Questions tagged [absolute-continuity]

Use this tag for questions related to absolute continuity, which is a smoothness property of functions stronger than that of continuity and uniform continuity.

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18 views

Is $g(x)=\int_a^x (x-s)^{-\alpha}f(s) \, ds\in AC[a,b]$ if $f\in L_1[a,b]$ and $\alpha\in (0,1)$?

Let $f\in L_1[a,b]$ and $\alpha\in(0,1).$ Is it true that the function $$ g(x)=\int_a^x \frac{f(s)}{(x-s)^\alpha}\,ds $$ is absolutely continuous on $[a,b]$? First, I thought it is true and I tried ...
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BV[a,b]$\cap$C[a,b]$\neq$AC[a,b]

Take an $f\in$BV[0,1]$\cap$C[0,1] e.g. the Cantor function. I take the Lebesgue Stiltigies measure of $f$: $$ \mu_f((a,b])=f(b)-f(a). $$ Now I have a finite positive measure, so I can do the Radon-...
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1answer
26 views

Can a piecewise $C^1$ mapping can define an absolutely continuous function

Let $f,g\in C^1([0,1],\mathbb{R}^n)$, and then we define $h$ as a peicewise $C^1$ mapping as follows $$h(t)=\begin{cases} f(t) & t\in[0,\frac{1}{2}[\\ g(t) &t\in]\frac{1}{2},1] \end{cases} $$ ...
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23 views

Differentiable but not absolutely continuous cumulative distribution function

There are many examples of functions that are differentiable but not absolutely continuous. But these examples are unbounded oscillating functions (see for example some of the answers to this question ...
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1answer
23 views

Absolute continuity of probability measure

Let $([0,1), \mathcal{B}([0,1))$ be a probability space and define the filtration $\mathcal{F}_n=\sigma([0,\frac{1}{2^n}),[\frac{1}{2^n}\frac{2}{2^n}),\ldots,[\frac{2^n-1}{2^n},1))$. Let $\mu_n$ and $\...
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22 views

Suppose $f,g$ are two absolutely continous functions in an interval $[a,b]$. Show $fg \in AC[a,b]$.

Suppose $f,g$ are two absolutely continous functions in an interval $[a,b]$. Show $fg \in AC[a,b]$. My try: $$ \begin{align} \int_a^x (fg)'(t)dt & \stackrel{(1)}= \int_a^x \Big[f'(t)g(t) + f(t)g'(...
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1answer
42 views

product of $L^2$ function and its $L^2$ derivative

Let $f$ be absolutely continuous on $[0,x]$ for all $x >0$. $f(0) = 0$ and $f,f' \in L^2([0,\infty))$. Prove that $\int\limits_0^x |ff'| dm\leq \frac{1}{2}(\int\limits_0^x |f'| dm)^2$. The problem ...
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55 views

if $\lim_{h\to 0} \frac{1}{h}\int_0^1|f(x+h)-f(x)|=0$ then $f$ is constant a.e

Let $f\in L^1([-1,2])$ and suppose $\lim_{h\to 0} \frac{1}{h}\int_0^1|f(x+h)-f(x)|=0$. I want to prove that f is constant a.e. I want to check if my approach is correct and fill out any gaps. My idea ...
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94 views

The space of absolutely continuous functions is separable with norm $\|F\|_{AC}=\sup |F|+\int_a^b|F'|$ [closed]

Consider the space of absolutely continuous functions $AC([a,b])$ equipped with norm $$ \|F\|_{AC}=\sup |F|+\int_a^b|F'| $$ It can be shown that $AC([a,b])$ is a Banach space under this norm. Please ...
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1answer
49 views

Absolute continuity on $[a,b]$ implies mapping of sets of measure zero to sets of measure zero

I am reviewing some materials for real analysis and measure theory. I want to show that if we have an absolutely continuous function on a closed interval, the function maps sets of measure zero to ...
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1answer
23 views

The variation of a function as a function

Let $f:\mathbb{R}\rightarrow\mathbb{R^*}$ be a continuous function (when $\mathbb{R^*}=[-\infty,\infty]=2$ points compactification of $\mathbb{R}$, this is not so much necessary for the question). Let ...
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30 views

If $u(x):=\int_{x_0}^x v(t)\;dt$, then $u$ locally absolutely continuous.

