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l_notes24 - Lecture XXIV Measures Remember that an...

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Lecture XXIV Measures Remember that an elementary region R in E 2 is a region that has a simple, closed, piecewise smooth curve C as its boundary. A regular region R is a region that is either regular or can be divided into finitely many regular regions. In the second case, the boundary of R consists of two or more simple, closed, piecewise smooth curves, of which one is the outer boundary curve, denoted by C , and the others are interior boundary curves C 1 , . . . , C n . Then R consists of the points in the interior of C , excluding those in the interior of C 1 , . . . , C n . Similarly, in E 3 a region is regular if its boundary is a simple, closed, piece- wise smooth surface S . An elementary region in E 3 is either regular or can be divided into finitely many regular regions. The boundary of an elementary region that is not regular is formed by one or more closed, but not simple piece- wise smooth surfaces. In the case that the boundary is formed by two or more surfaces, one of them is the outer boundary surface S , and
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