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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Upper right Dini derivative and indefinite integral

I don't understand how the proof of the theorem below works. (Theorem 13.26, Real Analysis, J. Yeh, 2nd ed.) "Let $f$ be a real-valued continuous function on [a,b] such that $f'$ exists almost ...
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Density of the restriction of an absolutely continuous measure

Suppose we have a probability space $(X,\mathcal{M},\mu)$ and an absolutely continuous probability measure $\nu$ with respect $\mu$ (in symbols $\nu\ll\mu$ with density $\rho$). If I consider the ...
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Definition of local absolute continuity

I am studying the notion of absolute continuity. I understood the definition of absolute continuity. I found many references where they use the notion of "local absolute continuity", but I ...
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If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?

If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$? If is not true in general, please give some counter-...
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How can I show that the Sobolev space $W_p^1 (a,b)$ is closed?

We define the Sobolev space $W_p^1 (a,b)$ as: $$W_p^1 (a,b) = \left\{ u \in L_p (a,b) \ : \ u \in AC\left[a, b\right], \ u' \in L_p \left[a, b\right] \right\}$$ where $AC\left[a,b\right]$ means "...
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If $X$ is an absolutely continuous random variable, and $g$ is Borel measurable then when is $g\circ X$ absolutely continuous

We often encounter absolutely continuous RV's with PDF and are interested in the PDF of a function of the RV. At this point, textbooks usually provide the distribution function technique, which is all ...
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Applying a time change to force a positive semidefinite function to be absolutely continuous

I have a question while reading the following post : https://almostsuremath.com/2010/06/16/continuous-processes-with-independent-increments/ Let $\Sigma:\mathbb{R}_+ \to \mathbb{R}^{d^2}$ be a ...
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Link between Riesz p-variation and absolute continuity

Fix $p\in(1,+\infty)$ and for $f:[0,1]\to\mathbb{R}$ define $$ V_p(f)=\sup \sum_{i=0}^{n-1}\frac{|f(t_{i+1})-f(t_i)|^p}{(t_{i+1}-t_i)^{p-1}} $$ where the sup is taken over all $n\in\mathbb{N}$ and ...
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An increasing absolutely continuous function s.t. $f'(x)=\infty$ in $A$

The exercise 14.13 of the Bass' Analysis Real for Graduate Students asks for an increasing absolutely continuous function from $[0,1]$ to $\mathbb{R},$ s.t. $\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}=\infty$...
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A continuous bounded variation function in $[0,1]$ that is absolutely continuous in $(a,1]$, but is not in $[0,1]$

I am seeking for a continuous of bounded variation function in $[0,1]$ that is absolutely continuous in $(a,1]$ for all $a\in(0,1)$, but is not in $[0,1]$. The function $x\sin\left(\frac{1}{x}\right)$,...
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Is $f(x)=x\sin(1/x)$ with $f(0)=0$ of bounded variation on $[0,1]$? - Problem with abs. continuous

I am having the following trouble: From Is $f(x)=x\sin(\frac{1}{x})$ with $f(0)=0$ of bounded variation on $[0,1]$?, $x\sin(1/x)$ has not bounded variation in $[0,1]$. $x\sin(1/x)$ has derivative $-\...
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Are these functions Absolute Continuous? (continuous nowhere-differentiable functions)

I am trying to understand absolute continuity and continuous nowhere-differentiable functions, and since are more than one criteria for continuity I am a bit lost. It is the function $q(x) = \frac{x}{...
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Why is the maximal domain $D_\text{max}$ of a Sturm-Liouville operator defined the way it is?

Given a Sturm-Liouville type operator which acts on functions on the interval $(a,b)$ $$T:= \frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x) \frac{d}{dx}\right]+q(x)\right)$$ where $w$, $p^{-1}$ and $q \in ...
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Uniform continuity of f given conditions on its derivative

Let $f:[0,\infty)\to[0,\infty)$ be a differentiable function, and say $\lim\limits_{x\to\infty}f(x)+f'(x)$ exists. Prove that $f$ is uniform continuous in $[0,\infty)$. I want to use the fact that if $...
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6 votes
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Must a monotone, differentiable function $[a, b] \rightarrow \mathbb{R}$ be absolutely continuous?

