Lebesque measure of line on R² surface

There were question in exam, which I understood very close to correct, thus absolutely wrong.

Roughly translated: “Define real axis ℝ Lebesque outermeasure on ℝ² surface

Lets get into it. Bit intuitively first.. Suppose we have segment of a line (a,b)

segment (a,b) on real axis

segment (s) itself can be expressed as

Actually, lets forget about s, lets consider without segments, ie

Then outer measure can be expressed as union of open sets:

for all

Where intervals are:
Thus length of intervals are from one integer to another – ,.. all integers
Height of intervals are

Ie , where biggest height is , which is quater of epsilon.


or shorter: and for all

It is obvious from here

This article was inspired by Margarita.