# On the Causality and K-Causality between Measures

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Theorem 1**(Minguzzi).

## 2. Results

**Definition**

**2.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**2.**

- 1
^{●} - $\mu {\u2aaf}_{K}\nu $.
- 2
^{●} - For any compact subset $\phantom{\rule{4pt}{0ex}}C\subseteq \mathcal{M}$,$$\begin{array}{c}\hfill \mu \left({K}^{+}\left(C\right)\right)\le \nu \left({K}^{+}\left(C\right)\right).\end{array}$$
- 3
^{●} - For any Borel subset $\mathcal{X}\subseteq \mathcal{M}$ such that ${K}^{+}\left(\mathcal{X}\right)\subseteq \mathcal{X}$,$$\begin{array}{c}\hfill \mu \left(\mathcal{X}\right)\le \nu \left(\mathcal{X}\right).\end{array}$$
- 4
^{●} - For any time function $\mathcal{T}$ and any $\alpha \in \mathbb{R}$,$$\begin{array}{c}\hfill \mu \left({\mathcal{T}}^{-1}\left((\alpha ,+\infty )\right)\right)\le \nu \left({\mathcal{T}}^{-1}\left((\alpha ,+\infty )\right)\right).\end{array}$$
- 5
^{●} - For any bounded time function $\mathcal{T}$,$$\begin{array}{c}\hfill {\int}_{\mathcal{M}}\mathcal{T}d\mu \le {\int}_{\mathcal{M}}\mathcal{T}d\nu .\end{array}$$

**Theorem 3**(Suhr).

- (i)
- $\mu {\u2aaf}_{K}\nu $.
- (ii)
- For any Borel $\mathcal{B}\subseteq \mathcal{M}$,$$\begin{array}{c}\hfill \mu \left(\mathcal{B}\right)\le \nu \left({K}^{+}\left(\mathcal{B}\right)\right)\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\mu \left({K}^{-}\left(\mathcal{B}\right)\right)\ge \nu \left(\mathcal{B}\right).\end{array}$$

**Lemma**

**1.**

**Proof of Theorem**

**2.**

- 3
^{′•} - For any Borel subset $\mathcal{Y}\subseteq \mathcal{M}$ such that ${K}^{-}\left(\mathcal{Y}\right)\subseteq \mathcal{Y}$,$$\begin{array}{c}\hfill \mu \left(\mathcal{Y}\right)\ge \nu \left(\mathcal{Y}\right).\end{array}$$

**Remark**

**1.**

**Proof.**

**Remark**

**2.**

- 4
^{′•} - For any time function $\mathcal{T}$ and any $\alpha \in \mathbb{R}$,$$\begin{array}{c}\hfill \mu \left({\mathcal{T}}^{-1}\left([\alpha ,+\infty )\right)\right)\le \nu \left({\mathcal{T}}^{-1}\left([\alpha ,+\infty )\right)\right).\end{array}$$

**Proof.**

## 3. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A

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Miller, T. On the Causality and *K*-Causality between Measures. *Universe* **2017**, *3*, 27.
https://doi.org/10.3390/universe3010027

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Miller T. On the Causality and *K*-Causality between Measures. *Universe*. 2017; 3(1):27.
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Miller, Tomasz. 2017. "On the Causality and *K*-Causality between Measures" *Universe* 3, no. 1: 27.
https://doi.org/10.3390/universe3010027