If we assume that the derivative is continuous the answer is clearly yes. I suspect however that now the answer is no. This question arose when I wanted to investigate for which non-decreasing, right-...
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If $f$ is continuous and real valued and $f'$ integrable on $[a, b],$ and $\int_a^b f' = f(b) - f(a)$, must $f$ be absolutely continuous

Here is a question I had as I read about absolutely continuous functions. If $f:[a,b] \rightarrow \mathbb{R}$ is continuous and real valued, $f'$ integrable on $[a, b],$ and $\int_a^b f' = f(b) - f(a)$...
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For probability measures $P,Q$ with $Q\ll P$, does it hold that $Q[A]=1\Rightarrow P[A]=1$?

We work on a probability space $(\Omega,\mathcal{F},P)$ and suppose that $Q$ is a probability measure s.t. $Q\ll P$, i.e. for $A\in\mathcal{F}$ with $P[A]=0$, also $Q[A]=0$. Suppose $A\in\mathcal{F}$ ...
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1 answer
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Absolute continuity implies the limit goes to zero

If $f=f(x)$ is absolutely continuous on $[0,1]$ such that $f' \in L^2([0,1])$ and $f(0)=0$. Prove then that $$\lim_{x \rightarrow 0^+}\frac{f(x)}{x^{\frac{1}{2}}}=0$$ The hint says to use Fundamental ...
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When solving this differential equation why should we look for absolutely continuous solutions?

Why does the author here suggest that we look for absolutely continuous curves as solutions to the following differential inclusion (or LESS generally a differential equation)? Here the $\partial F$ ...
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$\int_0^1f(t)\phi'(t)dt=-\int_0^1g(t)\phi(t)dt$, for all smooth $\phi\in[0,1]$ implies $f$ is absolutely continuous and $f'=g$ a.e.

I'm trying to solve the following problem. Let $f,g\in L^1[0,1]$ such that for all $\phi\in C^\infty[0,1]$ with $\phi(0)=\phi(1)$, $$\int_0^1f(t)\phi'(t)dt=-\int_0^1g(t)\phi(t)dt.$$ Show that $f$ is ...
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Quadratic variation and measure change

Let $W_t$ be a Brownian motion defined on probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and assume $X_t$ is a process given by SDE $$ dX_t=W_tdW_t, W_0=0 $$ i.e. $X_t=\int_0^tW_sdW_s$. With ...
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Quantile of the alpha level for an absolute continuous random variable

Absolutely continuous random variable X can take values only in the interval [4,9]. On this segment, the distribution density of the random variable $X$ has the form: $f (x) = C (1 + 7x^{0.5} + 8x^{0....
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Function $f\in L^1(\mathbb{R})$ which is absolutely continous but $f'\notin L^1(\mathbb{R})$

I need to find a function $f$ with the properties required in the title, so $f\in AC(\mathbb{R})\cap L^1(\mathbb{R})$ such that $f'\notin L^1(\mathbb{R})$, where $AC(\mathbb{R})$ denotes absolutely ...
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Composition of absolutely continuous functions with monotonic condition

$f$ is absolutely continuous on $\mathbb{R}$, and $g$ is absolutely continuous on $[a,b]$ and strictly monotone. Show $f \circ g$ is absolutely continuous. I'm using the $\epsilon-\delta$ definition ...
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Absolute continuity of a measure in the intervals implies absolute continuity of borel measures?

Suppose I have two positive Borel measures on $[0,\infty]$, say $\mu$ and $\nu$ and I know that $\nu([0,x]) = 0$ implies $\mu([0,x]) = 0$ for any $x$ and that $\nu((a,b]) = 0$ implies $\mu((a,b]) = 0$ ...
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Are differentiable functions of absolutely continuous random variables themselves absolutely continuous random variables

Let assume $n$ absolutely continuous random variables $X_1, \dots X_n$, let $g: \mathbb{R}^n \to \mathbb{R}; (x_1,\dots,x_n) \mapsto g(x_1,\dots,x_n)$ be a differentiable. I am wondering whether the ...
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Is the set $\{ \int_0^x f\,\mathrm d\lambda\mid f(x)=0\}$ a Lebesgue-null set for $f\geq0$?

