User:J D Bowen/Math710 HW7

Please do the following problems from Chapter 6 of the text:

10,11,12,13,16,17,20

10) Suppose $$f_n \in L^\infty \ $$. We aim to show that $$f_n\to_{L^\infty} f \iff \exists E: mE=0, \ f_n\to_{\text{uniformly on complement of E}} f \ $$.

$$\Rightarrow \ $$

Suppose $$f_n\to f \ $$ in $$L^\infty \ $$, ie, $$|f_n-f|\to 0 \ $$.

Define $$B_{M,n} = \left\{{x:|f_n(x)-f(x)|>M }\right\} \ $$.

By definition, $$\forall \epsilon \ \exists N: \ n\geq N \implies \text{inf}\left\{{M:mB_{M,n}=0}\right\}<\epsilon \ $$.

So $$mB_{\epsilon,n}=0 \ $$ and $$f_n\to f \ $$ uniformly on $$B_{\epsilon,n}^c \ $$.

$$\Leftarrow$$