This seminar by somiaj took place on 20th July 2008 20:00 UTC, in #mathematics channel of

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Seminar Edit

21:00:00 ChanServ changed the topic of #mathematics to: SEMINAR IN PROGRESS: If you want to ask a
                  question say "!" and wait to be called
21:04:22 somiaj: so anyways we'll see how this goes. First off just a bit on prereqs.
                 I plan to be abstract and general and not relay to heavly on any paticular
                 example, but some understanding of the topology of R^n (specifically the
                 reals R) and anylsis is helpful.
21:05:19 somiaj: So to start, measure is a generalization of volume. For example in the
                 reals R, we can give intervals a length.
21:05:58 somiaj: We can say the length of the closed interval [a,b] is b-a (a<b).
21:06:29 somiaj: In R^2 we have area, R^3 we have volume, an so forth. So what we want to
                 do is given any subest of R^n define a volume for it.
21:08:41 somiaj: Now instead of just looking at R^n I'm going to generalize this to any
                 set Omega. So suppose we have a set Omega, we want to define a non-negitive real
                 valued function which gives us the measure (or volume) of a subset of Omega.
21:09:27 somiaj: Unfortunately due to the axiom of choice and other bizar collections it might
                 not be possible to measure any subsest but only paticular ones. So the first
                 actual construction is the limit of the sets we are able to measure.
21:10:27 somiaj: So let Sigma be a collection of subest of Omega. We call Sigma a sigma-field if
21:10:53 somiaj: 1) Omega in Sigma (the whole space is in our collection)
21:11:10 somiaj: 2) if A is in Sigma then A^c (Omega - A, the complement) is in Sigma
21:11:55 somiaj: 3) if {A_n}_{n a natural number} is a subset of Sigma then union_n A_n is in Sigma
                    and intersection_n A_n is in Sigma
21:13:09 somiaj: the sigma-field is our collection of measurable sets. So we have our whole space
                 is measurable, complements of measurable sets are measurable and countable unions
                 and intersections of measurable sets are measurable.
21:13:38 somiaj: a few things that follow from this is the empty set is measurable since
                 Omega^c = empty set has to be in Sigma
21:14:46 somiaj: on the other hand since ( intersection_n A_n )^C = union_n (A_n)^c it is reasonable
                 just to ensure your sigma-field is closed under countable unions or countable
                 intersections, you don't need to check for both.
21:14:53 somiaj: yes
21:16:07 ichor: !
21:16:27 somiaj: as for Sigma, Omega those are just the captial greek symbols. and sigma is the
                 lowercase greek symbol if you see it in books.
21:16:31 somiaj: ichor: yes
21:16:36 ichor: Does the empty set always have measure 0?
21:17:11 somiaj: ichor: it does, but that can be proven, we'll get to that. So far we haven't
                 defined the function to assign a measure but are only looking at the limitation of
                 sets which are measurable.
21:17:28 ichor: ok
21:18:25 somiaj: so lets call (Omega, Sigma) the space and this sigma-field together
                 a measurable space.
21:19:09 somiaj: One common example that is used to create measurable spaces is to create it from
                 some known collection.
21:20:13 somiaj: So let C be any collection of subests of Omega. We can define
                 sigma(C) = intersection_a Sigma_a, where {Sigma_a}_{a in some indexset} is
                 the collection of all sigma-fields that contain C.
21:21:16 somiaj: It can be proven that the intersection of any number of sigma-fields is itself
                 a sigma-field. So from any collection C, we take the intersection of all possible
                 sigma-fields containing C to generate the smallest sigma-field containing C.
21:22:10 somiaj: For example if you have a topology on your set Omega (again think of Omega being
                 R^n for starters but it can be any set)
21:23:07 somiaj: The topology or open sets are only closed under finite intersections but arbitary
                 unions. So we will define the Borel sigma-field to be sigma(U) where U is the
                 collection of open sets on Omega (the topology)
21:24:19 somiaj: So there is our first construction the sigma-field, it is the collection of
                 measurable sets. A common place to start with is look at the sigma-field known as
                 the Borel sigma-field, or the extensions of the toplogy to the smallest sigma-field
                 that contains it.
21:24:46 somiaj: Next we need to define a non-negitive real valued function on Sigma to generate our
                 measure, but first any questions?
21:25:12 _llll_: is there an easy way to describe sigma(U) in terms of U?
21:26:29 somiaj: _llll_: not really, sigma(U) contains all open sets, all closed sets (since complements
                 have to be in there) and any type of set you can generate by countable unions and
                 intersections of the open and closed sets.
