Seminar on Measure Theory 1

This is the first in a three part lecture course. The course will be a basic introduction to measure theory and the lebesgue integral, including motivation and basic theorems (some of them will only have sketch proofs presented, as per usual in my lecture courses).

It should be fairly light on neccesary background - a first course in analysis will suffice, though a little bit of mathematical maturity would be useful. Also, if you don't know what countable means then go read this.

I intend to approximately follow the following schedule, but I can't promise I won't alter the order somewhat:

This lecture will be about measure theory. I don't know exactly what it will include yet, but it will give an introduction to lebesgue measure on R, product measures, the connection with probability theory and a number of useful or interesting theorems and lemmas (with particular focus on ones I'll need for doing integration later).

The second lecture will be the following week and will introduce the Lebesgue integral. More details to follow on that later.

The third lecture will be on something else within the subject that I deem to be sufficiently cool to give a talk on. The current most likely subject is Hilbert spaces and the connection to fourier series via L².

Seminar Log

15:59 <@Syzygy-> Hrm. Silence? Silence please? *tinkling of glass and fork*
15:59 <@Chandra> ok guys, If you have a question, please say "!" and patiently wait for lecturer's attention.
15:59 <@Syzygy-> I have the great honour tonight to welcome for the pleasure of our hearing, Mr. <DRMacIver> from freenode.
15:59 <@Syzygy-> Also known as Kit, he stems from Cambridge, where he's hopelessly infatuated with analysis and no hope of waking up again.
16:00 <@zeno> "The speaker will begin shortly; he's just looking in his personal null set for some notes."
16:00 <@DRMacIver> Ok. Thanks for the introduction Syzygy. If everyone is ready then I guess I'll begin. :)
16:01 <@DRMacIver> Today I'm here to talk about measure theory. Some of what I'm doing may seem a little dry - the real point (in my mind. The probability theorists may disagree) of measure theory is to do integration.
16:01 <@DRMacIver> Which is what next week's talk is about. :)
16:02 <@DRMacIver> So, I'll be giving an introduction to the theory of measures, proving some basic tools for getting nice measure spaces and using them to prove useful results about measures on R^n.
16:03 <@DRMacIver> I suppose the first question to ask is "What's a measure?"
16:03 <@DRMacIver> I'm going to turn this around and show you the type of question which asking will naturally lead us to formulating measure theory in the first place.
16:03 <@DRMacIver> "What's the area of a circle?"
16:03 <@DRMacIver> (Or more precisely a disk of radius 1)
16:03 <@DRMacIver> You all know it's pi I'm sure.
16:04 <@DRMacIver> But a) What do we actually mean by this question? and b) Once we know what we mean, how do we actually go about answering the question?
16:04 <@DRMacIver> (This is something that bothered the ancient Greeks an awful lot).
16:04 <@DRMacIver> I'm going to answer b first. :)
16:05 <@DRMacIver> Lets start with things we know the area of. Squares.
16:05 <@DRMacIver> Err. Rectangles.
16:05 <@DRMacIver> It's obvious what the area of a rectangle should be.
16:05 <@DRMacIver> Just the product of its two sides.
16:06 <@DRMacIver> Having said that, it's obvious what the area of a right angled triangle should be - half that.
16:06 <@DRMacIver> From this we can then extend it to more complicated polygonal shapes.
16:06 <@DRMacIver> You can put two right angled triangles together to get any triangle you like. (Draw a picture if you don't believe me).
16:06 <@DRMacIver> You can then piece these triangles together to get regular n-gons, etc.
16:07 <@DRMacIver> So what we've done is we've taken something we know the area of (right angled triangles) and used this to extend it to a larger class of shapes (finite unions of triangles).
16:07 <@DRMacIver> Can we perhaps do something like this to get the area of a circle?
16:08 <@DRMacIver> Well, obviouisly the circle is not a finite union of triangles - it doesn't have any corners. (Yes, this is handwavy and geometric. Sue me. :) )
16:08 <@DRMacIver> But, we can approximate it with polygons.
16:08 <@DRMacIver> Inscribe a regular n-gon inside the circle with its points on the boundary.
16:09 <@DRMacIver> This is made up of n triangles. Each of these have an inner angle of 2pi/n and two sides of length 1
16:09 <@DRMacIver> So using standardness each of the triangles has area 1/2 sin(2pi/n)
16:10 <@DRMacIver> Thus the area of the inscribed polygon is n/2 sin(pi / (n/2) )
16:10 <@DRMacIver> Thus using the standard result that sin(x)/x -> 1 as x -> 0 we have that the area of the circle is pi.
16:10 <@DRMacIver> Yay!
16:10 <@DRMacIver> Err. But what were we actually doing there?
16:11 <@DRMacIver> Why was it ok to pass to the limit like that?
16:11 <@DRMacIver> This is what we're trying to justify in producing measure theory.
