1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Lebesgue measure proof for a set in R^2

  1. Dec 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove (using Lebesgue outer measure) that $$m_{2}^{*}(\{(x,y)\in\mathbb{R}^{2}\colon x>1,0<y<x^{-2}\})=m_{2}^{*}(A)<\infty$$

    3. The attempt at a solution

    I'm not sure if this is valid proof but I'd have done it like this:
    $$\int_{1}^{k}\left|x^{-2}\right|\, dx <\infty \forall k\in\mathbb{N} \Rightarrow \int_{[1,\infty[}\left|x^{-2}\right|\, dx<\infty \Rightarrow A $$is measurable and$$ m_{2}^{*}(A)=\int_{1}^{\infty}x^{-2}\, dx<\infty.\qquad\square$$

    What else should I include to make sure the proof is valid?
  2. jcsd
  3. Dec 11, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    No, this implication is not valid. If you replace [itex]x^{-2}[/itex] with [itex]x^{-1}[/itex], then your left hand inequality would still be true, but the right hand inequality would be false.

    Similarly, the partial sums of an infinite series may all be finite, but that doesn't imply that the series converges. However, it is true that the series [itex]\sum_{n=1}^{\infty}n^{-2}[/itex] converges. Can you use this series to bound your integral?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook