# Approximating Information Measures for Fields

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Theorem**

**1.**

- 1.
- $I(\mathcal{A};\mathcal{B}|\mathcal{C})=I(\mathcal{A};\sigma \left(\mathcal{B}\right)\left|\mathcal{C}\right)=I(\mathcal{A};\mathcal{B}|\sigma \left(\mathcal{C}\right))$ (invariance of completion);
- 2.
- $I(\mathcal{A};\mathcal{B}\wedge \mathcal{C}|\mathcal{D})=I(\mathcal{A};\mathcal{B}\left|\mathcal{D}\right)+I(\mathcal{A};\mathcal{C}|\mathcal{B}\wedge \mathcal{D})$ (chain rule).

## 2. Proofs

**Theorem**

**2.**

- 1.
- $I(\mathcal{A};\mathcal{B}|\mathcal{C})=I(\mathcal{B};\mathcal{A}\left|\mathcal{C}\right)$;
- 2.
- $I(\mathcal{A};\mathcal{B}|\mathcal{C})\ge 0$ with the equality if and only if $P(A\cap B|\mathcal{C})=P(A\left|\mathcal{C}\right)P\left(B\right|\mathcal{C})$ almost surely for all $A\in \mathcal{A}$ and $B\in \mathcal{B}$;
- 3.
- $I(\mathcal{A};\mathcal{B}|\mathcal{C})\le min(H\left(\mathcal{A}\right|\mathcal{C}),H(\mathcal{B}\left|\mathcal{C}\right))$;
- 4.
- $I(\mathcal{A};{\mathcal{B}}_{1}|\mathcal{C})\le I(\mathcal{A};{\mathcal{B}}_{2}|\mathcal{C})$ if ${\mathcal{B}}_{1}\subset {\mathcal{B}}_{2}$;
- 5.
- $I(\mathcal{A};{\mathcal{B}}_{n}|\mathcal{C})\uparrow I(\mathcal{A};\mathcal{B}|\mathcal{C})$ for ${\mathcal{B}}_{n}\uparrow \mathcal{B}$.

**Theorem**

**3**

**.**For any field $\mathcal{K}$ and any event $G\in \sigma \left(\mathcal{K}\right)$, there is a sequence of events ${K}_{1},{K}_{2},\cdots \in \mathcal{K}$ such that

**Proof.**

- We have $\mathsf{\Omega}\in \mathcal{K}$. Hence, $\mathsf{\Omega}\in \mathcal{G}$.
- For $A\in \mathcal{G}$, consider ${K}_{1},{K}_{2},\cdots \in \mathcal{K}$ such that ${lim}_{n\to \infty}P(A\u25b5{K}_{n})=0$. Then, $A\u25b5{K}_{n}={A}^{c}\u25b5{K}_{n}^{c}$, where ${K}_{1}^{c},{K}_{2}^{c},\cdots \in \mathcal{K}$. Hence, ${A}^{c}\in \mathcal{G}$.
- For ${A}_{1},{A}_{2},\cdots \in \mathcal{G}$, consider events ${K}_{i}^{n}\in \mathcal{K}$ such that $P({A}_{i}\u25b5{K}_{i}^{n})\le {2}^{-n}$. Then,$$\begin{array}{c}\hfill P\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">\bigcap _{i=1}^{n}{A}_{i}\u25b5\left(\right)open="("\; close=")">\bigcap _{i=1}^{n}{K}_{i}^{i+n}\\ \le \sum _{i=1}^{n}P({A}_{i}\u25b5{K}_{i}^{i+n})\le {2}^{-n}.\end{array}$$$$\begin{array}{c}\hfill P\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">\bigcap _{i=1}^{\infty}{A}_{i}\u25b5\left(\right)open="("\; close=")">\bigcap _{i=1}^{n}{A}_{i}\\ =P\left(\right)open="("\; close=")">\bigcap _{i=1}^{n}{A}_{i}\end{array}.$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& P\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">\bigcap _{i=1}^{\infty}{A}_{i}\u25b5\left(\right)open="("\; close=")">\bigcap _{i=1}^{n}{K}_{i}^{i+n}\hfill \end{array}$$

**Theorem**

**4**

**.**Fix an $\u03f5\in (0,{e}^{-1}]$ and a field $\mathcal{C}$. For finite partitions $\alpha ={\left(\right)}_{{A}_{i}}^{}i=1I$ and ${\alpha}^{\prime}={\left(\right)}_{{A}_{i}^{\prime}}^{}i=1I$ such that $P({A}_{i}\u25b5{A}_{i}^{\prime})\le \u03f5$ for all $i\in \left(\right)open="\{"\; close="\}">1,\cdots ,I$, we have

**Proof.**

**Proof**

**of**

**Theorem**

**1.**

**1 (invariance of completion):**Consider some measurable fields $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$. We are going to demonstrate

**Theorem**

**5**

**.**Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{D}$ be subfields of $\mathcal{J}$. We have

**Proof.**

**Theorem**

**6**

**.**Let $\alpha ={\left(\right)}_{{A}_{i}}^{}i=1I$ be a finite partition and let $\mathcal{C}$ be a field. For each $\u03f5>0$, there exists a finite partition ${\gamma}^{\prime}\subset \sigma \left(\mathcal{C}\right)$ such that for any partition $\gamma \subset \sigma \left(\mathcal{C}\right)$ finer than ${\gamma}^{\prime}$ we have

**Proof.**

**Proof**

**of**

**Theorem**

**1.**

**2 (chain rule):**Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{D}$ be arbitrary fields, and let $\alpha $, $\beta $, $\gamma $, and $\delta $ be finite partitions. The point of our departure is the chain rule for finite partitions [9] (Equation 2.60)

## 3. Applications

## Funding

## Conflicts of Interest

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Dębowski, Ł.
Approximating Information Measures for Fields. *Entropy* **2020**, *22*, 79.
https://doi.org/10.3390/e22010079

**AMA Style**

Dębowski Ł.
Approximating Information Measures for Fields. *Entropy*. 2020; 22(1):79.
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**Chicago/Turabian Style**

Dębowski, Łukasz.
2020. "Approximating Information Measures for Fields" *Entropy* 22, no. 1: 79.
https://doi.org/10.3390/e22010079