Introductory Measure Theory

Seminar
 21:00:00 ChanServ changed the topic of #mathematics to: SEMINAR IN PROGRESS: If you want to ask a                 question say &quot;!&quot; 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&lt;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 U_lambda=sigma(U)? 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 -&gt; [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                infinite. 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 ) &lt;= 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 -&gt; [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 ) &lt;= 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                 definition. 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) &lt;= 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) &gt;= mu(A) [since by definition                mu(B-A) &gt;= 0] 21:54:43 somiaj: so since emptyset subset A for all sets A, we have mu(emptyset) &lt;= 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) &lt; 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) &lt; 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) &lt; 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) &lt; 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 -&gt; [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) &lt; infity for all n 22:12:05 somiaj: If mu(Omega) &lt; 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 Sunday 27 July 20:00 UTC | Transcript of last seminar: http://www.freenode-math.com/index.php/Introductory_Measure_Theory | Future Seminars: http://www.freenode-math.com/index.php/Seminars