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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Characterization of Lipschitz continuity

A function $f$ satisfies the Lipschitz condition on $[a,b]$ iff for all $\epsilon >0$ there exists $\delta>0$ for which the following is true: For all families $\{[a_k,b_k]\}_{k=1}^n$ of closed ...
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Does the product rule always hold in one dimensions? [duplicate]

Let $f \in W^{1,p}\bigl([0,1], \mathbb{R} \bigr)$ with $p \in (1,\infty)$. Then, we can assume that $f$ is absolutely continuous with the classical derivative a.e. equal to the weak derivative, which ...
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uniqueness of minimum of absolutely continuous stochastic process

I am trying to understand the arguments in the suggested proof of Remark 1.2 in this paper https://link.springer.com/article/10.1007/BF00536298 In essence, the authors consider the process $X(t)=B(t)+\...
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If f is monotone on a closed interval then it is absolutely continuous

How do I prove that if a function f is monotone on a closed interval then it is absolutely continuous, I know that if it is monotone that it has bounded variation, but that doesn't imply that $$\sum_{...
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Does continuously differentiable in the interior imply absolute continuity?

Suppose $f: [0, 1] \rightarrow \mathbb{R}$ is continuous, and is continuously differentiable on $(0, 1)$. Can we conclude that $f$ is absolutely continuous over $[0, 1]$? On the one hand, we clearly ...
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Determine bounded variation and absolute continuity for different parameters

Determine for which parameters $\alpha, \beta \in [0, +\infty[$ the function $$f_{\alpha,\beta}:[0,1] \to \mathbb{R}, \quad f_{\alpha, \beta}(x) := \begin{cases} x^{\alpha}\ \text{sin}(\frac{1}{x^{\...
MathGeek's user avatar
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Absolute continuity of the push forward of the Haar measure

Let $K$ be a compact group with Haar measure $m$. Let $K^2 = \{k^2 : k\in K\}$ and suppose that $K = K^2 \cup wK^2$ for some $w\in K$. Let $\mu$ be the push-forward of $m$ under the map $k\mapsto k^2$....
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Prove $\nu$ is absolutely continuous to Lebesgue measure if and only if $f$ is absolutely continuous.

Let $\nu$ be a finite Borel measure on $[0,1]$. Define $f : [0,1] \to \mathbb R$ by $f(x) = \nu ([0,x))$. Prove $\nu$ is absolutely continuous to Lebesgue measure (\mu) if and only if $f$ is ...
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How to prove that an absolutely continuous function space on a finite closed interval is a separable space in this norm sense?

$\quad$ Let a finite closed interval be $[a,b]$. Consider the absolutely continuous function space $\mathrm{AC}([a,b])$ on it, with a norm as follows: $$\|f\|_{\mathrm{AC}}=\sup_{x\in[a,b]}|f(x)|+\|f'\...
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Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?

The whole theorem goes as follows: Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying: $$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
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Characterization of absolute continuity for finite additive functions

Let $\nu:\mathbb{X}\longrightarrow \mathbb{R}$ and additive function, where $(X,\mathbb{X},\mu)$ is a measure space. That is, $\nu(\emptyset)=0$, and for any finite disjoint family of measurable sets $...
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Convergence of the derivatives implies convergence of the original functions on $1$-dimension.

Let $f_n, f \in W^{1,p}([0,1])$ for some fixed $p \in (1,\infty)$ and all $n \in \mathbb{N}$. By modification on null sets, we may then assume that $f_n, f \in C[0,1]$. Now, if $f_n(0)=f(0)$ for all $...
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How to show that an absolutely continuous increasing real function is an indefinite integral of an L1 function

By an indefinite integral, I don't mean a primitive (even though the concepts are essentially equivalent, an equivalence easily proven in the 'classical' case) because the equivalence in a more ...
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If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integtal and time derivative?

