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Can anyone explain the connections among ring, semiring, algebra, sigma-algebra in the scope of measure theory and why are these concepts important?

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A Ring is a family of sets $\mathcal{R}$ such that:

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

A Semiring is a non-empty collection $\mathcal{S}$ of sets such that:

  • $\emptyset \in \mathcal{S}$
  • If $E \in \mathcal{S}$ and $F \in \mathcal{S}$ then $E \cap F \in \mathcal{S}$.
    • If $E \in \mathcal{S}$ and $F \in \mathcal{S}$, then there is a sequence of mutually disjoint sets $C_i \in S$ for $i=1,\ldots,n$ such that $E \setminus F = \bigcup_{i=1}^n C_i$.

An Algebra is a non-empty collection $\mathcal{A}$ of subsets of $X$ such that: - If $E \in S$ and $F \in S$ then there exists a finite number of

  • $X \in \mathcal{A}$
  • If $A, B \in \mathcal{A}$, then $A\setminus B \in \mathcal{A}$.

A $\sigma$-algebra is a non-empty collection $\mathcal{F}$ of subsets such that:

  • $\emptyset \in \mathcal{F}$
  • If $E \in \mathcal{F}$, then $E^C \in \mathcal{F}$
  • If $E_1, E_2, \dots$ is a collection of sets in $\mathcal{F}$, then $\cup_{i=1}^{\infty}E_i \in \mathcal{F}$.

The point of these structures is that you can work with from the simpliest structure ultil the "richest" structure. There are some theorems that construct one structure from other.

The concept of $\sigma$-algebra is essential to understand many others concepts in measure theory and in probability!

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  • $\begingroup$ Thanks for the intuition. I do know the definition of these terms. I guess what I am asking is the connection among them. You mentioned that "you can work with from the simpliest structure ultil the "richest" structure". How is that accomplished? $\endgroup$ – user1559897 May 19 '16 at 0:56

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