Re: simple measure theory question


William Elliot wrote:
On Wed, 22 Mar 2006, Fatou wrote:

im trying to prove that the 2 elementary sets are elementary under
union, intersection, difference, symmetric difference and complement.
(set A is elementary if A = U (from k=1 to n) P_k where P_k are
disjoint rectangles)

Open, closed or semi-open/semi-closed rectangles
or does disjoint mean measure zero intersection?

rectangles can be any as you have described
disjoint does mean 0 intersection however


im having a slight problem with this line in my proof of the difference
of 2 elementary sets

P, E are rectangles => P\E is elementary.
is that obvious, or not?

More or less depending upon how you answer the above.
Now if the complement of an elementary set is elementary, then P\E would
be elementary for being the intersection of two elementary sets. However:

If the space is R^2, then the complement of the unit square is either
union of infinitely many rectangles or union of infinite rectangles.

Thus you need to clarify if infinitely many rectangles or infinite
rectangles are allowed or if the space itself is finite rectangle.

I should have mentioned it is a finite space which im dealing with.
it feels obvious..but im not sure if i need to write a little more to
prove its true..

thank you

.

Relevant Pages

  • Re: simple measure theory question
    ... union, intersection, difference, symmetric difference and complement. ... E are rectangles => P\E is elementary. ... be elementary for being the intersection of two elementary sets. ...
    (sci.math)
  • Re: fmincon
    ... The rectangles are alligned to axes so the code of calculating the intersection is o'k i guess. ... Excuse me for maybe stupid question: -A as sign to empty intersection is given to implement say constraint c< 0? ... As far as I understood from Alex, ...
    (comp.soft-sys.matlab)
  • Re: fmincon
    ... The rectangles are alligned to axes so the code of calculating the intersection is o'k i guess. ... Excuse me for maybe stupid question: -A as sign to empty intersection is given to implement say constraint c< 0? ... You can transform it into a equivalent linear constraints, however the equivalent number of linear equations should be much larger. ...
    (comp.soft-sys.matlab)
  • Re: Problem with "IntersectsWith" function in GDI+
    ... exercise to do your own intersection test with whatever boundary rules you ... Doug Forster ... strange problem w.r.t IntersectsWith function which is used to determine ... whether two given rectangles intersect with each other not. ...
    (microsoft.public.dotnet.framework.drawing)
  • Re: Find the intersection of two rectangles
    ... Unless the rectangles are axis-aligned, ... the general case of Convex Polygon Intersection. ... And the intersection is empty unless ...
    (comp.lang.c)