Archive for 7 helmikuun, 2020

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.

Therefore

or shorter: and for all

It is obvious from here

This article was inspired by Margarita.

Opiskelu.org – reborn

I think I’d better come back to good old opiskelu.org, instead of spamming facebook page.

Oh yea, it all seems to work now.. kinda enjoy new editor for WordPress. So – here are plans for the future:

  • Lebesque integral and measure, few notes regarding course and exam (english/finnish)
  • Cellular Automata (english)
  • Bitcoin shop with tor in mind (english)
  • Немножко политики/russian politics (russian/english)

There. Lebesque integral notes&measure is coming today.
CA – on this weekend, based on university lecture.
Bitshop – next week.
Russian politics – when feel like it.