# Decomposition of Finitely Additive Markov Chains in Discrete Space

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## Abstract

**:**

## 1. Introduction

## 2. Definitions, Notation and Some Information

- $ba(X,\Sigma )$ is the space of finitely additive measures, and
- $ca(X,\Sigma )$ is the space of countably additive measures.
- If $\mu \ge 0$, then norm $\left|\right|\mu \left|\right|=\mu \left(X\right)$.

**Definition**

**1**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Remark**

**1.**

**Theorem**

**1**

- ${V}_{ba}=\{\mu \in ba(X,\Sigma ):\mu \left(X\right)\le 1\},$
- ${V}_{ca}=\{\mu \in ca(X,\Sigma ):\mu \left(X\right)\le 1\},$
- ${V}_{pfa}=\{\mu \in pfa(X,\Sigma ):\mu \left(X\right)\le 1\}.$

**Definition**

**2.**

- 1.
- $0\le P(x,E)\le 1,\forall x\in X,\forall E\in \Sigma $;
- 2.
- $P(\xb7,E)\in B(X,\Sigma ),\forall E\in \Sigma $;
- 3.
- $P(x,\xb7)\in ca(X,\Sigma ),\forall x\in X$;
- 4.
- $P(x,X)=1,\forall x\in X.$

- $T:B(X,\Sigma )\to B(X,\Sigma ),\left(Tf\right)\left(x\right)=Tf\left(x\right)=\underset{X}{\int}f\left(y\right)P(x,dy),$
- $\forall f\in B(X,\Sigma ),\forall x\in X;$
- $A:ca(X,\Sigma )\to ca(X,\Sigma ),\left(A\mu \right)\left(E\right)=A\mu \left(E\right)=\underset{X}{\int}P(x,E)\mu \left(dx\right),$
- $\forall \mu \in ca(X,\Sigma ),\forall E\in \Sigma .$

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**2.**

**Definition**

**5.**

- (${3}^{\prime}$) $P(x,\xb7)\in ba(X,\Sigma ),\forall x\in X$, and
- (${3}^{\u2033}$) $P(x,\xb7)\in pfa(X,\Sigma ),\forall x\in X$, respectively.

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

## 3. Finitely Additive Markov Kernels in Discrete Space

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**2.**

**Remark**

**3.**

**Definition**

**8.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

- $P(x,\left\{{x}_{n}\left(x\right)\right\})={\alpha}_{n}\left(x\right)>0,n\in N$,
- and ${\sum}_{n}{\alpha}_{n}\left(x\right)=P(x,D\left(x\right))=P(x,X),P(x,X\setminus D\left(x\right))=0$.

**Corollary**

**1.**

**Example**

**1.**

**Example**

**2.**

**Corollary**

**2.**

**Example**

**3.**

**Theorem**

**6.**

**Proof.**

**Proposition**

**2.**

**Corollary**

**3.**

- 1.
- ${A}_{ca}\xb7{A}_{ca}\left[ca(X,{\Sigma}_{d})\right]\subset ca(X,{\Sigma}_{d});$
- 2.
- ${A}_{pfa}\xb7{A}_{pfa}\left[ba(X,{\Sigma}_{d})\right]\subset pfa(X,{\Sigma}_{d});$
- 3.
- ${A}_{pfa}\xb7{A}_{ca}\left[ba(X,{\Sigma}_{d})\right]\subset pfa(X,{\Sigma}_{d}).$

**Remark**

**4.**

## 4. Invariant Measures of Markov Operators

**Theorem**

**7.**

**Theorem**

**8.**

**Proof.**

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Example**

**4.**

- ${P}_{ca}(x,E)=\frac{1}{2}{\delta}_{0}\left(E\right)$ for all $x\in X$ and $E\subset X$, where ${\delta}_{0}$ is the Dirac at point 0;
- ${P}_{pfa}(x,E)=\frac{1}{2}\eta \left(E\right)$ for all $x\in X$ and $E\subset X$, where $\eta $ is some fixed purely finitely additive measure from ${S}_{pfa}$. For clarity, we take the measure $\eta $ from the family of purely finitely additive measures satisfying the condition $\eta \left(\right(0,\epsilon \left)\right)=1$ for any $\epsilon >0$.

**Example**

**5.**

## 5. Norms of Components in the Decomposition of a Markov Sequence of Measures and Their Asymptotic Behavior

**Remark**

**5.**

**Theorem**

**11.**

**Proof.**

**Remark**

**6.**

**Corollary**

**4.**

**Remark**

**7.**

**Theorem**

**12.**

**Proof.**

**Remark**

**8.**

**Corollary**

**5.**

**Corollary**

**6.**

**Remark**

**9.**

**Remark**

**10.**

**Theorem**

**13.**

- 1.
- the condition $\left({H}_{1}\right)$ is satisfied, and
- 2.
- the assertion of Theorem 11 is true.

**Proof.**

**Theorem**

**14.**

- 1.
- the condition $\left({H}_{2}\right)$ is satisfied, and
- 2.
- the assertion of Theorem 12 is true.

**Proof.**

**Remark**

**11.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Zhdanok, A.; Khuruma, A.
Decomposition of Finitely Additive Markov Chains in Discrete Space. *Mathematics* **2022**, *10*, 2083.
https://doi.org/10.3390/math10122083

**AMA Style**

Zhdanok A, Khuruma A.
Decomposition of Finitely Additive Markov Chains in Discrete Space. *Mathematics*. 2022; 10(12):2083.
https://doi.org/10.3390/math10122083

**Chicago/Turabian Style**

Zhdanok, Alexander, and Anna Khuruma.
2022. "Decomposition of Finitely Additive Markov Chains in Discrete Space" *Mathematics* 10, no. 12: 2083.
https://doi.org/10.3390/math10122083