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,
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
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) =
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 general
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 cons
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
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 measu
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!) si
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)
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
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
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 >
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) P
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
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,
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 theore
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
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
otherwis