I want to show that if $$u(x):= \int_{x_0}^x v(t) \; dt$$, that then $u$ is locally absolutely continuous. The proof in Lemma 3.31 in A First Course in Sobolev Spaces suggests that we first want to ...
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sequence of absolute continuous functions converging pointwise to y=x whose derivative converge pointwise to 0

Let $f_n: [0,1] \rightarrow \mathbb{R}$ be a sequence of absolutely continuous functions satisfy the following properties: $f_n \rightarrow f$ $a.e$ where $f(x) = x$. $f_n' \rightarrow 0$ $a.e$ and $...
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1answer
18 views

About absolute continuity of sigma finite measures

Let $\mu_1,\mu_2$ be sigma finite measures. Prove that $\mu_1\ll \mu_2$ and $\mu_2\ll \mu_1$ $\iff \exists f$, where $f$ is strictly positive such that $\nu(A)=\int_A fd\mu$ for all measurable $A$. It ...
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1answer
39 views

Show that the following set has measure 0

Let $g:[a,b] \rightarrow \mathbb{R}$ a strictly increasing and absolutly continous function with $g(a)=c$ and $g(b)=d$. Let $H=\{x: g'(x) \neq 0\}$ and $E \subseteq [c,d]$ with $|E|=0$. Show that $|g^{...
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34 views

Regularization theorem :$ AC[0,1]$ to $\mathcal{C}^1[0,1]$

For a given absolutely continuous function $x\in AC([0,1],\mathbb{R}^d)$ i.e : there is $f\in L^1[0,1]$ such that : $$x(t)= x(0)+ \int_0^tf(s)ds.$$ Can I find a continuously differentiable $y$, such ...
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200 views

The properties of convex function on the closed unit interval $[0,1]$.

Consider a continuous and convex function $F(x):[0,1]\longrightarrow\mathbb{R}$. I am wondering if $F(x)$ is continuously differentiable in $[0,1]$ $F(x)$ is of bounded variation in $[0,1]$ $F(x)$ ...
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1answer
27 views

Intuition behind the difference between absolute continuity and $\mu(A) = 0 \implies \mathcal H^{d-1}(A)<+\infty$

In the book Optimal Transport for Applied Mathematicians, the author states that the condition that a measure $\mu \in \mathcal P(\mathbb R^d)$ is absolutely continuous relative to the Lebesgue ...
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1answer
25 views

Prove that $\sup_{(x,y) \in D} \frac{|f(x)-f(y)|}{|x-y|^{\alpha}} < \sup_{(x,y) \in D} (|x-y|^{1-\alpha}(|\log|y-x|| + C) )$

I am working on the following qual prep question. Suppose $f:[0,1] \to \mathbb{R}$ is absolutely continuous and there exists $C>0$ such that $$\int_0^{1}e^{|f'(x)|}dx \leq C$$ Let $D=\{(x,y) \in [0,...
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24 views

Could monotone increasing but not continuous function send sets of measure $0$to measure $0$?

Suppose F is a monotone increasing function on [a,b], which sends sets of measure $0$ to sets of measure $0$. Let $G(x) = x +F(x)$, show that $G$ also sends sets of measure $0$ to measure $0$. It's ...
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21 views

How to prove that the dual of $L^\infty$ is the set of bounded finitely additive measures?

How to prove that the dual of $L^\infty(X,\mu)$ is the space of finitely additive, signed measures on $X$ that are absolutely continuous with respect to $\mu$? I only find that it's a "well known ...
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1answer
34 views

Necessity of probability in $P \ll \mu \iff \forall \varepsilon > 0 \exists \delta > 0 : \forall A, \mu(A) < \delta , P(A) \leq \epsilon$

Given the following proposition : Let $P$ be a probability and $\mu$ a $\sigma$ finite measure. Then $P \ll \mu \iff \forall \hspace{0.1cm} \varepsilon > 0 \hspace{0.1cm} \exists\hspace{0.1cm} \...
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42 views

Is the differential of an absolute continuous curve on a manifold measurable

I am working with an absolute continuous curve $c:[a,b]\rightarrow M$ on a smooth manifold $M$. The manifold is equipped with a metric $d$ and the curve is absolute continuous with respect to $d$. For ...
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1answer
42 views

Distribution function that is not absolutely continuous

Find, with proof, an example of a distribution function $F$ in $\mathbb{R}^d$ with absolutely continuous margins or component functions that is not absolutely continuous. I know that if $F$ is ...
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1answer
77 views

Is Radon-Nikodym derivative with respect to a finite measure real-valued a.e.?

