Lecture 7 Notes: Approximations
Review. The steps to defining Lebesgue measure. (1) measure of
rectangles (2) measure of special polygons (3) measure of open
sets: (G) = supcfw_(P ) P G, P special polygon. (4) measure of
compact sets: (K) = infcfw_(G) 
Lecture 5 Notes: Finite Functions
1
1
1
[
Definition of L . Let f : X
f is in L

and Xf
d
+
] be measurable. We say that
L () or simply f L )
(written f
1
,
X
X
f
X f d <
+

. Define
d <
f d = f d f d
X
X
when at least one of the terms on the rightha
Lecture 4 Notes: Simple Functions and Graphs
Integral is additive for simple functions.
Proposition 0.1. Let s and t be nonnegative measurable simple functions. Then X (s + t)d = X s d + X t d.
Proof. Let E M and define (E) = E s d. First we show that is
Lecture 2 Notes: Economic Models and Mathematics
Proposition 0.1. Let M be a algebra on X, let Y be a topological space, and
let f:XY .
(a)
1
Let be a collection of sets EY such that f (E) M. Then is a algebra
on Y .
1
(b) If f is measurable and EY is B
Lecture 6 Notes: Open Set Protocol
n
Lebesgue measure on R . We will define the Lebesgue measure
n
: cfw_subsets of R [0,] through a series of steps.
(1) () = 0.
(2) Special rectangles: rectangles with sides parallel to axes.
n = 1: ([a, b]) = b a
n =
Lecture 3 Notes: Riemann Intervals
Riemann integral. If s is simple and measurable then
N
X
where s =
N
f d = sup
X
i=1
. If f 0, then
i=1iEi
sd = i(Ei),
X
sd 0 s f, s simple & measurable .
Recall the Riemann integral of function f on interval [a, b]. De
Lecture 9 Notes: Measurable Sets
n
n
Invariance of Lebesgue measure. Given A R and z R , let z + A =cfw_z + x  x
Abe thetranslateof A by z. Given t >0, let tA =cfw_tx  x Abe thedilationof A by t.
Let I = [a1, b1] [an, bn] and z = z1 zn. Then z
+ I = [z
Lecture 10 Notes: Graphical Representations
Integration as a linear functional. A complex vector space is a set
V with two operations: addition (+) and scalar multiplication (). Addition: For all x, y, z V ,
x + y = y + x.
x + (y + z) = (x + y) + z.
un
Lecture 8 Notes: Open Macroeconomic Markets
More properties of L.
(1) All open sets and closed sets are in L. (In particular, L
contains the Borel algebra B.)
(2) If (A) = 0, then A is measurable and (A) = 0. (All sets of
measure zero are measurable.)
n
Lecture 1 Notes: Intro to Course
Preliminaries. We need to know how to measure the size or vol ume
of subsets of a space X before we can integrate functions f : X R or f : X
C.
n
Were familiar with volume in R . What about more general spaces X? We need