Corrections and comments to Assignment 5

PROBLEM 2: 
The conclusion should be that f is nonnegative _almost everywhere_ 
(the integral does not see sets of measure zero).

PROBLEM 4:
Assume additionally that \int |f_n| \le C for some constant C all n.  
In Part (b), I mean the simple version of Fatou's Lemma with "lim"

Remark: The result you proved, originally due to Brezis and 
Lieb [1983], is often used to show that a minimization problem 
has a solution.  In such applications, (f_n) is a minimizing 
sequence, and the pointwise limit f is a candidate for the 
minimizer.  Typically, the problem is to show that f is not 
the zero function.

The corresponding result holds for (f_n), f in L^p 
when 1<p<\infty (see Lieb & Loss, Section 1.9).
Notice that they prove a stronger identity.

PROBLEM 6, Part (b): Consider liminf A_n, and use the inequality
we proved in Assignment 1, Problem 4.

(Choosing a "fast" Cauchy subsequence will also work but is not
necessary.)