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Let's say I have a 2d rectangle defined by $ [0,x_0] \times [0,y_0]$. Now lets say I cut out the middle rectangle $[\frac{1}{3} x_0, \frac{2}{3} x_0] \times [\frac{1}{3} y_0, \frac{2}{3} y_0]$. Now suppose I take the hyperreal extension of this rectangle. I then "fill" back up the middle rectangle. I increase the rectangle to $ [0,x_0 + \varepsilon] \times [0,y_0 + \varepsilon]$ where $\varepsilon \in \mathbb{R}_{\varepsilon}$ (the infinitesimals). I then cut out the analagous middle rectangle of this square. I then proceed to take the standard part of this figure. Has the standard square gotten any bigger? If I repeat this process $ N \in \mathbb{N}^*$ number of times, what can be said about the standard part? Will I see entire figure continuously increase in measure? Will I see nothing at all?

The goal here is to make something like this process that is continuous. I want to be able to define something that increases the overall measure of the figure while still preserving the structure of the rectangle that has been cut out.

Any thoughts?

I essentially just want to be able to increase the measure, while preserving structure. Need more exposure on this question...

Offering 300 rep bounty for proper answer.

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It is not clear what you are looking for. How is the size of the big square affected by your cutting out the middle? Apparently you could ask the same thing without cutting out the middle. – Mikhail Katz Feb 26 '14 at 14:39
@user72694 Say I have something like the Sierpinski Carpet. I want to be able to continuously make it larger (or smaller) while still preserving the structure. – Anthony Peter Feb 26 '14 at 16:19
I really don't understand your question. If you are just interested in making something smaller/larger continuously, why don't you juest scale? I guess you want something different... – user126154 Mar 4 '14 at 11:57
@user126154 I want a continuous increase in measure, preserving structure, for example I'd like to map the menger sponge on the unit cube to the menger sponge on an arbitrary cube, by continuously increasing the measure of the unit menger – Anthony Peter Mar 5 '14 at 4:05
I honestly do not understand the question but I am quite confident that Lie álgebras have nothing to do with it. – Mariano Suárez-Alvarez Mar 8 '14 at 6:38

As far as looking for hyperreal approaches to constructing the carpet, which is how I understood your idea, I would suggest looking first at hyperreal approaches to constructing nowhere differentiable functions. This was dealt recently in a paper by McGaffey here:

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