16:11 <@DRMacIver> Sigh.
16:11 <@DRMacIver> As I was saying.
16:12 <@DRMacIver> What we want is an area function which satisfies certain properties.
16:12 <@DRMacIver> It's >= 0 obviously.
16:12 <@DRMacIver> When we stick pieces together we get the right thing. i.e. if we have two disjoint sets then the area of their union is the sum of their areas.
16:12 <@DRMacIver> (Or by induction any finite collection of pairwise disjoint sets)
16:13 <@DRMacIver> This is what we had with the polygons.
16:13 <@DRMacIver> And what we were doing in extending the area from the triangles to general unions of triangles.
16:13 <@DRMacIver> (finite unions)
16:13 <@DRMacIver> But this isn't enough for our purposes. We also want to be able to take limits.
16:13 <@DRMacIver> In the above example we had a nested sequence of sets A_1 <= A_2 <= ...
16:14 <@DRMacIver> And we wanted to say that the area of union A_n is the limit of the area of A_n
16:14 <@DRMacIver> Combining this with the above, the two conditions become the following:
16:15 <@DRMacIver> Suppose we've got a disjoint collection of sets A_n. Then the Area( union A_n ) = sum Area(A_n)
16:15 <@DRMacIver> This property is called countable additivity.
16:15 <@DRMacIver> A 'measure' is just a countably additive function from a collection of sets to [0, inf]
16:15 <@DRMacIver> Oh. Technicality there: We want to allow sets with infinite area.
16:16 <@DRMacIver> Because, for example, the entire plane doesn't have finite area.
16:16 <@DRMacIver> But unless we have 'enough' sets this isn't very useful.
16:16 <@DRMacIver> Do we want to insist that the measure is defined on all of P(X)?
16:16 <@DRMacIver> For various reasons this turns out to be inconvenient.
16:18 <@DRMacIver> Firstly, the natural ways of extending the measure from, say, the polygonal sets, turn out to not extend it to the whole of R^2
16:19 <@DRMacIver> Secondly, it turns out that we can prove there are no 'nice' measures on R^2 - ones with good properties which we'll find useful.
16:19 <@DRMacIver> So like our original example this measure isn't going to be defined everywhere.
16:19 <@DRMacIver> But it will be defined on every set we're really likely to care about.
16:19 <@DRMacIver> The measure of strange pathological sets isn't really that interesting.
16:20 <@DRMacIver> We'll insist that the sets it's defined on satisfy certain closure properties:
16:20 <@DRMacIver> It's defined for the empty set of course.
16:20 <@DRMacIver> If it's defined for a sequence of sets A_1, ..., A_n, ... then it's defined for union A_n
16:20 <@DRMacIver> And finally if it's defined for A then it's defined for A^c
16:20 <@DRMacIver> We call a set which is closed under these operations a sigma-algebra.
16:21 <@DRMacIver> So, properly, measure theoyr is the study of measures on sigma-algebras.
16:21 <@DRMacIver> I'm going to give you some nice ways of getting them today.
16:21 <@DRMacIver> First, a note: It's an easy check that an intersection of sigma-algebras is again a sigma algebra. So given any collection of sets there is a unique smallest sigma-algebra containing them.
16:22 <@DRMacIver> We call this the sigma-algebra generated by them. This will be important.
16:22 < supremum> !
16:22 <@DRMacIver> supremum: Yes?
16:22 < supremum> What is algebras good for?
16:22 < supremum> ordinary algebras not sigma algebras.
16:23 <@DRMacIver> Oh. Stuff. :)
16:23 <@DRMacIver> That's not relevant here.
16:23 <@DRMacIver> Finitely additive measures I suppose.
16:23 < supremum> ok.
16:23 <@DRMacIver> ok. So what we want to do is start with our measure on the polygonal sets and extend it to a measure on the sigma-algebra generated by them.
16:24 <@DRMacIver> The polygonal sets have certain nice properties:
16:24 <@DRMacIver> The empty set is polygonal.
16:24 <@DRMacIver> A union of two polygonal sets is polygonal
16:24 <@DRMacIver> If A, B are polygonal then so is A \ B
16:24 <@DRMacIver> (set minus)
16:24 <@DRMacIver> We call such a collection a ring.
16:24 <@DRMacIver> This situation is standard: We have a measure on a ring. We want a measure on the sigma-algebra generated by it.
16:25 <@DRMacIver> There are a number of ways to do this. The one I'm going to give is the way I consider most natural.
16:25 <@DRMacIver> Hmm. This is taking longer than I thought. Sorry. We may run over the hour limit slightly. :)
16:25 <@DRMacIver> The basic problem is that constructing measures is hard.
16:26 <@DRMacIver> We're going to construct something which is almost a measure, and then apply a clever trick to get an actual measure out of it.
16:26 <@DRMacIver> We have our ring of subsets of some set X.
16:26 <@DRMacIver> And a measure \mu on it.
16:26 <@DRMacIver> What we do is ask how large an extension of this measure can possibly be.
16:27 <@DRMacIver> Suppose we've got a set B <= X and B <= Union A_n for some sequence of sets A_n in the ring.
16:27 <@DRMacIver> Then we know that for any extension the measure of B must be <= the measure of union A_n, which is <= sum mu(A_n)
16:28 <@DRMacIver> Sorry. I actually haven't proved that yet. Oops. :)
16:28 <@DRMacIver> Let me back track slightly.
16:28 <@DRMacIver> Suppose we have a ring of sets. Say S.
16:28 <@DRMacIver> Let A, B be sets in S and let mu be a measure on S.
16:28 <@DRMacIver> If A <= B then B = A union B \ A, and this union is disjoint.
16:28 <@DRMacIver> So mu(B) = mu(A) + mu(B \ A) >= mu(A)
16:29 <@DRMacIver> We say that measures are monotone: If A <= B then mu(A) <= mu(B)
16:29 <@DRMacIver> This is pretty much what you'd expect. If A is contained in B then it has area less than or equal to that of B.
16:30 <@DRMacIver> Also, regardless of whether the A_n are disjoint, mu( Union A_n ) <= sum mu(A_n)
16:30 <@DRMacIver> Because you can always remove bits from the A_n to make them disjoint and this will only decrease the sum on the right hand side.
16:31 <@DRMacIver> So. Having said that we know that for any extension of the measure we must have the measure of B <= sum mu(A_n)
16:31 <@DRMacIver> So we define the following:
16:31 <@DRMacIver> mu*(B) = inf { \sum mu(A_n) : A_n in S and B <= U A_n }
16:32 <@DRMacIver> i.e. the largest possible value for the measure of B in any given extension
16:32 < davidhouse> !
16:32 <@DRMacIver> (If the set on the right is empty then we define mu*(B) = inf )
16:32 <@DRMacIver> davidhouse: Yes?
16:32 <@DRMacIver> err. infty
16:32 < davidhouse> sorry, just a quickie: what do you mean by 'extension of the measure'? is that some topic i should be reading up on?
16:32 <@DRMacIver> No, not really.
16:32 <@DRMacIver> What I mean is that we have a measure on the ring.
16:33 <@DRMacIver> And we want a measure on some sigma algebra containing it
16:33 < davidhouse> yes
16:33 <@DRMacIver> And this measure has to agree with the measure on the ring.
16:33 <@DRMacIver> i.e. we're extending the measure on the ring to a measure on a bigger collection of sets (the sigma algebra generated by it)
16:33 <@DRMacIver> Make sense?
16:33 < davidhouse> aha.
16:33 < davidhouse> thanks :)
16:34 <@DRMacIver> No problem. Sorry that wasn't clear. :)
16:34 <@DRMacIver> Anyway, this mu* we have is *not* a measure.
16:34 <@DRMacIver> But it's something very like one.
16:35 <@DRMacIver> Theorem: mu* as defined above has the property that mu* ( Union A_n ) <= \sum \mu* (A_n)
16:35 <@DRMacIver> Sorry. I've randomly started pseudotexing. Ignore it.
16:35 <@DRMacIver> We call this property subadditivity.
16:36 <@DRMacIver> I'm only going to very quickly sketch a proof of this. When I'm running late proofs suffer. :)
16:36 <@DRMacIver> The basic trick is to show that the left hand side is <= right hand side + epsilon
16:36 <@DRMacIver> For epsilon > 0 arbitrary
16:36 <@DRMacIver> Then because epsilon was arbitrary we can pass to the limit eps -> 0 and get rid of it.
16:36 <@DRMacIver> The reason for this is that the epsilon gives us a bit of extra 'cush' to play around with, so it's an easier inequality to prove.
16:39 <@DRMacIver> Sorry. SLight pause as I screamed at my worthless ISP. :(
16:39 < supremum> !
16:40 <@DRMacIver> What we do is for each of the A_n we pick a cover of it by sets B_mn such that sum_m mu(B_mn) < mu*(A_n) + eps/2^n
16:40 <@DRMacIver> supremum: Go on.
16:40 < supremum> Are you going to cover integration teory also?
16:40 <@DRMacIver> I said this at the beginning.
16:40 <@DRMacIver> Next week.
16:41 <@DRMacIver> And now I've lost my train of thought. One moment...
16:42 <@DRMacIver> Right. The basic idea is that this cover has sum mu(B_mn) <= sum mu*(A_n) + eps
16:43 <@DRMacIver> So mu* (union A_n) <= sum mu* A_n + eps
16:43 <@DRMacIver> Now pperform previously mentioned trick about getting rid of eps
16:44 <@DRMacIver> So, mu* is subadditive as promised.
