define an open set $U\supset A$ to be the union of all balls $B'(x,r)$ such that $x\in A$ and $r=\operatorname{dist}'(x,B)/(2K)>0$, where $K$ is the constant in the triangle inequality for the distance $d'$. Defining similarly an open neighbourhood $V\supset B$, we claim that $U\cap V=\varnothing$. Suppose that, on the contrary, there exists a point $z\in U\cap V$. Then we find $x\in A$ and $y\in B$ such that This implies that which is a contradiction, and so $U\cap V=\varnothing$. It just remains to observe that $A\Subset U$ and $A\Subset V$ with respect to both the distances $d$ and $d'$.
The lemma is proved. It is known in geometric measure theory (see, for example, [36], Theorem 1.11) that if in a topological measurable space $(X,\mu)$ every open set is of type $F_\sigma$ and the measure $\mu$ is open $\sigma$-finite (that is, $X$ is a countable union of open sets of finite measure), then
$$
\begin{equation}
\mu(E)=\inf_{\substack{G\text{ is open}\\ G\supset E}}\mu(G)
\end{equation}
\tag{3.2}
$$
for any Borel set $E$. Any space of homogeneous type $X$ has these properties since every open set is of type $F_\sigma$ by Lemma 1, and $X$ can be written as a countable union of $d'$-balls, which are open and have a finite measure. Hence we can state the following result. Lemma 3. Any space of homogeneous type $X$ enjoys the property (3.2). Definition 3. A sequence $\Omega=\{\omega_k\colon k=1,2,\dots\}$ of Borel sets in a topological space $X$ is said to be (1) a partition of an open set $B\subset X$ if
$$
\begin{equation}
B=\bigcup_k\omega_k\quad\text{and} \quad \omega_k\cap \omega_{k'}=\varnothing \quad\text{if } k\neq k';
\end{equation}
\tag{3.3}
$$
(2) locally finite if there are open sets $V_k\supset \omega_k$ such that each of them has only a finite number of intersection with others, that is,
$$
\begin{equation}
\#\{j\in \mathbb N\colon V_j\cap V_k\neq \varnothing\}<\infty \quad\text{for } k=1,2,\dots;
\end{equation}
\tag{3.4}
$$
(3) a regular partition of an open set $B$ if it is a partition of $B$ and there are open sets $V_j$ satisfying (3.4) (so that the sequence is locally finite) such that Lemma 4. Let $\omega_k$, $k=1,2,\dots$, be a locally finite family of measurable sets in a topological measurable space $X$. Then for any measurable set $E\subset \bigcup_k\omega_k$ and any positive numbers $\varepsilon_k $ there exists an open set $G\supset E$ such that Proof. Let $V_k\supset \omega_k$, $k=1,2,\dots$, be open sets such that each $V_k$ intersects only a finite number of elements $V_j$, and we denote this number by $l_k$. One can check that
$$
\begin{equation}
\varepsilon'_k=\min_{j\colon V_j\cap V_k\neq \varnothing}\frac{\varepsilon_j}{l_j}>0 \quad\text{for all } k=1,2,\dots,
\end{equation}
\tag{3.6}
$$
because only a finite number of indices $j$ under this minimum sign satisfy the relation $V_j\cap V_k\neq \varnothing$. Thus, applying Lemma 3 we can define open sets $G_k$, $k=1,2,\dots$, such that
$$
\begin{equation}
E\cap \omega_k\subset G_k\subset V_k, \qquad \mu(G_k\setminus E)<\varepsilon'_k.
\end{equation}
\tag{3.7}
$$
Hence $G=\bigcup_kG_k\supset E$ and
$$
\begin{equation}
\begin{aligned} \, \notag \mu((G\setminus E)\cap \omega_k) &\leqslant \sum_{j\colon V_j\cap \omega_k\neq \varnothing} \mu((G_j\setminus E )\cap \omega_k) \leqslant \sum_{j\colon V_j\cap V_k\neq \varnothing} \mu(G_j\setminus E) \\ &<\sum_{j\colon V_j\cap V_k\neq \varnothing} \varepsilon'_j< \sum_{j\colon V_j\cap V_k\neq \varnothing}\frac{\varepsilon_k}{l_k}=\varepsilon_k. \end{aligned}
\end{equation}
\tag{3.8}
$$
This completes the proof. Proof. For any open set $B\subset X$ we can find closed sets $F_k$, $k=1,2,\dots$, such that
$$
\begin{equation}
B=\bigcup_kF_k, \qquad F_k\Subset \operatorname{Int}(F_{k+1}),
\end{equation}
\tag{3.9}
$$
where $\operatorname{Int}(E)$ denotes the interior of the set $E\subset X$. Indeed, first we write ${B\!=\!\bigcup_k A_k}$, where the $A_k\Subset B$ are closed sets (see Lemma 1). Since ${\operatorname{dist}(A_1,B^{\rm c})\!>\!0}$, by Lemma 2 the sets $A_1$ and $B^{\rm c}$ have disjoint open neighbourhoods $D_1\Supset A_1$ and $G_1\Supset B^{\rm c}$. So we have $A_1\Subset D_1$ and $\overline {D_1}\subset (G_1)^{\rm c}\Subset B$. Then we do the same for the closed sets $A_2\cup \overline {D_1}\Subset B$ and $B^{\rm c}$ and obtain an open set $D_2\Supset A_2\cup \overline {D_1}$ such that $\overline {D_2}\Subset B$. Continuing this procedure without limit we finally obtain a sequence of open sets $D_k$ such that $D_k\Supset A_k\cup \overline{D_{k-1}}$ and $\overline {V_k}\subset B$. Let us see that $F_1=A_1$ and $F_k=A_k\cup \overline{D_{k-1}}$, $k\geqslant2$, are closed sets satisfying (3.9). Clearly, we have $B=\bigcup_kF_k$ and, moreover, $F_k\Subset D_k\subset F_{k+1}$. Since $D_k$ is open, we obtain $F_k\Subset \operatorname{Int}(F_{k+1})$.
