Sunday, 12 November 2017

functional analysis - fnrightarrow0 in L1 impliessqrtfnrightarrow0 also?



Let (X,Σ,μ) be a finite measure space, and let {fn:nN} be a sequence of non-negative measurable functions converging in the L1 sense to the zero function. Show that the sequence {fn:nN} also converges in the L1 sense to the zero function.



So I have to somehow show that




limnX|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 = \ ...


No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...