16:44 <@DRMacIver> Which isn't really much use on it's own - we wanted it to be additive.
16:45 <@DRMacIver> Anyway, we call a subadditive function defined on the power set to be an outer measure.
16:45 <@DRMacIver> Sorry for allt hese arcane terms. HOpefull you at least see where they're coming from. :
16:46 <@DRMacIver> (Also sorry for the horrendous typoing. My ISP is screwing up my SSH session with lag)
16:46 <@DRMacIver> The trick is now to take an outer measure and find some large enough collection of sets on which it is a measure.
16:46 <@DRMacIver> What we need is the following observation:
16:47 <@DRMacIver> If we have a subadditive function, all we need to do to make it additive is to have mu(A union B) = mu(A) + mu(B) for A, B disjoint.
16:47 <@DRMacIver> We don't need countable unions.
16:48 < supremum> !why?
16:48 <@DRMacIver> The reason for this is that we can show that if A_n are disjoint then mu( Union A_n ) > = mu ( Union_1^N A_n ) = sum_1^N A_n
16:48 <@DRMacIver> supremum: I'm about to explain.
16:49 <@DRMacIver> So, taking the limit as N -> inf, we have that mu (Union A_n ) >= sum mu(A_n)
16:49 <@DRMacIver> And we already know that it's <= by subadditivyt.
16:49 <@DRMacIver> So we have that mu ( union A_n ) = sum mu(A_n) as desired.
16:49 <@DRMacIver> So what we want to do is to restrict to subsets for which the above finite additivity formula holds.
16:50 <@DRMacIver> Lets rewrite it as follows:
16:50 <@DRMacIver> We want mu(A) = mu(A int B) + mu(A int B^c)
16:50 <@DRMacIver> (Easy check: This is the same after relabelling)
16:51 <@DRMacIver> Now, it's still not easy to see how to extract the neccesary sets until we rewrite it one more time.
16:52 <@DRMacIver> Given an outer measure mu* we say that B is mu* measurable if for *every* set A we have mu(A) = mu(A int B) + mu(A int B^c)
16:52 <@DRMacIver> Not just A measurable.
16:52 <@DRMacIver> And now we've got a definition we can work with.
16:52 <@DRMacIver> Theorem: The mu* measurable sets form a sigma-algebra.
16:52 <@DRMacIver> And then the above observation shows that mu* is countably additive on this sigma-algebra and so a measure on it.
16:53 <@DRMacIver> I'm going to skip the proof of this theorem entirely. It's fiddly and unenlightening.
16:53 <@DRMacIver> I'll link a version of a proof afterwards.
16:53 <@DRMacIver> Phew. Recap.
16:53 <@DRMacIver> We've got a way from going from a measure on a ring to an outer measure.
16:54 <@DRMacIver> We've got a way of going from an outer measure to a measure on a sigma-algebra.
16:54 <@DRMacIver> We want to show these work well together.
16:55 <@DRMacIver> Specifically we want to show that every element of the ring is mu* measurable, and that they mu = mu* on the ring.
16:56 <@DRMacIver> Ok. Again, it's enough to show that mu*(A) >= mu*(A int B) + mu*(A int B^c)
16:56 <@DRMacIver> Because the other direction follows from subadditivity.
16:56 <@DRMacIver> Actually I'm just going to throw away giving the proof of this as I'll just get flustered midway through. :)
16:57 <@DRMacIver> It's basically the same 'introduce an epsilon for cush' proof as before.
16:58 <@DRMacIver> Ad the extension is easy. Subadditivity of mu gives that mu <= mu*
16:58 <@DRMacIver> And A is a collection of elements of the ring covering A, so mu(A) >= mu((A)
16:58 <@DRMacIver> Hence they agree on the ring.
16:58 <@DRMacIver> Phew.
16:58 <@DRMacIver> Anyway, I'm about 20 minutes behind where I wanted to be.
16:58 <@DRMacIver> And I was meant to be ending now. :)
16:59 <@DRMacIver> So I'm going to call a 5 minute break and resume afterwards.
16:59 <@DRMacIver> Any questions?
16:59 < supremum> !
16:59 <@DRMacIver> Go ahead.
16:59 < supremum> is tehre a definition of inner measures aswell?
17:00 <@DRMacIver> Sortof.
17:00 <@DRMacIver> There's something called lebesgue inner measure.
17:00 < pelli> !
17:00 < brett1479> !
17:00 <@DRMacIver> For now just ask your questions. I'm not talking. :)
17:00 <@DRMacIver> But there isn't a good general definition of an inner measure that I know of. If there is one it's not too widely used.
17:01 < brett1479> Does sup of approximation from the inside by closed intervals not work?
17:01 <@DRMacIver> No
17:01 <@DRMacIver> For example consider the irrationals in R
17:01 <@DRMacIver> The only closed intervals they contain are points.