With (3.9) at hand, we define the Borel sets $\omega_1=F_1$ and $\omega_k=F_k\setminus F_{k-1}$, $k\geqslant 2$, which clearly satisfy the definition of a partition (3.3). One can also see that the open sets $V_1=\operatorname{Int}(F_2), V_2=\operatorname{Int}(F_3)$ and $V_k= \operatorname{Int}(F_{k+1})\setminus F_{k-2}$, $k>2$, satisfy (3.4). Also, by (3.9) we have
$$
\begin{equation*}
\omega_k\subset F_k\setminus \Subset \operatorname{Int}(F_{k+1})\subset \operatorname{Int}(F_{k-1})\subset V_k,
\end{equation*}
\notag
$$
which implies (3.5).
The lemma is proved. The notation $f_n(x)\rightrightarrows f(x)$ is used to denote the uniform convergence of a sequence of functions on a set. Lemma 6. Let $(X,d,\mu)$ be a space of homogeneous type, and let the operator sequence (1.4) satisfy conditions (U1) and (U3). If $G\subset X$ is an open set of finite measure and $F\Subset G$ is measurable, then Proof. First we note that by property (U3) we have
$$
\begin{equation}
U_n(x,f\cdot \mathbf 1_F)\rightrightarrows 0 \quad\text{on } G^{\mathrm c} \text{ as } n\to\infty
\end{equation}
\tag{3.11}
$$
for each $f\in L^1(X)$, since $f\cdot \mathbf{1}_F\in L^1(X)$ is identically equal to zero on the open set $F^{\rm c}$ and $G^{\rm c}\Subset F^{\rm c}$. Suppose that, on the contrary, $d_G(F)=\infty$. The boundedness property (U1) of the operators (1.4) implies that
$$
\begin{equation}
\begin{aligned} \, \notag d_{G,m}(F) &=\max_{1\leqslant n\leqslant m}\sup_{x\in G^{\mathrm c}}\sup_{\|f\|_1\leqslant 1}|U_n(x,f\cdot \mathbf 1_F)| \\ &\leqslant \max_{1\leqslant n\leqslant m}\rho_n<\infty \quad\text{for any } m\in \mathbb N, \end{aligned}
\end{equation}
\tag{3.12}
$$
and by the assumption to the contrary we have Applying induction, one can find integers $n_k\in\mathbb{N}$, points $x_k\in G^{\rm c}$ and functions $f_k\in L^1(X)$, $k=1,2,\dots $, such that
$$
\begin{equation}
\| f_k\|_1\leqslant 1, \qquad \operatorname{supp} f_k\subset F, \quad k\geqslant 1,
\end{equation}
\tag{3.14}
$$
$$
\begin{equation}
|U_{n_k}(x_k,f_k)|>k^3\Bigl(1+\max_{1\leqslant i<k}\rho_{n_i}\Bigr), \qquad k\geqslant 1,
\end{equation}
\tag{3.15}
$$
and
$$
\begin{equation}
\sup_{1\leqslant i<k}|U_{n_k}(x,f_i)|<1, \qquad x\in G^{\mathrm c}, \quad k>1.
\end{equation}
\tag{3.16}
$$
Indeed, using (3.13), first we define $f_1(x)$, $n_1\in\mathbb{N}$ and $x_1\in G^{\rm c}$ satisfying
$$
\begin{equation*}
\| f_1\|_1\leqslant 1, \qquad \operatorname{supp} f_1\subset F\quad\text{and} \quad |U_{n_1}(x_1,f_1)|>1,
\end{equation*}
\notag
$$
which is the basis of induction. Then suppose that we have already defined ${f_k\in L^1(X)}$, $n_k\in\mathbb{N}$ and $x_k\in G^{\rm c}$ satisfying conditions (3.14)–(3.16) for $k=1,2,\dots,p$. Once again, using (3.13) we find $f_{p+1}(x)$, $n_{p+1}\in\mathbb{N}$ and $x_{p+1}\in G^{\rm c}$, satisfying (3.14) and (3.15) for $k=p+1$. Moreover, by (3.12) the number $n_{p+1}$ can be chosen sufficiently large, and so, using (3.11), we can also ensure (3.16) for $k=p+1$. This completes the induction process. Let us also add the bound
$$
\begin{equation}
|U_{n_k}(x,f_i)|\leqslant \rho_{n_k}, \qquad x\in X, \qquad i=1,2,\dots,
\end{equation}
\tag{3.17}
$$
which immediately follows from (U1) and the bound $\|f_i\|_1\leqslant 1$. Now consider the function
$$
\begin{equation}
f(x)=\sum_{k=1}^\infty \alpha_kf_k(x), \qquad \alpha_k=\frac{1}{k^2(1+\max_{1\leqslant i<k}\rho_{n_i})}.
\end{equation}
\tag{3.18}
$$
Clearly, $f\in L^1 (X)$ and, moreover, $\operatorname{supp} f \subset F$. Using (3.14)–(3.18) we obtain
$$
\begin{equation*}
\begin{aligned} \, |U_{n_k}(x_k,f)| &\geqslant \alpha_k|U_{n_k}(x_k,f_k)|-\sum_{i=1}^{k-1}\alpha_i|U_{n_k}(x_k,f_i)|- \sum_{i=k+1}^\infty \alpha_i |U_{n_k}(x_k,f_i)| \\ &\geqslant k-\sum_{i=1}^{k-1}\frac{1}{i^2}-\sum_{i=k+1}^\infty \frac{1}{i^2}\geqslant k-2. \end{aligned}
\end{equation*}
\notag
$$
This leads to a contradiction since $\operatorname{supp} f \subset F$ and $U_n(x,f)\rightrightarrows 0$ on $G^{\rm c}$ according to (3.11).
