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Densities and different dimensions

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 ofis finite,
-outer regular:is open
-inner regular:is compact
for all open subsets
- The
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.
Thusis 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 within
(check that it is homeomorphism)
Left translation: given by the matrixand the vector
since the matrix multiplication is given by
which is precisely given by
.
Henceis 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 sphereand the special unitary group
:
where for
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)
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
(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 measure
- 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.
- It is easy to check that
is Borel regular.
- Fix
. Choose a Borel cover
of
such that
. Then
is a Borel set containing
with
.
- Fix
- 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
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:
- Measure 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.
- 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
is called Borel measure if the Borel sets are measurable.
- A Borel measure
is Borel regular if for any
, there exists a Borel set
such that
and
.
- 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 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….)