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11 - MAT 473 27 11 Lebesgue Measure Closed Boxes and...

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MAT 473 27 11. Lebesgue Measure: Closed Boxes and Special Polygons Recall that relatively few functions f : R R are Riemann integrable: the continuous functions, and in a sense not too many more. Also, the Riemann integral behaves badly with respect to limit operations — even pointwise limits of continuous functions need not be Riemann integrable. The Lebesgue integral improves on this. Very roughly speaking, the di ff erence between the two is that the Riemann integral chops up the domain of the function, while the Lebesgue integral chops up the range. A Riemann sum for f looks at a subinterval of the domain of f and asks how big f ( x ) is on that subinterval; a “Lebesgue sum” for f looks at a value y in the range of f and asks how big the set is on which f ( x ) is near y . This means we first need to know how to suitably “measure” sets in R n . Definition 11.1. Let X be a set. A σ -algebra of subsets of X is a collection M of subsets of X such that (i) ∅ ∈ M . (ii) i =1 A i M for any countable collection { A i } M . (iii) A c M for any A M . It is immediate from the definition that if M is a σ -algebra of subsets of X , then X M , A \ B M for any A, B M , and i =1 A i M for any countable collection { A i } M .
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