Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
1answer
23 views

Absolute continuity of convex function

Let $f:(a,b) \to \mathbb{R}$ be a convex function. A convex function is absolutely continuous on each closed subinterval of $(a,b)$. Also, the right and left-hand derivatives exist at each point of $(...
2
votes
1answer
39 views

Taylor's Theorem Remainder with Unbounded Derivative

According to the Wikipedia entry and a few I've seen online, the remainder form with a $(n+1) \text{th}$ derivative can be used as long as $f: \mathbb R \to \mathbb R$, is $n+1$ times differentiable ...
0
votes
0answers
18 views

Name for functions with absolutely continuous derivatives

Does the set of functions which as absolutely continuous derivatives have a name? If $f'\in AC(I,\mathbb R^d)$, is there a notation for the space containing $f$?
0
votes
1answer
46 views

Absolute continuity, implies bounded variation.

I am trying to directly prove from the definition of absolute continuity, that if a function $f:[a,b]\rightarrow\mathbb{R}$ is absolutely continuous, then it is of bounded variation, i.e $Vf$ is ...
0
votes
1answer
51 views

Is there an invariant measure absolutely continuous wrt to the lebesgue measure for the map f

Let $f:[0,1]\rightarrow[0,1]$ where $f(x)=x/2$ $(1-x)$, and let $\lambda$ be the lebesgue measure on [0,1]. Is there a probability measure $\mu$ that is invariant and absolutely continuous wrt to the ...
0
votes
1answer
35 views

Is a Cauchy distribution absolutely continuous with respect to a Gaussian distribution?

This stack exchange answer showed that Kullback-Leibler divergence between a Cauchy distribution and a Gaussian distribution is infinite. Formally, $$KL(P||Q)=\infty$$, where $P$ is a Cauchy ...
0
votes
1answer
20 views

Why finite condition is needed in absolute continuity for a function?

I heard that absolute continuity for measure $\nu$ w.r.t. to $\mu$ where $\nu$ is a signed measure and $\mu$ is a positive measure, is as follows. If $\nu (E) = 0 $ for every $ E \in \mathcal M $ for ...
0
votes
1answer
38 views

Difference between absolutely continuous measures

Suppose we are given two probability measures $\mathbb{G}^{H}$ and $\mathbb{G}^{L}$ with same support $\text{supp}(\mathbb{G})$. Suppose as well that there exists an integrable function $\gamma$ ...
0
votes
0answers
39 views

Epsilon-delta definition of continuity in two dimensions

Usually the continuity of multi-variate function is defined through something like "$\delta$-disk." Can we define such continuity with the similar one-dimension definition? As follow: $f:\mathbb R^2\...
2
votes
2answers
147 views

If $f$ is integrable in $\mathbb R$, Is it true $F$ is Absolutely continuous?

If $f$ is integrable in $\mathbb R$, define $$ F(x)=\int_{-\infty }^xf(t) \, \mathrm dt $$ Is it true that $F$ is Absolutely continuous in $\mathbb R$?
3
votes
1answer
145 views

Equivalence condition of Absolute Continuity

Let me restate my point. The intuition behind this construction is straightforward. Let $f$ be a continuous increasing function. Take out pieces of the graph of $f$ which is correspond to a collection ...
0
votes
0answers
23 views

Absolutely continuous measure and equality to $0$

In the context of proving that the data processing inequality for $f$-divergences hold for any Markov kernel I am interested in the following statement If $\mu$ and $\nu$ are two probability ...
0
votes
1answer
32 views

Absolutely continuous Banach space valued function

Let $X$ be a Banach space and $F:[a,b] \to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f \in L^1([a,b],X)$, is the ...
0
votes
1answer
42 views

When is a random variable $\mathit{X}:\Omega\rightarrow\mathbb{R}^n$ absolutely continuous?