The lemma is proved. For a sequence of pairwise disjoint measurable sets $\Omega=\{\omega_k\}$ in a space of homogeneous type $X$ and a mapping $\nu\colon \Omega\to \mathbb{N}$ we set
$$
\begin{equation}
U_{(\Omega,\nu)}(x,f)=\sum_{k\geqslant 1} U_{\nu(\omega_k)}(x,f)\cdot \mathbf 1_{\omega_k}(x).
\end{equation}
\tag{3.19}
$$
Lemma 7. Let the operator sequence $U_n$ satisfy (U3) and the functions ${f_k\,{\in}\, L^1(X)}$, $k=1,2,\dots,n$, be constant on an open set $G\subset X$, which has a partition $\Omega=\{\omega_k\}$ such that $\omega_k\Subset G$. Then for any $\varepsilon >0$ and $R>0$ there exists a mapping $\nu\colon \Omega\to \mathbb{N}$ for which and Proof. From condition (U3) it follows that $U_n(x,f_i)\rightrightarrows f_i(x)$ on each set $\omega_k$, for any $i=1,2,\dots n$. Hence for every $\omega_k$ we can find an integer $\nu(\omega_k)\in \mathbb{N}$ such that $\nu(\omega_k)>R$ and
$$
\begin{equation*}
|U_{\nu(\omega_k)} (x,f_i)-f_i(x)|<\varepsilon, \qquad x\in \omega_k, \qquad i=1,2,\dots ,n.
\end{equation*}
\notag
$$
Thus we obtain (3.20) and (3.21).
The lemma is proved. Lemma 8. Let the operator sequence $U_n$ satisfy (U1) and (U3), let $G\subset X$ be an open set with a regular partition $\Omega=\{\omega_k\}$, and let $\nu\colon \Omega\to \mathbb{N}$ be a mapping. If ${C\subset G}$ is a measurable set and a function $f\in L^\infty(X)$ satisfies Proof. For $x\notin G$ inequality (3.23) holds trivially since $U_{(\Omega,\nu)}(x,f)=0$ (see (3.19)). By the definition of a regular partition we can fix open sets $V_k\supset \omega_k$, $k=1,2,\dots$, satisfying (3.3). For $x\in G$ we can write
$$
\begin{equation}
x\in \omega_k\quad \text{for a unique } k\geqslant1,
\end{equation}
\tag{3.24}
$$
and then
$$
\begin{equation}
\begin{aligned} \, \notag |U_{(\Omega,\nu)}(x,f)| &\leqslant \sum_{j\colon V_j\cap V_k=\varnothing} |U_{\nu(\omega_k)}(x,f\cdot\mathbf 1_{\omega_j})| \\ &\qquad +\sum_{j\colon V_j\cap V_k\neq \varnothing} |U_{\nu(\omega_k)}(x,f\cdot\mathbf 1_{\omega_j})|=S_1(x)+S_2(x). \end{aligned}
\end{equation}
\tag{3.25}
$$
The condition $V_j\cap V_k=\varnothing$ implies that $x\in\omega_k\Subset V_k\subset (V_j)^{\rm c}$. So by applying Lemma 6 and relations (3.22), for the first sum we obtain
$$
\begin{equation}
S_1(x)\leqslant \sum_{j\colon V_j\cap V_k=\varnothing}\|f\cdot\mathbf 1_{\omega_j}\|_{1}d_{V_j}(\omega_j) \leqslant \sum_{j\colon V_j\cap V_k=\varnothing}d_{V_j}(\omega_j)|C\cap \omega_j|.
\end{equation}
\tag{3.26}
$$
Using (U1) and (3.22) we have
$$
\begin{equation}
\begin{aligned} \, \notag S_2(x) &\leqslant \sum_{j\colon V_j\cap V_k\neq \varnothing}\rho_{\nu(\omega_k)}|C\cap \omega_j| \\ &\leqslant \sum_{j\colon V_j\cap V_k\neq \varnothing}\max\{\rho_{\nu(\omega_i)}\colon V_i\cap V_j\neq \varnothing \}|C\cap \omega_j|. \end{aligned}
\end{equation}
\tag{3.27}
$$
Combining (3.25)–(3.27) we obtain (3.23) with constants
$$
\begin{equation}
c(\omega_j)=\max\bigl\{\max\{\rho_{\nu(\omega_i)}\colon V_i\cap V_j\neq \varnothing \},d_{V_j}(\omega_j)\bigr\}.
\end{equation}
\tag{3.28}
$$
It just remains to note that by (U1) and Lemma 6 we have $c(\omega_j)\mkern-1mu<\mkern-1mu\infty$ for ${j\!=\!1,2,\dots}$ .
The lemma is proved. For an open set $A\subset X$ of finite measure we set
$$
\begin{equation*}
\lambda(A)=A\cup \Bigl\{x\in X\colon \limsup_{n\to \infty}|U_n(x,\mathbf 1_A)| >0\Bigr\}.
\end{equation*}
\notag
$$
If an operator sequence $\{U_n\}$ satisfies (U4), then
$$
\begin{equation*}
\lim_{n\to \infty}U_n(x,\mathbf 1_A)=0\quad\text{ a.e. on } X\setminus A,
\end{equation*}
\notag
$$
so we obtain Lemma 9. Let the operator sequence (1.4) satisfy (U1), (U3) and (U4). If two open sets $A$ and $B$ in $X$ have a finite measure and $\lambda(A)\subset B$, then for any $\varepsilon>0$ there exists an open set $G\subset B$ such that and Proof. Applying Lemma 5, for the open set $B$ we find a regular partition ${\Omega=\{\omega_k\colon k=1,2,\dots\}}$. From Lemma 6 and (3.5) we conclude that $d_B(\omega_k)<\infty$ for $k=1,2,\dots$ . Applying Lemma 4 we find an open subset $G$ of $B$ such that and
$$
\begin{equation}
\mu((G\setminus A)\cap \omega_k)<\varepsilon_k=\frac{\varepsilon}{2^k(1+d_B(\omega_k))}.
