Math Problems Vault

Problem 6.34

(i)(ii)(iii)

Let $f:[0,1] \rightarrow \mathbb {R}$ be continuous, and be absolutely continuous on $[\epsilon , 1]$ for each $0 < \epsilon < 1$ .

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(i)

Show that $f:[0,1] \rightarrow \mathbb {R}$ may not be absolutely continuous.

(ii)

Show that $f:[0,1] \rightarrow \mathbb {R}$ is absolutely continuous if it is increasing.

(iii)

Show that the function $f:[0,1] \rightarrow \mathbb {R}$ defined by $f(x) = \sqrt{x}$ for $0 \leq x \leq 1$ is absolutely continuous, but not Lipschitz.

Problem 6.35

Verify that the inclusions in (19) are strict.

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Problem 6.36

Show that a Lipschitz function $f:[a, b] \rightarrow \mathbb {R}$ is the difference of increasing Lipschitz functions.

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Problem 6.37

Assume that $f:[a, b] \rightarrow \mathbb {R}$ is increasing. Show that $f$ is absolutely continuous if and only if for each $\epsilon > 0$ , there is a $\delta > 0$ such that for every countable, disjoint collection $\left\{ \left(a_{k}, b_{k}\right)\right\}_{k = 1}^{\infty }$ of open intervals in ( $a, b$ ),

\sum _{k = 1}^{\infty }\left[f\left(b_{k}\right)-f\left(a_{k}\right)\right] < \epsilon \text{ if } \sum _{k = 1}^{\infty }\left[b_{k}-a_{k}\right] < \delta

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Problem 6.38

Show that if $f:[a, b] \rightarrow \mathbb {R}$ is increasing and absolutely continuous, then it maps sets of measure zero to sets of measure zero.

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Problem 6.39

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous. Conclude from the continuity of $f$ and the compactness of $[a, b]$ that $f$ maps closed sets to closed sets and therefore maps $F_{\sigma }$ sets to $F_{\sigma }$ sets.

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Problem 6.40

Show that if $f:[a, b] \rightarrow \mathbb {R}$ is increasing and absolutely continuous, then $f$ maps measurable sets to measurable sets.

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Problem 6.41

Show that if $f:[a, b] \rightarrow \mathbb {R}$ is continuous and increasing, then $f$ is absolutely continuous if and only if for each $\epsilon$ , there is a $\delta > 0$ such that for each measurable subset $E$ of $[a, b]$ ,

m(f(E)) < \epsilon \text{ if } m(E) < \delta .

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Problem 6.42

Show that the Cantor-Lebesgue function $\varphi$ is not absolutely continuous on $[0,1]$ by arguing that the function $\psi$ , defined by $\psi (x) = x+\varphi (x)$ for $0 \leq x \leq 1$ , maps the Cantor, a set of measure zero, to a set of measure 1.

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Problem 6.43

Show that both the sum and product of absolutely continuous functions are absolutely continuous.

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Problem 6.44

(i)(ii)(iii)

Define the functions $f$ and $g$ on $[-1,1]$ by $f(x) = x^{\frac{1}{3}}$ for $-1 \leq x \leq 1$ and

g(x) = \begin{cases} x^{2} \cos (\pi / 2 x) & \text{ if } x \neq 0, x \in [-1,1] \\ 0 & \text{ if } x = 0.\end{cases}

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(i)

Show that both $f:[-1,1] \rightarrow \mathbb {R}$ and $g:[-1,1] \rightarrow \mathbb {R}$ are absolutely continuous.

(ii)

For the partition $P_{n} = \left\{ -1,0,1 / 2 n, 1 /[2 n-1], \ldots , 1 / 3,1 / 2,1\right\}$ of $[-1,1]$ , examine $V\left(f \circ g, P_{n}\right)$ .

(iii)

Show that $f \circ g:[-1,1] \rightarrow \mathbb {R}$ fails to be of bounded variation, and hence also fails to be absolutely continuous.

Problem 6.45

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be Lipschitz and $g:[a, b] \rightarrow \mathbb {R}$ be absolutely continuous. Show that the composition $f \circ g:[a, b] \rightarrow \mathbb {R}$ is absolutely continuous.

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Problem 6.46

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be absolutely continuous and $g:[a, b] \rightarrow \mathbb {R}$ be absolutely continuous and strictly increasing. Show that the composition $f \circ g:[a, b] \rightarrow \mathbb {R}$ is absolutely continuous.

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Problem 6.47

Prove the converse of the implication in Theorem 11.

Theorem 11: If the function $f: \left[a, b+1\right] \rightarrow \mathbb {R}$ is absolutely continuous, then the collection of divided difference functions $\mathcal{F} = \left\{ \operatorname {Diff}_{h} f: \left[a, b\right] \rightarrow \mathbb {R}\right\}_{0 < h \leq 1}$ is uniformly integrable.

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