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)