Let (X,Σ,μ) be a finite measure space, and let {fn:n∈N} be a sequence of non-negative measurable functions converging in the L1 sense to the zero function. Show that the sequence {√fn:n∈N} also converges in the L1 sense to the zero function.
So I have to somehow show that
limn→∞∫X|√fn(x)|dμ(x)=0
If I'm honest I don't really know where to start. I think it's an easy question, but I'm new to this stuff. Any help appreciated!
Answer
You have the right choice of p = q = 2. However, choose u = 1, v = \sqrt{f_n}, then
\int_X \left|\sqrt{f_n}\right| \ d\mu = \left\| 1.\sqrt{f_n} \right\|_1 \ \leq \ \left\| 1 \right\|_2 \ \ \left\| \sqrt{f_n}\right\|_2 = \ ...
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