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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 $T_1$ group, we focus on locally compact Hausdorff group.

Let us quickly recall some preliminaries:

• Locally compact Hausdorff Topological group:
• $(X,\tau)$ – Topological space ($X$ is a set and $\tau$ is a collection of empty set, subsets of $X$ and the set $X$ itself such that $\tau$ is closed under arbitrary union and finite intersection.)
• $(X,\tau)$ – 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.
• $X$ is called a topology group, if it is a group and also a topological space along with the continuous group actions $X \times X \rightarrow X: (x_1,x_2)\rightarrow x_1x_2$ and $X\rightarrow X: x\rightarrow x^{-1}$
• Borel regular measure:
• The $\sigma$-algebra $\sigma(\Sigma)$ is a collection of subsets of $X$ that is closed under complement, countable union and intersection generated by $\Sigma\subset 2^X$. Any set in $\sigma(\Sigma)$ is called a Borel set.
• A measure $\mu$ on a topological space $(X,\tau,\Sigma,\mu)$ with an underlying $\sigma$-algebra is called Borel measure.
• A Borel measure that satisfies the following are called regular measure.
-measure of any compact subset of $X$ is finite,
-outer regular: $\mu(A)=inf\{\mu(U):A\subset U, U$ is open $\}\ \forall A\in \Sigma$
-inner regular: $\mu(U)=sup\{\mu(K):K\subset U, K$ is compact $\}\$ for all open subsets $U$

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

• a Borel regular measure
• a left invariant measure, that is, $\mu(gA)=\mu(A)$ for all $g\in X$ and all measurable sets $A$.

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 $\mathbb{R}^n$; Lebesgue measure $\frac{1}{(2\pi)^n}d\theta_1\cdots d\theta_{n}$ on $S^1\times\cdots\times S^1$, where $S^1$ denotes the circle group.

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

Suppose $X$ is homeomorphic to an open subset $U\subset \mathbb{R}^n$ in such a way that if we identify $X$ with $U$, left translation is an affine map in the sense that $xy=A_xy+b_x$, where $A_x$ is an invertible $n\times n$ matrix and $b_x$ is a vector in $\mathbb{R}^n$. Then $|det\ A_x|^{-1}dx$ is a left Haar measure on $X$.

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:

1. Multiplicative Group: $((0,\infty),\cdot)$
This is an open subset of the Euclidean line $\mathbb{R}$.
Left translation: $xy=A_xy+b_x$ where $A_x=(x)$ and $b_x$ is zero.
Thus $x^{-1}dx$ is the left Haar measure on the multiplicative group.
In fact, it is the right Haar measure as well.
2. The upper half plane or the set
$X=\bigg\{$ $\begin{pmatrix} x_1 & x_2 \\ 0 & 1 \end{pmatrix}$, for all $x_1>0,x_2\in\mathbb{R}\bigg\}$
Identify each entry with $\begin{pmatrix} x_1\\x_2 \end{pmatrix}$ in $(0,\infty)\times\mathbb{R}$ (check that it is homeomorphism)
Left translation: given by the matrix $A_{x=(x_1,x_2)}:=$ $\begin{pmatrix} x_1 & 0 \\ 0 & x_1 \end{pmatrix}$ and the vector $b_x:=(0,x_2)$ since the matrix multiplication is given by
$\begin{pmatrix} x_1 & x_2\\ 0 & 1 \end{pmatrix}$ $*$ $\begin{pmatrix} y_1&y_2\\0 & 1 \end{pmatrix}$ $=$ $\begin{pmatrix} x_1y_1\\x_1y_2+x_2 \end{pmatrix}$ which is precisely given by $A_xy+b_x$.
Hence $x_1^{-2}dx_1dx_2$ is the left Haar measure (and also the right Haar measure).However, the right translation is given by $yx=A_yx+b_y$ where $A_y=$ $\begin{pmatrix} y_1 & 0\\ y_2 & 1 \end{pmatrix}$ and $b_y$ is zero. Hence $x^{-1}dx_1dx_2$ is the right Haar measure.

