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Zhivago

Member

Registered: 2008-11-21

Posts: 5

Sigma-algebra & measurable sets

I think I understand sigma-algebra and measurable sets to some extent.  I have these unresolved doubts:

1) Given a sample space, does it mean that a subset of the power set of the sample space, which is not a sigma-algebra is not measurable?

2) How is a sigma-algebra over a sample space always measurable?  How does the property of being closed under complementation and   countable unions lead to the measurable criteria?

Maybe, I'm not asking the right questions.  But if someone can set my understanding right, that'll be great.

Cheers!
Zhivago...

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Sigma-algebra & measurable sets

1) Given a sample space, does it mean that a subset of the power set of the sample space, which is not a sigma-algebra is not measurable?

Be careful here.  A subset of the power set will be a collection of sets.  All you want is a subset of the sample space.  Also, I believe you meant "which is not in the sigma-algebra is not measurable", in which case you're right.

2) How is a sigma-algebra over a sample space always measurable?  How does the property of being closed under complementation and   countable unions lead to the measurable criteria?

Here is what I think you may be asking: If we have a sigma-algebra, can we always define a measure on it?  The answer to this is yes.  There are two measures that work on every sigma-algebra for any sample space: the zero measure (everything gets measure 0) and the counting measure.  There are others like this as well (everything is infinity but the empty set measure).

However, as you should have noticed, there is not a unique way to define a measure on any given sigma-algebra.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Sigma-algebra & measurable sets

On reflection, I think I may see what's going on.  I think you are confusing "measurable set" with "sigma-algebra".

A measure is defined on a sigma-algebra, which is a subset of the power set of your sample space (with certain properties).  If a subset of the sample space is not inside your sigma-algebra, then we call such a set "unmeasurable".

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Zhivago

Member

Registered: 2008-11-21

Posts: 5

Re: Sigma-algebra & measurable sets

That was a dead-on analysis of my confused mind!  I want you to run through my thoughts to see if I'm getting it alright.  Here we go... (I've used the definition of measure from Wikipedia)

A measure μ is a function that is defined on a σ-algebra Σ over a set X and taking values in the extended interval [0,∞] such that the following properties are satisfied:

1) The empty set has measure zero
2) Countable additivity or σ-additivity: if E1, E2, E3, ... is a countable sequence of pairwise disjoint sets in Σ, the measure of the union of all the Ei is equal to the sum of the measures of each Ei.

Why is the measure defined on a σ-algebra, and not on any arbitrary set?

The properties  of a σ-algebra facilitate the fulfillment of the conditions placed on any measure:

1) A σ-algebra guarantees the existence of the empty set and hence the measure of the empty set can be defined.  It depends on the application to assign a value of zero to the measure of the empty set for condition (1) to be satisfied.

2) The property of a σ-algebra being closed under countable unions guarantees that the sum of the measures of a countable number of disjoint sets in Σ can be represented as the measure of another set (which will be the result of the countable unions) in Σ.  Whether the condition (2) is satisfied or not solely depends on the way the measure is defined for the various (elementary) sets in Σ.

Here are some lingering doubts:

1) Is there any more relationship between σ-algebra and measure that is basic, and cannot be derived?

2) Suppose I look at a probability problem and construct a sample space Ω and assign probabilities to all the elementary events in Ω, satisfying the properties of μ(Ø) = 0 and sum of all elementary events being equal to 1, i.e. μ(Ω) = 1.  Does it mean that I am safe now and wouldn't have to worry about whether an event (a subset of Ω) is measurable or not?  How about the case of an infinite sequence of events?  Are there any concerns lurking out there?

Thanks Ricky!