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 $$B_{\epsilon,n}^c \ $$, since we have explicitly removed all points where the convergence is not uniform.

$$\Leftarrow$$

Suppose $$\exists E \ $$ such that $$mE=0 \ $$ and $$f_n\to f \ $$ uniformly on $$E^c \ $$. Then for all $$\epsilon>0, \ \exists N \ $$ such that $$|f_n(x)-f(x)|<\epsilon \ $$ when $$n\geq N \ $$ for all $$x\in E^c \ $$. So $$B_{\epsilon,n}\subset E \ $$, and so $$|f_n-f|\to 0 \ $$.

11) Let $$f_n \ $$ be a Cauchy sequence of functions in $$L^\infty \ $$. Then for all $$\epsilon \ $$ there exists an $$N \ $$ such that $$m,n>N \implies ||f_m-f_n||_<\epsilon \ $$.

Then

$$ \text{max}(f_n,f_m)<\epsilon^p \ $$

12) Define $$x_n=\left\{{y_{n1},y_{n2},\dots }\right\} \ $$. Suppose $$\left\{{x_n}\right\} \ $$ is a Cauchy sequence.  Then for all $$\epsilon>0 \ $$, there is an $$N \ $$ such that for $$m,n\geq N \ $$, we have

$$||x_m-x_n||_p<\epsilon \ $$.

But $$||x_m-x_n||_p = \left({ \sum_{i=1}^\infty |y_{mi}-y_{ni}|^p }\right)^{1/p} \ $$, so this is just

$$\sum_{i=1}^\infty |y_{mi}-y_{ni}|^p < \epsilon^p \ $$.

Since $$|y_{mi}-y_{ni}|^p>0 \ $$, this implies $$|y_{mi}-y_{ni}|^p<\epsilon^p \implies |y_{mi}-y_{ni}|<\epsilon \ $$, and so there is a sequence $$x \ $$ such that $$x_n\to x \ $$.

Observe that for all $$\epsilon>0 \ $$, there is an $$N \ $$ such that $$n>N\implies ||x_N-x_n||_p < \epsilon \ $$. Since $$||x_n||_p \ $$ is a Cauchy sequence, it converges and so $$||x_n||_p \to ||x||_p \ $$.

13) Suppose $$f_n \in C[0,1] \ $$ is a Cauchy sequence. Then $$\exists f: \ f_n \to f \ $$ pointwise, and for all $$\epsilon \  $$ there is a $$N \ $$ such that $$n>N\implies ||f_n-f||<\epsilon \ $$, so $$f_n\to f \ $$ uniformly.

Since a uniformly convergent sequence of continuous functions converges to a continuous function, $$f \ $$ is continuous.

Now define $$a_n=\text{max}_{i\leq n} ||f_i|| \ $$. Since $$f_n\to f \ $$, for all $$\epsilon>0 \ $$ there is an $$N \ $$ such that $$n>N \implies |f_n-f|<\epsilon \ $$, so $$a_n \ $$ is a Cauchy sequence of real numbers and $$a_n\to a \ $$ satisfies $$a-a_n<\epsilon \ $$. Therefore, $$\text{max}(f) \ $$ exists and is the limit of $$a_n=\text{max}_{i\leq n} ||f_i|| \ $$.

16) Suppose $$f_n \to f \ $$. Then

$$f_n\to f \implies f_n^p\to f^p \implies \int_0^1 |f_n^p|\to\int_0^1 |f^p|\implies \int_0^1 |f_n|^p \to \int_0^1 |f|^p \ $$

$$\implies \left({ \int_0^1 |f_n|^p }\right)^{1/p} \to \left({ \int_0^1 |f|^p }\right)^{1/p} \implies ||f_n||_p\to ||f||_p \ $$.

Now suppose $$||f_n-f||_p\to 0 \ $$. This is true if and only if $$\int \left|{f_n-f}\right| \to 0 \ $$.

So let $$g_n, f_n, g, f \ $$ be as in the theorem and note $$|f_n|\leq g_n \implies |f|\leq g \ $$. Therefore, $$|f-f_n|\leq |f_n|+|f|\leq g_n+g \ $$, and so $$ h_n=g_n+g-|f_n-f| \geq 0 \ $$ is a sequence of nonnegative measurable functions.

By Fatou, we have $$\int \lim h_n \leq \lim \inf \int h_n \ $$,

which becomes

$$\int 2g \leq \int 2g - \lim \sup \int |f_n-f| \ $$.

This gives $$\lim \sup \int |f_n-f| \leq 0 \leq \lim \inf \int |f_n-f| \implies |f_n-f|\to 0 \ $$.

17)

20) Fix $$y \in \mathbb{R} \ $$ and $$f:A \to A \ $$, where $$A \ $$ is a bounded subset of $$\mathbb{R} \ $$. Define $$E=\left\{{x :f(x)>y}\right\} \ $$.

For a partition $$\Delta=\left\{{\xi_0,\dots,\xi_n}\right\}, \ \xi_j-\xi_{j-1}\leq \delta \ \forall j \ $$ of $$A \ $$, define $$F_\Delta = \left\{{x: \varphi_\Delta(x)>y }\right\} \ $$.

Suppose $$(\xi_j,\xi_{j+1})\subset E \ $$. Then $$f(x)>y \ \forall x \in (\xi_j,\xi_{j+1}), \ $$ and so

$$\varphi_\Delta(x)=\frac{1}{\xi_{j+1}-\xi_j}\int_{\xi_j}^{\xi_{j+1}} f(t)dt > \frac{1}{\xi_{j+1}-\xi_j}\int_{\xi_j}^{\xi_{j+1}} ydt=y \ $$.

Hence $$(\xi_j,\xi_{j+1})\subset E \implies (\xi_j,\xi_{j+1})\subset F_\Delta \ $$.

Now suppose $$(\xi_j,\xi_{j+1})\subset E^c \ $$. Then $$f(x)<y \ \forall x \in (\xi_j,\xi_{j+1}), \ $$ and so

$$\varphi_\Delta(x)=\frac{1}{\xi_{j+1}-\xi_j}\int_{\xi_j}^{\xi_{j+1}} f(t)dt < \frac{1}{\xi_{j+1}-\xi_j}\int_{\xi_j}^{\xi_{j+1}} ydt=y \ $$.

Hence $$(\xi_j,\xi_{j+1})\subset E^c \implies (\xi_j,\xi_{j+1})\subset F_\Delta^c \ $$.

Therefore, $$E-F_\Delta \subseteq \Delta\cup\bigcup (\xi_k,\xi_{k+1}) \supseteq F_\Delta-E \ $$,

ie, the symmetric difference $$E\ominus F_\Delta\subseteq \Delta\cup\bigcup (\xi_k,\xi_{k+1}) \ $$.

where the union is taken over all $$(\xi_k, \xi_{k+1}) \ $$ such that $$\exists x_1, x_2 \in (\xi_k, \xi_{k+1}): \ f(x_1)\geq y, f(x_2)\leq y \ $$.

Since $$\Delta \ $$ is a finite number of points, it has measure zero, and hence

$$m(E\ominus F_\Delta) \leq \Sigma (\xi_{k+1}-\xi_k) \leq m\delta $$,

where $$m \ $$ is the number of intervals such that $$\exists x_1, x_2 \in (\xi_k, \xi_{k+1}): \ f(x_1)\geq y, f(x_2)\leq y \ $$.

Note that since $$f\in L^P(A) \ $$, it is either constant $$=y \ $$ or it