21:26:53 _llll_: ok
21:27:02 kommodore: !
21:27:03 somiaj: for example if we restrict to R with the usual topology, then you have all open intervals
                 (a,b), all closed intervals [a,b] and even all half-open intervals (a,b] and (a,b]
                 inside your sigma(U)
21:27:57 somiaj: one thing that is of interest is the borell sets on R are equivlant to
                 sigma( (-infty,x] ) where x is in R (you might even be able to restrict to x rational
                 cause of countable limits and the completeness of R)
21:28:01 somiaj: kommodore: yes.
21:28:40 kommodore: Taking the example of R, then in the usual constructive construction of sigma(U),
                    does it take all the way to the first uncountable ordinal to get sigma(U)?
21:29:40 somiaj: kommodore: well U itself is a subset of sigma(U), and the topolgy itself on R is an
                 uncountable collection.
21:30:23 somiaj: just consider all open sets of the form (0,x) where x is real. Have an uncountable sets
                 there and only have a 'small' example of all possible open sets.
21:31:25 kommodore: i mean let U_0=U, U_{alpha+1}={sets in U_alpha, complement of sets in U_alpha,
                    countable intersections, countable union of sets in U_alpha}, and limit ordinal
                    collects all previous ones; does there exists a countable ordinal lambda so that
21:35:44 somiaj: I am not able to answer your question in that terms, All the work I have seen is going
                 the other way
21:36:01 kommodore: nm, thanks
21:37:07 somiaj: Ok, back to the direction, we have our collection of measurable sets Sigma, and
                 you can think of the restrictions on Sigma as being the fact that you can build
                 measurable sets from measurable sets via complements and countable intersections
                 or unions.
21:37:41 somiaj: Next we need to define a measure on this space. Thus we want a function
                 mu: Sigma -> [0,infity]
21:38:32 somiaj: here [0,infity] is the extended non-negative reals, we want to include infity, for example
                 we say the measure of the reals is infinite, or the length of the real number line is
21:39:05 somiaj: The major property we want this function to follow is countably additive. First let's start
                 with the finite case
21:39:29 somiaj: we say the function mu is finitely sub-additive if
21:40:00 somiaj: mu( union_{k-1}^n A_n ) <= sum_{k=1}^n mu(A_n)
21:40:28 somiaj: i.e. the measure of a union of sets is less than the sum of the measure of the sets itself.
21:40:50 somiaj: we then say the function mu is finitely additive if mu is finitely sub-additive and
21:41:31 somiaj: mu( union_{k=1}^n A_n ) = sum_{k=1}^n mu(A_n) where {A_n} is a disjoint collection of
                 subsets of Omega.
21:42:15 somiaj: i.e. if we can write a set as the disjoint union of known sets then the measure of the
                 full set is just the sum of the measures of the disjoint parts.
21:43:53 somiaj: Now of course we want this same idea to apply to any measureable set, so we can extend
                 this and we acually want mu: Sigma -> [0,infity] to be countaably additive. So let
                 {A_n} be a countable collection of subsets
21:44:22 somiaj: then we want mu( union_n A_n ) <= sum_n mu(A_n) and equality to occur if {A_n} is a
                 disjoint collection.
21:45:23 somiaj: Thus we have our full space, Omega, Our collection of measurable sets, Sigma, and our
                 measure function, mu. All with properties of how we want measure to work.
21:45:53 somiaj: Here is a good place to add a theorem with some work, just to see how we acually work
                 within our sigma-field.
21:46:10 somiaj: I'll just finish this by proving some basic results from my definitions.
21:46:20 somiaj: 1) mu(emptyset) = 0.
21:46:51 somiaj: Note emptyset = emptyset union emptyset, and this is a disjoint union since the 
                 emptyset is disjoint form itself
21:47:22 somiaj: thus mu(emptyset) = mu(emptyset union emptyset) = mu(emptyset) + mu(emptyset)
                  [since we have finite additivity]
21:47:40 somiaj: and for the real numbers we know that if c = c + c = 2c, then c=0. Thus
                 mu(emptyset) = 0.
21:47:47 kommodore: !
21:48:00 somiaj: yes
21:48:19 kommodore: are you rejecting the possibility mu(A)=infinity for all A?
21:49:23 somiaj: ahh yes, I belive so. most books define mu, such that mu(emptyset)=0, as part of its
21:49:40 kommodore: ok
21:50:09 somiaj: Though I guess you could live in a space where mu(A) = infinity, but there wouldn't
                 be much of interest if everything had infinite volume.
21:50:36 somiaj: here, look at 2) If A subset B, then mu(A) <= mu(B).
