Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Measure of limit of decreasing sequence

  1. Apr 20, 2008 #1
    If E_1, E_2, ... is a sequence (of subsets of R^n) that decreases to E
    (i.e. E_m+1 is a subset of E_m for all m, and E = intersection of all the E_m's)
    and some E_k has finite (lebesgue) measure, i.e. lambda(E_k) is finite
    it is a known result that the measure of E is equal to the limit of the measure of E_m.

    But now if we are given some bounded set E
    and we define E_m = { x : d(x,E) < 1/m }
    where d(x,E) = minimum distance from x to any point in set E,
    then howcome we have lambda(E) = lim_m->oo lambda( E_m ) when E is closed
    but not when E is open?

    Doesn't the fact that E is bounded imply some E_k has finite measure, and hence the above result applies, regardless whether E is open or closed or neither?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted