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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<r<\delta 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<y<\infty\} 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<a_{i+1} 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<a_{i+1} 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….)


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