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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

1 Answers1

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According to Samuel James Taylor (Introduction to Measure and Integration) the first definition is due to Von Neumann, who was the creator of the concept, however over time authors realized that the weaker condition 3. is not only more general, but also easier to work with.

The only disadvantage of the current definition is that an additive function defined on a semi-ring need not be finitely additive. This isn't a big issue because the only cases that matter are those where the generated outer measure is a genuine extension, so that you can get a measure by Carathéodory.

wannabecapablanca
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