COMMENTS TO ASSIGNMENT 5


PROBLEM 2

Correction: The conclusion is that f is nonnegative
_almost everywhere_ (note that changing f on a set
of measure zero has no effect on the assumption.)


PROBLEM 3

Hint: Given a Cauchy sequence (A_n), extract a "fast" Cauchy 
subsequence and show that it converges to A = limsup A_n.

Estimate separately m(A_n\A) and m(A\A_n), using continuity
from above and below. Don't forget to check convergence of
the entire sequence !

Correction: I should have included the definition of
the symmetric difference:
   A "triangle" B = A\B union B\A 


PROBLEM 4

Correction: Assume that the set E is non-empty.


PROBLEM 5

Note, in contrast, that the composition of Borel measurable 
functions is Borel measurable.  Therefore the set B you 
constructed cannot be a Borel set.


PROBLEM 6

Hint: Every interval contains Cantor-type sets of positive 
measure. Use this to construct a pair of disjoint sets A, B 
that both meet each non-empty open subinterval I of [0,1] 
in a set of positive measure.

There are many ways to proceed. 
You could, for example, argue that it suffices to consider 
intervals I with rational endpoints.

Useful fact: A finite union of Cantor sets is again
totally disconnected, compact, and nowhere dense. Hence 
its complement is dense in any interval (and consists of
open intervals).