We are given a non-negative, integrable function $f: [0, H]\rightarrow \mathbb R_{\geq0}$ and we define the absolutely continuous and non-decreasing function $F(x):=\int_0^x f\,\mathrm d\lambda$ where ...
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Is independence preserved under change of measure?

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $P,Q$ be two probability measures on $(\Omega, \mathcal{F})$ s.t. $Q<<P$, i.e. $Q$ is absolutely continuous w.r.t. $P$. Furthermore, let $X,...
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Is the following application of Ito's formula correct?

Suppose that $|\Lambda_t|$ is an $\mathbb{R}^d$-stochastic process with differential of $|\Lambda_t|^2$ being \begin{align*} d |\Lambda_t|^2 & = 2 e^{\delta t } \langle G(U_t), \Lambda_t\...
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How to prove the limit of minimizing sequence of measures is again absolutely continuous(w.r.t. Lebesgue) in the minimizing movement scheme?

I am considering the minimizing movement scheme related to the gradient of entropy functional in 2-Wasserstein space. The problem is to minimize the following functional for each fixed $\eta$ which is ...
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1 answer
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Prove Absolutely Continuous Function is L1 Differentiable

A function $f \in L_1(\mathbb{R})$ is said to be absolutely continuous if there exists $g \in L_1(\mathbb{R})$ such that: $$f(b)-f(a) = \int_a^b g(t) dt \tag{1}$$ for all $a, b \in \mathbb{R}$. The ...
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Necessary conditions for $f$ such that $f(X)$ has an absolutely continuous distribution when $X$ has one

I was wondering if anyone could give me a reference for necessary conditions for a function $f$ such that $f(X)$ has an absolutely continuous distribution when $X$ has one.
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Jordan decomposition of differentiable functions

Let $f:[a,b]\to \mathbb{R}$ be a function of bounded variation. A well known result, known as Jordan's decomposition, states that we can write $f=W_1-W_2$ where $W_1,W_2$ are increasing and minimal in ...
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$F*f$ is absolutely continuous in $\mathbb{R}$

Let $F:\mathbb{R}\rightarrow [0, 1]$ be a increasing function in $\mathbb{R}$ hat is strictly increasing in $[0, \infty[$ such that $F(0)=0$. Suppose that $F$ is absolutely continuous in $\mathbb{R}$ ...
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If $\hat{F}$ - estimated empirical CDF, then what does $(\hat{F}) \ll F_{empirical}$ mean? (Absolutely continuous on $F_{emp}$)

Here is the part of the definition I don't get: Given $X^n$, let $\hat{F}$ be a CDF that is absolutely continuous on $F_e \ \ (\hat{F}\ll F_e)$. What does it mean?
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$g$ is of bounded variation but is not absolutely continuous.

Let $\chi_{(0,t]}$ denote the characteristic function of $(0, t]$ and $X=\operatorname{span}\left\{\chi_{(0,t]}: t \in(0,1]\right\}$ with sup norm. If $f=k_{1} c_{t_{1}}+\cdots+k_{n} c_{t_{n}}$, $0<...
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If $f$ is absolutely continuous then $PV(f,[a,b])=\int_{[a,b]}(f')^+$

Let $f:[a,b]\to\Bbb R$. Define $PV(f,[a,b])$ to be the positive variation of $f$ on $[a,b]$. I need to show that $PV(f,[a,b])=\int_{[a,b]}(f')^+$. I've already shown that $TV(f,[a,b])=\int_{[a,b]} |...
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Does an absolutely continuous distribution function imply continuity of conditional expectation?

