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## Definition of Haar Measure

The name “Haar measure” came into existence after Alfred Haar in 1933 introduced invariant measures (invariant with respect to the group operation) on topological groups. Although Haar measure can be defined on locally compact group, we focus on locally compact Hausdorff group.

Let us quickly recall some preliminaries:

**Locally compact Hausdorff Topological group:**- – Topological space ( is a set and is a collection of empty set, subsets of and the set itself such that is closed under arbitrary union and finite intersection.)
- – Hausdorff topological space if the underlying topology is Hausdorff, that is, any two distinct points have disjoint neighborhood.

Most of the spaces I consider in practice are Hausdorff spaces. Simple example would be Euclidean space. One of the simplest counter example is cofinite topology and one of the other rich counter example is Pseudometric spaces. - is called a topology group, if it is a group and also a topological space along with the continuous group actions and

**Borel regular measure:**- The -algebra is a collection of subsets of that is closed under complement, countable union and intersection generated by . Any set in is called a Borel set.
- A measure on a topological space with an underlying -algebra is called Borel measure.
- A Borel measure that satisfies the following are called regular measure.

-measure of any compact subset of is finite,

-outer regular: is open

-inner regular: is compact for all open subsets

Throughout, let be a locally compact Hausdorfff topological group (also a measure space). **A non-zero measure on is called left Haar measure if it is**

**a Borel regular measure****a left invariant measure, that is, for all and all measurable sets .**

Similarly we define right Haar measure. The construction of Haar measure and proving its uniqueness are beautiful, which we will save it for later. In this note, let us consider only examples.

Some trivial examples are cardinality acts as Haar measure on finite groups (can be normalized) and for infinite discrete groups; the usual Lebesgue measure on ; Lebesgue measure on , where denotes the circle group.

The following formula is one of the nice one that I cherish.

*Suppose is homeomorphic to an open subset in such a way that if we identify with , left translation is an affine map in the sense that , where is an invertible matrix and is a vector in . Then is a left Haar measure on .*

Similarly we can state it for right Haar measure with the appropriate right translation.

Let us recall at first a few examples applying the above theorem.

## Examples:

- Multiplicative Group:

This is an open subset of the Euclidean line .

Left translation: where and is zero.

Thus is the left Haar measure on the multiplicative group.

In fact, it is the right Haar measure as well. - The upper half plane or the set

, for all

Identify each entry with in (check that it is homeomorphism)

Left translation: given by the matrix and the vector since the matrix multiplication is given by

which is precisely given by .

Hence is the left Haar measure (and also the right Haar measure).However, the right translation is given by where and is zero. Hence is the right Haar measure.

**In general, if is an open subset in and we have with where all the partial derivatives of exists and continuous, and then is the left Haar measure and is the right Haar measure where denotes the modulus of the determinant of the Jacobian of the function . **

- Special unitary group: (set of all unitary matrices with determinant 1.)

Homeomorphism between the sphere and the special unitary group :

where for

is the left and right Haar measure.

**References: **

**References:**

S. C. Bagchi, S. Madan, A. Sitaram and U. B. Tiwari, A first course on Representation theory and linear Lie groups, Universities Press, Hyderabad (2000)

## Hausdorff dimension of certain sets

## 1. Generalized Cantor set:

Fix . Choose (small) and (large) such that . Fix .

Let be the union of intervals of length . Let be the natural measure (normalized one dimensional Lebesgue measure restricted to ) arises from .

Let be the union of intervals of length and be its natural measure.

Similarly for each let be the union of intervals of length and be its natural measure.

We consider the generalized Cantor set .

(Exercise: Then is compact perfect and nowhere dense set.)

Also note that,

- weakly converges to a measure supported in .
- is locally compact and , the space of compactly supported continuous function on is a separable topological vector space.
- By the application of Riesz representation theorem, space of continuous function vanishing at infinity is same as the space of Radon measures. (All the above ‘s are Radon measure, that is locally finite and inner regular measure)
- By the application of Banach-Alaoglu theorem, is a weakly convergent sequence.

- (enough to prove ).
- By the definition of , for there exists large such that . Then

and hence .