Let $f(x,t) : S^1 \times [0,\infty) \to \mathbb{R}$ be a function such that $f(x,t) \in L^q_t\bigl([0,\infty), L^p_x(S^1)\bigr)$ $\partial_t f(x,t) \in L^{q'}_t\bigl([0,\infty), L^{p'}_x(S^1)\bigr)$. ...
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Stochastic differential equation with Bernoulli random process as a solution

Let us assume that $T$ is an absolutely continuous random variable such that $T>0$ almost surely and $\mathbb{E} e^T < \infty$. Let us define $$ X_t = \mathbb{I}_{ [ t <T ] }, \ t \geq 0 .$$ ...
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Show that $f(g(x))g′(x)$ is measurable where $f$ is bounded and measurable

Let $g$ be absolutely continuous and increasing on $[a,b]$ with $g(a)=A$ and $g(b)=B$. Suppose $f$ is any measurable and bounded function on $[A,B]$. Show that $f(g(x))g'(x)$ is measurable on $[a,b]$. ...
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Absolute continuity of two distributions

Consider a real-valued $Y$ and a Hilbert-space valued $X$ such that $Y|X\sim N(0,1)$. Let $(Y+c_X,X)\sim P_1$ and $(Y,X)\sim P_0$, where $c_X$ is a function of $X$. Show that $P_1\ll P_0$. My attempt: ...
John Smith's user avatar
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Proof about Markov kernels and absolute continuity

Assumptions: $(\mathsf{X}, \mathcal{X})$ is a measurable space. $M_n$ and $L_{n-1}$ are Markov probability kernels for $n=2, \ldots, P$. $\mu_n$ be probability measures on $(\mathsf{X}, \mathcal{X})$ ...
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For an a.e. differentiable continuous function $f$ with $f'$ integrable, is it true in general that $f(b)-f(a) \geq \int_a^b f'(x) dx$?

I am aware that the fundamental theorem of calculus fails for functions which are a.e. differentiable but not absolutely continuous, such as the Cantor function. However, I wonder if a general ...
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Absolute Continuity of Signed Square Root of Gaussian

Suppose $X$ is standard Gaussian. The random variable $Y$ is $sgn(X)\sqrt{|X|}$. Is $Y$ absolutely continuous? The issue is obtaining a density. The problem is $sgn(X)$ since the derivative is ...
Will's user avatar
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Absolutely continous function on an interval?

I suddenly remembered a fact about absolutely continous functions, but I can't remember the proof or the book where I read about this. Could you help me? An absolutely continous function on $(a,b)$ ...
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if $f : [0,1] \to \mathbb{R}$ is differentiable almost everywhere and right-continuous almost everywhere, is it absolutely continuous?

Let $f : [0,1] \to \mathbb{R}$ be a function defined everywhere and differentiable almost everywhere on $[0,1]$. Moreover, the derivative $f'$ is integrable. Assume further that $f$ is right-...
Keith's user avatar
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Absolutely continuous function and distribution theory

Let $f$ and $g$ be a locally integrable function on $[0,1]$. For any $\phi\in C^1[0,1]$, $f$ and $g$ satisfies $\displaystyle \int_{0}^{1}\phi'(t)f(t)dt+\int_{0}^{1}\phi(t)g(t)dt = \phi(1)f(1)-\phi(0)...
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A Holder-continuous function differentiable a.e. is absolutely continuous?

Let $f : [0,1] \to \mathbb{R}$ be a Holder-continuous function of an exponent $\alpha \in (0,1)$ and differentiable a.e. at the same time. Assume further that the derivative $f'$ is integrable on $[0,...
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Let $f$ be a real-valued, everywhere differentiable function on $[0,1]$. Suppose $f \in BV[0,1]$. Prove $f \in AC[0,1]$. [duplicate]

I tried to prove this by showing that the derivative $f'$ must be bounded. But I am not sure this is true, as the Extreme Value Theorem cannot be invoked ($f'$ is not guaranteed to be continuous) nor ...
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Showing that if $\mu$ is an ergodic probability measure and $\nu\ll\mu$ then $\nu(A)=\int_A fd\mu$ with $f$ being constant $\mu$-a.e.

