21-356 Lecture 30

I1 then p1 and p2 are pluri rectangles p1 sf p2 and

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Unformatted text preview: n E Ui . i=1 Then P1 and P2 are pluri-rectangles, P1 ) Sf ) P2 and measN +1 P2 ' measN +1 P1 = = n + measN +1 Ui ' i=1 n +, i=1 n + measN +1 Ti i=1 n +, sup f ' 0 measN Ri ' inf f ' 0 measN Ri Ri i=1 ( Ri ( = U (f, P ) ' L (f, P ) % (. It follows from Theorem 137 that the set Sf is Peano—Jordan measurable with A measN +1 Sf = f (x) dx. R Conversely, assume that the set Sf is Peano—Jordan measurable. Let R 1 [0, M ] be a rectangle containing Sf . Then /Sf is Riemann-integrable over R 1 [0, M ] with A measN +1 Sf = /Sf (x, y ) d (x, y ) . R For every x ! R, the function...
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This document was uploaded on 03/31/2014.

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