- Since is compact, if is a Borel cover of then we can assume that is a finite Borel cover. Also since is totally disconnected, can be assumed to be finite Borel cover, with ‘s disjoint open intervals and their end points are in the complement of .Let be such a cover. Then for each let be the smallest integer such that has at least one interval of length . Note that there cannot be more than consecutive intervals of length in (otherwise, it contradicts the choice of to be the smallest). Let be the number of consecutive intervals of length in . Then . Hence diameter of .
We know that each interval of length has intervals of length and intervals of length . Similarly each interval of length has intervals of length for . Thus intervals of length has intervals of length . There are totally intervals of length and since ‘s are disjoint and finite, we have for large (for all ).

Also we have . Hence

Thus .

- By the definition of , for there exists large such that . Then

(to be updated) This blog is loosely based on the notes prepared while giving a lecture series during January 2017-April 2017 at IISER - Bhopal. The definitions, the notations, proofs, theorems are from the references: 1) W. F. Donoghue; Distributions and Fourier Transforms, Acad. Press, 1969. MR3363413 2) Pertti Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR1333890 3) P. Mattila. Fourier Analysis and Hausdorff Dimension, Cambridge University Press, Cambridge, 2015. 4) Wolff T.H.: Lectures on Harmonic Analysis. University Lecture Series, 29. Amer. Math. Soc., Providence, RI (2003)

## Equivalent definitions of Hausdorff dimension

### Notations:

- – Measurable space, (, ) where denotes the -algebra
- We mean measure, to be outer measure, that is function on to such that it is monotone, countably sub-additive and assumes value zero for the empty set.
- A measure is called Borel measure if the Borel sets (sets formed by open/closed sets) are measurable, that is in .
- A measure is Borel if and only if for any sets with we have . ( denotes the distance between two sets and )

- A Borel measure is Borel regular if for every there exists a Borel set such that and .
- If a Borel measure is not Borel regular, then defined as Borel , is Borel regular.

## Equivalent definitions of Hausdorff dimension

### 1. Definition of Hausdorff dimension:

(By Hausdorff via Caratheodory’s construction)

For any set we define the -dimensional Hausdorff measure of as,

where denotes the infimum of taken over all the collection of Borel cover of with diameter of , .

- Note that for all .
- When is an integer, denotes the volume of the -dimensional unit ball in (for ).
- For convenience, we assume to be one when we consider to be non-integer.
- is counting measure.

We observe the following:

- is Borel measure
- It is easy to check that is monotone, countably sub-additive non-negative measure that assumes value zero for the empty set, that is is a (outer) measure for all . Hence for any ,
- If with , then choose and by the definition of infimum choose a Borel cover of with , that is diameter of is . Then can either intersect or , but not both. So we have

and thus .

- That is, and hence . Thus is Borel measure.

- is Borel regular.
- Fix . Choose a Borel cover of such that . Then is a Borel set containing with .

- For and , implies . (In other words, implies )
- Note that for , by the definition of infimum, there exists a Borel cover of such that where denotes the diameter of . Hence

which goes to zero as goes to zero.

- Note that for , by the definition of infimum, there exists a Borel cover of such that where denotes the diameter of . Hence

The * Hausdorff dimension* of a set is defined as

*We have computed the Hausdorff dimension of generalized Cantor set here.*

(to be updated) This blog is loosely based on the notes prepared while giving a lecture series during January 2017-April 2017 at IISER - Bhopal. The definitions, the notations, proofs, theorems are from the references: 1) W. F. Donoghue; Distributions and Fourier Transforms, Acad. Press, 1969. MR3363413 2) Pertti Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR1333890 3) P. Mattila. Fourier Analysis and Hausdorff Dimension, Cambridge University Press, Cambridge, 2015. 4) Wolff T.H.: Lectures on Harmonic Analysis. University Lecture Series, 29. Amer. Math. Soc., Providence, RI (2003)

## Some measures and some sets

This blogpost is loosely based on a series of talk in IISERB during August-December 2016..

We try to learn the relation between geometric measure theory and Fourier analysis on . We will be concentrating on those parts of Fourier analysis on where Hausdorff dimension plays role. However, we discuss a few topics in geometric measure theory in detail. Let us see how restriction conjecture is related to Kakeya type problems.

