# UMD Analysis Qualifying Exam/Jan09 Real

## Problem 1

 (a) Let ${\displaystyle f,g}$  be real valued measurable functions on ${\displaystyle [0,1]}$  with the property that for every ${\displaystyle x\in [0,1]}$ , ${\displaystyle g}$  is differentiable at ${\displaystyle x}$  and ${\displaystyle g'(x)=(f(x))^{2}.}$  Prove that ${\displaystyle f\in L^{1}[0,1]}$  (b) Suppose in addition that ${\displaystyle f}$  is bounded on ${\displaystyle [0,1].}$  Prove that ${\displaystyle 2\int _{0}^{1}g(x)f^{2}(x)\,dx=g^{2}(1)-g^{2}(0).}$

## Problem 3

 Let ${\displaystyle f\in L^{1}(-\infty ,\infty )\!\,}$  and suppose ${\displaystyle \alpha >0\!\,}$ . Set ${\displaystyle f_{n}(x)={\frac {f(nx)}{n^{\alpha }}}\!\,}$  for ${\displaystyle n=1,2,\ldots \!\,}$ . Prove that for almost every ${\displaystyle x\in (-\infty ,\infty )\!\,}$ , ${\displaystyle \lim _{n\rightarrow \infty }f_{n}(x)=0\!\,}$

### Change of variable

By change of variable (setting u=nx), we have

${\displaystyle \int |f_{n}(x)|dx=\int {\frac {|f(u)|}{n^{\alpha +1}}}du\quad \quad (*)\!\,}$

### Monotone Convergence Theorem

Define ${\displaystyle u_{n}(x)=\sum _{i=1}^{n}|f_{i}(x)|\!\,}$ .

Then, ${\displaystyle u_{n}\!\,}$  is a nonnegative increasing function converging to ${\displaystyle \sum _{i=1}^{\infty }|f_{i}(x)|\!\,}$ .

Hence, by Monotone Convergence Theorem and ${\displaystyle (*)\!\,}$

{\displaystyle {\begin{aligned}\int \sum _{i=1}^{\infty }|f_{i}(x)|dx&=\sum _{i=1}^{\infty }\int |f_{i}(x)|dx\\&=\sum _{i=1}^{\infty }\int {\frac {|f(x)|}{i^{\alpha +1}}}dx\\&=\left(\int |f(x)|dx\right)\left(\sum _{i=1}^{\infty }{\frac {1}{i^{\alpha +1}}}\right)\\&<\infty \end{aligned}}\!\,}

where the last inequality follows because the series converges (${\displaystyle \alpha >0\!\,}$  ) and ${\displaystyle f\in L^{1}\!\,}$

### Conclusion

Since

${\displaystyle \int \sum _{i=1}^{\infty }|f_{i}(x)|dx<\infty \!\,}$ ,

we have almost everywhere

${\displaystyle \sum _{i=1}^{\infty }|f_{i}(x)|<\infty \!\,}$

This implies our desired conclusion:

${\displaystyle \lim _{i\rightarrow \infty }f_{i}(x)=0\quad {\mbox{a.e.}}\!\,}$