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Measure theory  Help 
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#1
May212, 07:25 PM

P: 176

question 1: if f is a countably additive set function (probability measure) defined on σalgebra 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 σalgebra? for this question in the second part do we have to write the definition & properties part of http://en.wikipedia.org/wiki/Sigmaa...and_properties or something else? Plus, can anyone please help me that what is countably additive? 


#2
May212, 08:11 PM

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P: 39,318

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, [itex]\{A_i\}[/itex], of disjoint sets, [itex]f(\cup A_i)= \sum f(A_i)[/itex]. 


#3
May312, 06:22 AM

P: 74

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



#4
May312, 01:15 PM

Sci Advisor
P: 3,248

Measure theory  Help
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. 


#5
May312, 01:27 PM

P: 25

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 


#6
May312, 04:56 PM

P: 176

OK, so by the "algebra" we mean that the algebra with binary operations on sets. 


#7
May412, 09:29 AM

P: 176

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. 


#8
May412, 01:06 PM

Sci Advisor
P: 1,169

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.



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