A question from Rudin RCA

Question

I'm studying Rudin's RCA chapter 7 and get in trouble with exercise 8.

Let $V = (a, b)$ be a bounded segment in $R^1$. Choose segments $W_n \subset V$ in such a way that their union $W$ is dense in $V$ and the set $K = V - W$ has $m(K) > 0$. Choose continuous functions $\phi_n$ so that $\phi_n(x) = 0$ outside $W_n$, $0 < \phi_n(x) < 2^{-n}$ in $W_n$. Put $\phi = \sum\phi_n$ and define $$ T(x) = \int_a^x\phi(t)dt\qquad(a<x<b) $$ Prove the following statements:

(a) $T$ satisfies the hypotheses of that $T$ is one-to-one on $X$, and $T$ is differentiable at every point of $X$, with $X = V$.

(b) $T'$ is continuous, $T'(x) = 0$ on $K$, $m(T(K)) = 0$.

(c) If $E$ is a nonmeasurable subset of $K$ (see Theorem 2.22) and $A = T(E)$, then $\chi_A$ is Lebesgue measurable but $\chi_A \circ T$ is not.

(d) The functions $\phi_n$ can be so chosen that the resulting $T$ is an infinitely differentiable homeomorphism of $V$ onto some segment in $R^1$ and (c) still holds.

Can anyone point me out about the b $m(T(K)) = 0$ and $d$? I would be very thankful for any answer.

Answer 1

For (b), to prove that $m(T(K))=0$, one need the followings:

1. $T(K)$ is Lebesgue measurable (Why?) (Hint: prove that $T(W)$ is $F_{\sigma}$ set, and theorem 2.10 could be helpful)

2. Construct a Lebesgue measurable function $$f(T(y))=\chi_{T(K)}(T(y))=\begin{cases}1, & y\in K\\0, & y\in K^C\end{cases}$$ (Check why this holds. Hint: for $=1$, use the fact that $T$ is one-to-one; for $=0$, use the same fact and argue by contradiction)

By theorem 7.26 in the text, since $K\subset V\Rightarrow T(K)\subset T(V)$, we have $$m(T(K))=\int_{T(V)}\chi_{T(K)}(x)dm(x)=\int_{V}\chi_{T(K)}(T(y))|J_T|dm(y)=\int_{V}\chi_{K}(y)|J_T(y)|dm(y)$$ $$=\int_{K}\chi_{K}(y)|J_T|dm(y)=\int_{K}0dm(y)=0$$ Note that this is a consequence of (iii).

For (d), choose $W_n=(\alpha_n,\beta_n)$ that are disjoint, construct $$\varphi_n(x)=\begin{cases}2^{-n}\exp\left(\frac{(\frac{\beta_n-\alpha_n}{2})^2}{(x-\frac{\alpha_n+\beta_n}{2})^2-(\frac{\beta_n-\alpha_n}{2})^2}\right), &x\in(\alpha_n,\beta_n)\\0, &\text{otherwise}\end{cases}$$

Then check that $\varphi_n(x)$ satisfies all the conditions.

Hope this helps.