338 CHAPTER 12. THE CONSTRUCTION OF MEASURES

of the form U ×V where U and V are open in Rn and Rm respectively. Next verifyouter regularity by considering A×B for A,B measurable. Argue sets of R definedabove have the property that they can be approximated in measure from above byopen sets. Then verify the same is true of sets of R1. Finally conclude using anappropriate lemma that µ×ν is inner regular as well.

9. Let (Ω,S ,µ) be a σ finite measure space and let f : Ω→ [0,∞) be measurable.Define

A≡ {(x,y) : y < f (x)}

Verify that A is µ×m measurable. Show that∫f dµ =

∫ ∫XA (x,y)dµdm =

∫XAdµ×m.