\end{equation}
\tag{3.33}
$$
The first relation in (3.30) immediately follows from (3.32). Then we have
$$
\begin{equation}
\mu(G\setminus A)\leqslant \sum_{k=1}^\infty \mu((G\setminus A)\cap \omega_k)<\sum_{k=1}^\infty \frac{\varepsilon}{2^k}=\varepsilon.
\end{equation}
\tag{3.34}
$$
It remains to prove that $\lambda(G)\subset B$. We choose an arbitrary point $x\notin B$. The relation $\lambda(A)\subset B$ implies that Thus, by the linearity of $U_n$ and the pairwise disjointness of the sets $\omega_k$ we can write
$$
\begin{equation}
\begin{aligned} \, \notag \limsup_{n\to\infty }|U_n(x,\mathbf 1_G)| &=\limsup_{n\to\infty }|U_n(x,\mathbf 1_{G\setminus A})| \\ &\leqslant \limsup_{n\to\infty }\sum_{k=0}^\infty |U_n(x,\mathbf 1_{(G\setminus A)\cap \omega_k})|. \end{aligned}
\end{equation}
\tag{3.36}
$$
By property (U3) we have
$$
\begin{equation}
\lim_{n\to\infty}U_n(x,\mathbf 1_{(G\setminus A)\cap \omega_k})=0 \quad\text{for every } k\geqslant 1.
\end{equation}
\tag{3.37}
$$
In addition, by Lemma 6 and (3.33) we can write
$$
\begin{equation}
|U_n(x,\mathbf 1_{(G\setminus A)\cap \omega_k})|\leqslant d_B(\omega_k)\|\mathbf 1_{(G\setminus A)\cap \omega_k}\|_1< \varepsilon_k d_B( \omega_k)\leqslant \frac{\varepsilon}{2^k}.
\end{equation}
\tag{3.38}
$$
Combining (3.36)–(3.38) we obtain that is, $x\notin \lambda(G)$ and therefore $\lambda(G)\subset B$.
The lemma is proved. Lemma 10. Let the operator sequence (1.4) satisfy (U1), (U3) and (U4). If $A$ and $B$ are open sets of finite measure in $(X,\mu)$ such that $\lambda(A)\subset B$, then there exists a family of open sets such that and Proof. We set $G_1=A$ and $G_0=B$ and apply Lemma 9 to the pair of open sets $G_1$, $G_0$. Then we find an open set $G=G_{1/2}$ satisfying (3.30) and use induction. Set
$$
\begin{equation*}
\mathcal D_k=\biggl\{\frac{i}{2^k}\colon 0\leqslant i\leqslant 2^k\biggr\}
\end{equation*}
\notag
$$
and suppose that we have already defined $G_r$ for each $r\in\mathcal{D}_k$ so that (3.40) holds whenever $r,r'\in \mathcal{D}_k$. Applying Lemma 9 to each pair of sets $G_{i/2^k}$, $G_{(i+1)/2^k}$ we obtain intermediate sets $G_{(2i+1)/2^{k+1}}$, $0\leqslant i\leqslant 2^k-1$. Obviously, the new family of sets $\{G_r\colon r\in\mathcal{D}_{k+1}\}$ also satisfies (3.40). Continuing the recursive procedure we finally obtain sets $G_r$ defined for all $r\in\mathcal{D}$ and satisfying (3.40) for the full range of dyadic indexes $r$ and $r'$.
The lemma is proved. Lemma 11. Let the operators (1.4) satisfy (U1)–(U4). If $\varepsilon >0$, $G\subset X$ is an open set of finite measure and $E\subset G$ is a nullset, then there exists an open set $A$ satisfying $E\subset A \subset G$ and a function $h(x)$, $x\in X$, such that Proof. Applying Lemma 5 we find a regular partition $\Omega=\{\omega_k\colon k\in \mathbb{N} \}$ of $G$. Then, using Lemma 4 we define an open set $B$ satisfying
$$
\begin{equation}
\mu(B\cap \omega_k)<\varepsilon_k=\frac{\varepsilon}{2^k(1+d_G(\omega_k))}.
\end{equation}
\tag{3.47}
$$
Applying Lemma 9 we obtain an open set $A$ satisfying $E\subset A$ and $\lambda(A)\subset B$. The bound (3.47) implies that $\mu(B)<\varepsilon $, and so (3.41) holds. According to Lemma 10, there exists a family of open sets $\{G_r\colon r\in \mathcal{D}\}$ satisfying (3.39) and (3.40). Consider the function
$$
\begin{equation}
h(x)= \begin{cases} \sup\{r\colon x\in G_r\} &\text{if } x\in B, \\ 0 &\text{if } x\in X\setminus B. \end{cases}
\end{equation}
\tag{3.48}
$$
Obviously, $h(x)$ satisfies conditions (3.42) and (3.43) and, moreover, $\operatorname{supp} (h) \subset B$. Using Lemma 6 and (3.47), for any $x\in G^{\rm c}$ we obtain
$$
\begin{equation}
|U_n(x,h)| \leqslant \sum_{k=1}^\infty |U_n(x, h\cdot \mathbf 1_{\omega_k})| <\sum_{k=1}^\infty d_G(\omega_k)|B\cap \omega_k|\leqslant \sum_{k=1}^\infty\frac{\varepsilon}{2^k}=\varepsilon,
\end{equation}
\tag{3.49}
$$
which yields (3.44). It remain to check condition (3.45). To this end we consider the function
$$
\begin{equation}
p(x)=\mathbf 1_{G_{r_0}}(x)+\sum_{k=0}^{m-1}r_k\mathbf 1_{G_{r_{k+1}}\setminus G_{r_k}}(x),
\end{equation}
\tag{3.50}
$$
where the dyadic rational numbers $r_k\in \mathcal{D}$ satisfy
$$
\begin{equation}
1=r_0>r_1>\dots>r_m=0\quad\text{and} \quad r_{k}-r_{k+1}<\varepsilon.
