7  Semirings and Rings

In the previous section, countably additive, nonnegative functions on rings were extended to such functions on \(\sigma\)-algebras, i.e. measures. Length was shown to be countably additive on the collection \(\mathcal C := \{(a, b]: a < x \le b\}\); however, there is still a missing link since \(\mathcal C\) is not a ring. We will introduce the concept of semirings, which will be useful in extending measures.

7.1 Semirings and its connection

Definition 7.1 For any set \(X\), a collection \(\mathcal D \subset 2^X\) is called a semiring if

  • \(\varnothing \in \mathcal D\)
  • \(A, B \in \mathcal D\) implies
    • \(A \cap B \in \mathcal D\)
    • \(A \setminus B = \bigcup_{1 \le j \le n} C_j\) for some finite \(n\) and disjoint \(C_j \in \mathcal D\)

Proposition 7.1 The collection \(\mathcal C := \{(a, b]: a < x \le b\}\) of half intervals is a semiring.

  • We have \(\varnothing = (a, a] \in \mathcal C\)
  • The intersection of two intervals in \(\mathcal C\) is in \(\mathcal C\).
  • \((a, b] \setminus (c, d]\) is either in \(\mathcal C\) or a union of two disjoint intervals in \(\mathcal C\).

Another kind of semiring will be useful in constructing measures on Cartesian products: if we think of \(\mathcal A\) and \(\mathcal B\) as semirings of intervals, then \(\mathcal D\) will be a semirings of rectangles.

Proposition 7.2 Let \(X = Y \times Z\), let \(\mathcal A\) and \(\mathcal B\) be semirings of subsets of the sets \(Y\) and \(Z\), respectively. Let \(\mathcal D := \{A \times B: A \in \mathcal A, B \in \mathcal B\}\). Then \(\mathcal D\) is a semiring.

First, \(\varnothing = \varnothing\times\varnothing \in \mathcal D\).

For intersections, we have \[ (A \times B) \cap (E \times F) = (A \cap E) \times (B \cap F) \] for any \(A, E \in \mathcal A\) and \(B, F \in \mathcal B\).

For differences, \[ D := (A \times B) \setminus (E \times F) = (A \times B) \setminus ((A \cap E) \times (B \cap F)) \] so in evaluating \(D\) we may assume \(E \subset A\) and \(F \subset B\). Then

\[ \begin{align*} D &= ((A\setminus E)\times B)\cup (E\times(B\setminus F)) \\ &= \bigg(\bigg(\bigcup_{1\le j \le m} G_j\bigg)\times B \bigg)\cup \bigg(E \times \bigg(\bigcup_{1\le k \le n} H_k\bigg) \bigg) \\ &= \bigg(\bigcup_{1\le j \le m}(G_j\times B)\bigg) \cup \bigg(\bigcup_{1\le k \le n}(E \times H_k)\bigg) \end{align*} \] for some finite \(m, n\), disjoint \(G_j\in \mathcal A\) and disjoint \(H_k\in\mathcal B\). Thus \(D\) is a union of sets in \(\mathcal D\), which are disjoint (since \(G_j \cap E = \varnothing\) for all \(j\)).

Note that a semiring \(\mathcal R\) is a ring if for any \(A, B \in \mathcal R\) we have \(A \cup B \in\mathcal R\). Any semiring gives a ring as follows:

Proposition 7.3 For any semiring \(\mathcal D\), let \(\mathcal R\) be the set of all finite disjoint unions of members of \(\mathcal D\). Then \(\mathcal R\) is a ring.

Example 7.1 Any union of two intervals \((a, b]\) and \((c, d]\) either is another left-open, right-closed interval \((\min(a, c), \max(b, d)]\) if the two intervals overlap, or it’s a disjoint union.

For any collection \(\mathcal A \subset 2^X\), the intersection of all rings including \(\mathcal A\) is a ring including \(\mathcal A\), called the ring generated by \(\mathcal A\). If \(\mathcal A\) is a semiring, then the ring generated by it is given by Proposition 7.3.

Proposition 7.4 Let \(\mathcal D\) be ay semiring and \(\alpha\) an additive function from \(\mathcal D\) to \([0, \infty]\). For disjoint \(C_j \in \mathcal D\), let \[ \mu\bigg(\bigcup_{1\le j\le m}C_j\bigg) := \sum_{1 \le j \le m} \alpha(C_j) \] Then \(\mu\) is well-defined and additive on the ring \(\mathcal R\) generated by \(\mathcal D\).

If \(\alpha\) is countably additive on \(\mathcal D\), so is \(\mu\) on \(\mathcal R\), and then \(\mu\) extends to a measure on the \(\sigma\)-algebra generated by \(\mathcal D\) or \(\mathcal R\).

An algebra is a ring \(\mathcal A\) such that the complement of any set in \(\mathcal A\) is in \(\mathcal A\). To get from a ring \(\mathcal R\) to the smallest algebra including it, we clearly have to put in the complements of sets in \(\mathcal R\). This turns out to be enough since, for example, unions of complements are complements of intersections.

