# Definition of a semiring of sets

I'm reading Halmos's Measure Theory and his definition of semiring seems to disagree with the ones that I find on the internet.

Halmos's definition (p. 22):

A semiring is a non empty class $$\mathbf{P}$$ of sets such that

• if $$E\in\mathbf{P}$$ and $$F\in\mathbf{P}$$. then $$E\cap F\in\mathbf{P}$$. and
• if $$E\in\mathbf{P}$$ and $$F\in\mathbf{P}$$ and $$E\subset F$$, then there is a finite class $$\{C_0, C_1, \cdots, C_n\}$$ of sets in $$\mathbf{P}$$ such that $$E=C_0\subset C_1\subset\cdots\subset C_n=F$$ and $$D_i=C_i-C_{i-1}\in\mathbf{P}$$ for $$i=1,\cdots,n$$.

Wikipedia's definition (for example):

A semiring (of sets) is a non-empty collection $$S$$ of sets such that

1. $$\emptyset \in S$$

2. If $$E\in S$$ and $$F\in S$$ then $$E\cap F\in S$$.

3. If $$E\in S$$ and $$F\in S$$ then there exists a finite number of mutually disjoint sets $$C_{i}\in S$$ for $$i=1,\ldots ,n$$ such that $$E\setminus F=\bigcup _{i=1}^{n}C_{i}$$.

These definitions are not equivalent! For example, the collection $$\{\emptyset,\{a\},\{b\}, \{c\}, \{a,b,c\}\}$$ is a semiring under the second definition, but not the first.

Questions:

1. History question: why does Halmos use a different definition than we do today? Was the definition weakened at some point in order to be more general?
2. Math question: what are the advantages/disadvantages of these two definitions from the standpoint of measure theory?
• Also, the collection of products of half-open intervals $\Pi_{i=1}^n[a_i, b_i)\subset\mathbb{R}^n$ is not a semiring in Halmos's definition and it is in the modern definition. This really seems to favor the modern definition. – Joshua Meyers Oct 21 '18 at 16:04
• One advantage of Halmos' definition is that additive implies finitely-additive. This is not the case with the modern definition. See math.stackexchange.com/q/1414094 and math.stackexchange.com/q/3493674 – JasonJones Jan 1 at 0:33
• To add to the comment above, if $n > 1$, then the collection of products of left-closed right-open intervals $\prod_{i=1}^{n} [a_i,b_i) \subseteq \mathbb{R}^n$ is not a semiring in Halmos' definition, but it is in the modern definition. However, if $n=1$, then it is a semi-ring in both definitions. – JasonJones Jan 1 at 0:42
• As stated in the question, the collection $\{\emptyset,\{a\},\{b\}, \{c\}, \{a,b,c\}\}$ is a semi-ring under the modern definition but not under Halmos' definition. However, $\{\emptyset,\{a\},\{b\}, \{c\} \}$ is a semi-ring under both definitions. – JasonJones Jan 1 at 0:48