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I also got some student notes and while going through them, two examples caught my eye. I want to learn how to make valid mathematical arguments for such examples.

**Example 1.**

__Definition.__

We say that C has an area if the function ##\chi _C ## is integrable on C, i.e. on some rectangle that contains C. In that case, the area of C is ## \nu (C) = \int _C \chi _C ## where:

$$\chi _C (x,y) = \begin{cases} 1, (x,y) \in C \\ 0, (x,y) \notin C \end{cases} $$and C is a bounded subset of ##\mathbb{R}^2 ##.

The notes I've got ask such questions as "Does a disk have an area?", "Does a triangle have an area?". (in terms of the definition above)

**Example 2.**

$$ C =\{ (x,x) | x\in\mathbb{R} \} $$

Claim: C has a (Lebesgue) measure of zero.

The notes say that the argument "C is just a rotated x-axis" is not valid because ##d(k , k+1) = (k+1) - k = 1 < d(f(x_{k_1}), f(x_k)) ## so we have a rotation and "stretching".

Thanks in advance for your answers!