UMD Analysis Qualifying Exam/Jan09 Real

Problem 1

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

Problem 3

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

Change of variable

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

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

Monotone Convergence Theorem

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

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

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

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

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

Conclusion

Since

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

we have almost everywhere

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

This implies our desired conclusion:

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