Absolutely Continuous Functions

Prove that $f(x)=\sqrt{x}$ is absolutely continuous on $[0,1]$ but not Lipschitz.

Let $f, g:[a, b] \rightarrow \mathbb {R}$ be absolutely continuous and let $\alpha , \beta \in \mathbb {R}$. Prove that $\alpha f+\beta g$ is absolutely continuous on $[a, b]$.

Let $f, g:[a, b] \rightarrow \mathbb {R}$ be absolutely continuous. Prove that $f g$ is absolutely continuous on $[a, b]$.

Let $f:[a, b] \rightarrow \mathbb {R}$ be absolutely continuous and suppose that $f(x) \neq 0$ for all $x \in [a, b]$. Prove that $1 / f$ is absolutely continuous on $[a, b]$.

Let $g$ be a strictly increasing absolutely continuous function on $[a, b]$ and let $f$ be an absolutely continuous function on $[g(a), g(b)]$. Prove that $f \circ g$ is absolutely continuous on $[a, b]$.

Prove that $f$ is absolutely continuous on $[a, b]$ if and only if for every $\varepsilon >0$ there exists $\delta >0$ such that, for every collection $\left\{ \left[a_{k}, b_{k}\right]\right\}_{k=1}^{\infty }$ of nonoverlapping closed intervals in $[a, b]$,

Let $\left\{ G_{n}\right\}$ be a decreasing sequence of open subsets of $[0,1]$ such that $m\left(G_{n}\right)<1 / 2^{n}$, and let $f(x)=\sum_{n=1}^{\infty } m\left(G_{n} \cap [0, x]\right)$. Prove that:

Let $\left\{ f_{n}\right\}$ be a sequence of absolutely continuous functions on $[0,1]$ that converge to a function $f$ uniformly on $[0,1]$. Prove or disprove: $f$ is absolutely continuous.

Let $f$ be a Lipschitz function on $[a, b]$ with constant $M$ and let $E \subset [a, b]$. Prove that $m(f(E)) \leq M m(E)$. Conclude that if $f$ is a Lipschitz function on $[a, b]$ then it has the Luzin N property. Is the converse true?

Definition 9.4.20: A real-valued function defined on $[a, b]$ has the Luzin N property, if it maps every set of measure zero to a set of measure 0.

Prove that every Lipschitz function maps measurable sets to measurable sets.

Give an example of a function without the Luzin N property.

Definition 9.4.20: A real-valued function defined on $[a, b]$ has the Luzin N property, if it maps every set of measure zero to a set of measure 0.

(Banach-Zarecki) Prove that a function $f$ on the interval $[a, b]$ is absolutely continuous if and only if it is continuous, is of bounded variation, and has the Luzin N property.

Definition 9.4.20: A real-valued function defined on $[a, b]$ has the Luzin N property, if it maps every set of measure zero to a set of measure 0.

Prove that an absolutely continuous function maps a measurable set to a measurable set.

Prove that if the word nonoverlapping is deleted in Definition 9.4.1 the function must be Lipschitz. More precisely, let $f:[a, b] \rightarrow \mathbb {R}$ be a function with the property that for every $\varepsilon >0$ there exists $\delta >0$ such that, for every collection $\left\{ \left[a_{k}, b_{k}\right]\right\}_{k=1}^{n}$ of closed intervals in $[a, b]$,

Then $f$ is Lipschitz on $[a, b]$.

Definition 9.4.1: A real-valued function $f$ on $[a, b]$ is absolutely continuous on $[a, b]$ if for every $\varepsilon >0$ there exists $\delta >0$ such that, for every collection $\left\{ \left[a_{k}, b_{k}\right]\right\}_{k=1}^{n}$ of closed nonoverlapping intervals in $[a, b]$, $\sum_{k=1}^{n}\left(b_{k}-a_{k}\right)<\delta \Rightarrow \sum_{k=1}^{n}\left|f\left(b_{k}\right)-f\left(a_{k}\right)\right|<\varepsilon$. Here nonoverlapping means with disjoint interiors.

Let $f:[0,1] \rightarrow \mathbb {R}$ be an increasing continuous function that is absolutely continuous on $[\varepsilon , 1]$ for each $\varepsilon \in (0,1)$. Prove that $f$ is absolutely continuous on $[0,1]$.

Let $f(x)=\sqrt{x}$ and

Prove that $f$ and $g$ are absolutely continuous on $[0,1]$ but $h:=f \circ g$ is not even of bounded variation on $[0,1]$.

Suppose that $f$ is a function on $[a, b]$, and let $V_{a}^{x}(f)$ be the total variation function for $f$. Prove that $V_{a}^{x}$ is absolutely continuous on $[a, b]$ if and only if $f$ is. Conclude that every absolutely continuous function is the difference of increasing absolutely continuous functions.

Prove that a function $f$ is Lipschitz on $[a, b]$ if and only if it is absolutely continuous function on $[a, b]$ with $\left|f^{\prime }\right|$ essentially bounded. (A function is essentially bounded if it is equal a.e. to a bounded function.)

Give an example of a function of bounded variation that is not absolutely continuous.

Let $f$ be a function of bounded variation on $[a, b]$. Prove that $V_{a}^{b}(f)= \int_{a}^{b}\left|f^{\prime }\right|$ if and only if $f$ is absolutely continuous on $[a, b]$.

