Haar measures + Plancherel for LCA groups

Hinzugefügt: 2026-05-09

[00:00:00]Welcome back mathematicians.

[00:00:02]We're talking about fora analysis on groups a bit more today.

[00:00:07]Uh in particular I want to go into a little bit more detail of some of the subtleties behind the planter theorem.

[00:00:15]Uh and I'll talk about a couple examples and stuff. But before I get into that, I do want to sort of ask you about what direction this might go in. This video got a lot of views. The first video got a lot of views, uh, which I'm happy to see, but I'm wondering, uh, what people are are hoping for next. I'll let you know. Here are some things that I'm thinking about. I'm thinking about talking about Forier theory on the piatics and number fields and the Adele's and moving over to some Tate's thesis sort of stuff. I'm also thinking about maybe uh the forier transform for finite nonabelian groups and maybe some you know planter theorem da da da da le group stuff. Um I don't know. So so let me know in the in the comments below if you have a feeling of uh somewhere you'd like to to a particular direction you'd like to see us to go in. Um but okay so so you know we talked about plant trail theorem last time and uh so what do we need to make sense of all this? Well we need topological groups but we also need uh a measure right. So perhaps I mentioned this last time but uh you know we want to do integration on a group. How can you possibly integrate a function on a group? Well you need some way to to do integrals. And the main mathematical theory for that is to have a measure.

[00:01:48]And the reason why we consider groups which are locally compact. So, oh gosh, I should have looked up the exact definition uh before I started. But if I'm recalling correctly, it says that uh locally compact means that so locally, you know, it always means for every element of your topological space, there exists an open set containing it such that blah blah blah.

[00:02:15]So in this case, it's maybe a little weirder than than what you're used to.

[00:02:20]So it it is that there's an open set uh which contains G. uh but the such that is now uh such that uh it sits inside a compact uh subset of our topological space. So that's what it means to be locally compact. So so essentially it says that you can uh find a compact open set around every point. Um and so why do we care about uh locally compact?

[00:02:55]because locally compact groups have a god-given measure on them. Uh which is called a HAR measure. So I'm not going to go into the technical definition. You can look it up. You know it's in uh it's in har wait is har I always get confused. Two A's I think.

[00:03:15]HAR measure. So this is a particularly nice measure on your group. So first off I mean there should be a measure space right of course this is a topological group so that means it has a collection of open sets and whenever you have a topological space you have a sigma algebra called the burell algebra which is just generated by the open sets of your topological space right so it's with respect to that measure space um and the harm measure well it's not quite totally well defined but uh some properties of of a harm measure. There's there's a few definitions um or a few a few things that go into it, but some properties are that if you have any subset of your group, uh the harm measure is supposed to be uh invariant under translation.

[00:04:07]Now, uh normally you pick ahead of time is it left invariant, is it right invariant? Um doesn't really matter as long as you pick one and stick to it. I mean there's like conventions and and as long as you're consistent that's what matters. So uh let's say you know it's it's it's invariant under translation which by the way I mean this is just the the measure that we take on the real numbers by the way I mean this is one of the things that we want. So a if if a here is a subset of real numbers you know this is one of the properties that we want I mean it's true for the lebeg measure right the standard measure we do integrals with on the real numbers um but you know this is also one of the properties in in the the proof that says that not all sets subsets of the real numbers are measurable uh you know using the axiom of choice.

[00:04:59]Well, it's in in that proof you're you're basically giving a list of demands you want for the measure and this and this is one of the things um uh that you want. Okay. Uh so, oh I didn't write down the other features of it. Um there's two other features though and that is uh basically that the measure if you take some arbitrary set s um this will be equal to the limb I want to say it's the limb in the measures of the open sets uh which which contain uh your subset s.

[00:05:44]So maybe I'm getting this backwards. Um and and likewise it will be the limb soup of the compact sets um compact subsets uh which are contained in your set. So you can approximate it from above and above. So these are some of the nice features that that define the har measure. So uh what does this look like?

[00:06:13]Well, I just I I well Okay. Uh uh there's there's another theorem. So another theorem that basically says uh if G is locally compact uh then it has a harm measure, right? So there's some other niceness words but basically it's you know it has some measure which is invariant under left translation and it satisfies both of these properties.

[00:06:52]Moreover, how many harm measures though because I said there's supposed to be like a god-given one. Uh that's not entirely true. Moreover, uh for any harm measures, for any pair of measures mu1 and mu2, uh there exists some constant uh I guess what are we taking? We could take this valued in the complex numbers, I guess.

