Analysis II: Measure, Integration and Banach Spaces
MATH 114

Spring 2016
Vacaciones
Hans Prakash
Mi familia y yo somos de la India. India esta en
Asia. Hace dos semana mi madre, mi padre y yo
fuimos a la India. Para vacaciones fui a la India.
Paintball, ping pong, gokarti
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Homework 9 Solutions
Xiaoyu He
November 19, 2014
1. If X is finite dimensional then the ball is a closed and bounded set in a Euclidean
space, hence compact. We directly exhibit an infinite d
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
The danger of difference: nonmeasurable
sets A A with A R measurable
Curtis T. McMullen
17 September, 2014
1
Introduction
A standard homework exercise in real analysis is to show that AA contains
a no
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
MATH 114 HOMEWORK 4 SOLUTIONS
Sahana Vasudevan
10/08/14
1. By Lusins theorem there exist continuous gn : R R such that gn f
in measure. However this means a subsequence gnk f pointwise a.e.
2. Since F
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Analysis II Midterm: Solutions
Math 114 Fall 2014
1. Let f : R R be a monotone increasing function. Prove that for any
Rb
a < b, we have a f (x) dx f (b) f (a). (You may assume f (x) exists
a.e.)
Proo
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Homework 6
Math 114: Analysis II
Measure, Integration and Banach Spaces
Due Tuesday, 14 October 2014
1. Let f (x) = x sin(1/x ) with , > 0 (and set f (0) = 0).
For what values of and does f have b
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
MATH 114 HOMEWORK 10 SOLUTIONS
Sahana Vasudevan
11/25/14
1. Note that `1 is separable, as the sequences with rational entries, with
only finitely many nonzero entries form a countable dense subspace.
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
MATH 114 HOMEWORK 6 SOLUTIONS
Sahana Vasudevan
10/19/14
1. First, note that if the given f has bounded variation on [1, 1] then
as we
in class f 0 L1 ([1, 1]). If f 0 L1 ([1, 1]) then let g =
R x show
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Homework 3
Math 114: Analysis II
Measure, Integration and Banach Spaces
Due Tuesday, 23 September 2014
1. Let En [0, 1] satisfy m(En ) > c > 0. Prove that lim sup En , the set
of points belonging to i
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Homework 9
Math 114: Analysis II
Measure, Integration and Banach Spaces
Due Tuesday, 11 November 2014
1. Let X be a Banach space and suppose the closed unit ball B = cfw_x
X : kxk 1 is compact. Show
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 TakeHome Final
Analysis II: Measure, Integration and Banach Spaces
Due by 5 pm, Wednesday, 10 December 2014
Email to [email protected] with Math 114 as the Subject line.
Instructions. Put
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Homework 5 Solutions
Xiaoyu He
October 14, 2014
1. (i) To see that f 6 L1 , note that
Z
0
N
Z
N
sin x X
1
 sin x,
x
n 0
n=1
where on each interval (n 1), n] we bound the integral by
harmo
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 TakeHome Final: Solutions
1. Let us say f : [0, 1] R is pretty continuous if the points where f is continuous form
a dense subset of [0, 1]. Show that the sum of two pretty continuous functi
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Spring 2016
Hans Prakash
4/7/16
IB HL Biology
Mr. Ajerman
Transport Lab DialysisOsmosisDifusion
Introduction
In this lab we will observe the process of diffusion and osmosis in a model of a
membrane system. Thi
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Spring 2016
Name: _
IBSL Lang and Lit  1
Short Cuts Reading Log
Title:
Summary and Significance
1. In your own words, explain what happens in this story. Who were the main
characters and events? What was the cli
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Spring 2016
The Apartheid Laws
Task:
1. Read the details about each Act in your textbook and use the information to complete the second
column with the correct titles from this list:
Population Registration Act,
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Problem Set 1 Solutions
Xiaoyu He
September 2014
Problem 1
Let I denote the set of all nonempty open intervals (a, b) R (we allow a = and b = +). A choice
function is a map c : I Q such that
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Homework 3 Solutions
Xiaoyu He
October 2, 2014
1. We will prove the stronger statement that m(lim sup En ) c as well, given that all of
the individual sets have more than measure c. Recalling
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
MATH 114 HOMEWORK 8 SOLUTIONS
Sahana Vasudevan
11/12/14
R
1. It is easily verified that hem , en i = 0 (2/)1/2 sin(mx)(2/)1/2 sin(nx) =
mn . Hence the cfw_en form an orthonormal
set. We may write 1 =
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
MATH 114 HOMEWORK 2 SOLUTIONS
Sahana Vasudevan
09/21/14
1. It suffices to show that every open set U R is F , since complements
of F sets are G . If U is open, then U is a countable union of open
inte
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Problem Set 11 Solutions
Xiaoyu He
December 2014
1. In the interior, we have to check the Laplace equation. The partial derivatives are Txx = k 2 T and
Tyy = k 2 T so Txx + Tyy = 0. If y = 0
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Homework 7 Solutions
Xiaoyu He
November 3, 2014
1. We may assume N (f ), N (g) > 0, or the problem is trivial. As in lecture, it is possible
to normalize so that N (f ) + N (g) = 1. Write t =
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Homework 2
Math 114: Analysis II
Measure, Integration and Banach Spaces
Due Tuesday, 16 September 214
1. Prove that every closed set F R is a G . (That is, F is a countable
intersection of open sets.)
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Midterm (with solutions)
October 16, 2013
(1) Let C = cfw_(x, y ) R2 : x2 + y 2 = 1 be the unit circle. Show that C has measure zero (when regarded
as a subset of the Euclidean plane R2 ).
Fo
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Assignment Four Solutions
Stephen Mackereth
Problem One.
Suppose that f = 0 a.e. on E . Let E be any set E E , m(E ) < , and let g be any
bounded measurable function g : E R with 0 g f (since
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Assignment Two Solutions
Stephen Mackereth
Problem One.
Let S Rn be a measurable set with m(S ) < and let > 0 be a positive real number.
Show that there exists a compact subset K S such that
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Assignment 3 Solutions
Stephen Mackereth
Problem 3.
Composition of Borel measurable functions is Borel measurable
Proof. It suces to show that, if
t R,
f 1 (, t]) = cfw_x R : f (x) t is Borel
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
PROBLEM SET III, PROBLEMS I, II
PATRICK RYAN
Problem 1. Let E Rn be a measurable set with (E ) < 1. Show that for each
> 0, there exists a set E 0 Rn which is a nite disjoint union of open boxes
sati