In general, if $X$ is an open subset in $\mathbb{R}$ and we have $x=(x_1,x_2\cdots x_n)$ with $xy=F(x_1,\cdots,x_n,y_1,\cdots,y_n)$ where all the partial derivatives of $F$ exists and continuous, $\sigma_x(y)=xy$ and $\delta_{y}(x)=xy$ then $|J_{\sigma_x}|^{-1}dx$ is the left Haar measure and $|J_{\delta_x}|^{-1}dx$ is the right Haar measure where $|J_{\sigma_x}|$ denotes the modulus of the determinant of the Jacobian of the function $\sigma_x$

• Special unitary group: $SU(2)$ (set of all $2\times 2$ unitary matrices with determinant 1.)
Homeomorphism between the sphere $S^3$ and the special unitary group $SU(2)$:
$(x_1,x_2,x_3,x_4) \sim$ $\begin{pmatrix} x_1+ix_2 & -x_3+ix_4\\ x_3+ix_4 & x_1-ix_2 \end{pmatrix}$
where for $0\leq\theta\leq\pi,\ 0\leq\phi\leq \pi\ 0\leq\psi\leq 2\pi,$
$\begin{array}{rcl} x_1 & = & cos\ \theta \\ x_2 & = & sin\ \theta \ cos \ \phi\\ x_3 & = & sin\ \theta \ sin\ \phi\ cos\ \psi\\ x_4 & = & sin\ \theta \ sin\ \phi\ sin\ \psi \end{array}$
$\frac{1}{2\pi^2}sin^2\theta\ sin\phi\ d\theta d\phi d\psi$ is the left and right Haar measure.

### 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)

## Equivalent definitions of Hausdorff dimension

### Notations:

• $(X,\Sigma)$ – Measurable space, ($X=\mathbb{R}^n$, $n\geq 1$) where $\Sigma$ denotes the $\sigma$-algebra
• We mean measure, to be outer measure, that is function on $\{A: A \subset \mathbb{R}^n\}$ to $[0,\infty]$ such that it is monotone, countably sub-additive and assumes value zero for the empty set.
• A measure $\mu$ is called Borel measure if the Borel sets (sets formed by open/closed sets) are measurable, that is in $\Sigma$.
•  A measure $\mu$ is Borel if and only if for any sets $A_1,A_2\subset \mathbb{R}^n$ with $d(A_1,A_2)>0$ we have $\mu(A_1\cup A_2)=\mu(A_1)+\mu(A_2)$. ($d(A_1,A_2)=\inf\{d(a_1,a_2):a_1\in A_1, a_2\in A_2\}$ denotes the distance between two sets $A_1$ and $A_2$)
• A Borel measure $\mu$ is Borel regular if for every $A\subset \mathbb{R}^n$ there exists a Borel set $B$ such that $A\subset B$ and $\mu(A)=\mu(B)$.
• If a Borel measure $\mu$ is not Borel regular, then $\tilde{\mu}$ defined as $\tilde{\mu}(A)=\inf\{\mu(B):A\subset B: B-$ Borel $\}$, is Borel regular.

## Equivalent definitions of Hausdorff dimension

### 1. Definition of Hausdorff dimension:

(By Hausdorff via Caratheodory’s construction)

For any set $A\subset \mathbb{R}^n$ we define the $\alpha$-dimensional Hausdorff measure of $A$ as,

$\mathcal{H}^{\alpha}(A)=\lim_{\delta\downarrow 0}\mathcal{H}_{\delta}^{\alpha}(A)$

where $\mathcal{H}_{\delta}^{\alpha}(A)$ denotes the infimum of $\{c_{\alpha}2^{-\alpha}\sum_{i}d(E_i)^{\alpha}\}$ taken over all the collection of Borel cover $\{E_i\}_i$ of $A$ with diameter of $E_i$, $d(E_i)\leq \delta$.