17:01 < supremum> i saw in a book a definition of area and then used inner emasure and outer measure and said that iff they where equal the set was measurable
17:01 <@DRMacIver> So you just get sup of a lot of 0s.
17:01 <@DRMacIver> supremum: Yes. That's the original approach, and giving a bit of faffing it's really more or less the same as this one.
17:02 <@DRMacIver> This way works a little better.
17:02 < supremum> does this realte to general measure theory?
17:02 < supremum> so its eqvivalent?
17:03 <@DRMacIver> More or less.
17:03 <@DRMacIver> What I'm doing is easier to do in a general setting.
17:03 < supremum> ok
17:03 <@DRMacIver> pelli: Did you have a question?
17:03 < brett1479> my question was actually: I vaguely remember a definition of measurable involving arbitrarily good approximation via outer measure.
17:04 < brett1479> Is that also equivalent to this method?
17:04 <@DRMacIver> Umm.
17:04 <@DRMacIver> I'm less convinced of their equivalence.
17:04 <@DRMacIver> I suspect it ends up as broadly speaking the same thing.
17:04 <@DRMacIver> But I wouldn't swear to it.
17:04 <@DRMacIver> The 'broadly speaking' may turn out to be very broad.
17:04 < toad-> what's this P(X) you mention towards the beginning?
17:04 < brett1479> I think its Given e > 0 there is an open set O containing E such that m*(O \ E) < e.
17:05 <@DRMacIver> Sorry. P(X) means the power set of X>
17:05 <@DRMacIver> i.e. the collection of all subsets of X.
17:05 < toad-> ah, right
17:05 < supremum> hmm
17:05 < supremum> brett1479 isn't that only for reimanstyle measure?
17:05 < brett1479> The reason I mention it, is that I always liked that definition better
17:05 < brett1479> supremum, nope
17:05 < supremum> ok
17:06 < supremum> so it was in a book on general measure theory?
17:06 <@DRMacIver> brett1479: I think it's an open set O containing E and a closed set F contained in E with m*(O \ F) < e
17:06 < brett1479> I learned it in the context of lebesgue measure
17:06 < brett1479> (Royden)
17:06 < supremum> ok
17:06 < supremum> i have royden here
17:06 <@DRMacIver> Right. I'm really going to take that 5 minute break now. After the break I'll resume with a bit on uniqueness of measures.
17:06 <@DRMacIver> Sorry I'm running late.
17:06 < supremum> its ok ;)
17:07 < davidhouse> who's logging?
17:07 < brett1479> That definition feels a lot more like you are measuring something
17:07 < brett1479> Whereas this definition seems cooked to make things smooth (no problem with that)
17:07 < supremum> the definition with outer and innermeasure seams quite nice also
17:07 < brett1479> Right, they both seem to have a good feel
17:08 < brett1479> but I think even Royden mentions that Kit's def, which Royden calls Caratheodory's def makes the proofs go through easier
17:08 < supremum> but i think kit motivated quite nicley the definition he used
17:08 <maherarar> I'm logging.
17:08 < davidhouse> maherarar: okay. just checking :)
17:09 < supremum> but my book didn't motivate it
17:09 <@Chandra> i am logging too...but my timestamp is: 1134252535 17:08:55 <blah> blah
17:09 < davidhouse> eww
17:09 <@DRMacIver> ok. Back.
17:10 <@DRMacIver> brett1479: Yeah, this method is heavily cooked for smoothness. :)
17:10 <@DRMacIver> It's the only way I can skip all the proofs and have people go away thinking they've learned something. ;)
17:11 <@DRMacIver> Ok. I'm ready to resume.
17:11 < supremum> ok move in gogogo
17:12 <@DRMacIver> Now, the question we want to ask is how canonical is our extension? i.e. is it the only possible one.
17:12 <@DRMacIver> Because if it is then we've got a good chance of being able to calculuate it just based on what we know about measures and the ring we started with.
17:12 <@DRMacIver> If it isn't then we're probably going to have to mess around with calculating infimums. And that would be annoying.
17:13 <@DRMacIver> Unfortunately the answer is that in general it's *not* the only one.
17:13 <@DRMacIver> Example:
17:13 <@DRMacIver> Take some uncountable set X.
17:13 <@DRMacIver> Let S be the set of all countable subsets of X.
17:13 <@DRMacIver> Easy check: S is a ring and the 0 function is a measure on it.
17:14 <@DRMacIver> The set of all countable or cocountable (their complement is countable) subsets of X is a sigma algebra containing this.
17:14 <@DRMacIver> And mu(A) = 0 if A is countable, t if A^c is countable, is a measure on this sigma algebra extending the one on S.
17:14 <@DRMacIver> For any value of t we like.
17:14 <@DRMacIver> So it very badly fails to be unique.