Let $g:[a, b] \rightarrow \mathbb {R}$ be a strictly increasing, absolutely continuous function. Prove that for any measurable set $E \subset [a, b]$

Let $f:[a, b] \rightarrow \mathbb {R}$, be integrable and let $g:[\alpha , \beta ] \rightarrow \mathbb {R}$ be a strictly increasing, absolutely continuous function for which $g(\alpha )=a, g(\beta )=b$. Prove that

Prove Corollary 9.4.14.

Corollary 9.4.14: Let $f: [a,b] \rightarrow \mathbb {R}$ be a function that is differentiable a.e. with an integrable derivative. If $\int_{a}^{x} f' = f(x) - f(a)$ for all $x \in [a,b]$, then $f$ is absolutely continuous on $[a,b]$.

Prove Corollary 9.4.16.

Corollary 9.4.16: A function $f$ is absolutely continuous on $[a,b]$ if and only if it is an indefinite integral over $[a,b]$. (A function $F$ is the indefinite (Lebesgue) integral of an integrable function $f$ over $[a,b]$ if $F(x) = F(a) + \int_{a}^{x} f$ for all $x \in [a,b]$.)

If $f:[a, b] \rightarrow \mathbb {R}$ is continuous and real-valued, $f^{\prime }$ integrable on $[a, b]$, and $\int_{a}^{b} f^{\prime }=f(b)-f(a)$, must $f$ be absolutely continuous? What if $f$ is monotone increasing?

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be a function that is absolutely continuous on each finite interval, and suppose that both $f$ and $f^{\prime }$ are integrable over $\mathbb {R}$. Prove that:

Let $\left\{ f_{n}\right\} \subset A C[0,1]$ and suppose that

1. $f=\sum_{n=1}^{\infty } f_{n}$ converges for every $x \in [0,1]$;

2. $\int_{0}^{1} \sum_{n=1}^{\infty }\left|f_{n}^{\prime }\right|$ is finite.

Prove that $f \in A C[0,1]$.

The purpose of this exercise is to establish the following result: If $f$ is differentiable for all $x \in \left[a, b\right]$ and $f^{\prime }$ is integrable over $\left[a, b\right]$ then (9.30) holds.

Formula (9.30), from Theorem 9.4.15: $\int_{a}^{x} f^{\prime }=f(x)-f(a)$, for $a \leq x \leq b$.

Let $f$ be differentiable on $[a, b]$ with derivative $f^{\prime }$ integrable over $[a, b]$. Prove that

Let $f$ be a function on $[a, b]$ for which $D^{+} f$ is integrable over $[a, b]$. Prove or disprove: $V_{a}^{b}(f)=\int_{a}^{b}\left|D^{+} f\right|$.

Let $f$ and $g$ be absolutely continuous functions on $[a, b]$. Show that

Let $f$ be integrable over $\mathbb {R}$. Prove that for a.e. $x \in \mathbb {R}$

Let $f$ be integrable over $\mathbb {R}$ and suppose that $f$ is continuous at $x=a$. Prove that the indefinite integral of $f$ is differentiable at $x=a$.

Let $A$ be a measurable subset of $\mathbb {R}$ and suppose that there is $\alpha >0$ such that $m(A \cap [a, b]) \geq \alpha (b-a)$, whenever $a<b$. Prove that $m\left(A^{C}\right)=0$

Let $f$ be absolutely continuous and singular on $[a, b]$. Show that $f$ is constant.

When Lebesgue defined a singular function it was the following way: $F_{s}$ is the singular function associated with a function of bounded variation $F$, if the difference $F-F_{s}$ is absolutely continuous and among all the functions $G_{s}$ such that $F-G_{s}$ is absolutely continuous, $F_{s}$ has the smallest total variation, and $F_{s}(a)=0$. Prove that the decomposition $F=\left(F-F_{s}\right)+F_{s}$ is the same as the one provided by Theorem 9.4.19.

Theorem 9.4.19: A continuous function $f$ of bounded variation on $[a, b]$ can be represented in a unique way as $f=g+h$ where $g$ is absolutely continuous on $[a, b]$ satisfying $g(a)=f(a)$, and $h$ is either a singular function or the zero function. (Definition 9.4.17: a function $f:[a, b] \rightarrow \mathbb {R}$ is singular if it is continuous and of bounded variation on $[a, b]$, its derivative vanishes almost everywhere, and it is non-constant on $[a, b]$.)

Let $A$ be a measurable subset of $[0,1]$ such that $0<m(A \cap I)<m(I)$ for every interval $I \subset [0,1]$ (see Exercise 5.3.47), and let $F(x):=m([0, x] \cap A)$, $G(x):=m([0, x] \cap A)-m\left([0, x] \cap A^{C}\right)$. Prove that:

Show that the condition imposed on the set $A$ in Exercise 9.4.37 (a) is not necessary, i.e., that there is a set that does not satisfy it, yet $F(x):= m([0, x] \cap A)$ is an absolutely continuous, strictly increasing function on $[0,1]$ such that $F^{\prime }=0$ on a set of positive measures.

(Vitali) Let $f$ be an increasing function on a closed and bounded interval $[a, b]$. Prove that $f$ is singular if and only if for any $\varepsilon , \delta >0$ there is a finite disjoint collection $\left\{ \left(a_{k}, b_{k}\right)\right\}_{k=1}^{n}$ of open intervals in ( $a, b$ ) for which

Give an example of a function $f:[a, b] \rightarrow \mathbb {R}$ that is differentiable on $[a, b]$ but its derivative is not integrable over $[a, b]$.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be an increasing function such that $f(-\infty )=0$ and $f(\infty )=1$. Prove that $f$ is absolutely continuous on every closed finite interval if and only if $\int_{\mathbb {R}} f^{\prime }=1$.