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 (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.
Therefore
or shorter: and for all
It is obvious from here
This article was inspired by Margarita.