Limit of a sequence of sets: hypercubes.


by benorin
Tags: hypercubes, limit, sequence, sets
benorin
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#1
Dec20-05, 03:30 PM
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Ok, so I would like to formally settle this one. I have a sequence of sets [itex]C_{N}^{n}\subset\mathbb{R} ^{n}, N\in\mathbb{Z} ^{+}[/itex] defined as follows

[tex]C_{N}^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \sum_{j=1}^{n} x_{j}^{2N} \leq n \right\} [/tex]

I would like to prove that [itex]C_{N}^{n}\rightarrow\mbox{ The Hypercube with verticies at } (\pm 1, ..., \pm 1) \mbox{ as }N\rightarrow \infty[/itex] (through integer values.)

I have done some study of limits of sequences of sets: most fruitful has been Measure Theory, by Halmos, in which the limit of a sequence of sets is defined as the set [itex]A = \liminf A_{n} = \limsup A_{n}[/itex], where the upper and lower limits of a sequence of sets [itex]\left\{ A_{n} \right\} [/itex] are defined by

[tex]\liminf A_{n} = \bigcup_{j=1}^{\infty} \bigcap_{k=j}^{\infty} A_{k}[/tex]

and

[tex]\limsup A_{n} = \bigcap_{j=1}^{\infty} \bigcup_{k=j}^{\infty} A_{k}[/tex]

But, I hadn't heard of that when I started playing with the above sets. It is easy to see that, as "limit equations," the following are tennable:

[tex]Q^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \lim_{N\rightarrow\infty} \sum_{j=1}^{n} x_{j}^{2N} \leq n \right\}= \mbox{ The Hypercube with verticies at } (\pm 1, ..., \pm 1) [/tex]

and, in fact, the sets

[tex]Q_{d}^{n}:=\left\{ \left( x_{1}, ..., x_{n}\right) \in\mathbb{R} ^{n}: \lim_{N\rightarrow\infty} \sum_{j=1}^{n} x_{j}^{2N} \leq n-d \right\}[/tex]

possess that property of describing precisely the d-dimensional content (less lower dimensional boundaries) of the hypercube with verticies at [itex](\pm 1, ..., \pm 1) [/tex]; that is to say that the sets [itex]Q_{0}^{n},Q_{1}^{n},...,Q_{n-2}^{n},Q_{n-1}^{n},\mbox{ and }Q_{n}^{n}[/itex] describe the vertices, edges..., ridges, facets, and hypervolume of said n-dimensional hypercube, respectively. And since [itex]Q_{i}^{n}\cap Q_{k}^{n}=\emptyset, \mbox{ for }j\neq k[/itex], and [itex] \bigcup_{k=1}^{n} Q_{k}^{n} = Q^{n}[/itex] the above sets provide a strataification of said hypercube.

So my quesion is: "Is the later "limit equation" interpetation given consistent with the notion of limits of sequences of sets given by Halmos?"
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