17:15 <@DRMacIver> But what if we fix the measure on the whole space, X? In this case it fixes uniqueness. Is that enough in general?
17:15 <@DRMacIver> Well... almost.
17:15 <@DRMacIver> Lets restrict for now to the case where mu(X) < infty
17:15 <@DRMacIver> We'll finesse the infinite measure case away. :)
17:16 <@DRMacIver> Suppose we've got two measures mu_1, mu_2
17:16 <@DRMacIver> What can we say about { A : mu_1(A) = mu_2(A)
17:16 <@DRMacIver> }
17:16 <@DRMacIver> It would be really nice if this was a sigma algebra.
17:16 <@DRMacIver> Because then if it contains a ring it contains the sigma algebra generated by that ring
17:17 <@DRMacIver> So two extensions from that ring are equal on the generated sigma algebra.
17:17 <@DRMacIver> This is alas rather hard to prove (and may even be false in general).
17:17 <@DRMacIver> So, lets see what we can prove:
17:17 <@DRMacIver> We've fixed mu_1(X) = mu_2(X), so X is in this set.
17:18 <@DRMacIver> If A, B are in this set with A <= B then we know that mu(B \ A) = mu(B) - mu(A)
17:20 <@DRMacIver> (Sorry, ISP conkout again)
17:20 <@DRMacIver> So, if A, B are in this set with A <= B then B \ A is in the set.
17:21 <@DRMacIver> Also if we have an increasing collection A_1 <= A_2 <= ...
17:21 <@DRMacIver> Then their union is in the set
17:21 <@DRMacIver> Because mu( Union A_n ) = lim mu(A_n)
17:21 <@DRMacIver> It occurs to me I haven't proved this.
17:21 <@DRMacIver> So just take it on faith. :)
17:22 <@DRMacIver> It's not hard. It's just a question of rewriting the union as union A_n \ A_{n-1} and showing how the resulting sum turns into a limit.
17:22 <@DRMacIver> Ok. We call such a set a D-system.
17:22 <@DRMacIver> And now we pull the following magical wonderful lemma out of our hats:
17:22 <@DRMacIver> Lemma: Suppose a D system contains a ring. It also contains the sigma algebra generated by that ring.
17:22 <@DRMacIver> Proof: Unenlightening. Skipped! :)
17:23 <@DRMacIver> This lemma lets us prove the theorem we wanted.
17:23 <@DRMacIver> Suppose we have two measures mu_1, mu_2 with mu_1(X) = mu_2(X).
17:23 <@DRMacIver> ANd suppose these measures agree on a ring S.
17:23 <@DRMacIver> Then they agree on the sigma-algebra generated by this ring.
17:24 <@DRMacIver> Ok. Now I'm actually going to do something with all this machinery I've built up.
17:24 <@DRMacIver> I'm going to give you lebesgue measure on R.
17:24 <@DRMacIver> (Lebesgue measure on R^n follows as a corollary by something called the product measure. THis is far more work than I want to go into today)
17:24 <@DRMacIver> What we do is we find a convenient ring to work with.
17:24 <@DRMacIver> In this case it will be finite unions of sets of the form [a, b)
17:25 <@DRMacIver> The reason we take them to be closed on the left and open on the right is so we have that closure under set minus property.
17:25 <@DRMacIver> We define the obvious measure on this ring:
17:25 <@DRMacIver> mu( Union_1^N [a_n, b_n) ) = sum b_n - a_n
17:26 <@DRMacIver> (Where the union is disjoint)
17:26 <@DRMacIver> Exercise: This is a measure.
17:26 <@DRMacIver> We extend by the above magic extension theorem.
17:26 <@DRMacIver> We call the sigma algebra generated by these sets the borel algebra of R.
17:26 <@DRMacIver> Written B(R)
17:27 <@DRMacIver> Theorem: THere is a unique measure on B(R) such that mu([a, b)) = b - a
17:27 <@DRMacIver> Existence follows by what I've just said.
17:27 <@DRMacIver> Uniqueness.
17:27 <@DRMacIver> We only have uniqueness for finite measures. We want an infinite measure.
17:27 <@DRMacIver> What we do is we break R up into countably many sets.
17:28 <@DRMacIver> Specifically [n, n+1) for n in Z
17:28 <@DRMacIver> (the integers)
17:28 <@DRMacIver> On each of these the measure must give measure 1
17:28 <@DRMacIver> So we can apply the uniqueness theorem from above to give a unique measure on [n, n+1)
17:29 <@DRMacIver> And then a measure on R is defined by it's restriction to each of these, because mu(A) = sum mu(A int [n, n+1) )
17:29 <@DRMacIver> (by additivity)
17:29 <@DRMacIver> So uniqueness on each interval gives uniqueness on R.
17:29 <@DRMacIver> Yay. :)
17:29 <@DRMacIver> This is lebesgue measure on R.