Let $(\theta_1,\theta_2) \in [0,1]^2$ be a two-dimensional random variable with a absolutely continuous joint distribution $F$ with full support. Can we say $\mathbb{E}(\theta_2|\theta_1)$ is ...
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3 votes
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Integration by parts when the two functions are absolutely continuous and continuously differentiable respectively

I am doing Exercise 1.6.51 in Terence Tao's introduction to measure theory. Let $F : [a, b]\to\mathbb R$ be an absolutely continuous function, and let $\phi: [a, b]\to\mathbb R$ be a continuously ...
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3 votes
1 answer
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Absolute continuity of $g(x)=\int_a^x f(t) dt.$

Let $f$ be a nonnegative and Lebesgue measurable function. And difine $g(x):=\displaystyle\int_a^x f(t) dt \ (a\leqq x)$. Then, prove that $g$ is absolutely continuous. Let $\epsilon>0.$ I have to ...
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Define the following group action in terms of matrix representation and show its differentiable

I don't have much idea about groups acting on sets and so I have been having trouble trying to approach such kind of questions. If i have group $G$ = $GL$($V$) and the space $S$ of symmetric bilinear ...
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1 answer
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To show absolute continuity of measures.

Let $\delta_x$ denote the measure defined by $$\delta_x(E)=\begin{cases} 1, & x\in E \\ 0, & x\not\in E \\ \end{cases} $$ Let $\mu:=m+\delta_0+\delta_1$, where $m$ denotes the ...
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4 votes
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70 views

Showing that $\sum_{n=1}^\infty e^{in^2x}/{n^3}$ is absolutely continuous

This paper claims that the function $\sum_{n=1}^\infty \frac{e^{2\pi in^kx}}{n^\alpha}$ is absolutely continuous if $\alpha > k+\frac{1}{2}$. Whilst it makes that claim, it never further shows/...
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8 votes
1 answer
303 views

Why is it necessary for density functions to be absolutely continuous with respect to a measure in order for the cross entropy to be defined?

In the Wikipedia page describing cross entropy, the following expression is written down to denote the cross entropy $H$ between two densities $p(x)$ and $q(x)$: $H(p,q) = - \int_\mathcal{X}p(x)\log q(...
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Integrand of convergent integral - absolute continuity is necessary?

I think the following proposition is true (proof offered below) Proposition. Let function $f(x)$ be real-valued and nonnegative on $(a,\infty)$, and suppose that for $r > -1$ the integral $\int_a^...
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2 votes
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48 views

Showing that $f(x) = \sqrt{x}\sin(\sqrt{x})$ is absolutely continuous on $[0,1]$

This is an old qualifying exam problem, and I'm wondering if my proof is correct. (I'm still not too comfortable with absolute continuity.) We know that $$f'(x) = \frac{\sin(\sqrt{x})}{2\sqrt{x}} + \...
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1 vote
2 answers
197 views

Cantor curve is not absolutely continuous

Definition : A curve $\omega : [0,1]\to X$ is defined absolutely continuous whenever there exists $g\in L^1([0,1])$ such that $d(\omega(t_0),\omega(t_1))\le\int_{t_0}^{t_1}g(s)ds$ for every $t_0<...
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Bounding the total variation of an Absolute Continuous function

Theorem: Suppose $f \in AC([a,b])$. Prove that $$ V_a^b(f) \leq ||(f')^+||_{L^1([a,b])} + ||(f')^-||_{L^1([a,b])} $$ Proof (my attempt): If $f \in AC([a,b])$ then calculus formula holds and we can ...
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Proving absolute continuity/uniform continuity of $-\log^{-1}(x)$

My question is how to prove that the function $$ f(x)=-\log^{-1}(x) $$ for $x>0$ and $f(0)=0$ on $[0,1/2]$ is absolutely continuous/uniform continuous. I feel like both proofs would be similar, but ...
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Can we prove almost everywhere absolute continuity?

This problem arises in my research in the field of information theory, any reference to some similar work, any comment about style or imprecisions are most welcome. Let $\mathcal X$ be some space and $...
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