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missavvy
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Homework Statement
Show that the outer area of S = outer area of the closure of S
Homework Equations
The Attempt at a Solution
I don't really understand 100% the difference between the set S and the closure of S.
I know the closure is S [tex]\cup[/tex] [tex]\partial[/tex]S (the boundary of S), but then what does that mean S is? Since S is not the same as the interior of S or the closure..
Anyways,
Let A denote the outer area.
I'm starting by showing that A(S) [tex]\leq[/tex] A(Cls
Let a partition of a rectangle which contains R, P= {xi, yj} Then
S = upper sum, 1 = characteristic function
Sp(1s) = [tex]\sum[/tex] Mij A(Rij)
Where Mij = sup{1s(x,y); (x,y) belong to Rij}
I'm trying to figure out how the rectangles are defined in S, is it just Rij [tex]\cap[/tex]S?
Then
Sp(1s) [tex]\leq[/tex] [tex]\sum[/tex] Mij A(Rij) = Sp1[Cls(S)]
Rij [tex]\cap[/tex] S & Rij [tex]\cap[/tex] [tex]\partial[/tex]S
Anyways once I show that A(S)[tex]\leq[/tex] A(Cls
If possible, could someone maybe describe it as a picture? I find I understand this stuff much better visually..
Thanks :)
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