Mathematics Department Stanford University
Math 205A Autumn 2013, Lecture Supplement #2
Product measures and Fubinis theorem
Let (X, A, ) and (Y, B , ) be arbitrary measure spaces.
Denition: By an A, B -rectangle we mean any set of the form A B with A A a
Mathematics Department Stanford University
Math 205A Homework Assignment 6
Solutions
1. Let f : [a, b] R be increasing. If
each x [a, b].
b
af
(x) dx = f (b) f (a), prove f (x) f (a) =
x
x
af
(x) for
b
Solution: f is increasing so by a result of lecture w
Mathematics Department Stanford University
Math 205A Homework Assignment 6
Due at Lecture Nov. 5
1. Let f : [a, b] R be increasing. If
for each x [a, b].
b
af
(x) dx = f (b) f (a), prove f (x) f (a) =
x
af
(x)
2. (i) If A is a Lebesgue measurable subset o
Mathematics Department Stanford University
Math 205A Homework Assignment 7
Due at Lecture Nov. 12
1
1. (i) If f (x) = x sin x , 0 < x 1, f (0) = 0, prove that f is not BV on [0, 1].
(ii) Prove generally that f : [a, b] R is BV the graph map G : [a, b] R2
Mathematics Department Stanford University
Solutions to Math 205A Homework Assignment 7
1
1. (i) If f (x) = x sin x , 0 < x 1, f (0) = 0, prove that f is not BV on [0, 1].
2
2
2
Solution: For N 2 consider the partition 0, N , (N 1) , , 22 , , 1. Thats a p
Mathematics Department, Stanford University
205A Lecture Supplement #1, 2013
Lebesgues theorem on the Riemann integral
We let R = [a1 , b1 ] [an , bn ] be any closed interval in Rn . Recall that by a partition P of
R we mean the collection of closed inter
Mathematics Department Stanford University
Solutions to Math 205A Homework Assignment 8
1. Let X be any space and A any -algebra of subsets of X and suppose is a real valued
(no allowed!) signed measure on A such is absolutely continuous with respect to ,
Mathematics Department Stanford University
Math 205A Homework Assignment 8
Due at Lecture Nov. 19
1. Let X be any space and A any -algebra of subsets of X and suppose is a real valued
(no allowed!) signed measure on A such is absolutely continuous with re
Mathematics Department Stanford University
Math 205A Homework Assignment 5
Solutions
x
1. Let Cn , C be as in question 3 of homework 1. Let gn = (3/2)n Cn , and fn (x) = 0 gn (t) dt. Prove that
fn (1) = 1, fn+1 fn on [0, 1] \ Cn and the common value remai
Mathematics Department Stanford University
Math 205A Homework Assignment 5
Due at Lecture Oct. 29
x
1. Let Cn , C be as in Q.3 of homework 1. Let gn = (3/2)n Cn , and fn (x) = 0 gn (t) dt. Prove
that fn (1) = 1, fn+1 fn on [0, 1] \ Cn and the common value
Mathematics Department Stanford University
Math 205A Homework Assignment 2
Due at Lecture Oct. 8
1. Let (X , A, ) be a measure space and for A X dene (A) = inf (B ), where the inf
is over all B A with B A.
(i) Prove that is an outer measure on X and (A) =
Mathematics Department Stanford University
Math 205A Homework Assignment 1
Due at Lecture Oct. 1
1. Suppose f : [0, 1] R is dened by
f (x) =
1/q
if x = p/q with p, q relatively prime integers and q > 0
0
otherwise.
Prove that f is Riemann integrable on [0
Mathematics Department Stanford University
Math 205A Homework Assignment 2
Solutions
1. Let (X, A, ) be a measure space and for A X dene 0 (A) = inf (B ), where the inf is over
all B A with B A.
(i) Prove that 0 is an outer measure on X and 0 (A) = (A) fo
Mathematics Department Stanford University
Math 205A Homework Assignment 3
Due at Lecture Oct. 15
1. From lecture we know that that if fj is a sequence of non-negative extended real-valued Ameasurable functions on a space X then f (x) = supj 1 fj (x) is a
Mathematics Department Stanford University
Math 205A Homework Assignment 4
Solutions
1. (Application of the dominated convergence theorem: the dierentiation under the integral
sign theorem.) If (X, A, ) is an arbitrary measure space, if g (x, y ) is real-
Mathematics Department Stanford University
Math 205A Homework Assignment 4
Due at Lecture Oct. 22
1. (Application of the dominated convergence theorem: the dierentiation under the integral
sign theorem.) If (X, A, ) is an arbitrary measure space, if g (x,
Mathematics Department Stanford University
Math 205A Homework Assignment 3
Solutions
1. From lecture we know that that if fj is a sequence of non-negative extended real-valued Ameasurable functions on a space X then f (x) = supj 1 fj (x) is also A-measura
Mathematics Department Stanford University
Math 205A Homework Assignment 1
Solutions
1. Suppose f : [0, 1] R is dened by
f (x) =
1/q
if x = p/q with p, q relatively prime integers and q > 0
0
otherwise.
Prove that f is Riemann integrable on [0, 1]. In fac