• Note that $\mathcal{H}_{\delta}^{\alpha}(A)\leq \mathcal{H}_{\epsilon}^{\alpha}(A)$ for all $0<\epsilon\leq \delta\leq \infty$.
• When $\alpha$ is an integer, $c_{\alpha}$ denotes the volume of the $\alpha$-dimensional unit ball in $\mathbb{R}^n$ (for $0\leq \alpha\leq n$).
• For convenience, we assume $c(\alpha)2^{-\alpha}$ to be one when we consider $\alpha$ to be non-integer.
• $\mathcal{H}^0$ is counting measure.

We observe the following:

• $\mathcal{H}^s$ is Borel measure
• It is easy to check that $\mathcal{H}_{\delta}^s$ is monotone, countably sub-additive non-negative measure that assumes value zero for the empty set, that is $\mathcal{H}_{\delta}^s$ is a (outer) measure for all $\delta$. Hence for any $A_1,A_2\subset\mathbb{R}^n$,$\mathcal{H}_{\delta}^s(A_1\cup A_2)\leq \mathcal{H}_{\delta}^s(A_1)+\mathcal{H}_{\delta}^s(A_2).$
• If $A_1,A_2\subset\mathbb{R}^n$ with $d(A_1,A_2)>0$, then choose $0<\delta<(1/2)d(A_1,A_2)$ and by the definition of infimum choose a Borel cover $\{E_i\}_i$ of $A_1\cup A_2$ with $d(E_i)$, that is diameter of $E_i$ is $\leq \delta$. Then $E_i$ can either intersect $A_1$ or $A_2$, but not both. So we have
$\mathcal{H}_{\delta}(A_1)+\mathcal{H}_{\delta}(A_2)$$\ \ \ \ \ \ \leq \sum_{A_1\cap E_i \neq \emptyset}d(E_i)^s+\sum_{A_2\cap E_i \neq \emptyset}d(E_i)^s$$\ \ \ \ \ \ \leq \sum d(E_i)^s$

and thus $\mathcal{H}_{\delta}^s(A_1)+\mathcal{H}_{\delta}^s(A_2)\leq \mathcal{H}_{\delta}^s(A_1\cup A_2)$.

• That is, $\mathcal{H}_{\delta}^s(A_1\cup A_2)= \mathcal{H}_{\delta}^s(A_1)+\mathcal{H}_{\delta}^s(A_2)$ and hence $\mathcal{H}^s(A_1\cup A_2)= \mathcal{H}^s(A_1)+\mathcal{H}^s(A_2)$. Thus $\mathcal{H}^s$ is Borel measure.
• $\mathcal{H}^s$ is Borel regular.
• Fix $i$. Choose a Borel cover $\{E_{ij}\}_j$ of $A$ such that $\sum d(E_{ij})\leq \mathcal{H}_{1/i}^s(A)+1/i$. Then $B=\cap_i\cup_jE_{ij}$ is a Borel set containing $A$ with $\mathcal{H}^s(B)=\mathcal{H}^s(A)$.
• For $0\leq s and $A\subset \mathbb{R}^n$, $\mathcal{H}^s(A)<\infty$ implies $\mathcal{H}^t(A)=0$. (In other words, $\mathcal{H}^t(A)>0$ implies $\mathcal{H}^s(A)=\infty$)
• Note that for $\delta>0$, by the definition of infimum, there exists a Borel cover $\{E_i\}_i$ of $A$ such that $\sum d(E_i)^s\leq \mathcal{H}_{\delta}^{s}(A)+1$ where $d(E_i)$ denotes the diameter of $E_i$. Hence$\mathcal{H}_{\delta}^t(A)$
$\ \ \ \leq \sum_i d(E_i)^t$
$\ \ \ \leq \delta^{t-s}\sum_i d(E_i)^s$
$\ \ \ \leq \delta^{t-s}(\mathcal{H}_{\delta}^{s}(A)+1)$which goes to zero as $\delta$ goes to zero.