Given a probability space $\Gamma:=\langle\Omega,\Sigma,P\rangle$ and a random variable $\mathit{X}:\Omega\rightarrow\mathbb{R}^n$, I do not fully understand the definition of absolute continuity of $\...
3
votes
1answer
36 views

Absolutely continuous function not differentiable on uncountable set

From real analysis one knows that an absolutely continuous function is differentiable a.e.. Is there a function showing that this statement cannot be made into "every AC function is differentiable ...
0
votes
0answers
12 views

Can a continuous time Markov processes have transition densities only at certain times?

Suppose that $X=(X_t)_{t\ge 0}$ is a Markov process on some state space $E\subset\Bbb R^n$ with transition semi-group $(P_t)_{t\ge 0}$. It is often nice to know, whether the transition measures have ...
0
votes
0answers
29 views

What are sufficient conditions for the boundedness of a Radon-Nikodym derivative of a pull back measure?

Framework. Let $(X,\mu)$, and $(Y,\nu)$ be probability spaces on compact Hausdorff sets. Let $T:X\longrightarrow Y$ be a measurable function. Assume $\fbox{$T^{\ast}\mu\ll\nu$}$, where $T^{\ast}\mu$ ...
0
votes
1answer
41 views

Is composition of an absolutely continuous function and a $C^1$ function still absolutely continuous?

Suppose $y:[a,b]\to \mathbb{R}^n$ is absolutely continuous, where $[a,b]\subset \mathbb{R}$ is a compact interval. Let $\phi:\mathbb{R}^n\to \mathbb{R}$ be a $C^1$ function. Is it true that $$\phi \...
1
vote
1answer
84 views

Why do we need absolute continuity of $\langle M \rangle_t(\omega)$ with repect to the Lebesgue measure?

I am trying to understand the proof of proposition 3.2.6 in Stochastic Calculus and Brownian Motion by Karatzas and Shreve. For $X$ bounded they use Lemma 3.2.4 in the same book and eventually claim(...
2
votes
1answer
33 views

$f\in L^1$ , $F_h(x) = \frac{1}{2h} \int_{-h}^h f(x-y)d\lambda(y)$ is absolutely continuous

$f\in L^1(\mathbb{R})$ I wish to show that $F_h(x) = \frac{1}{2h} \int_{-h}^h f(x-y)d\lambda(y)$ is absolutely continuous over any close interval $[a,b]$ What I tried so far: $|F_h(d)-F_h(c)| = |\...
0
votes
0answers
29 views

Proof verification on integral function and absolute continuity

I've tried to prove something I was interested in, hope someone could spend a couple of minutes to review it. I'm sorry if it is too long, but I tried to be as clear as possible. Let $f:[0,+\infty)\...
0
votes
1answer
36 views

Is the lebesgue integral of a measurable function continuous?

I was wondering if the lebesgue integral of a measurable function at least continuous? What kind of regularity on the integrand do we need for it to be absolutely continuous so that we can say its ...
0
votes
0answers
42 views

Let $f$ be Lebesgueable integrable, then $\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)| = 0$

I must show that Let $f$ be Lebesgueable integrable, then $$\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)|dy = 0$$ for almost every $x \in \mathbb{R}$. Tentative proof: $$|f(x) - f(y)| \geq ...
1
vote
1answer
49 views

Let $f$ be a continuous monotone function. Show that $f$ must be absolutely continuous on [0,1]

Let $f:[0,1]\to\mathbb{R}$ be a continuous monotone function such that $f$ is differentiable everywhere on (0,1) and $f'(x)$ is continuous on $(0,1)$. Show that $f$ must be absolutely continuous on $[...
0
votes
0answers
18 views

The function defined by the sum of non-decreasing absolutely continuous functions on $\mathbb{R}$ is continuous and differentiable a.e.

True or False? If $\{f_n\}_{n\ge 0}$ is a sequence of non-decreasing absolutely continuous functions on $\mathbb{R}$, and if $f:=\sum_{n\ge0}f_n(x), x\in\mathbb{R},$ is finite at each $x$, then $f$ is ...
2
votes
2answers
32 views

Rewriting an integral of an absolutely continuous function

Let $(X,M, \mu)$ be a $\sigma$-finite measure space and $f: X \rightarrow [0, \infty)$ be measurable. For each $\alpha \ge 0$ Define $E_\alpha = \{ x \in X : f(x) > \alpha \}$ and $\lambda(\alpha) =...
0
votes
1answer
41 views

The mutual density of $X,Y$ in $\{|t|+|s|<1\}$ is constant, are $X,Y$ independent?