Let $(X,B,\mu)$ be a measure space and $T:X\to X$ be a transformation such that $\mu$ is ergodic w.r.t. $T$. See the Wikipedia article for further details of the ergodicity setting. I am looking to ...
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Extending the notion of absolute continuity and Fundamental theorem of calculus for Lebesgue integral to multivariable cases

In "Real Analysis" (1999) by Folland, p.105-106, it is stated that a function $F(x) : \mathbb{R} \to \mathbb{R}$ defined by \begin{equation} F(x)=\int^x_{-\infty} f(t)dt \end{equation} for ...
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$\int_{F^{-1}(O)}F'(x)dx = m(O)$, $F$ is absolutely continuous

I'm solving the following question. $F$ is absolutely continuous on $[a,b]$ and increasing. Let $A = F(a)$ and $B = F(b)$. Show that $\int_{F^{-1}(O)}F'(x)dx = m(O)$, where $O$ is any open set in $[A,...
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Why components functions absolutely continuous does not imply F_{X} absolutely continuous?

I'm studying the Probability from the book Probability, Alan F. Karr. Proposition 2.29. If X = (X1 , .. . , Xd) is absolutely continuous, then for each i, Xi is absolutely continuous, and $f_{X_i}(x)...
TY FIRE's user avatar
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Conditional probability with absolute continuous random variable

Given a probability space $(\Omega, \mathcal{A}, \mathbb{P})$. Let $Y$ be a random variable taking the values 0 or 1. Let $X$ be some absolutely continuous random variable taking values in say $[0,1]$....
guest1's user avatar
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If $f\in L^{1}([0,1]\times[0,1])$ and $F(x)=\int_{[0,x]\times[0,x]} f$ for$x\in [0,1]$, prove $F$ is absolutely continuous from the definition.

I've seen there was a similar question. This question was only for $L^{1}[0,1]$. I want to extend it to $L^{1}([0,1]\times[0,1])$. The question is: If $f\in L^{1}([0,1]\times[0,1])$ and for $x\in [0,...
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Some additional questions about Lebesgue decomposition

I'm solving exercise 24 of chapter 3 in Stein's Real Analysis, and I've found a good material regarding the exercise in here. Below is the given exercise. In this link, I've got 2 additional ...
jason 1's user avatar
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How can I prove that if $\int_{[0,1]} f~d\nu\leq C\int_{[0,1]} f~d\mu$ then the measures are absolutely continuous?

Let us consider $T:[0,1]\rightarrow [0,1]$ be a continuous map. Let $\mu, \nu$ be $T$ invariant probability measures on $\mathcal{B}([0,1])$. I want to show that if there exists $D>0$ s.t. $\int_{[...
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Are absolutely continuous functions with values in a Hilbert space differentiable almost everywhere?

I have some questions regarding absolutely continuous Hilbert space valued functions defined on the reals. Define the function $$x:[t_0, +\infty) \to \mathcal{H},$$ with $\mathcal{H}$ a real Hilbert ...
AverageJoe's user avatar
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"$x^p \sin(\frac{1}{x^q})$" absolutely continuous / of bounded variation

$f(x)\ =\ x^p \sin(\frac{1}{x^q})\ (f(0)=0)$ (1) For $x$ on $[0,1]$, Find the condition of $p,q>0$ that the function f is absolutely continuous. (2) For $x$ on $[0,1]$, Find the condition of $p, q&...
Hobby's user avatar
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A sufficient condition for convexity: $f$ absolutely continuous on $[a,b]$ and $f'$ increasing a.e. imply $f$ is convex

Problem: I am asked to prove the claim in the title: that if $f$ is absolutely continuous on $[a,b]$, with $f'$ increasing almost everywhere, then $f $ is a convex function. The problem comes with the ...
PrincessEev's user avatar
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Contiguity between Probability Measures

Suppose $P_n=\mathscr{U}([0,1])$ and $Q_n=\mathscr{U}([0,1+\frac{1}{n}])$ on $(\mathbb{R}, \mathscr{B}(\mathbb{R}))$. How can I show if the probability measures are contiguous or not to each other? ...
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Absolute continuity of a $W^{1,1}(\mathbb{R}^d)$-function along smooth curves