What is measure theory? The study of measures, generalizing the intuitive notions of length, area, volume.. We are aware of the Lebesgue measures. In general it is the study of measures on any general space, not just Euclidean spaces.

**Lebesgue Measure**: : Given a set , we define the measure (outer measure)

where infimum is taken over , the countable collection of boxes whose union covers . In other words instead of collection of boxes we can consider the countable collection of balls of radius whose union covers :

We recall a very beautiful measure:

Hausdorff Measure: : Given a set , we define the measure

with ,

where the infimum is taken over the countable collection of balls of radius whose union covers .

Recall the following:

on a set $X$, we mean usually by outer measure $\mu$: a non-negative, monotone, countably subadditive (that is, ) function on that gives value zero for the empty set.*Measure*is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.*Borel set*- A measure is called
if the Borel sets are measurable.*Borel measure* - A Borel measure is
if for any , there exists a Borel set such that and .*Borel regular* - A Borel measure is
if compact sets have finite measure.**locally finite** - Locally finite Borel measures are often called
.**Radon measures**

*Example*: Lebesgue measure is Radon measure. Counting measure on any metric space is Borel regular but it is Radon only if the space is discrete. Hausdorff measure is not Radon.

**Vitali type covering theorem for Lebesgue measure**: Let and be a family of closed balls such that every point of is contained in an arbitrarily small ball in , that is, for . Then there are disjoint balls such that .

*Consider the example*: Let be a Radon measure on : , that is, denotes the length measure on axis. The family covers in the sense of the above theorem. But fails the conclusion, since for any countable collection we have .

For Radon measures, the theorem holds if we assume to be a family of closed balls such that each point of A is the centre of arbitrarily small balls of .

Geometric measure theory is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth.

When we mean smoothness, on a careful observation, what we see is the smoothness of the boundary. For a given curve in , can we find a surface of minimal area with that curve as the boundary? This is nothing but the Plateau’s problem, posed in 1760 and completely solved in 1930. While studying this, geometric measure theory was developed.

How about sets which has ‘roughness’ at every point? Can we still study geometry of these sets? We will start with some examples of non-smooth sets:

**Generalized Cantor set**: Let . Fix a set of finite numbers in with for all . Fix small such that for all and ; also such that . We construct a sequence of sets :

is the union of intervals of length with starting points in the set . In other words we have removed in the interval . We call the removed intervals as black intervals and the considered intervals as white intervals. Now on each interval in we have points: . By the assumption on we can see that . Hence we continue to construct as the set of intervals of length . Proceeding like this, at step we have intervals of length . The generalized Cantor set is $Elatex =\cap_nE_n$.

It is an easy ‘Basic real analysis’-exercise to check that this set is nowhere dense perfect non-empty uncountable set.

**Salem sets**: Similar to generalized Cantor sets, we have Salem sets (constructed by Salem 1950). The purpose of this set will be described later. Let . Fix a set of finite numbers in with for all . Fix small such that for all and ; also such that . Choose a sequence such that . We construct a sequence of sets :

is the union of intervals of length with starting points in the set . In other words we have removed intervals in the interval . We call the removed intervals as black intervals and the considered intervals as white intervals. Now on each interval in $E_1$ we have $N$ points: . By the assumption on we can see that . Hence we continue to construct as the set of intervals of length . Proceeding like this, at step we have intervals of length . The Salem set is .

**Brownian Motion**: Consider the space of all continuous functions with such that the increments and $\omega(t_4)-\omega(t_3)$ are independent of and .

For Example: For any fixed , consider the random variable given by with the probability density function , that is with ). We call the random variables defined by , Brownian motion. Zeroes of the Brownian motion has Hausdorff dimension when we consider . (What are zero sets of the Brownian motion? For each , “Hitting back at the starting point”). We then have the probability measure on .

Does these sets have any geometry to it? We can right away observe that there is an in-built measure to these sets. Are these measures absolutely continuous with respect to the measures like Hausdorff measures? Is there an analogue of ‘smoothness’ to be spoken on these sets?

The main references are : Fourier analysis and Hausdorff dimension and Geometry of sets and measures in Euclidean Spaces by Pertti Mattila; Decay of the Fourier transform by Alex Iosevich and Elijah Liflyand.

(to be continued….)