Understanding Measure Theory: Countably Additive Functions and σ-Algebras

[woundedtiger4]

question 1: if f is a countably additive set function (probability measure) defined on σ-algebra A of subsets of S, then which of the probability space "(f, A, S) is called events?
question 2: define what we mean by algebra and σ-algebra? for this question in the second part do we have to write the definition & properties part of http://en.wikipedia.org/wiki/Sigma-algebra#Definition_and_properties or something else?
Plus, can anyone please help me that what is countably additive?

 

[HallsofIvy]

These questions are asking you to apply basic definitions. Do you know the definitions?

For one, as set function, f, is said to be "countably additive" if and only if, for every countable collection, $\{A_i\}$, of disjoint sets, $f(\cup A_i)= \sum f(A_i)$.

 

[camillio]

HallsofIvy is right, you are asking about the basic definitions. I suggest:

1. Read well what algebra is and make own example;
2. Read well what sigma-algebra is, make own example;
3. Compare algebra and sigma algebra and find out differences;
4. Proceed to probability space.

Until you know these basics, you can't understand what you were asked. Furthermore, ignorance propagates, if you do not get this, you will a.s. fail to get the next.

 

[Stephen Tashi]

which of the probability space "(f, A, S) is called events?

That's a good question. The opinion of the web is that the individual elements of the sigma algebra A are the "events" and the individual elements of the set S are the "outcomes". I'm careless about this terminology, myself.

question 2: define what we mean by algebra and σ-algebra? for this question in the second part do we have to write the definition & properties part of http://en.wikipedia.org/wiki/Sigma-algebra#Definition_and_properties or something else?

My interpretation of the question is that you must give two definitions, a definition of "an algebra" an a definition of "a sigma algebra".

The Wikipedia link that you gave defines "sigma algebra".

Defining "algebra" is a harder matter. I recall seeing a book on measure theory that did define "an algebra of sets", but I don't recall the definition. Although you can find many hits on "the algebra of sets", I don't see any that define "an algebra of sets". Your best bet is to see how your instructor or textbook defined this.

 

[Hawkeye18]

The definition of the algebra of sets is almost the same as of sigma algebra, with the only difference that property 3 is replaced by

3' Ʃ is closed under FINITE unions

Also, sometimes an equivalent to property 1 statement is used:
1' ∅ ad X belong to Ʃ

 

[woundedtiger4]

That's a good question. The opinion of the web is that the individual elements of the sigma algebra A are the "events" and the individual elements of the set S are the "outcomes". I'm careless about this terminology, myself.



My interpretation of the question is that you must give two definitions, a definition of "an algebra" an a definition of "a sigma algebra".

The Wikipedia link that you gave defines "sigma algebra".

Defining "algebra" is a harder matter. I recall seeing a book on measure theory that did define "an algebra of sets", but I don't recall the definition. Although you can find many hits on "the algebra of sets", I don't see any that define "an algebra of sets". Your best bet is to see how your instructor or textbook defined this.

Thanks a tonne, this is exactly I was thinking that the measurable sets are events.
OK, so by the "algebra" we mean that the algebra with binary operations on sets.

HallsofIvy is right, you are asking about the basic definitions. I suggest:

1. Read well what algebra is and make own example;
2. Read well what sigma-algebra is, make own example;
3. Compare algebra and sigma algebra and find out differences;
4. Proceed to probability space.

Until you know these basics, you can't understand what you were asked. Furthermore, ignorance propagates, if you do not get this, you will a.s. fail to get the next.

I was confused about the "algebra" I thought that is the one I studied in 10th grade (at school level) but thanks to Stephen Tashi who cleared that it is algebra of sets.

 

[woundedtiger4]

A ring of sets with a unit is called an algebra whereas a unit of ring is E (belongs to to "S" the system of sets), and A intersection E = A, for every A belongs to S, unit of S is the maximal set of S
example: Given a set A, the system M(A) of all subsets of A is an algebra of sets, with unit E=A.
P.S. Please correct me if I am wrong.

 

[Bacle2]

I think I saw the definition of algebra of sets, also ring of sets, in Kolmogorov's Intro. to Real Analysis. If you don't have it with you, maybe check out Google books.