[00:07:16]Or maybe the real numbers is just real valued measure. That's normally what we do. uh um but there's some some uh number such that uh you know for all subsets one is just a scalar multiple of of the other right so in other words in other words uh one is just a scalar multiple of the other so it's well defined so there exists one and only one up to scaling so usually what we do then is we pick some sort of nice subset set and we demand that the measure of that subset be equal to one and that normally uh that then fixes the choice of hard measure that we care about. So for example, the real numbers is a locally compact group under addition and the har measure uh if if we choose the har measure that assigns uh the unit interval one then this is just what's called the lebeg measure right and it just recovers ordinary integrals um on r uh um for example in the piotic numbers which you know I I need to make some videos um about that. A very special subset is the piatic integers. You know, it's common to give that volume one. Um if you have a league group and you choose you, you know, you choose some maximal compact subgroup, you know, it's not uncommon to normalize it so that that maximal compact uh has has volume one. Um so speaking of some other examples I mean uh by the way if you take this to be r to the n and you know you take this the the unit uh box uh then uh again this har measure just recovers the ordinary bag measure on r to the n.

[00:09:16]Um if you work instead on the multiplicative group uh of real numbers then the har measure can be defined uh in the following way. So it is the on some subset of the real numbers it becomes you take the integral over s with respect to now this is the har measure on the additive real numbers. uh divided by uh divided by uh of of of this function I should say.

[00:09:55]So let me put it let me put it here.

[00:10:02]Um uh what are some other examples for GLN for GLNR which is a league group uh the har measure on GLNR maybe I'll just I don't know I'll just write it as mu uh this is going to be the integral over s but remember I mean this is n byn matrices right so we can view this naturally as a subset of r the n^2 and so it's going to be the integral over um one over the determinant of x absolute value to the power n with respect now to the har measure on on r to the n and okay this is great because now you know you can use this to get hard measures for for all sorts of groups um as well. So these are exampam these are some examples of of hard measures. And by the way um so you know I like I said if you want to go read the actual construction of this you know you can go read it in any of these books you know these books of Deepmar I've been following Rama Krishnan Valenza um where else is it it's it's in a bunch of places but there's actually a lot of uh there's a lot uh of computations that you can do without knowing the precise rigorous definition uh I I mean I already told you that that for the real numbers you know we just recover the league measure that you know and love.

[00:11:36]Um and here are some other cases where we can you know relate measure the the hard measures that we're interested in to to measures that we know. But even like in this piatic case uh you know maybe eventually I'll I'll do some videos where um you know just having this fact just knowing that this is a hard measure and it's normalized to this and knowing some things about the topology there's actually a bunch of integrals that you can compute um what their actual values are just based on these things. So you can get by with just these properties alone. Um okay so that is why we care uh about locally compact groups right so so going back up what are we really interested in we want to talk about the planter theorem uh which means we need to talk about integrals um which means that we need some way of uh measuring subsets we need some way of integrating uh shocking so uh now here's an issue here's an issue with what I said uh last video. I mean uh the planel theorem definitely does say that we have uh an isometry between these two L2 spaces. Um but there's a little bit of a problem if we try to approach this naively. So consider the real function uh f ofx is 1 / 1 + the absolute value of x to the power of 3 over 4.

[00:13:10]So I claim that this is an L2 function.

[00:13:14]Uh why is that? Well, if we integrate the square norm of this function over the entirety of the real numbers, uh this is just well I mean this is already a positive function. So we're just integrating it square. So this is something like you know the the integral of 1 + x uh over you know 3 halves right so this is this is vaguely you know this is approximately something on the order of right it's about x to the minus uh uh 1.5 right so so this this looks something like the integral over all real numbers of you know x to the um well x x x to the minus 1.5 I guess right so you know if you think about like your p series test your comparison test like this this converges right so great now let's go back uh to the forier transform and actually uh brace yourselves I'm going to some whiplash is going to happen on stream so so remember how we wrote the forier transform for the the real numbers up above. So the forier transform was uh we interpreted it as this uh as this character.

[00:14:37]Actually I lied to you a little bit.

[00:14:39]Notice I put the minus sign in both places here. Um the actual forier transform that we want to define uh would be would be uh without this minus sign.

[00:14:53]So uh uh so that is let me let me I can erase some of this and clarify the notation a bit.

[00:15:03]So we want something like f hat at y uh to be the integral over r of fx e to the two and then the minus sign is still here as I said right but uh the way that we typically label the characters of the real numbers because remember now we want to um from the from the previous video we want to interpret this this y as being the label of the character. We normally associate that for the label of the character uh given by this right that sends x to to this quantity. So what's actually going on here is the value of the forier transform at the character determined by y is the integral over all of this given by uh now we take this character and evaluate it at x but we're taking the conjugate the complex conjugate of that character and so this is actually the pattern that we're going to repeat here.

[00:16:15]So, so the way that we would like to so mimicking this pattern uh the way that we're going to define say the the forier transform so we suppose I have some character on the real numbers or really I mean in general I have a group I have a character uh of this uh of its dual group then the forier transform of a function you know l well some sufficiently nice function uh on the group this should be the integral over the entire group of f gi g bar dg. So this is going to be uh our definition of the forier transform.

