JHiN8y-12

JHiN8y-12 - 30 SPRING 2012 12. Lebesgue Measure: Open Sets...

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

View Full Document Right Arrow Icon
30 SPRING 2012 12. Lebesgue Measure: Open Sets and Compact Sets So far we have deFned λ ( A )fo r A empty, for any closed box A , and more generally for any special polygon A R n . Defnition 12.1. If G R n is a nonempty open set, let λ ( G )=sup { λ ( P ) | P is a special polygon such that P G } . Since every nonempty open set contains an open ball in R n , and every open ball contains anondegeneratec losedbox ,wehave0 < λ ( G )foranynonemptyopenset G . Examples 12.2. (i) λ ( R n )= . To see this, Fx M> 0andcons iderthec losedbox I M =[0 ,M ] × [0 , 1] × ··· × [0 , 1]. Then I M is a special polygon and I M R n , λ ( R n ) λ ( I M )= M . Since M> 0wasarb itrary ,wehave λ ( R n )= . (ii) If G is a bounded nonempty set, then λ ( G ) < . To see this, by deFnition there exists K R such that G B K ( 0 ). So if we let J K be the closed box [ K,K ] n , then G J K . ±or any special polygon P with P G we thus have P J K ,and therefore λ ( P ) λ ( J K )=(2 K ) n < . Taking the sup over all such P ’s gives λ ( G ) (2 K ) n < . Exercise 12.1. Show that for any special polygon P ,w ehav e λ ( P )= λ ( P o ), where P o denotes the interior of P (that is, the largest open set contained in P ). Hint: ±irst consider ac lo sedbox I , and argue that for any ° > 0, we can construct a closed box J such that J I 0 I and λ ( I ) λ ( J ) < ° . Proposition 12.3. Let G i R n be open for each i N . (i) If G 1 G 2 , then λ ( G 1 ) λ ( G 2 ) . (ii) λ ( i =1 G i ) ° i =1 λ (
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

JHiN8y-12 - 30 SPRING 2012 12. Lebesgue Measure: Open Sets...

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