\end{equation}
\tag{3.51}
$$
Observe that Indeed, we have and
$$
\begin{equation}
h(x)=p(x)=0, \qquad x\in X\setminus G_0=X\setminus B.
\end{equation}
\tag{3.54}
$$
If $x\in G_0\setminus G_1$, then we have $x\in G_{r_{k+1}}\setminus G_{r_k}$ for some $k=0,1,\dots,m-1$. Then from the definition of $h(x)$ it follows that
$$
\begin{equation}
r_k\geqslant h(x)\geqslant r_{k+1}\quad\text{and} \quad p(x)=r_k.
\end{equation}
\tag{3.55}
$$
Thus we obtain (3.52). From (3.52) and the bound $\|U_n\|_{L^\infty\to M}\leqslant \varrho$ (see property (U2)) it follows that Consider the following cases.
Case 1: $x\in G_0=G_{r_0}$. We have $h(t)=1$ for $t\in G_0$. By Lemma 1 there exists a closed set $F$ such that $x\in F\Subset G_0$. Thus, from property (U3) we obtain Case 2: $x\in B\setminus G_0$. In this case we have for some $k=0,1,\dots ,m-1$. Again, from (U3) it follows that
$$
\begin{equation}
\lim_{n\to\infty}U_n(x,\mathbf 1_{G_{r_i}})=1, \qquad i\geqslant k+1.
\end{equation}
\tag{3.58}
$$
On the other hand, using the property $\lambda(G_{r_i})\subset G_{r_{i+1}}$ we can write
$$
\begin{equation}
\lim_{n\to\infty}U_n(x,\mathbf 1_{G_{r_i}})=0, \qquad i\leqslant k-1,
\end{equation}
\tag{3.59}
$$
whenever $k\geqslant 1$. Hence
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{n\to\infty}U_n(x,\mathbf 1_{G_{r_{i+1}}\setminus G_{r_i}}) \\ &\qquad=\lim_{n\to\infty}U_n(x,\mathbf 1_{G_{r_{i+1}}})- \lim_{n\to\infty}U_n(x,\mathbf 1_{G_{r_i}})=0, \qquad i\in \mathbb N,\quad i\neq k,k-1. \end{aligned}
\end{equation}
\tag{3.60}
$$
Thus, first using (3.55) and (3.60) and then (3.58), (3.59) and the boundedness property (U2), in the case $k>0$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\limsup_{n\to\infty}|U_n(x,p)-p(x)| \\ \notag &\qquad=\limsup_{n\to\infty}|r_{k-1}U_n(x,\mathbf 1_{G_{r_{k}}\setminus G_{r_{k-1}}})+r_kU_n(x,\mathbf 1_{G_{r_{k+1}}\setminus G_{r_k}})-r_k| \\ \notag &\qquad\leqslant \varrho \varepsilon+\limsup_{n\to\infty}|r_{k}U_n(x,\mathbf 1_{G_{r_{k}}\setminus G_{r_{k-1}}})+r_kU_n(x,\mathbf 1_{G_{r_{k+1}}\setminus G_{r_k}})-r_k| \\ &\qquad= \varrho \varepsilon+r_k\limsup_{n\to\infty}|U_n(x,\mathbf 1_{G_{r_{k+1}}}) -U_n(x,\mathbf 1_{G_{r_{k-1}}}) -1| = \varrho \varepsilon. \end{aligned}
\end{equation}
\tag{3.61}
$$
If $k=0$, then we similarly have
$$
\begin{equation}
\limsup_{n\to\infty}|U_n(x,p)-p(x)|\leqslant \varrho \varepsilon+r_0\limsup_{n\to\infty}|U_n(x,\mathbf 1_{G_{r_1}}) -1|= \varrho \varepsilon.
\end{equation}
\tag{3.62}
$$
Combining the last two estimates with (3.56) we conclude that $U_n(x,h)\to h(x)$.
Case 3: $x\in X\setminus B$. From the relations $\lambda(G_{r_i})\subset G_{r_m}=B$, $i=1,2,\dots,m-1$, we have
$$
\begin{equation*}
\lim_{n\to\infty}U_n(x,\mathbf 1_{G_{r_{i+1}}\setminus G_{r_i}})=0,\qquad i=1,2,\dots ,m-2,
\end{equation*}
\notag
$$
and therefore using also (3.50) and the inequality $r_{m-1}<\varepsilon$ (see (3.51)) we obtain
$$
\begin{equation*}
\limsup_{n\to\infty}|U_n(x,p)|=\limsup_{n\to\infty}r_{m-1}|U_n(x,\mathbf 1_{G_{r_m}\setminus G_{r_{m-1}}})|<\varepsilon \varrho,
\end{equation*}
\notag
$$
which completes the proof of (3.45).
