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)