Weak ergodic theorem for Markov chains in the absence of invariant countably additive measures

General Markov chains (MC) with a countably additive transition probability on a normal topological space are considered. We extend the Markov operator from the traditional space of countably additive measures to the space of finitely additive measures. We study the Cesaro means for the Markov sequence of measures and their asymptotic behavior in the weak topology generated by the space of bounded continuous functions. It is proved ergodic theorem that in order for the Cesaro means to converge weakly to some bounded regular finitely additive (or countably additive) measure, it is necessary and sufficient that all invariant finitely additive measures (such always exist) are not separable from the limit measure in the weak topology. Moreover, the limit measure may not be invariant for a MC, and may not be countably additive. The corresponding example is given and studied in detail. Key words: Markov chain, Markov operators, Cesaro means, weak ergodic theorem, absence of invariant countably additive measures, invariant finitely additive measure, purely finitely additive measures.Comment: 15 pages, 1 figur

The inapproximability for the (0,1)-additive number

An {\it additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ is joined to $y$). The {\it additive number} of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. The {\it additive choosability} of a graph $G$, denoted by $\eta_{\ell}(G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone (2012) \cite{a80} conjectured that for every graph $G$, $\eta(G)= \eta_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_{\ell}(G)- \eta(G) \geq k$. A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\{0,1\}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is $\mathbf{NP}$-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in V(G)}\ell(v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\varepsilon >0$, approximating the $(0,1)$-additive number within $n^{1-\varepsilon}$ is $\mathbf{NP}$-hard.Comment: 14 pages, 3 figures, Discrete Mathematics & Theoretical Computer Scienc