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Open sets convergence


Consider the sequence of open sets of [0,1] defined by
a) \displaystyle \Omega_n:= \bigcup_{k=0}^{2^{n-1}-1} \left(\frac{2k}{2^n},\frac{2k+1}{2^n}\right).

b) \displaystyle \Omega_n:= \bigcup_{k=0}^{2^{n}-1} \left(\frac{k}{2^n},\frac{k+1}{2^n}\right).

Study the convergence of the sets \Omega_n in the following types of convergence:

  • weak convergence of characteristic functions;
  • convergence in the topology defined by the Hausdorff distance;
  • convergence in the way of compact sets.

(Each of the previous types of convergence will be presented below.)

weak convergence of characteristic functions

The sequence of measurable sets ( \Omega_n ) of \Bbb{R}_N is said to converge in the way of characteristic functions towards the measurable set \Omega if the sequence of characteristic functions \chi_{\Omega_n} converges to \chi_{\Omega} in L^p_\text{loc}(\Bbb{R}^N), \ \forall p \in [1,\infty).

The first remark is that p \in [1,\infty) doesn’t matter since 0^p=0,\ 1^p=1. The second remark is that if the sequence \chi_{\Omega_n} converges in the weak-* topology of L^\infty (\Bbb{R}^N)  to \chi_{\Omega} then (\Omega_n) converges in the way of characteristic functions to \Omega.

convergence in the topology defined by the Hausdorff distance

Consider a fixed (large enough) compact of K \subset \Bbb{R}^N. For two compacts K_1,K_2 in K, and consider the following definitions:

  1. d(x,K_1):=\inf_{y \in K_1} d(x,y);
  2. \rho(K_1,K_2)=\sup_{x \in K_1}d(x,K_2);
  3. d^H(K_1,K_2)=\max(\rho(K_1,K_2),\rho(K_2,K_1)).

A sequence of open sets (\Omega_n) and \Omega included in B. We say that \Omega_n converges in the Hausdorff distance to \Omega if d^H(B\setminus \Omega_n, B\setminus \Omega) \to 0 for n \to \infty. We denote \Omega_n \xrightarrow[]{H} \Omega. We can define the same type of definition for compact sets from K.

convergence in the way of compact sets

We say that the sequence of open sets (\Omega_n) of \Bbb{R}^N converges in the way of compact sets towards \Omega, and we denote \Omega_n \xrightarrow[]{K} \Omega if we have

  • \forall K compact \subset \Omega, we have K\subset \Omega_n for n large enough;
  • \forall L compact \subset \overline{\Omega}^c we have L \subset \overline{\Omega_n}^c for n large enough.

Solution: a) For the characteristic functions convergence we will show that \chi_{\Omega_n} \to 1/2\ weakly in L^p(0,1), which gives us an example of sequence of characteristic functions whose limit is not a characteristic function.
We need to show that \displaystyle \lim_{n\to \infty} \int_{(0,1)} \chi_{\Omega_n} \phi \to \int_{(0,1)} \frac{1}{2} \phi, for all \phi \in L^1(0,1).

For proving this, we use a trick used in many real analysis proofs. Start with step functions, then extend to positive function, and finally to measurable functions. I will propose a similar, more detailed approach:

  • \phi=\chi_{[a,b]} where a,b are fractions with denominators powers of 2. This is quite easy, given the form of the sets \Omega_n.
  • extend the last step to all compact intervals, $latex \phi=\chi_{[a,b]}$, (using Beppo Levi’s Theorem)
  • extend to step functions
  • then to positive functions
  • finally to integrable functions

 

    b) \chi_{\Omega_n}=\chi_{[0,1]}\ a.e., therefore, the convergence in the way of characteristic functions is clear enough.

    Denote K_n:=[0,1]\setminus \Omega_n =\bigcup_{k=0}^{2^n} \{\frac{k}{2^n}\} which converges in the Hausdorff distance to the interval [0,1], which means by taking complements that \Omega_n \xrightarrow[]{H} \emptyset.

    Finally, it is clear that \Omega_n does not converge to any open set in the way of compacts. Suppose \Omega_n \xrightarrow[]{K} \Omega. If \Omega=\emptyset. Pick K \subset [0,1]. Then it would mean that K\subset \Omega_n for n large enough. But the “holes” in \Omega_n become “denser” in [0,1], which means that if we pick some compact containing an open set, that compact will not satisfy the given definition. On the contrary, if the sequence would converge to a nonempty open set, some compact within it will not belong to all \Omega_n.

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    1. November 25, 2010 at 5:38 pm

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