Ring of Sets in Measure Theory

Question

I'm studying Measure Theory course in my faculty and a have an problem... If i have E, and E is a class of sets like E={A, B, H,...}.

When i can said that: 1- E is a Ring of Sets. 2- E is a Semi-Ring of Sets. 3- E is a Sigma-Ring of Sets.

In other way...
What is the definition and properties of the following: 1- Ring of Sets.
2- Semi-Ring of Sets.
3- Sigma-Ring of Sets.

And what are the conditions that the ring and Semi-Ring are closed under it, i know that the ring of sets is closed under union and intersection.. But this true or not??

Finally, Which is the bigger between them, Is any ring is a Semi-Ring or, Any Semi-Ring is a Ring or WHAT??

Answer 1

This is a case when looking for answers will work before asking questions. All three of these definitions even appear in Wikipedia (paraphrased slightly for consistency):

• Let $\mathcal{R}$ be a nonempty collection of sets. Then $\mathcal{R}$ is a ring of sets if:

  • $A\cup B \in \mathcal{R}$ if $A,B \in \mathcal{R}$
  • $A \setminus B \in \mathcal{R}$ if $A, B \in \mathcal{R}$

• Let $\mathcal{R}$ be a nonempty collection of sets. Then $\mathcal{R}$ is a $\sigma$-ring if:

  • $\bigcup_{n=1}^{\infty} A_{n} \in \mathcal{R}$ if $A_{n} \in \mathcal{R}$ for all $n \in \mathbb{N}$
  • $A \setminus B \in \mathcal{R}$ if $A, B \in \mathcal{R}$

• Let $\mathcal{S}$ be a nonempty collection of sets. Then $\mathcal{S}$ is a semiring of sets if:

  • $\emptyset \in \mathcal{S}$
  • $A \cap B \in \mathcal{S},$ if $A, B \in \mathcal{S}$
  • $A \setminus B = \bigcup_{i=1}^nC_i$ for some disjoint $ C_1,\dots,C_n \in \mathcal{S}$ if $A, B \in \mathcal{S}$

The terminology is meant to suggest that these "containments" are in order: $\sigma$-rings $\subseteq$ rings $\subseteq$ semirings

That $\sigma$-rings are rings is almost too trivial to talk about. The wiki page even gives you your answer that rings are semirings, albeit without explanation, not completely trivial, but still not hard. Just notice that $A\cap B=A\setminus (A\setminus B)$.