17:29 <@DRMacIver> I'll use this uniqueness trick to prove a few basic properties, then I'm going to call it a night.
17:30 <@DRMacIver> Theorem: Let mu be lebesgue measure on R. Then mu(A + x) = mu(A)
17:30 <@DRMacIver> Where by A + x we mean { y + x : y \in A }
17:30 <@DRMacIver> Proof: First we have to show that this makes sense and that A + x is in B(R)
17:30 <@DRMacIver> This is easy.
17:30 <@DRMacIver> We show that { A : A + x in B(R) } is a sigma algebra.
17:31 <@DRMacIver> And it clearly contains our original ring.
17:31 <@DRMacIver> So it contains the generated sigma algebra, and we're done.
17:31 <@DRMacIver> Now, the map A -> mu(A + x) may be seen to be a measure on B(R)
17:32 <@DRMacIver> Further mu([a + x, b + x) ) = b - a
17:32 <@DRMacIver> Hence it agrees with lebesgue measure on these sets.
17:32 <@DRMacIver> So, by our uniqueness result, it must be equal to lebesgue measure.
17:32 <@DRMacIver> A similar trick shows that mu(-A) = mu(A)
17:33 <@DRMacIver> Final theorem: Lebesgue measure is the unique measure on B(R) such that mu([0, 1)) = 1 and mu(A + x) + mu(A)
17:33 <@DRMacIver> Proof: NOte that [0,1) = union [r/2^n, (r+1)/2^n)
17:34 <@DRMacIver> FOr 0 <= r <= 2^n
17:34 <@DRMacIver> These are all translates of eachother so have the same measure.
17:34 <ramanujan> ! you meant mu(A + x) = mu(A) ?
17:34 <@DRMacIver> Yes
17:34 <@DRMacIver> Sorry
17:34 <@DRMacIver> Typoing as I make a mad dash to the finish line. :)
17:34 <@DRMacIver> So, we have that mu([0, 1/2^n)) = 1/2^n
17:35 <@DRMacIver> Now filling up [a, b) with a lot of copies of [0, 1/2^n) and passing to the limit, we get that mu([a, b)) = b - a
17:35 <@DRMacIver> Now apply previous theorem.
17:35 <@DRMacIver> A similar result applies to R^n and we can use this to prove useful things about how linear maps affect lebesgue measure.
17:37 <@DRMacIver> But I won't.
17:37 <@DRMacIver> Because I'm already running too bloody late as it is. :)
17:37 <@DRMacIver> The End.
17:38 * ramanujan applauds
17:38 <@DRMacIver> Thanks for coming, and sorry I screwed this one up so badly.
17:38 * Syzygy- applauds
17:38 <@DRMacIver> At least it was better than the ordinals talk. ;)
17:38 < davidhouse> thankypou :)
17:38 <@DRMacIver> (I blame my ISP. It kept disconnecting me and totally derailing my train of thought)
17:39 < davidhouse> i followed most of that, until you started mentioning borel sets. i'll work through the logs and make sure i understand it all
17:39 < davidhouse> what time next week?
17:39 <@DRMacIver> Questions?
17:39 <@DRMacIver> Same time next week.
17:39 < davidhouse> okay.
17:39 <ramanujan> It's almost as if the ISPs are conniving against us. first nerdy2's talk suffered on account of a flaky wifi connection, now yours on account of a silly ISP
17:40 <Syzygy-> Bah. I hope more ppl logged, since I cannot seem to get my irssi to cooperate.
17:40 < davidhouse> will someone be able to put the log onto the efnet-math.org wiki tonight?
17:41 <@DRMacIver> Unless someone else volunteers, probably not tonight. :)
17:41 <@DRMacIver> I can send you an unedited #mathematics log from today if you want.
17:42 < davidhouse> unedited is fie.
17:42 < davidhouse> *fine
17:42 <@Chandra> i will upload the logs
17:43 <@DRMacIver> Thanks Chandra
17:43 <@DRMacIver> Oh. I almost forgot.
17:43 < wtbw> thank you DRMacIver :)
17:44 <@DRMacIver> The presentation was strongly influenced by Geoffrey Grimmett's Probability and Measure course
17:44 <@DRMacIver> See http://www.statslab.cam.ac.uk/~grg/teaching/probmeas.html
17:44 <@DRMacIver> It has a lot of the proofs I skipped
17:44 * maherarar applauds belatedly, due to an aunt's call.
17:45 <annie_tyr> banachTarski paradox, havnt thought about that one in a while
17:46 <annie_tyr> i remember a calc prof complaining that a lot of things were like the bourbaki argument deliniating the diferences between a ball and a sphere
17:46 <@DRMacIver> I'm going to switch back to my efnet account.
