Understanding Measure Theory with Rudin's Principles of Mathematical Analysis

[OhMyMarkov]

Hello everyone, I needed to know more about measure theory so I'm reading in Rudin's Principle's of Mathematical Analysis, somewhere in the chapter, he says:

We let E denote the family of all elementary subsets of $R^p$... E is a ring, but not a $\sigma$-ring.

According to my understanding of what a $\sigma$-ring is, it is the union of infinitely many sets $A_i$, each belongs to this ring. In "slang" mathematical terms, given any subset in $R^p$, we can describe it as the union of infinitely many subsets in $R^p$.

Any help is appreciated!

 

[tarantella]

How are elementary subsets defined?

 

[Evgeny.Makarov]

According to my understanding of what a $\sigma$-ring is, it is the union of infinitely many sets $A_i$, each belongs to this ring.

First, you probably mean the family of such unions. Each union is a subset of the universal set, while a $\sigma$-ring is a family of subsets of the universal set. Second, you are describing a $\sigma$-ring generated by a ring. The standard definition just says that $\sigma$-ring is closed under countable unions.

In "slang" mathematical terms, given any subset in $R^p$, we can describe it as the union of infinitely many subsets in $R^p$.

This is too "slang" and does not give much information. You need to specify which subsets are representable as countable unions and which subsets participate in the union. Of course, any subset is the union of infinitely many copies of itself, or the union of singletons of all its elements.

Finally, what is your question?

 

[OhMyMarkov]

How are elementary subsets defined?

I don't know to be honest, it's the first time I come across this term. I tried searching for them too.My question is, why is it a ring, but not a $\sigma$-ring?

 

[Evgeny.Makarov]

I don't know to be honest, it's the first time I come across this term. I tried searching for them too.

My question is, why is it a ring, but not a $\sigma$-ring?

So, you are trying to prove something about a concept whose definition you don't know? Not a good idea...

My guess is that elementary sets are finite unions of rectangles. They form a ring because the family of elementary sets are closed under finite union and relative complements. However, they are not closed under countable unions and intersections. For one, the whole plane is a countable union of finite rectangles, but it has infinite area. For another, the set of rational number inside a segment and a circle can be formed from rectangles using countable unions and intersections.