[00:17:12]Um but so I didn't have to tell you all of that. Now let's go and look at you know this is an L2 function. Let's go see what happens when we try to take its fora transform. Uh and in particular we look at the forier transform at zero.

[00:17:28]Right? So that's that's the uh its value at the character which sends x to 2 pi i 0 * x which is just one. Right? So this is the identity function everywhere.

[00:17:42]Um well that should be the integral over the real numbers of our function 1 + <unk>x uh 3/4s times the value of our character the the complex conjugate of this character but the complex conjugate of one is one. So this is just uh this is just this uh integral. Uh but now we have a problem right? Now we have a problem because uh this is a exponent that's smaller than one. So again you think about your your P series test for uh convergence uh you know this is this is something on the order of the integral um of of x to the 3/4s uh and that's no bueno. So uh this is going to be infinite. And so actually our forier transform uh if we try to in this naive sense uh forier transform uh is undefined uh doesn't seem to be defined everywhere. So we run into a little bit of a problem. Uh so here's the fix that one normally does. Um so you instead take so so for the moment we're going to assume I'm working with a locally compact aelion group an LCA group right uh like the real numbers and I'm going to define um if I have uh uh a function which is in L1 BCA so so this means it is uh it is bounded And this means it has compact support.

[00:19:28]uh then on these functions I can define uh I can define the fora transform exactly as as I mentioned right so this will be the integral over a of f a times the complex conjugate uh here okay so there's no l2 in this theory yet however uh because f is l1 uh well uh any given f is bounded right any given f in in L1 BCA is bounded. And so in fact if I were to look at the uh the L2 norm of f well this is this is f squared which is you know f uh absolute value f absolute value f d mu and of course this is going to be you know f is bounded above by m. So uh so this is going to be what should I say here um uh you know f bounded above by by m so I'll bound one of the factors here and uh the thing is this m is also uh I mean it's it's it's well we can just we can just pull it out um we can just pull it out here right so this is uh This is simply given by by this right. Um but what did we assume? We assumed that that that f is in uh l1 uh bc. So this uh converges.

[00:21:12]So actually our function uh our function is in uh L2 and so we find that uh L bc1 uh is contained uh in in L2.

[00:21:29]In fact more is true and this is what's going to allow us to uh rectify the situation. It turns out that this L1 BC is dense in L2, right? So everything in L2 can actually be approximated by L1 BC. So now here's the fix that we're going to do. And uh I'll call this let's go purple again. Maybe it was hard to read, but I like Oh no, where did purple go? I lost purple. Okay, well that's fine. I guess we'll just do blue. Um so theorem And this is theorem uh 8.4 uh 2 from Deepmar's uh a first course in harmonic analysis. Although he doesn't prove the full theorem there. He actually points to a different source.

[00:22:22]Um but here's what it is. There is a unique harm measure on uh the dual group such that for all functions in uh L1 BCA the forier transform is in uh L2 uh BCA hat. Yeah. And moreover uh the L2 norm in fact of f is the L2 norm of uh f hat. So this is the isometry part of the condition. And the final thing I want to mention in in today's video then is uh we can now extend we can this is the idea that we use to extend to get a forier transform between the L2 spaces uh I guess I've been calling this a forelian group right is because basically now take now if you take an element of of L2 two. I mean this is approximated by some uh sequence of functions that converges to f. And so therefore we can take um we can take we can define a map maybe call it script f between these L2 spaces which is going to take f and it will send it uh to the limit of the forier transforms of the individual functions.

[00:24:17]And of course there's you know there's work that you need to do there's things that you need to check here. Um but it turns out that this is a well- definfined map and this is actually what defines the um isometry. So so this is a is a linear uh isomorphism of uh vector spaces uh hilbert spaces. In fact I mean this this gives you uh this here gives you um an isometry. Okay. So I think that does it for today's video. I think maybe oh something else I forgot to mention. uh posson uh posson summation because this is something you know maybe I could talk a little bit about um you know this has to do with maybe the number theory places that we could go um I don't know uh we could look at examples of dual groups you know uh finite aelion groups let let me know what you guys think let let me know what you want to see you seem to like part one um I'm interested to know what you're hoping for for future installments in here. Uh, I should mention I think my Patic hat is now the sale is done. I think it's still available if you click the link below.

[00:25:29]Let me know if you want a Patic hat.

[00:25:30]Where do I put my Pic hat? Uh, you know, I have YouTube memberships. I have uh a Patreon with a Discord that no one ever says anything in. Uh, you know, and I offer tutoring services. Uh, you can email me at k Theoretutoringgmail.com.

[00:25:52]And for now, all that's left to say is, I know you know it. Say it with me.

[00:26:00]Q E