MATH 605, HW 1 SOLUTIONS
Follands Real Analysis; Chapter 1:
4.) This follows since any countable union can be written as an increasing countable union:
Ej = j =1 Ek ;
j =1
j =1
k
note that j =1 Ek is a nite union of sets in the algebra and is hence in th
MATH 605, HW 2 SOLUTIONS
Follands Real Analysis; Chapter 1:
18.) (a) By denition, given > 0, there exist Aj A (j = 1, 2, . . . ) with E Aj and
1
0 (Aj ) (E ) + . Subadditivity of and Proposition 1.13 give
(Aj )
(Aj ) =
0 (Aj ) .
(b) To get one way, use
MATH 605, HW 6 SOLUTIONS
Follands Real Analysis; Chapter 5:
1.) Let X be a normed v.s. over R.
(a) Show that vector addition and scalar multiplication are continuous from X X and
X R to X : Ill just do this for addition. Dene T : X X X by T (x, y ) = x +
MATH 605, HW 4 SOLUTIONS
Follands Real Analysis; Chapter 2:
19.) Suppose fn L1 () and fn f uniformly.
(a) Show that if (X ) < , then f L1 () and fn f : f is automatically measurable
since fn f . Given > 0, there exists N so that |fn f | < /(X ) for all n
MATH 605, HW 5 SOLUTIONS
Follands Real Analysis; Chapter 2:
39.) Suppose fn f almost uniformly.
(a) Show fn f a.e.: This is immediate. Let B be the bad set where fn does not converge
to f (we say in an earlier assignment that this was measurable). It foll
MATH 605, HW 7 SOLUTIONS
Follands Real Analysis; Chapter 6:
7.) If f Lp L , then show: |f | = limq |f |q . We can assume |f |p = 0 and
|f | = 0 (otherwise f = 0 a.e. so its obvious).
(a) Proposition 6.10 gives |f |q |f |p/q |f |1p/q . Since limr0 ar = 1 a
MATH 605, HW 3 SOLUTIONS
Follands Real Analysis; Chapter 2:
2.) f, g : X R are measurable.
(a) Show that f g is measurable: Since
cfw_x | f g (x) > a = bQ+ [cfw_x | f (x) > b cfw_x | b g (x) > a]bQ [cfw_x | f (x) < b cfw_x | b g (x) > a] ,
we can write (f
MATH 605, HW 8 SOLUTIONS
Follands Real Analysis; Chapter 3:
22.) Suppose f L1 (Rn ) and |f |L1 = 0. Show there exist C, R > 0 so (the maximal function) Hf (x) C |x|n for |x| > R: Since |f |L1 = 0, we can choose R > 0 so
BR/2 (0) |f | > > 0 for some . For