This question comes from this question. The answer therein missed an argument that the Radon-Nikodym derivative is real-valued a.e. Without this, the proof in that answer has flaw because either the ...
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1answer
56 views

Conditions for equivalent definition of absolutely continuous measures

Let $\lambda, \mu$ be two real valued measures in the measurable space $(X, \mathcal{A})$. Bartle's "Elements of Integration" Lemma 8.8 says that, if both are finite measures, then $\lambda \...
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30 views

Finiteness in the Proof of Lemma on Absolute Continuity of Measures.

To prove that absolute continuity implies $\epsilon-\delta$ criterion, it is said that finiteness is required. The proof involves construction of sequences of measurable sets, where their upper bound ...
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1answer
51 views

Justifying the change of variables formula $\int_{g(a)}^{g(b)} f(y)dy = \int_a^b f(g(x))g'(x)dx$ for Lebesgue Integration

This is a problem from Royden & Fitzpatrick 4th ed, page 129 problem 59. I am struggling proving it and was wondering if someone can help prove it please? Thank you For a nonnegative integrable ...
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149 views

Continuity of the Random Variable Defining the Occupation Measure of Gaussian Process

Suppose $Z:\Omega \times [0,1] \to \mathbb{R}$ is a continuous Gaussian process with mean $\mu(t)$ and covariance kernel $C(t,s)$. Consider the random variable $$ X_\alpha = \lambda( \{t \; : \; Z(t) &...
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1answer
91 views

Proving $f(x)$ is absolutely continuous on $[a,b].$

Can someone please help me prove the following? I am having difficulties proving it. Let $f_n(x) (n=1,2,\cdots)$ be increasing absolutely continuous functions on $[a,b].$ Assume $f(x) = \sum_{n=1}^\...
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1answer
42 views

Does weak convergence preserve absolute continuity when $ F_n(B) \leq M \lambda(B)$ for all measurable sets $B$?

Suppose $\{F_n\}$ is a sequence of probability measures on a compact set $X$ such that there exists a constant $M>0$ that for all $n$ and for any measurable sets $B$, we have $$ F_n(B) \leq M \...
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14 views

why the resulting distribution is absolutely continuous?

I just read a book whose author proposes a function: $ Y = Q(X, \eta, U)$, where $X, \eta$ and $U$ are random variables. Then he says that conditional on $(X,\eta)$, if $Y$ is strictly increasing in $...
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71 views

Absolute Continuity of $(X+Y, Z)$ where $X \perp Y, Z$

Let $X$ be an absolutely continuous $n_1$-dimensional random variable which is independent from $Y, Z$ where are $n_1$- and $n_2$-dimensional random variables, respectively. Further, assume $Z$ is an ...
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1answer
82 views

Show that $\lambda$ is absolutely continuous w.r.t. the Lebesgue measure $\mu$

Let $F(x)$ be nondecreasing and absolutely continuous function on $[0,1]$ with $F(0)=0$ and $F(1)=1$. Let $\lambda$ be the measure on the Borel $\sigma$-field $\mathcal{B}$ s.t. $\lambda([a,b])=F(b)-F(...
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23 views

How are usually denoted functions whose derivative is absolutely continuous?

Given a real interval $I$, are there standard name and notation for the vector space $\{f\in C^1(I)\,:\,f'\,\mathrm{is}\,\mathrm{absolutely}\,\mathrm{continuous}\}$? I would guess $C^1_{a.c.}$ but I'm ...
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28 views

Are non-increasing absolutely continuous functions dense in non-increasing, $L^1([a,b])$ functions?

I'd like to know whether the set of non-increasing absolutely continuous function is dense in the space of non-increasing functions in $L^1([a,b])$, equipped with the metric $d(f,g)=\int_{[a,b]}\vert ...
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40 views

If $\nu \ll \mu_1 \otimes \mu_2$, for $d\nu(x,y) = \nu_1(x) \nu^x_2(y)$, is it true that $v_1 \ll \mu_1$ and $\nu_2^x \ll \mu_2$?

I'm trying to prove the chain rule for relative entropy using measure theory, and the following problem showed up. Assume that $\mathcal X_1, \mathcal X_2$ are bosh polish. Let $\mu_1:\mathcal X_1 \...
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34 views

absolute continuity to differentiability a.e.

