1. Lebesgue measure: Lebesgue measure m0 is a measure on i.e., 1. 2.
disjoint
It generalizes the concept of length on

2. Lebesgue integral Example

3. Lebesgue integral Example
In other words, the value of the integral is independent of the representation of the simple functions in this example

4. The Lebesgue integral:
The Lebesgue integral is defined using Lebesgue measure
- For indicator functions,
- For simple funcitons,
- For non-negative functions,
- For general functions,

5. General Probability Spaces:
(W, P) is a probability triple
1. W, a nonempty set, called the sample space, which contains
all possible outcomes of the same random experiment.
a s-algebra of subsets of W.
3. P, a probability measure on (W, ), i.e., a function which
assigns to each set a number
representing the probability that the outcome of the
random experiment lies in the set A.

6. Integration using general probability measure:
Let X be a random variable on
1. Indicator function:
2. Simple function:

7. Integration of random variables
3. X is nonnegative:
(Xn: simple function)
4. General:

8. Expectation and other properties:
c: constant if
if A and B are disjoint

9. Monotone Convergence Theorem:
Let Xn, n = 1, 2,…. be a sequence of functions converging
almost surely to a random variable X, i.e.,
a.s.
Assume that
a.s.
Then
or equivalently,

10. Probability measure induced (引導)by a random variable:
X is a random variable on (W, P)
We could define the expectation of X as
This does not look like familiar in the old definition
… at least to me
A more familiar density formula can be derived from the
so-called induced measure using X (random variable)

11. Induced measure: For a random variable X on we write So
The induced measure of B is a measure on s.t.
In fact, the induced measure LX is a probability measure,
because

12. Expectation using density formula:
We now have two measures on
LX(A); the induced measure, m0; Lebesgue measure
The two measures on are connected through
a “density” (if exists) satisfying
i.e., under a certain condition, there exists j s.t.
where j is the Radon-Nikodym derivative of LX wrt. m0:

13. Suppose f is a function on
Then we have
Density formula
Expectation of f
To prove this, we use the “standard machine.”

14. Standard machine: Prove !!
Step 1: Start with the assumption that f: indicator function.
Step 2: Then extend to the simple function case.
Step 3: Construct a sequence of nonnegative simple functions
which converges to a nonnegative function f. Use the
Monotone Convergence Theorem to get the integral.
Step 4: For a general (integrable) function f, first split into
positive and negative parts, and integrate them
separately.

15. Probability distributions using Lebesgue measure:
Uniform distribution on [0, 1]
For W = [0, 1], B([0, 1]), let
Then LX is a probability measure because m0([0, 1]) = 1.
Standard normal distribution
For
let
To compute LX(A), one can also use the Riemann integral,

16. Independence: Definition 1.8: We say that two sets are independent if
Definition 1.9: We say that two s-algebras
are independent if
Definition 1.9: We say that two random variables, X
and Y, are independent if s-algabra generated by
these random variables are independent, i.e.,
s(X) and s(Y) are independent.

17. Independence of two functions:
If two random variables, X and Y, are independent, then
two functions, g(X) and h(Y), are also independent.
Proof: At first, recall that
For each
there exists
s.t.
Therefore
Similarly,
Since s(X) and s(Y) are independent, we conclude s(g(X))
and s(h(Y)) are independent.

18. Variance, covariance, and correlation:
If two random variables X and Y are independent,
More generally,
if X and Y are independent

19. Theory
Application
Discrete
Continuous
Understand the important concept of
conditional expectation

20. Definition of conditional expectation:
Probability triple
G: sub-s-algebra of
X: random variable on
The conditional expectation of X given G is a G-measurable
random variable Y satisfying
We write Y=E( X | G ).
Conditional expectation always exists, if
Conditional expectation is unique.

21. Illustrative example:
Binomial process given by 3 coin tosses
: stock price at time k
p is the probability of H
q=1-p is the probability of T
W={HHH,HHT,HTH,HTT,….}
s-algebra of subsets of W
The coin tosses are independent
filtration

22. Expectation and partial averages:
where
We will compute a partial average of X on a sub-s-algebra
of in the 3 coin toss example.

23. and unions of these sets}
Let X = S3(w) and

24. Let
s(S2)-measurable random variable
Similarly, one can show that holds for every
set in s(S3) with
We also write instead of

25. N coin tosses: Start from S0 : deteministic t=k+1 p q t=k Sk uSk dSk
p is the probability of H
q=1-p is the probability of T
W: sample space
s-algebra of subsets of W
The coin tosses are independent
filtration
We compute

26. First of all, note that, if
(k+1)-th entry
there is always

27. Let -measurable random variable
Conditional expectation (denoted by )

28. Conditional expectation:
t=k+1 p q
t=k Sk
uSk
dSk
In this case, becomes
a function of Sk which gives an
estimate based on the information
of Sk.