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First, an apology for the completely speculative nature of this question. It doesn't come directly from a textbook or lectures. I am not exactly sure if it has to do with measure theory, but if it doesn't feel free to edit.

Suppose you have two real intervals $[0,1]$ and $[0,2]$. Suppose I want to measure them, I couldn't use bijectivity because there is the bijection $f(x)=2x$ and I would need to conclude that both sets have the same measure which is absurd (at least for some applications). I could try to find a measure by comparing the number of rational points in these intervals, but it would yield the same result.

Now suppose I count the number of integer coordinates, then I get: $M([0,1])=2$, $M([0,2])=3$ which seems much more reasonable. Suppose also I take the number of coordinates with integer multiples of $\frac{1}{2}$, then: $M([0,1])=3$, $M([0,2])=5$. I could also take the number of positive integers inside the interval and hence: $M([0,1])=1$, $M([0,2])=2$ which is what one would customarily expect.

So, I have given three examples of sets that do enable us to measure in a certain reasonable way by just counting the number of coordinates inside the interval. But what is the general property that defines those sets? I believe that it is the absence of density but I'm not sure about it.

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    What exactly do you want to measure? Which properties do you want or expect your measure to have? Your question seems really vague. If you wanted your measure to correspond to our usual notion of *length* of an interval, then there's the Lebesgue measure. – 2017-02-05
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    I was thinking in terms of isometries. (Not sure if this makes it clearer.) – 2017-02-05
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    [Any translation-invariant measure on $\mathbb{R}^n$ is a multiple of the Lebesgue measure](https://proofwiki.org/wiki/Translation-Invariant_Measure_on_Euclidean_Space_is_Multiple_of_Lebesgue_Measure). So, if your measure is translation-invariant and it assigns the measure $1$ to $[0,1]^n$, then it is the Lebesgue measure, and the Lebesgue measure *is* invariant under isometries. Notice in particular that translations are isometries. – 2017-02-05

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