MATH 564: Assignment 2
. (problem 2.1 in Rudin)
' upper semioontinuous and lsc For lower semicontinuous.
at (a),(b),(d) are true and (c) is False.
he topological space whose underlying set is ER and whose open sets are intervals of the form
Lagota Math 564: Advanced Real Analysis 1
Homework 1 Solutions
J- and G the sets
Pi = Ej =infcfw_Ej :j e cfw_runF 1, -
G? m U EJ m SUPcfw_Ej 3 j E cfw_71.77. + 1, _,
73+; and GS; g2 Gj+1 for all j. Hence,
1.u.(lirninf4E)_;,-:) m ,u, (U 1+;
Lemma 1. Let b 2 O and a 2 1. Than,
111 (1 + b) S ab.
Proof. By Minkowski Inequality we have
(1 + bar/a .-. (1 + 0) + (0 + mar/a g (1 + 0)0 + (0 + barf ._ 1 + b.
So (1 + b) 2 1 + b and hence
69" x (4) 2 (1 + b) 2 1 +1) :> ab 21n(1+ (3).
MATH564 Assignment no2-Oct 8th due on (or before) Oct 26th
Each question is worth 10 points.
(Problem 2.1 in Rudin (third edition) )
(Problem 2.3 in Rudin )
(Problem 2.9 in Rudin )
Let X be a topological space for w
MATH564 Assignment no1-Oct 8th due on (or before) Oct 26th
Given cfw_Ej jN P(X), set lim sup Ej :=
j=0 n=j En .
j=0 n=j En and lim inf Ej :=
If (X, M, ) is a measure space and cfw_ES
lim inf (Ej ). Also
EXERCISES, MATH 570, FALL 2012
I suggest solving all the exercises, as soon as they get posted. Periodically, I will ask you to submit
selected exercises, in LaTeX (or at any rate, as a pdf file).
Note carefully: The pdf file should be called as follows