Let $X,Y$ absolutely continuous random variables with density finctions $f_X,f_Y$. Assume that the mutual density $f_{X,Y}$ equals to a constant $c$ in $\{(t,s)\in\mathbb{R}^2:|t|+|s|<1\}$. Are $X,...
0
votes
1answer
34 views

$\operatorname{supp}(f_{X,Y})=\{(x,y)\in\mathbb{R}^2||x|+|y|\leq1\}$ then $X,Y$ are not independent

Let $Z=(X,Y)$ be a absolutely coninuous random variable such that $$ \\\operatorname{supp}(f_Z)=\{(x,y)\in\mathbb{R}^2||x|+|y|\leq1\} \ $$ Show that $X,Y$ are not independent. I don't have a good ...
0
votes
0answers
34 views

Singular continuous

I am trying to construct a probability measure which is absolute continuous, singular constinuous and discrete. How can I do? I have not been able to find any example os such a measure. Might you help ...
0
votes
0answers
31 views

If $\mu_F\ll\beta_1$ and $\mu_F$ is finite then $F$ is absolutely continuous

If $\mu_F\ll\beta_1$ and $\mu_F$ is finite then $F$ is absolutely continuous. Note: here $\beta_1$ is the Lebesgue measure restricted to the Borel $\sigma$-algebra in the real line. Im not sure if ...
1
vote
1answer
140 views

Radon-Nikodym Derivative of a Limit of a Sequence of Measures

Let $\nu$ be a sigma-finite measure, let $\mu$ be a measure absolutely continuous with respect to $\nu$, and let $(\mu_n)$ be a sequence of measures such that $\mu_n(E)\uparrow\mu(E)$ for all ...
1
vote
2answers
88 views

A counterexample to the epsilon-delta criterion for Absolute Continuity of Measures

Let $p>0$, and let $\mu$ be a Borel measure on $[0,\infty)$ defined by $\mu(E)=\int_Ex^pd\lambda$ where $\lambda$ denotes Lebesgue measure. Show that $\mu$ is absolutely continuous with respect to ...
2
votes
0answers
126 views

Absolute continuity of increasing functions on an interval

I have been using Real Analysis by Royden and Fitzpatrick. I am currently stuck on problem 39 of chapter 6. The problem is stated as follows: "Use the preceding problem to show that if f is ...
1
vote
0answers
50 views

Measure theory: on absolutely continuous measures

Let F be a random variable defined on a probability space $(\Omega,\mathcal F, \mathbb P)$ with values on $(E,\mathcal E, \lambda)$, where $E=(-1,1)$ and $\lambda$ is the Lebesgue measure. To prove ...
1
vote
0answers
74 views

Chapter 3, Problem 3.5 of Stein's Real Analysis

The problem states the following. Suppose that $F$ is continuous on $[a,b]$, $F'(x)$ exists for every $x\in(a,b)$, and $F'(x)$ is integrable. Then $F$ is absolutely continuous and $$F(b)-F(a)=\int_a^...
-3
votes
1answer
43 views

Absolute continuous functions property. Is it true? [closed]

Let $f,g:[0,T]\to [0,\infty)$ be two absolute continuous functions such that: $f(0)=g(0)>0$. We know that there is a sequence $(x_n)_n$ converging to $0$ such that: $$f(x_n)\neq g(x_n)$$ Can we ...
2
votes
0answers
50 views

Why is this measure absolutely continuous?

Let $U \subset \mathbb R^3$ be an open, bounded and connected set with a $C^2-$regular boundary $\Gamma$. If $f \in \mathcal M_{+}(\Gamma)$ i.e belongs to the space of non negative, Radon ...
0
votes
1answer
54 views

How to prove that $\lambda$ is not absolutely continuous with respect to $\mu$?

