8601: Math REAL ANALYSIS.
Fall 2010
Homework #7 (due on Friday, December 10). 50 points are divided between 5 problems, 10 points each. #1. Let f L1 (R1 ). Show that 1 Sn (x) = n in L1 (R1 ) as n , i.e. |Sn (x) - S(x)| dx 0 as n . Hint. Using Theorem 2.26, show that (h) := sup
|t|h R1 n-1
j S(x) = f x+ n j=0
x+1
f (t) dt
x
|f (x + t) - f (x)| dx 0 as h 0+ .
Moreover, writing Sn - S in the form
n-1 1/n
Sn (x) - S(x) =
j=0 0
f x+
j j -f x+ +t n n
dt,
one can get |Sn (x) - S(x)| dx (1/n).
R1
#2. Let f, f1 , f2 , . . . , fn , . . . be measurable functions on a measure space (X, M, ) with (X) = 1, such that fn (x) f (x) as n for all x X. Suppose that |fn |1+ d C Show that |fn - f | d 0 as n . Hint. For A > 0, introduce the functions t FA (t) := A -A 1 for all n, with some constants > 0 and C > 0.
if |t| < A, if t A, if t -A.
Then the uniformly bounded functions gn,A (x) := FA (fn (x)) gA (x) := FA (f (x)) as n for all X. x Moreover, |gn,A - fn | d
X {|fn |A}
|fn | d,
and a similar estimate is true for the difference gA - f . You need to show that the right sides in these inequalities are uniformly small for large A > 0. #3. Show that any non-negative measurable function f (x) on a measurable space (X, M) can be represented in the form
f=
n=1
1 En , n
where En (x) :=
1 0
if x En , if x En . /
Hint. This representation is not unique. One can try to find the "maximal possible" E1 , then E2 , etc. #4. Evaluate
/2
sin6 x cos8 x dx
0
in terms of the constant and rational numbers. Hint. By substitution t = sin2 x, the integral is reduced to the Beta function
1
B (a, b) :=
0
ta-1 (1 - t)b-1 dt =
(a) (b) . (a + b)
#5. Let be a signed measure on a measurable space (X, M), such that m := sup (E) < .
EM
Let {Ej } be a sequence of sets in M, such that (Ej ) > m - 2-j for all j. Show that
m = (P ),
where P := lim sup Ej :=
j k=1 j=k
Ej .
2