Class 2
Chap.1. Measure theory
X set
En subsets of X
Def. limEn x X : x belongs to infinitely many En 's
En
k 1n k
limEn x X : x belongs to all but finitely many En 's
En .
k 1n k
Note: limEn lim En
Def. En has a limit if limEn = limEn
In this cas
Class6
Section 1.7. Metric space
X set
Def. : X X is a metric if
Then (X , ) metric space.
(1) ( x, y ) 0
(2) ( x, y ) 0 x y
(3) ( x, y ) ( y, x)
(symmetry)
( A, B)
(4) ( x, y ) ( x, y ) ( y, z ) ( ).
Ex.
Def.1. A, B X
(A, B) inf xA, yB ( x, y ). (dis
Class 3
Sec.1.2. Measure
X set, a -algebra on X
Def. u : a 0, is a measure if
(1) u ( )=0;
(2) u is countably additive:
n 1
n 1
u ( En ) u ( En ) for En a , Ei E j for i j
Def. additive: u ( E F ) u ( E ) u ( F ) for E , F a , E F
finitely additive: u
Class4
Thm. u *outer measure
Let a u * -mesurable subsets
Then(i) a is -algebra;
(ii) u * |a is measure.
Ex. X set. (Ex.1.3.2)
0 if
Define u* ( E )=
for E X .
1 if E
Then u* outer measure
a = , X
u*|a is u* ( )=0
u* (X )=1.
Pf. of Thm.
(1) a
Check:
Class 7
Thm. u* metric outer measure on metric space (X, )
closed (open) sets are measurable.
Note: " " true (cf. Ex.1.8.1)
Pf. of Thm.
Let F be closed set
Check: u* ( A) u* ( A F ) u* ( A \ F ) A X
Note: (A F , A \ F ) may not be 0, replace A \ F by s
Class5
Ex. (Ex.1.4.1) Let K ,X , singletons ;
( )=0, ( X ) # X , ( E )=1 if E singleton.
Then u* (A)=#A; outer measure
a =(X );
-algebra
u*measure on (X ) (counting measure); measure.
Homework: Ex.1.4.4, 1.4.5
Sec. 1.5.
(X ,a ,u ) measure space.
Def.
Class 9
Sec.1.10. Signed measure
a -algebra on X
Def: : a (, ] or [, ) is a signed measure if
(1) u ( ) 0,
(2) u countably additive
Note: u may not be monotone
Ex. u1 , u2 measures on a
Assume one of them is a finite measure.
Let u (E ) u1 (E ) u2 (E )
1
1.1
Chapter 12 Limits and Open Sets
Sequences of real numbers:limit, accumulation point and subsequences
Example 12.1 Some examples of a sequence of real numbers
a)cfw_1, 2, 3, 4, .
b) 1, 1 , 1 , . ,
2 3
c) 1, 1 , 4, 1 , 16, .
2
8
d) 0, 1 , 2 , 3 , 4 ,
Hello!
We are from the City of Rochester Hills Government Youth Council. In order to educate students from
the area, we are organizing a Youth Summit on March 10 th. The Summit will provide area students with
an opportunity to learn about the inner workin
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MATH 452 TESTING HYPOTHESIS
Hypothesis Testing is a systematic way of testing claims or ideas about a group or population.
Statistical Hypothesis: Suppose that a population pmf or pdf is given by
f ( x ; ) , where xX R
and is an unknown parameter which be
5. A 10 m by 14 m house is built on a 12 cm thick slab of concrete. What is the rate of heat loss through
the slab if the ground temperature is 5.00 C while the interior of the house is 22.0 C?
Reference info:
12 cm concrete slab
Concrete thermal conducti
PHYS310(O) iLab Experiment 7
Energy Conservation (Springs)
DATA SHEET
Name: Date: February 12, 2017
2 questions, 1 point each; 8 questions, 3 points each; and 1 question, 4 points; 30 points total
Prediction
1. How does the decrease in elastic potential e
Exercise on chapter 5:
Exercise 1:
A sinusoidal electromagnetic wave of frequency 40.0 MHz travels in free space in
the x direction, as in Figure below
(A)
Determine the wavelength and period of the wave.
(B) At some point and at some instant, the electri
Grade Details - All Questions
1.
Question :
(TCO 2) Which of the following statements is true?
Student Answer:
Object names are listed in the bottom section of a UML diagram.
Attribute names are listed in the bottom section of a UML diagram.
Method names
Exam 1, PH204, July 19, 2012,
With Answers
Question 1:
1) How long (in ns) does it take light to travel 2 m in vacuum?
Answer 6.6 ns
2) What distance does light travel in water during the time that it travels 2.0 m in vacuum? Use
1.33 for the index of ref
Physics 310(B) Experiment 6
Newtons Laws Answer Key
DATA SHEET
Name: Date: February 5, 2017
Prediction
1.
What happens to an object when you apply a net force to it? (1 point)
When the net force is applied to an object, accelerates towards the direction o