# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $$f'(x) = \frac{\sin(\sqrt{x})}{2\sqrt{x}} + \... • 1,765 1 vote 2 answers 197 views ### Cantor curve is not absolutely continuous Definition : A curve \omega : [0,1]\to X is defined absolutely continuous whenever there exists g\in L^1([0,1]) such that d(\omega(t_0),\omega(t_1))\le\int_{t_0}^{t_1}g(s)ds for every t_0<... • 1,006 1 vote 0 answers 38 views ### Bounding the total variation of an Absolute Continuous function Theorem: Suppose f \in AC([a,b]). Prove that$$ V_a^b(f) \leq ||(f')^+||_{L^1([a,b])} + ||(f')^-||_{L^1([a,b])} $$Proof (my attempt): If f \in AC([a,b]) then calculus formula holds and we can ... • 58 0 votes 1 answer 88 views ### Proving absolute continuity/uniform continuity of -\log^{-1}(x) My question is how to prove that the function$$ 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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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 \$...