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I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is what I had in mind.

Suppose that $(G,+,\cdot,0,1)$ is a compact semiring where $+$ and $\cdot$ are commutative and $X$ is a compact totally disconnected space. Let $\mathfrak{C}(X)$ be the algebra of all clopen sets. A $G$-valued measure on $X$ is a function $\mu:\mathcal{C}(X)\rightarrow G$ such that $\mu(\emptyset)=0$ and $\mu(A\cup B)=\mu(A)+\mu(B)$ whenever $A\cap B=\emptyset$.

1. If $f:X\rightarrow G$, and $L\in G$, then define $\int fd\mu=L$ if for each neighborhood $U$ of $L$ there is some partition $C_{1},...,C_{n}$ of $X$ into clopen sets where if $D_{1},...,D_{m}$ is a partition of $X$ into clopen sets that refines $C_{1},...,C_{n}$ and $x_{1}\in D_{1},...,x_{m}\in D_{m}$, then $\mu(D_{n})x_{1}+...+\mu(D_{n})x_{n}\in U$. A function $f:X\rightarrow G$ is said to be integrable if the limit exists. Are there some necessary and sufficient conditions on the semiring $G$ for which every continuous function $f:X\rightarrow G$ is integrable? What about when one slightly changes the definition of integrability? I am interested in a version of an integral or a kind of semiring where every continuous function is integrable.

2. Does anyone know of any reference related to integration for semiring valued functions?

I am interested in integration on semirings because measures and integration on a particular commutative semiring have popped up in my research (which up until recently had absolutely nothing to do with any kind of measures and integration). I would be interested in any information on measures and integration on rings even if it is not exactly the notion of integration and measures I had in mind in my research.

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is what I had in mind.

Suppose that $(G,+,\cdot,0,1)$ is a compact semiring where $+$ and $\cdot$ are commutative and $X$ is a compact totally disconnected space. Let $\mathfrak{C}(X)$ be the algebra of all clopen sets. A $G$-valued measure on $X$ is a function $\mu:\mathcal{C}(X)\rightarrow G$ such that $\mu(\emptyset)=0$ and $\mu(A\cup B)=\mu(A)+\mu(B)$ whenever $A\cap B=\emptyset$.

1. If $f:X\rightarrow G$, and $L\in G$, then define $\int fd\mu=L$ if for each neighborhood $U$ of $L$ there is some partition $C_{1},...,C_{n}$ of $X$ into clopen sets where if $D_{1},...,D_{m}$ is a partition of $X$ into clopen sets that refines $C_{1},...,C_{n}$ and $x_{1}\in D_{1},...,x_{m}\in D_{m}$, then $\mu(D_{n})f(x_{1})+...+\mu(D_{n})f(x_{n})\in U$. A function $f:X\rightarrow G$ is said to be integrable if the limit exists. Are there some necessary and sufficient conditions on the semiring $G$ for which every continuous function $f:X\rightarrow G$ is integrable? What about when one slightly changes the definition of integrability? I am interested in a version of an integral or a kind of semiring where every continuous function is integrable.

2. Does anyone know of any reference related to integration for semiring valued functions?

I am interested in integration on semirings because measures and integration on a particular commutative semiring have popped up in my research (which up until recently had absolutely nothing to do with any kind of measures and integration). I would be interested in any information on measures and integration on rings even if it is not exactly the notion of integration and measures I had in mind in my research.

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is what I had in mind.

Suppose that $(G,+,\cdot,0,1)$ is a compact semiring where $+$ and $\cdot$ are commutative and $X$ is a compact totally disconnected space. Let $\mathfrak{C}(X)$ be the algebra of all clopen sets. A $G$-valued measure on $X$ is a function $\mu:\mathcal{C}(X)\rightarrow G$ such that $\mu(\emptyset)=0$ and $\mu(A\cup B)=\mu(A)+\mu(B)$ whenever $A\cap B=\emptyset$.

1. If $f:X\rightarrow G$, and $L\in G$, then define $\int fd\mu=L$ if for each neighborhood $U$ of $L$ there is some partition $C_{1},...,C_{n}$ of $X$ into clopen sets where if $D_{1},...,D_{m}$ is a partition of $G$ into clopen sets that refines $C_{1},...,C_{n}$ and $x_{1}\in D_{1},...,x_{m}\in D_{m}$, then $\mu(D_{n})x_{1}+...+\mu(D_{n})x_{n}\in U$. A function $f:X\rightarrow G$ is said to be integrable if the limit exists. Are there some necessary and sufficient conditions on the semiring $G$ for which every continuous function $f:X\rightarrow G$ is integrable? What about when one slightly changes the definition of integrability? I am interested in a version of an integral or a kind of semiring where every continuous function is integrable.

2. Does anyone know of any reference related to integration for semiring valued functions?

I am interested in integration on semirings because measures and integration on a particular commutative semiring have popped up in my research (which up until recently had absolutely nothing to do with any kind of measures and integration). I would be interested in any information on measures and integration on rings even if it is not exactly the notion of integration and measures I had in mind in my research.

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is what I had in mind.

Suppose that $(G,+,\cdot,0,1)$ is a compact semiring where $+$ and $\cdot$ are commutative and $X$ is a compact totally disconnected space. Let $\mathfrak{C}(X)$ be the algebra of all clopen sets. A $G$-valued measure on $X$ is a function $\mu:\mathcal{C}(X)\rightarrow G$ such that $\mu(\emptyset)=0$ and $\mu(A\cup B)=\mu(A)+\mu(B)$ whenever $A\cap B=\emptyset$.

1. If $f:X\rightarrow G$, and $L\in G$, then define $\int fd\mu=L$ if for each neighborhood $U$ of $L$ there is some partition $C_{1},...,C_{n}$ of $X$ into clopen sets where if $D_{1},...,D_{m}$ is a partition of $X$ into clopen sets that refines $C_{1},...,C_{n}$ and $x_{1}\in D_{1},...,x_{m}\in D_{m}$, then $\mu(D_{n})x_{1}+...+\mu(D_{n})x_{n}\in U$. A function $f:X\rightarrow G$ is said to be integrable if the limit exists. Are there some necessary and sufficient conditions on the semiring $G$ for which every continuous function $f:X\rightarrow G$ is integrable? What about when one slightly changes the definition of integrability? I am interested in a version of an integral or a kind of semiring where every continuous function is integrable.

2. Does anyone know of any reference related to integration for semiring valued functions?

I am interested in integration on semirings because measures and integration on a particular commutative semiring have popped up in my research (which up until recently had absolutely nothing to do with any kind of measures and integration). I would be interested in any information on measures and integration on rings even if it is not exactly the notion of integration and measures I had in mind in my research.