21:51:36 somiaj: proof: B = A union (B-A) since A subset B, and this is a disjoint union.
21:52:24 somiaj: Thus mu(B) = mu( A union (B-A) ) = mu(A) + mu(B-A) >= mu(A) [since by definition
                 mu(B-A) >= 0]
21:54:43 somiaj: so since emptyset subset A for all sets A, we have mu(emptyset) <= mu(A) for all A,
                 so that in order for 1) to hold, ie mu(emptyset)=0, all we need is one set to have
                 finite measure. (cause as kommodore pointed out having all sets of infinite measure
                 would break 1)
21:55:49 somiaj: 3) if mu(A intersect B) < infity then mu(A union B) = mu(A) + mu(B) - mu(A intersect B)
21:57:27 somiaj: to see this, note that B = (B-A) disjoint-union (A intersect B)
21:58:36 somiaj: thus mu(B) = mu(B-A) + mu(A intersect B). Since mu(A interset B) < infity, we have
                 mu(B-A) = mu(B) - mu(A intersect B)
21:59:34 somiaj: finally A union B = A disjoint-union (B-A), so
                 mu(A union B) = mu(A) + mu(B-A) = mu(A) + mu(B) - mu(A intersect B)
22:01:33 somiaj: 4) if A subest B and mu(A) < infity, then mu(B-A) = mu(B) - mu(A). This follows from
                 the subresult in part 5), mu(B-A) = mu(B) - mu(A intersect B) which holds if
                 mu(A intesect B) < infity. And since A subset B, A intersect B = A. Thus we have
                 the result.
22:02:50 somiaj: so again the basic construction is start with a full space Omega, from there
                 define a collection of mesurable sets, or a sigma-field. On the collection of
                 measurable sets define a non-negtive function mu: Sigma -> [0,infity]
22:04:19 somiaj: I think of the definitions as being that we can do countable operations and want
                 to preserve our mesure, so thus we want the function mu to be countabally additive.
22:04:48 somiaj: Looks like I'm out of time, but from here one can then take this basic construction
                 and try to build the lebesgue measure.
22:05:15 somiaj: though building the lebesgue measure takes a lot of work to show that the function
                 we create obeys all the properties I have listed.
22:05:27 somiaj: any questions so far (almost feel as if I'm talking to myself)
22:05:49 _llll_: what sort of morphisms of measurable space do people use?
22:07:29 somiaj: I can't think of any, most of the studying I have done of measurable spaces quickly
                 developes to the general lebesgue intergral and then creating the L_p(Omega) spaces.
22:09:20 somiaj: The two main paths from here are going into the lebesgue Measure, or looking at R^n,
                 and generalizing volume there. In these cases mu(R^n) = infity, so we have an infite
                 measure space.
22:09:55 pyninja: Thanks somiaj, some of it went over my head but it was still interesting.
22:10:03 somiaj: Though in the case of R^n we can define it to be a sigma-finite measure.
22:10:03 _llll_: are you going to follow one of these paths in a follow-up seminar next week?
22:11:38 somiaj: so we say mu is sigm-finite if there exists a sequence
                 ( Omega_1 subset Omega_2 subset Omega_3 subset .... ) and
                 Omega = union_n Omega_n Such that mu(Omega_n) < infity for all n
22:12:05 somiaj: If mu(Omega) < infity we call mu a finite-measure, the most common one of these is
                 when mu(Omega) = 1.
22:12:31 somiaj: if mu(Omega) = 1, then (Omega, Sigma, mu) form a probability space, which is the
                 second main path you can go from here)
22:13:16 somiaj: would people be intersted, I could continue though my notes to develope outter measure,
                 and then the lebesgue measure and if we have time look at the cantor set. maybe get
                 our hands on some examples that are less definitions and abstract.
22:13:37 _llll_: id be interested
22:14:20 * ichor would be interested too.
22:14:29 |Steve|: And I.
22:15:07 ~DWarrior-: I'd definitely be interested next week
22:15:19 burned: I would be too
22:16:31 somiaj: we should try to limit these to an hour.
22:19:49 somiaj: though one note above, on analogy that I liked was thinking back to simple high-school
                 math problems where you want to find the area of an object by spliting it up into pieces
                 you know the area of (rectangles, triangles, circles) and then adding the desired areas
                 to get the result
22:20:23 somiaj: That is kinda what the sigma-field and countabally additive function do, but our shapes
                 can become far more bizar.
22:23:53 ChanServ changed the topic of #mathematics to: NEXT SEMINAR: The Lebesgue Measure by somiaj
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