17:47 <Kit> I didn't intend to bother mentioning BT. You'll note I barely bothered mentioning nonmeasurable sets.:)
17:47 <annie_tyr> and that one was high on his list
17:47 <wtbw> oh, it was you Kit. thought i recognized the surname :)
17:48 <Kit> Yeah. For uninteresting reasons I felt it prudent to use the freenode side of the link.
17:48 <Kit> And someone over there has already taken Kit. ;)
17:49 <annie_tyr> Kit : i was just following the wilkepedia links:)
17:49 <wtbw> hehe
17:50 <Kit> I'll try to make next week more concise.
17:50 <Syzygy-> Hehe
17:50 <Kit> Oh. I almost forgot.
17:50 * Syzygy- believes that as much as he wants to. :P
17:50 <Kit> Your homework. :)
17:50 <maherarar> What's on tap for next week?
17:51 <Kit> http://www.efnet-math.org/~david/limsupinf.pdf
17:51 <Kit> If you don't understand this, make sure you do. As I intend to wield lim sup and inf like a great big axe in next week's proofs. :)
17:51 <Kit> maherarar: I'm going to use some of the nonsense I developed today to introduce lebesgue integration.
17:51 <annie_tyr> Kit what degree are you working on
17:51 <Kit> The 'finding a job and making lots and lots of money' degree.
17:52 <Kit> I graduated this summer.
17:52 <maherarar> Will there be anything for people who've had two semesters of real analysis?
17:52 <ramanujan> so you've given up on grad school for good?
17:52 <maherarar> Tangents are highly recommended.
17:52 <Syzygy-> Kit: With which degree?
17:52 <Kit> ramanujan: Not for good. For now.
17:52 <annie_tyr> I got as far as grad level calculus when I took my masters
17:52 <annie_tyr> impressive kit
17:52 <Kit> maherarar: Does two semesters of real analysis include lebesgue integration?
17:52 <ramanujan> what is grad level calculus?
17:52 <maherarar> Yes.
17:53 <Kit> Syzygy-: Part III mathematics (roughly masters equivalent)
17:53 <Kit> maherarar: Then no, probably not. The third one likely will.
17:53 <Syzygy-> right.
17:53 <Kit> (Which isn't till january)
17:53 <annie_tyr> hmm, a bachelors here is the 4 year degree
17:53 <Kit> Given that the description of the third one is currently 'fun stuff you can do with the lebesgue integral' :)
17:53 <annie_tyr> ramanujan
17:53 <annie_tyr> and I have the degree beyond that
17:54 <annie_tyr> so its 5th year studies
17:54 <maherarar> i'll check that one out, then
17:54 <ramanujan> annie_tyr: i was asking what grad level calc comprises, not what the education system in your area of the world is :)
17:54 <Kit> (with a fourier series bent)
17:55 <annie_tyr> oh sorry
17:56 <annie_tyr> ramanujan its been a few years frankly, and I have not needed it since
17:57 <annie_tyr> it made sense ofthe 6 statistics courses I had after
17:58 <ramanujan> ok
17:58 <Kit> maherarar: Sorry if this wasn't very interesting. It was really intended as an introductory talk, so was probably never going to be too exciting for people who have already met the subject.
17:59 <delta_> Kit, it's the updated version? ;)
17:59 <Kit> delta_: Should be. :)
17:59 <delta_> ah :)
17:59 <delta_> weird, then ;)
18:00 <Kit> Oh dear. The error not fixed?
18:00 <Kit> Maybe you're getting a cached version?
18:00 <delta_> maybe. Let me try again.
18:01 <maherarar> Yeah, it's a good subject for people to know, though. And I'm not even an analyst. I'm not too far from suicide-bombing a bus of analysts.
18:02 <Kit> Heh
18:02 <Kit> I agree. I'm big on people learning it.
18:02 * Kit considers the Riemann integral to be an abomination which should not be taught. :)
18:02 <Kit> So the thought of people being able to grow up as mathematicians thinking that integration = Riemann integration sends shudders down my spine.
18:02 <@Chandra> done editing the log, let me upload it :)
18:03 <delta_> Kit, for example, "sup xn â\x89¤ x + eps" the sup is taken on m >= n, yes? is it correct?
18:03 <toad_> how does one pronounce lebesgue in english? like in french?
18:03 <Kit> I think it's taken on n >= m.
18:03 * Kit wonders if he can really be bothered to fix this tonight. :)
18:03 <delta_> I think so, but in this case, there exists an 'm'
18:03 <Kit> I suppose I should.
18:04 <delta_> same typo later.
18:05 <Kit> Hmm. Ok, thanks.
18:05 <Kit> I'll deal with it.
18:06 <maherarar> lehBAYG
18:07 <pseudoXh4> Those were some mighty interesting jokes that were given during the lecture.
18:07 <pseudoXh4> :}
18:07 <pseudoXh4> Three unknowns.. lol
18:07 <Kit> ok. Updated the file. Hopefully I've fixed those.