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Unformatted text preview: Lecture 12 So far, we have been studying only the Riemann integral. However, there is also the Lebesgue integral. These are the two basic integral theories. The Riemann inte- gral is very intuitive and is usually adequate for problems that usually come up. The Lebesgue integral is not as intuitive, but it can handle more general problems. We do not encounter these problems in geometry or physics, but we would in probability and statistics. You can learn more about Lebesgue integrals by taking Fourier Analysis (18.103) or Measure and Integration (18.125). We do not study the Lebesgue integral. Let S be a bounded subset of R n . Theorem 3.17. If the boundary of S is of measure zero, then the constant function 1 is R. integrable over S . The converse is also true. Proof. Let Q be a rectangle such that Int Q S . Define 1 if x S, 1 S ( x ) = (3.110) 0 if x / S . The constant function 1 is integrable over S if and only if the function 1 S is integrable over Q . The function 1 S is integrable over Q if the set of points D in Q where 1 S is discontinuous is of measure zero. If so, then 1 S = 1 . (3.111) Q S Let x Q . 1. If x Int S , then 1 S = 1 in a neighborhood of x , so 1 S is continuous at x ....
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- Fall '04