My question is related to this one Is a Sobolev function absolutely continuous with respect to a.e.segment of line? and its answer, but to my understanding it has not answered the question below. Let $...
PDEprobabilist's user avatar
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Folland lemma 3.7

I am struggling to understand the following theorem's proof on Folland's Real Analysis, page 89, lemma 3.7. Suppose that $\nu$ and $\mu$ are finite measures on $(X, \mathcal M)$. Either $\nu \perp \...
Squirrel-Power's user avatar
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A question about absolutely continuous measure

Let $(\Omega,\mathcal{F},\mu)$ be a finite measure space. Let $\nu$ be a finite measure such that $\nu << \mu$. My question is that, do we always have a constant $C>0$ such that $\nu(A)\leq C\...
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de la Valleé-Poussin's theorem on composition of absolutely continuous functions

I'm looking for a proof of the following classical result by de la Valleé-Poussin. Let $I=[a,b],\,J=[c,d]$ be two compact intervals. Assume that $u:I\mapsto J$ and $f:J\mapsto\mathbb{R}$ are ...
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How to use $L^2$ Lebesgue density of a measure $\mu$ to conclude that $\mu$ is absolutely continuous w.r.t. the Lebesgue measure?

I am looking for an answer to the question: Why $\hat{\mu}\in L^2(\mathbb{R}^n)$ implies $\mu$ absolutely continuous?. Namely, let $\mu$ be a compactly supported probability measure (or a finite ...
Cartesian Bear's user avatar
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Question regarding Lebesgue decomposition of a finite measure

Let $P$ and $Q$ denote finite measures. Let $\mu$ denote a finite measure that dominates them both, and $p, q$ the Radon-Nikodym derivatives of $P, Q$, respectively, with respect to $\mu$. The ...
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Generalization of absolute continuity to functionals of non-decreasing càdlàg functions

I am wondering if a non-decreasing càdlàg function $x$ can be recovered via "clever" integration when integrating over a general functional. We know that for all absolutely continuous $x\in ...
Florian Brück's user avatar
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Absolutely continuous functions send a set of positive derivative and positive measure to a set of positive measure

In P49 of Ziemer's "Weakly Differentiable Functions", it was used that "by classical considerations", if $u$ is an absolutely continuous function on $\mathbb R$, and let \begin{...
Tian LAN's user avatar
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When is a cdf absolutely continuous?

Let $\mu$ be a probability distribution on $(\mathbb R, \mathcal B(\mathbb R))$ and $X$ be a random variable with distribution $\mu$. According to the theory developed in Section 3.5 of Folland's Real ...
Marlou marlou's user avatar
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Questions about theorem 2.1 and lemma 2.2 in chapter 3 of Stein's Real Analysis

I have a difficulty dealing with some proofs in theorem 2.1 and lemma 2.2 in chapter 3 of Stein's Real Analysis. The theorem 2.1 and lemma 2.2 are as follows. For reference, pictures containing the ...
jason 1's user avatar
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absolutely continuous ($\sigma$-) finite measures

I got two questions: If I have two measures $\nu$ and $\mu$ with $\nu << \mu$ and we know that $\nu$ is (i) finite (e.g. a probability measure) or (ii) $\sigma$-finite. Can we draw any ...
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Show: If a density exists, then the involved measures are absolutely continuous

Let $(X, \mathcal{A})$ be a measure space. We can define a measure $\nu$ w.r.t. to another measure $\mu$ and a function $f:X\rightarrow \bar{\mathbb{R}}^+$ (called the density) as follows: $$\nu(A)=\...
guest1's user avatar
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Why do we write $P\ll Q$ and why is $P$ called absolutely continuous w.r.t. $Q$?

Let $P$ and $Q$ be two measures on a measurable space $(\Omega,\Sigma)$. My book, Stochastic Finance by Föllmer and Schied, says that if $$\forall A\in\Sigma: Q(A)=0\Rightarrow P(A)=0$$ then $P$ is ...
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