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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?
Joshua Meyers
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  • 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 https://math.stackexchange.com/q/1414094/ and https://math.stackexchange.com/q/3493674/ – JasonJones Jan 01 '20 at 00:33
  • 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 01 '20 at 00:42
  • 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 01 '20 at 00:48

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