Lemma 11 is proved. Lemma 12. If the operators (1.4) satisfy (U1)–(U4), then for every $G_\delta$-nullset $E\subset X$ and any $\varepsilon>0$ there exists a function $g\in M(X)$ satisfying and Proof. We have $E=\bigcap_{k=1}^\infty E_k$, where the $E_k\subset X$ are open sets and we can also suppose that for each $k$. We claim that there exist functions $h_k(x)$, $x\in X$, and open sets $A_k\subset X$, $k=1,2,\dots$, satisfying the following conditions (h1)–(h8):
For any open set $A_k$ there is a regular partition $\Omega_k$ and a function $\nu_k\colon \Omega_k\to\mathbb{N}$ satisfying
We implement this construction by induction. Applying Lemma 11 we find an open set $A_1$ satisfying $E\subset A_1\subset E_1$ and a function $h_1(x)$, $x\in X$, such that
$$
\begin{equation*}
\begin{gathered} \, h_1(x)=1, \qquad x\in A_1, \\ 0\leqslant h_1(x)\leqslant 1, \qquad x\in X, \end{gathered}
\end{equation*}
\notag
$$
and By Lemma 7 there exists a regular partition $\Omega_1$ of the open set $A_1$ and a function $\nu_1\colon \Omega_1\to\mathbb{N}$ such that This gives us the basis of induction. Now suppose that we have already chosen sets $A_k$ and functions $h_k(x)$, satisfying (h1)–(h8) for $k=1,2,\dots,p$. Let $c_k(\omega )$, $\omega\in\Omega_k$, be the constants from (3.23) corresponding to the regular partition $\Omega_k$ of $A_k$, $k=1,2,\dots,p$. For any $l=1,2,\dots,p$ we fix a collection of positive numbers $\{\varepsilon_l(\omega)>0\colon \omega\in \Omega_l\}$ satisfying
$$
\begin{equation}
\sum_{\omega\in \Omega_l}\varepsilon_l(\omega)<\frac{1}{(p+1)^2}.
\end{equation}
\tag{3.68}
$$
Then, applying Lemma 4, for any $l\leqslant p$ we choose an intermediate open set $C_l$ satisfying and
$$
\begin{equation}
|C_l\cap \omega |<\frac{\varepsilon_l(\omega)}{c_l(\omega )}, \qquad \omega \in\Omega_l,\quad l=1,2,\dots ,p.
\end{equation}
\tag{3.70}
$$
Set $C=\bigcap_{l=1}^pC_l$. Thus, applying Lemma 8, for any function $f$ satisfying (3.22) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag |U_{(\Omega_l,\nu_l)}(x,f)| &\leqslant \sum_{\omega \in\Omega_l}c_l(\omega )|C\cap \omega | \\ &\leqslant\sum_{\omega \in\Omega_l}c_l(\omega )|C_l\cap \omega |\leqslant \sum_{\omega \in\Omega_l}\varepsilon_l(\omega)< \frac{1}{(p+1)^2}, \end{aligned}
\end{equation}
\tag{3.71}
$$
which holds for every $x\in X$ and $l\leqslant p$. Then using Lemma 11 we find an open set $A_{p+1}$ and a function $h_{p+1}(x)$, $x\in X$, such that
$$
\begin{equation*}
\begin{gathered} \, E\subset A_{p+1}\subset C\subset A_p\cap E_{p+1}, \\ \operatorname{supp} h_{p+1}\subset C, \qquad h_{p+1}(x)=1,\,x\in A_{p+1}, \\ 0\leqslant h_{p+1}(x)\leqslant 1, \qquad x\in X, \\ |U_n(x,h_{p+1})|\leqslant 2^{-p-1} , \qquad x\in (A_p)^{\mathrm c}, \\ U_n(x,h_{p+1})\to h_{p+1}(x) \quad \text{for all } x\in X. \end{gathered}
\end{equation*}
\notag
$$
Thus we satisfy conditions (h1)–(h5) for $k=p+1$. From (3.71) we can also obtain the inequality
$$
\begin{equation*}
|U_{(\Omega_l,\nu_l)}(x,h_{p+1})|\leqslant \frac{1}{(p+1)^2}, \qquad x\in X, \quad l\leqslant p,
\end{equation*}
\notag
$$
which implies (h8) in the case $k=p+1$. Then, applying Lemma 7 we find a locally finite partition $\Omega_{p+1}$ of $A_{p+1}$ and a mapping $\nu_{p+1}\colon \Omega_{p+1}\to\mathbb{N}$ satisfying (h6) and (h7) for $k=p+1$. This completes the construction. From (h1) we obtain Set
$$
\begin{equation}
g(x)=\sum_{i=1}^\infty (-1)^{i+1}h_i(x) \quad\text{if } x\in X\setminus E,
\end{equation}
\tag{3.72}
$$
and $g(x)=0$ if $x\in E$. Observe that the series (3.72) converges in $L^1(X)$, since from conditions (h1)–(h3) and (3.67) it follows that $\|h_k\|_1<\varepsilon 2^{-k+1}$. Thus, from the boundedness of the operator $U_n\colon L^1(X)\to M(X)$ we conclude that where the series is uniformly convergent. Clearly, (h1) and (3.67) imply (3.63). Fix a point $x\in E^{\rm c}$. Then we have for some $k\geqslant 1$. Taking (h2) into account, this implies that and Thus we obtain
$$
\begin{equation}
g(x)= \sum_{i=1}^k(-1)^{i+1}h_i(x)=\sum_{i=1}^{k-1} (-1)^{i+1}+(-1)^{k+1}h_k(x).
\end{equation}
\tag{3.77}
$$
If $k$ is even, then $g(x)=1-h_k(x)$, and for odd $k$ we have $g(x)=h_k(x)$. From this and property (h3) of the function $h_k(x)$ we obtain condition (3.64). Then condition (h4) implies that Thus, using (3.73), (3.77) and (h5), for $m>k$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \limsup_{n\to\infty}|U_n(x,g)-g(x)| &=\limsup_{n\to\infty} \biggl|\sum_{i=m}^\infty (-1)^{i+1}U_n(x,h_i)\biggr| \\ &\leqslant \sum_{i=m}^\infty|U_n(x,h_i)|\leqslant\sum_{i=m}^\infty 2^{-i}=2^{-m+1}. \end{aligned}
\end{equation*}
\notag
$$
Since $m$ can be arbitrarily large, this implies condition (3.65). Now, to prove (3.66) assume that $x\in E$. Then we have $x\in A_k$ for all $k=1,2,\dots$ . Using this, for any $k=1,2,\dots$ we can fix a set $\omega_k\in\Omega_k$ such that $\omega_k\ni x$. By condition (h8)
$$
\begin{equation}
|U_{\nu_k(\omega_k)}(x,h_i)|\leqslant \frac{1}{i^2}, \qquad i>k,
\end{equation}
\tag{3.79}
$$
and from (h7) it follows that
$$
\begin{equation}
|U_{\nu_k(\omega_k)}(x,h_i)-1|\leqslant \frac{1}{k^2}, \qquad i\leqslant k.
