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 $g^{\prime }(x)=(f(x){)}^2.$ 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 $f\in L^1(-\mathrm{\infty },\mathrm{\infty })$ and suppose $\alpha >0$. Set $f_n(x)=\frac{f(nx)}{n^{\alpha }}$ for $n=1,2,\dots$. Prove that for almost every $x\in (-\mathrm{\infty },\mathrm{\infty })$, $\underset{n\to \mathrm{\infty }}{lim}{f}_n(x)=0$

Change of variable

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

$\int |f_n(x)|dx=\int \frac{|f(u)|}{n^{\alpha +1}}du(\ast )$

Monotone Convergence Theorem

Define $u_n(x)=\sum _{i=1}^n|f_i(x)|$.

Then, $u_n$ is a nonnegative increasing function converging to $\sum _{i=1}^{\mathrm{\infty }}|f_i(x)|$.

Hence, by Monotone Convergence Theorem and $(\ast )$

$\begin{array}{cc}\int \sum _{i=1}^{\mathrm{\infty }}|f_i(x)|dx& =\sum _{i=1}^{\mathrm{\infty }}\int |f_i(x)|dx\\ & =\sum _{i=1}^{\mathrm{\infty }}\int \frac{|f(x)|}{i^{\alpha +1}}dx\\ & =(\int |f(x)|dx)\left(\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i^{\alpha +1}}\right)\\ & <\mathrm{\infty }\end{array}$

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

Conclusion

Since

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

we have almost everywhere

$\sum _{i=1}^{\mathrm{\infty }}|f_i(x)|<\mathrm{\infty }$

This implies our desired conclusion:

$\underset{i\to \mathrm{\infty }}{lim}{f}_i(x)=0\text{a.e.}$