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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 f 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 )isonthatsub interva l ;a“Lebesguesum”for f looks at a value y in the range of f and asks how big the set is on which f ( x )isnear y . This means we Frst need to know how to suitably “measure” sets in R n . Defnition 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 deFnition 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 . Defnition 11.2. Let M be a σ -algebra of subsets of X .A measure on M is a function μ : M R such that (i) μ ( A ) 0fora l
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online