The Hausdorff dimension of a set $A$ is defined as

$dim_{\mathcal{H}}A$
$\ \ \ =\sup\{s:\mathcal{H}^s(A)>0\}$
$\ \ \ =\sup\{s:\mathcal{H}^s(A)=\infty\}$
$\ \ \ =\inf\{s:\mathcal{H}^s(A)=0\}$
$\ \ \ =\inf\{s:\mathcal{H}^s(A)<\infty\}$

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 $\mathbb{R}^n$. We will be concentrating on those parts of Fourier analysis on $\mathbb{R}^n$ 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: $\mathcal{L}^n$: Given a set $A\subset\mathbb{R}^n$, we define the measure (outer measure)

$\tilde{\mathcal{L}}^n(A)=inf\{\sum_{B\in\mathcal{B}}vol(B)\}$

where infimum is taken over $\mathcal{B}$, the countable collection of boxes $B$ whose union covers $A$. In other words instead of collection of boxes we can consider $\tilde{\mathcal{B}}$ the countable collection of balls $B_r$ of radius $r>0$ whose union covers $A$:

$\tilde{\mathcal{L}}^n(A)=C_n inf\{\sum_{B_r\in\tilde{\mathcal{B}}}r^n\}$

We recall a very beautiful measure:

Hausdorff Measure: $\mathcal{H}^{\alpha}$: Given a set $A\subset \mathbb{R}^n$, we define the measure

$\mathcal{H}^{\alpha}(A)=\lim_{\delta\rightarrow 0}\mathcal{H}^{\alpha}_{\delta}(A),$

with $\mathcal{H}^{\alpha}_{\delta}(A)=inf\{\sum_{B_r\in\tilde{\mathcal{B}}_{\delta}}r^n\}$,

where the infimum is taken over $\tilde{\mathcal{B}}_{\delta}$ the countable collection of balls $B_r$ of radius $0 whose union covers $A$.

Recall the following:

• Measure on a set $X$, we mean usually by outer measure $\mu$: a non-negative, monotone, countably subadditive (that is, $\mu(\cup_{k=1}^{\infty}A_k)\leq\sum_{k=1}^{\infty}\mu(E_k)$) function on $\{A: A\subset X\}$ that gives value zero for the empty set.
• Borel set 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.
• A measure $\mu$ is called Borel measure if the Borel sets are measurable.
• A Borel measure $\mu$ is Borel regular if for any $A\subset X$, there exists a Borel set $B$ such that $A\subset B$ and $\mu(A)=\mu(B)$.
• A Borel measure is locally finite if compact sets have finite measure.
• 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 $A\subset\mathbb{R}^n$ and $\mathcal{B}$ be a family of closed balls such that every point of $A$ is contained in an arbitrarily small ball in $\mathcal{B}$, that is, $inf\{d(B):x\in B\in\mathcal{B}\}=0$ for $x\in A$. Then there are disjoint balls $B_i\in\mathcal{B}$ such that $\mathcal{L}^n(A\backslash\cup_iB_i)=0$.

Consider the example: Let $\mu$ be a Radon measure on $\mathbb{R}^2$: $\mu(A)=\mathcal{L}^1(\{x\in\mathbb{R}:(x,0)\in A\})$, that is, $\mu$ denotes the length measure on $x-$axis. The family $\mathcal{B}=\{B_y((x,y)):x\in\mathbb{R},0 covers $A=\{(x,0):x\in \mathbb{R}\}$ in the sense of the above theorem. But fails the conclusion, since for any countable collection $B_i\in\mathcal{B}$ we have $\mu(A\cap \cup_iB_i)=0$.