Let $\lambda$ counting measure on $\mathbb{N}$, and let $\mu (E) = \sum_{n\in E}2^{-n}$ defined for $E \subset\mathbb{N}$. How to prove that $\lambda$ is not absolutely continuous with respect to $\...
1
vote
1answer
24 views

Why the Lemma below is true even if the measure $\mu$ is infinite?

A Bartle exercise proposes to show that lemma 8.8 is true even if the measure $\mu$ is infinite. What I can not see is any difference between the proof of lemma 8.8 and the solution for exercise. Can ...
4
votes
1answer
56 views

$T_{f}(a,x)$ is absolutely continuous in $[a,b]$ whenever $f$ is absolutely continuous $[a,b]$. (Proof verification)

Let $$T_{f}(a,x) = \sup \sum_{j=1}^{n}|f(t_{j}) - f(t_{j-1})|$$ be the total variation of $f$ on $[a,x]$. I want show that: $T_{f}(a,x)$ is absolutely continuous in $[a,b]$ whenever $f$ is ...
1
vote
0answers
44 views

Regularity in the trapezoidal rule

I tried to understand if the following sentence is true of false. Let $u$ be an absolutely continuous function defined in $[0,2\pi]$ such that $u(0)=u(2\pi)$. Let $n\in \mathbb{N}, h=\frac{2\pi}{n}$ ...
1
vote
1answer
105 views

Absolutely continuous and differentiable almost everywhere

I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it: "A strongly absolutely continuous function which is differentiable almost everywhere is ...
1
vote
1answer
64 views

A question about absolute continuity

This is an exercise from Royden, Flitzpatrick's Real Analysis (4th edition), chapter 6, Q.N. 39. Let $f:[a,b]\to \mathbb R$ be an increasing function. I have to show that $f$ is absolutely continuous ...
2
votes
2answers
110 views

$\{f_n\}$ absolutely continuous converging pointwise to $f$. Is $f$ of bounded variation and/or absolutely continuous?

I'm trying to solve the following problem: $\{f_n\}:[a,b] \to \mathbb{R}$ sequence of absolutely continuous functions converging pointwise to $f: [a,b] \to \mathbb{R}$. Say if $f$ is of bounded ...
0
votes
0answers
35 views

Composition of absolutely continuous function

if I have a function $x:[0,\infty)\rightarrow H$ (H is an hilbert space) that is absolutely continuous (AC), why should the function $\|x(t)-y\|^2$ be AC, for $y\in H$? Yet again, why should $\|x_{\...
1
vote
1answer
54 views

Hölder continuity of integral function

Let $[a,b] \subset \mathbb{R}$ and $p \in [1, \infty)$. Consider $f \in L^{p}([a,b])$ with respect to the Lebesgue measure and set $F(x) = \int_{a}^{x} f(t) dt $ $\ \ \forall x \in [a,b].$ Prove that ...
0
votes
1answer
49 views

differentiability of variation of absolutely continuous function

Let $X$ be a reflexive and separable Banach space. Assume that $f\colon [0,b]\to X$ is absolutely continuous, so it has bounded variation $$V(f)(b)=\sup \{V(f,P)\,|\, P\quad\text{is partition of}\quad ...
3
votes
2answers
75 views

Does absolute convergence imply integrability?

For a measurable function $f$ on $[1,+\infty)$, which is bounded on bounded sets, define $a_n=\int\limits_n^{n+1}f dm$ for each natural number $n$. Is is true that $ f $ is integrable over $[1,+\infty)...
5
votes
1answer
92 views

$x \sin({1 \over x})$ can be decomposed?

Let $f(x)=x\sin(1/x)$ for $0<x\le 1$, and $f(0)=0$. I was told that every continuous function on $[0,1]$ can be written as $a(x)+s(x)$, where $a(x)$ is absolute continuous and $s(x)$ is singular. ...
1
vote
0answers
160 views

If total variation function for f of bounded variation is absolutely continuous, then f is absolutely continuous

Suppose that $f$ is a function of bounded variation on [0, 1], and let $V(x)$ be the total variation function for $f$, i.e., for any $x \in [0, 1]$, $V(x)$ is the total variation of $f$ on the ...