Math 447
Solutions to Assignment 1
Spring 2009
Exercise 16.4 on page 271 Given any subset of E of R and any h R, show that m (E + h) = m (E), where E + h = cfw_ x + h : x E . Solution The statement, which is part (iii) of Proposition 16.2 on page 269, say
Math 447
Solutions to Assignment 2
n=1
Spring 2009
Exercise 16.22 on page 273 Let E = m (En ) = 0 for every n.
En . Show that m (E) = 0 if and only if
Solution Since En E, the monotonicity property of outer measure [part (ii) of Proposition 16.2 on page
Math 447
Solutions to Assignment 3
Spring 2009
Exercise 16.58 on page 284 Suppose that m (E) < . Prove that E is measurable if and only if, for every > 0, there is a finite union of bounded intervals A such that m (E A) < (where E A is the symmetric diffe
Math 447
Solutions to Assignment 4
Spring 2009
Exercise 17.6 on page 297 Suppose f : D R, where D is measurable. Show that f is measurable if and only if cfw_ f > is measurable for each rational . Solution If the function f is measurable, then by the def