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

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 $\mathbb{R}^3$, 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 $0<\alpha<1$. Fix a set $S=\{a_1,a_2,...a_N\}$ of $N$ finite numbers in $[0,1]$ with $a_i for all $1\leq i\leq N$. Fix $0<\eta<1$ small such that $\eta<|a_i-a_j|$ for all $i$ and $j$; also such that $N\eta^{\alpha}=1$. We construct a sequence of sets $E_n$:

$E_1$ is the union of $N$ intervals of length $\eta$ with starting points in the set $S$. In other words we have removed $N-1$ in the interval $[a_1,a_N]$. We call the removed intervals as black intervals and the considered intervals as white intervals. Now on each interval $[a_i,a_i+\eta]$ in $E_1$ we have $N$ points: $\{a_i+a_1\eta,a_i+a_2\eta,...a_i+a_N\eta\}$. By the assumption on $\eta$ we can see that $\eta^2<|(a_i+a_j\eta)-(a_i+a_{j-1}\eta)|$. Hence we continue to construct $E_2$ as the set of $N^2$ intervals of length $\eta^2$. Proceeding like this, at $n^{th}$ step we have $N^n$ intervals of length $\eta^{n}$. 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 $0<\alpha<1$. Fix a set $S=\{a_1,a_2,...a_N\}$ of $N$ finite numbers in $[0,1]$ with $a_i for all $1\leq i\leq N$. Fix $0<\eta<1$ small such that $\eta<|a_i-a_j|$ for all $i$ and $j$; also such that $N\eta^{\alpha}=1$. Choose a sequence $\{\eta_k\}$ such that $\eta(1-\frac{1}{(j+1)^2})\leq\eta_j<\eta$. We construct a sequence of sets $E_n$:

$E_1$ is the union of $N$ intervals of length $\eta_1$ with starting points in the set $S$. In other words we have removed $N-1$ intervals in the interval $[a_1,a_N]$. We call the removed intervals as black intervals and the considered intervals as white intervals. Now on each interval $[a_i,a_i+\eta_1]$ in $E_1$ we have $N$ points: $\{a_i+a_1\eta_1,a_i+a_2\eta_1,...a_i+a_N\eta_1\}$. By the assumption on $\eta$ we can see that $\eta_1\eta_2<|(a_i+a_j\eta_1)-(a_i+a_{j-1}\eta_1)|$. Hence we continue to construct $E_2$ as the set of $N^2$ intervals of length $\eta_1\eta_2$. Proceeding like this, at $n^{th}$ step we have $N^n$ intervals of length $\eta_1\eta_2..\eta_{n}$. The Salem set is $E=\cap_n E_n$.

Brownian Motion: Consider the space $\Omega_n$ of all continuous functions $\omega:[0,\infty)\rightarrow\mathbb{R}^n$ with $\omega(0)=0$ such that the increments $\omega(t_2)-\omega(t_1)$ and $\omega(t_4)-\omega(t_3)$ are independent of $0\leq t_1\leq t_2\leq t_3\leq t_4$ and $\omega(t+h)-\omega(t)~\mathcal{N}(0,h)$.

For Example: For any fixed $h>0$, consider the random variable $X_{\omega}$ given by $X_{\omega}(t)=\omega(t+h)-\omega(t)$ with the probability density function $f(x)=\frac{e^{-\frac{x^2}{2h^2}}}{\sqrt{2\pi h}}$, that is with $P(t:X_{\omega}(t)\in A)=\int_Af(x)dx$). We call the random variables $B_t$ defined by $B_t(\omega)=\omega(t)$, Brownian motion. Zeroes of the Brownian motion has Hausdorff dimension $\frac{1}{2}$ when we consider $\mathbb{R}^1$. (What are zero sets of the Brownian motion? For each $\omega$, $Z(\omega)=\{t:\omega(t)=0\}$ “Hitting back at the starting point”). We then have the probability measure $P_n$ on $\Omega_n$.

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….)