I'm working with the epsilon/delta formulatin of absolute continuity: A function $f:[a,b]\rightarrow R^{n}$ is called absolute continous, if for all $\epsilon>0$ there is $\delta>0$, such that ...
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1answer
50 views

A question about absolutely continuous functions

I know that the definition of absolute continuity is for a real-valued function $f$, which is defined on $[a,b]$, if $\forall\epsilon>0$, $\exists\delta>0$, such that whenever $\{(a_k, b_k)\}_{k=...
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80 views

Does there exist an absolutely continuous probability measure on every measure space?

Let $(\Omega,\mathcal F,\mu)$ be an arbitrary measure space, where $\mu$ is non-zero but does not need to be $\sigma$-finite or semi-finite. Does there necessarily exist a probability measure $P$ on $(...
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82 views

Continuous function is absolutely continuous on [0,1] if it is increasing

Let $f$ be a continuous function on $[0,1]$ that is absolutely continuous on $[\epsilon, 1]$ for $\epsilon \in (0,1)$. a. Show that $f$ may not be absolutely continuous on [0,1] b. Show that $f$ is ...
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2answers
43 views

$m^*(f(E))\leq\int_E|g'(x)|dx$ for absolutely continuous function $f$

Suppose $f$ is an absolutely continuous function on $[0,1]$, and suppose $E\subset (0,1)$ is any measurable set. I'd like to show that $m^*(f(E))\leq\int_E|f'(x)|dx$. I know that since $f$ is AC on $[...
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1answer
80 views

Absolutely continuous functions that fix zero and satisfies $f'(x)=2f(x)$

A past question from a qualifying exam at my university reads: Let $f$ be a continuous real-valued function on the real line that is differentiable almost everywhere with respect to the Lebesgue ...
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2answers
73 views

Absolutely continuous on every closed interval iff $\int_{\mathbb{R}}f'(t)dt = 1$

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a non-negative increasing function with $$\lim_{t\to-\infty}f(t) = 0, \lim_{t\to\infty}f(t) = 1$$ Prove that $f$ is absolutely continuous on every closed ...
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1answer
23 views

Proving absolute continuity of the laplace transform

Suppose $f \in L^\infty(\mathbb{R})$ and define the laplace transform $F:(0,\infty)\rightarrow \mathbb{R}$ by $$F(s) = \int_0^\infty f(t)e^{-st}dt.$$ Prove that $F$ is absolutely continuous on $[a,b]$ ...
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1answer
63 views

Approximation of Dirac measure by absolutely continuous measures (w.r.t. Lebesgue measure)

Let $\mathcal{P}$ be the set of probability measures on the Borel $\sigma$ Algebra on the unit circle $S^1$. Let $a\in S^1$ and $\delta_a$ the Diracmeasure in $a$. I want to find a sequence $(\mu_n)_{...
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38 views

$f$ absolutely continuous and $f'\in L^3[0,1]$, which values of $\alpha $ does $\lim_{x\rightarrow 0^+} x^{-\alpha}f(x)=0$?

I'm working my way through some old analysis quals at my university and I came across this question. Let $f$ be absolutely continuous on $[0,1]$ with $f(0)=0$ and $f'\in L^3([0,1])$. For which values ...
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3answers
86 views

If $f$ is absolutely continuous and monotonic on a compact interval, then the flat part of $f$ consists of at most a countable number of segments.

I have a question that is seemingly somewhat similar to Froda's theorem. Suppose $f : [0,1] \rightarrow [0,1]$ is absolutely continuous and monotonic with $f(0)=0$ & $f(1)=1$, and let $\Omega$ be ...
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3answers
42 views

Absolutely continuous on $[-1,1]$ of a function

Show that $f(x)=x^2 \cos\left(\dfrac{\pi}{2x}\right)$ when $x\neq 0$, and $0$ when $x=0$, is absolutely continuous on $[-1,1]$. I'm honestly not sure how to get this one off the ground. I thought ...
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15 views

How would one state the negation of the absolute continuity definition

The definition of absolute continuity I'm using is the following (taken from Royden); $f:[a,b] \rightarrow \mathbb{R}$ is absolutely continuous on $[a,b]$ provided that for all $\epsilon > 0$, ...

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