\end{equation}
\tag{3.80}
$$
Thus, using (3.73) we obtain
$$
\begin{equation*}
\begin{aligned} \, \biggl|U_{\nu_k(\omega_k)}(x,g)-\sum_{i=1}^k(-1)^{i+1}\biggr| &\leqslant \sum_{i=1}^k|U_{\nu(\omega_k)}(x,h_i)-1|+\sum_{i=k+1}^\infty |U_{\nu(\omega_k)}(x,h_i)| \\ &\leqslant k \frac{1}{k^2}+\sum_{i=k+1}^\infty \frac{1}{i^2}<\frac{2}{k}. \end{aligned}
\end{equation*}
\notag
$$
Since by (h6) we have $\nu_k(\omega_k)\to\infty$ and the sums $\sum_{i=1}^k(-1)^{k+1}$ take the values $0$ and $1$ alternatively, we obtain (3.66) for any $x\in E$, finalizing the proof of Lemma 12. § 4. Proof of theorems Proof of Theorem 1. Suppose $E$ is a $G_{\delta\sigma}$-nullset and we have $E=\bigcup_{k=1}^\infty E_k$, where the $E_k$ are $G_\delta$-nullsets and we can additionally suppose that $E_k\subset E_{k+1}$. Applying Lemma 12, to each $E_k$ we attach a function $g_k(x)$ such that
We claim that is the required function, where $\varrho$ is the constant from (U2). The boundedness of $f$ follows from condition (g2), and condition (g1) implies that $\mu(\operatorname{supp}(f))<\varepsilon$. Choose an arbitrary point $x\in E$. For some $k$ we have In combination with (g3) and (g4), this implies that
$$
\begin{equation}
\lim_{n\to\infty}U_n(x,g_i)=g_i(x) \quad\text{if } i\leqslant k-1,
\end{equation}
\tag{4.2}
$$
and
$$
\begin{equation}
\delta(x,g_i)\leqslant \sup_n|U_n(x,g_i)|+|g_i(x)|\leqslant \varrho+1, \qquad i=1,2,\dots\,.
\end{equation}
\tag{4.4}
$$
Thus, using also (U2) we obtain
$$
\begin{equation*}
\begin{aligned} \, \delta(x,f) &\geqslant \frac{\delta(x,g_k)}{(4\varrho+53)^k}-\sum_{i=k+1}^\infty \frac{\delta(x,g_i)}{(4\varrho+5)^i} \\ &\geqslant \frac{1}{2(4\varrho+5)^k}-\sum_{i=k+1}^\infty \frac{\varrho+1}{(4\varrho+5)^i} =\frac{1}{4(4\varrho+5)^k}>0, \end{aligned}
\end{equation*}
\notag
$$
which means that the sequence $U_n(x,f)$ diverges for any $x\in E$. Letting $x\notin E$, we have $x\notin E_i$ for any $i\in \mathbb{N}$ and so $U_n(x,g_i)\to g_i(x)$ as $n\to\infty$. This implies that
$$
\begin{equation*}
\begin{aligned} \, \delta(x,f) &=\delta\biggl(x,\sum_{i=k+1}^\infty (4\varrho+5)^{-i}g_i\biggr) \\ &\leqslant \sup_n\biggl|U_n\biggl(x,\sum_{i=k+1}^\infty (4\varrho+5)^{-i}g_i\biggr)\biggr| +\biggl|\sum_{i=k+1}^\infty (4\varrho+5)^{-i}g_i(x)\biggr| \\ &\leqslant (\varrho+1)\sum_{i=k+1}^\infty (4\varrho+5)^{-i}. \end{aligned}
\end{equation*}
\notag
$$
Since the last expression goes to zero, we obtain $\delta(x,f)=0$.
Theorem 1 is proved. Proof of Theorem 2. We make use of the same functions $h_k$ as constructed in the proof of Lemma 12, with the additional bound $\mu(A_k)<\alpha_k$, which can clearly be ensured. Then for sufficiently small numbers $\alpha_k<2^{-k}$ we have and $\mu(\operatorname{supp}(f))<\varepsilon$. By the construction of the functions $h_k$ the original set $E$ can be written in the form $E=\bigcap_kA_k$. Since $\|h_k\|_1<2^{-k+1}$, the series (4.5) converges in $L^1(X)$ and by the boundedness of the operator $U_n\colon L^1\to M$ we can write where the series converges uniformly on $X$. Letting $x\in E^{\rm c}$, we have $x\in A_{k-1}\setminus A_{k}$ for some $k\geqslant 1$ ($A_0=X$). Then we obtain (3.75) and (3.76), and therefore Moreover, for such a point $x\in A_{k-1}\setminus A_{k}$ we have (3.78). Thus, using (h5) in combination with (3.78), (4.6) and (4.7), for $m>k$ we obtain
$$
\begin{equation}
\limsup_{n\to\infty}|U_n(x,f)-f(x)| =\limsup_{n\to\infty}\biggl |\sum_{i=m}^\infty U_n(x,h_i)\biggr| \leqslant \sum_{i=m}^\infty2^{-i}.
\end{equation}
\tag{4.8}
$$
Since $m$ can be arbitrarily large, this implies the convergence condition (3) from the theorem. To prove condition (2), suppose $x\in E$ and so $x\in A_k$ for all $k=1,2,\dots$ . Using this we can fix sets $\omega_k\in \Omega_k$ such that $\omega_k\ni x$. Then using (3.79) and (3.80) we obtain which implies the unbounded divergence of $U_n(x,f)$ at the point $x\in E$.
