COMMENTS TO ASSIGNMENT 9


PROBLEM 2

Hint: Egoroff's theorem.


PROBLEM 3

Remark: This is a special case of the Strong Law of Large Numbers
--- think of x as the result of picking its decimal digits 
"independently, uniformly at random" from the set {0, ..., 9}. 
The value 0.1 represents the average frequency at which 
"7" appears in such a sequence.  So, if the limit
exists and is constant, that's what its value must be.


PROBLEM 4

I forgot to state the assumption that the  random variables
X_i are independent.

You may use that independence is equivalent to the property

     E ( \prod g_i(X_i) ) = \prod E( g_i(X_i) )

for every finite collection of functions g_i such that the 
composition g_i(X_i) is integrable.

Gratuitous remark: Kolmogorov's criterion can be used to 
provide another solution of Problem 6 in Assignment 5, 
as follows.  For x in [0,1], let x_n be the n-th binary
digit of x, similar to Problem 3 above (but with the binary 
in place of the decimal expansion). The series 

    g(x) = \sum_n  (\Chi_{x_n=1} - \Chi_{x_n=0}) / n

converges pointwise for almost every x. On the other hand, 
\sum 1/n diverges. Combining these two facts, one can show
that g is unbounded above and below on every interval.
Moreover, for each real number t, the level set

       A_t = { x : g(x)>t }

defines a subset of (0,1) that intersects every subinterval
of positive length in  subset of positive, but not full
measure. 


PROBLEM 5

For the first part, note that f * \phi = \phi * f 
(by changing variables y -> x-y). 
Use Dominated Convergence on the differential quotient.

For the limit in the second part, approximate f with really 
simple functions.


PROBLEM 6

(a) Hint: First consider f(0), f(1), f(n), and f(m/n), where m 
and n are natural numbers. What can you say if f is continuous?

Next, convolve e^{if} with a smooth nonnegative function
of compact support. The convolution is smooth (and not 
identically zero) !

(b) Consider R (the reals) as a vector space over Q (the 
field of rationals).