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Example of Lebesgue Measure and Lebesgue Integration

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Let f be the following function with domain C = [0, 1] X [0, 1] (in two-dimensional Cartesian space):

f(x, y) = 0 on the line segments x = 0, y = 0, and x = y

f(x, y) = -1/(x^2) if 0 < y < x <= 1

f(x, y) = 1/(y^2) if 0 < x < y <= 1

Compute each iterated Riemann integral of f on C (by integrating first over x and then over y, and vice versa), and show that the two iterated Riemann integrals are unequal to each other.

Also, for every positive integer k, let E_k = {(x, y) in C: k^2 <= |f(x, y)| < (k + 1)^2}, and compute the two-dimensional Lebesgue measure of |f| on E_k. Use the results of those computations to show that |f| is not Lebesgue integrable.

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The statement of the problem is given in an attached .jpg file (Lebesgue.jpg). A complete, very detailed solution of the problem is provided in an attached .doc file (Lebesgue.doc).

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Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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