The theorem is proved. Proof of Theorem 3. We construct open sets $G_k$, $k=1,2,\dots$, together with a regular partition $\Omega_k$ and a mapping $\nu_k\colon \Omega_k\to\mathbb{N}$, satisfying the following conditions:
We do this using induction. We define the open set $G_1$ arbitrarily. By Lemma 7 there exists a regular partition $\Omega_1$ of $G_1$ and a function $\nu_1\colon \Omega_1\to\mathbb{N}$ such that
$$
\begin{equation*}
|U_{(\Omega_1,\nu_1)}(x,\mathbf 1_{G_1})-1|<1, \qquad x\in G_1.
\end{equation*}
\notag
$$
Then suppose that we have already chosen the sets $G_k$ and the corresponding partitions $\Omega_k$ for $k=1,2,\dots,p$. Applying Lemma 4 we can choose open sets $C_l$, $l=1,2,\dots,p$, where $E\subset C_l\subset G_p$, satisfying
$$
\begin{equation*}
\sum_{\omega\in \Omega_l}c_l(\omega )|C_l\cap \omega |<\frac{1}{(p+1)^2}, \qquad l=1,2,\dots ,p,
\end{equation*}
\notag
$$
where $c_l(\omega )$ are the constants from (3.23). Then, setting $G_{p+1}=\bigcap_{l=1}^p C_l$ and applying Lemma 8 we obtain the inequality
$$
\begin{equation}
|U_{(\Omega_l,\nu_l)}(x,\mathbf 1_{G_{p+1}})|\leqslant \sum_{\omega \in\Omega_l}c_l(\omega )|G_{p+1}\cap \omega |\leqslant \frac{1}{(p+1)^2},
\end{equation}
\tag{4.10}
$$
which holds for every $x\in X$ and $l\leqslant p$. This implies (g4) for $k=p+1$. Then applying Lemma 7 we find a regular partition $\Omega_{p+1}$ of $G_{p+1}$ and a mapping $\nu_{p+1}\colon \Omega_{p+1}\to\mathbb{N}$ satisfying (g2) and (g3) for $k=p+1$. This completes the induction. Now we can define the required set by One can check that
$$
\begin{equation}
U_n(x,\mathbf 1_G)=\sum_{k=1}^\infty(-1)^{k+1}U_n(x,\mathbf 1_{G_k}).
\end{equation}
\tag{4.12}
$$
Suppose $x\in E$. Then we have $x\in G_k$, $k=1,2,\dots$ . Using this we can fix $\omega_k\in\Omega_k$ such that $\omega_k\ni x$. By condition (g4) we have
$$
\begin{equation}
|U_{\nu_k(\omega_k)}(x,\mathbf 1_{G_i})|\leqslant \frac{1}{i^2},\qquad i>k,
\end{equation}
\tag{4.13}
$$
and it follows from (g3) that
$$
\begin{equation}
|U_{\nu_k(\omega_k)}(x,\mathbf 1_{G_i})-1|\leqslant \frac{1}{k^2},\qquad i\leqslant k.
\end{equation}
\tag{4.14}
$$
Hence we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \biggl|U_{\nu_k(\omega_k)}(x,f)-\sum_{i=1}^k(-1)^{i+1}\biggr| &\leqslant \sum_{i=1}^k|U_{\nu(\omega_k)}(x,\mathbf 1_{G_i})-1|+\sum_{i=k+1}^\infty |U_{\nu(\omega_k)}(x,\mathbf 1_{G_i})| \\ &\leqslant k \frac{1}{k^2}+\sum_{i=k+1}^\infty \frac{1}{i^2}<\frac{2}{k}, \end{aligned}
\end{equation}
\tag{4.15}
$$
Thus we obtain the divergence of $U_{\nu(\omega_k)}(x,f)$ at all $x\in E$.
The theorem is proved. § 5. Open problemsSome open problems listed here can also be found in [38]–[40] and [7]. 1. Find a complete characterization of the $D$-sets ($UD$-sets) of ordinary trigonometric series and Fourier series of functions in $L^p$, $1\leqslant p\leqslant \infty$, or $C(\mathbb{T})$. Note that Körner [26] constructed in 1961 a $G_\delta$-set that is not the convergence set of any trigonometric series. This example shows that for the partial sums of Fourier or ordinary trigonometric series a pure topological characterization of $C$-$D$-sets can fail to exist (see [38] and [39]). 2. The same problems are also open for Walsh series. 3. An analogue of Kahane–Katznelson’s theorem [16] for Walsh system is an open problem (see [40]). Given a nullset $E$, it requires to construct a continuous function whose Walsh–Fourier series diverges at each point $x\in E$. Concerning this problem, we note that Harris [13] proved that for any compact nullset $e\subset [0,1]$ there exists a continuous function whose Walsh–Fourier series diverges at all $x\in e$. As we noted, the original proof of [16] essentially uses the technique of analytic functions, which hardly can be applied to the Walsh case. A real-function approach to Kahane–Katznelson’s theorem can be found in [21]. 4. Characterize sets that are the radial $D$-sets of univalent functions on the unit disc (see [7]). 5. In [21] the exceptional nullset problem was considered for the Hilbert transform
$$
\begin{equation}
Hf(x)=\lim_{\varepsilon\to 0}\int_{|t-x|>\varepsilon}\frac{f(t)}{x-t}\,dt.
\end{equation}
\tag{5.1}
$$
It is well known that this limit exists almost everywhere whenever $f\in L^1(\mathbb{R})$ (see, for example, [44], § 4.3). It was proved in [21] that for any closed nullset $e\subset \mathbb{R}$ there exists a continuous function $f\in C(\mathbb{R})\cap L^1(\mathbb{R})$ such that the limit in (5.1) does not exist at any point in $e$. We do not know whether $e$ in this statement can be an arbitrary nullset.
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\Bibitem{Kar24} |
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