Proposition 7.5 Let \(\mathcal R\) be any ring of subsets of a set \(X\). Let \(\mathcal A := \mathcal R \cup \{X \setminus B: B \in \mathcal R\}\). Then \(\mathcal A\) is an algebra.

Clearly \(\varnothing \in \mathcal A\), and \(A \in \mathcal A\) iff \(X \setminus A \in \mathcal A\). Let \(C, D \in \mathcal A\). Then \(C \cap D \in \mathcal A\) if

  • \(C, D \in \mathcal R\), so \(C \cap D \in \mathcal R\);
  • \(C \in \mathcal R\) and \(D = X\setminus B, B \in \mathcal R\), with \(C \cap D = C \setminus B \in \mathcal R\);
  • \(X \setminus C\) and \(X \setminus D \in \mathcal R\), for then \(X \setminus (C \cap D) = (X \setminus C)\cup (X\setminus D)\in\mathcal R\)

Recall our definitions, let \(G: \mathbb R\to \mathbb R\) be any nondecreasing function continuous from the right. On the semiring \(\mathcal C = \{(a, b]: a \le b\}\), we have \(\mu := \mu_G\) defined by \(\mu((a,b]) = G(b) - G(a)\).

The most important special case is when \(G(x) = x\), the identity function. In this case, \(\mu\) will be called \(\lambda\), for Lebesgue.

The ring generated by \(\mathcal C\) from Proposition 7.3 consists of all finite unions

\[ A = \bigcup_{j=1}^n (a_j, b_j], \text{ with } -\infty<a_1\le b_1 < ...\le b_n < \infty \] The complement of \(A\) is

\[ A^c = (-\infty, a_1]\cup (b_1,a_2]\cup ...\cup (b_n, \infty) \]

Let \(\mathcal B\) be the \(\sigma\)-algebra of subsets of \(\mathbb R\) generated by \(\mathcal C\). Note that:

  • Any open interval \((a,b)\) with \(b < \infty\) can be written as the union of intervals \((a, b-1/n]\).
  • The open intervals \((q, r)\), \(q,r\in\mathbb Q\) form a countable base for the topology in \(\mathbb R\).

Thus, all open sets are in \(\mathcal B\). Conversely:

  • Any interval \((a, b]\) is the intersectino of the open intervals \((a, b+1/n)\).

Thus, \(\mathcal B\) is also the \(\sigma\)-algebra generated by the topology of \(\mathbb R\).

Definition 7.2 In any topological space, the \(\sigma\)-algebra generated by the topology is called the Borel \(\sigma\)-algebra. Sets in this \(\sigma\)-algebra are called Borel sets.

Theorem 7.1 For any nondecreasing function \(G\) continuous from the right, \(\mu_G((a, b]) = G(b) - G(a)\) for \(a \le b\), on the collection \(\mathcal C\) of intervals \((a, b]\), \(a \le b\) extends uniquely to a measure on the Borel \(\sigma\)-algebra.

Specifically, the Lebesgue measure \(\lambda\) exists on the Borel \(\sigma\)-algebra in \(\mathbb R\) with \(\lambda((a, b]) = b-a\) for any real \(a \le b\).

7.2 Determining Class

More generally, when is a measure \(\mu\) on a \(\sigma\)-algebra \(\mathcal B\) determined uniquely by its values on the sets in a class \(\mathcal C\subset \mathcal B\)?

If this is true for every measure \(\mu\) which has finite values on all sets in \(\mathcal C\), then \(\mathcal C\) will be called a determining class.

Example 7.2 If \(X\) is a finite set, \(\mathcal B = 2^X\); then the set \(\mathcal C\) of all singletons \(\{x\}\) for \(x \in X\) is a determining class.

Here are two sufficient conditions for a class \(\mathcal C\) to be a determining class.

Theorem 7.2 If \(X\) is a set, \(\mathcal C \subset \mathcal B\) where \(\mathcal B\) is the \(\sigma\)-algebra of subsets of \(X\) generated by \(\mathcal C\), then \(\mathcal C\) is a determining class if

  • \(\mathcal C\) is an algebra, or
  • \(X\) is the union of countably many sets in \(\mathcal C\) and \(\mathcal C\) is a semiring.

One reason for talking about semirings, rings and algebras is that some such assumption is needed for the unique extension or determining class property; for \(\mathcal C\) just to generate \(\mathcal B\) is not enough, even if \(\mathcal C\) only contains two sets.

Example 7.3 Let \(X = \{1,2,3,4\}\) and \(\mathcal B = 2^X\).

Let \(C = \{\{1,2\},\{1,3\}\}\). Then \(\mathcal C\) generated \(\mathcal B\), but it is not a determining class.

Indeed, if \(\mu\{j\} = 1/4\) for \(j = 1,2,3,4\), and \(\alpha\{1\} = \alpha\{4\} = 1/6\), while \(\alpha\{2\} = \alpha\{3\} = 1/3\), then \(\alpha = \mu\) on \(\mathcal C\).