# Coincidences of the Concave Integral and the Pan-Integral

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

(1) | $\mu $(∅) = 0; | (vanishing at ∅) |

(2) | $\mu (A)\le \mu (B)$ whenever $A\subset B$ and $A,B\in \mathcal{F}$. | (monotonicity) |

- (i)
- subadditive if $\mu (A\cup B)\le \mu (A)+\mu (B)$ for any $A,B\in \mathcal{A}$;
- (ii)
- superadditive if $\mu (A\cup B)\ge \mu (A)+\mu (B)$ for any $A,B\in \mathcal{A}$ such that $A\cap B=$ ∅ [19];
- (iii)
- (iv)
- continuous from below (resp. from above), if ${lim}_{n\to \infty}\mu ({E}_{n})=\mu (E)$ whenever ${E}_{n}\nearrow E$ (resp. whenever ${E}_{n}\searrow E$ and $\mu ({E}_{1})<\infty $) ([20]).

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 3. The Main Results

**Theorem**

**5.**

**Theorem**

**6.**

- (i)
- if μ is superadditive, then ${\mathbf{Pan}}_{\mu}\le {\mathbf{Ch}}_{\mu}$, i.e., for each $f\in {\mathbf{F}}_{+}$, ${\mathbf{Pan}}_{\mu}(f)\le {\mathbf{Ch}}_{\mu}(f)$;
- (ii)
- if μ is subadditive, then ${\mathbf{Pan}}_{\mu}\ge {\mathbf{Ch}}_{\mu}$.

**Theorem**

**7.**

**Proof.**

**Remark**

**8.**

**Theorem**

**9.**

**Proof.**

**Example**

**10.**

**Theorem**

**11.**

**Proof.**

## 4. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Ouyang, Y.; Li, J.; Mesiar, R.
Coincidences of the Concave Integral and the Pan-Integral. *Symmetry* **2017**, *9*, 90.
https://doi.org/10.3390/sym9060090

**AMA Style**

Ouyang Y, Li J, Mesiar R.
Coincidences of the Concave Integral and the Pan-Integral. *Symmetry*. 2017; 9(6):90.
https://doi.org/10.3390/sym9060090

**Chicago/Turabian Style**

Ouyang, Yao, Jun Li, and Radko Mesiar.
2017. "Coincidences of the Concave Integral and the Pan-Integral" *Symmetry* 9, no. 6: 90.
https://doi.org/10.3390/sym9060090