By Fatou's Lemma, we get the upper bound is 1 and since all the functions $ f_{n}\frac{}{} $ are positive, we get the lower bound is 0. This is as good as it get. Examples:
Let $ \Omega=[0,1]\frac{}{} $, the $ \sigma- $algebra is the power set and counting measure.
Example 1:
For $ n $ odds, $ f_{n}(x)=1\frac{}{} $ if $ x=\frac{1}{n} $, $ 0\frac{}{} $ otherwise.
For $ n $ even, $ f_{n}(x)=3\frac{}{} $ if $ x=\frac{1}{n} $, $ 0\frac{}{} $ otherwise.
Example 2:
For $ n $ odds, $ f_{n}(x)=1\frac{}{} $ if $ x=1\frac{}{} $, $ 0\frac{}{} $ otherwise.
For $ n $ even, $ f_{n}(x)=1\frac{}{} $ if $ x=1\frac{}{} $, $ f_{n}(x)=2\frac{}{} $ if $ x=\frac{1}{n} $, $ 0\frac{}{} $ otherwise.
