Mathematics

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.

Absolute continuity is a smoothness property of functions stronger than that of and . The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration—expressed by the fundamental theorem of calculus in the framework of . Such generalizations are often formulated in terms of the . For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. Those two notions are generalized in different directions. The usual derivative of a function is related to the derivative, or density, of a measure.

We have the following chains of inclusions for functions over a compact subset of the real line:

absolutely continuous ⊆ uniformly continuous = continuous,

and

continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere.

422 questions
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Why is the Cantor function not absolutely continuous?

Is there an easy way to see that the Cantor function is not absolutely continuous that directly uses the definition of absolutely continuous?
user782220
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Continuous and bounded variation does not imply absolutely continuous

I know that a continuous function which is a BV may not be absolutely continuous. Is there an example of such a function? I was looking for a BV whose derivative is not Lebesgue integrable but I couldn't find one.
Mykie
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Image of set of measure zero has measure zero if the function is absolutely continuous

I have been trying to solve this problem from Bass. Let $f$ be a real valued absolutely continuous function defined on $[0,1].$ Denote $f(A)=\{f(x): x \in A\}$ for $A \subset [0,1].$ Prove that if $A$ has Lebesgue-measure zero, then so does $f(A)$ .…
pikachuchameleon
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Example of a function that has the Luzin $n$-property and is not absolutely continuous.

The Banach–Zaretsky theorem ( page 196 ) says that a continuous function $f:[a,b]\to\mathbb{R}$ of bounded variation is absolutely continuous if and only if $$E\subset I \text{ has zero Lebesgue measure }\Rightarrow f(E) \text{ has zero Lebesgue…
Pedro
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$u\in W^{1,p}(0,1)$ is equal a.e. to an absolutely continuous function?

I have a simple question on Sobolev space theory. Let $1\le p \le \infty$. How can one prove that every $u\in W^{1,p}(0,1)$ is equal a.e. to an absolutely continuous function and that $u'$ exists a.e. and belongs to $L^p(0,1)$? Thank you for your…
Pooya
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Example of continuous but not absolutely continuous strictly increasing function

Could one give an example of a strictly increasing, continuous but not absolutely continuous, function on $[0,1]$ into $[0,1]$ or on $[0,1)$ into $R$ or any of the related combinations of 1-d domain and range?
Hans
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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 called absolutely continuous w.r.t. $Q$ and we…
Filippo
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What is the most general Carathéodory-type global existence theorem?

I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$ $$ \begin{equation} \left\{ \begin{aligned} x'(t) &= f(t, x(t)), \qquad t \in [a,b] \\ x(a) &= x_0 \end{aligned} \right.…
A. Pesare
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Absolute continuity inside the interval extends to the endpoint of the interval under some constraints

The below problem appeared on the UW-Madison Analysis qualifying exam in January 2020. The proof that $f$ is absolutely continuous (AC) on the whole interval is still not posted, although the requested counterexample has been given. I don't seem to…
Deepak
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Composition of absolutely continuous function on $\mathbb{R}$

Is it true that the composition of two absolutely continuous functions on the real line is absolutely continuous? I feel like this should be a resounding no, however, I'm unsure of any quick counterexamples to this claim. Can anyone think of one?
mathgenesis22813
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Sobolev space and equivalence of norms

I'm considering the space $W\{n,p\}[0,1]$ of functions with $n-1$ continuous derivatives $f^{(n-1)}$ is absolutely continuous and $f^{(n)}$ is in $L^p[0,1]$. The usual norm is the sum of the $p$-norms of each derivative from $1$ to $n$ and the $p$…
Simon
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Finding Radon-Nikodym derivative

Let $m$ be Lebesgue measure on $\mathbb R_+=(0,\infty)$ and $\mathcal A = \sigma\left(( \frac 1{n+1} , \frac 1n ]:n=1,2,...\right)$. Define a new measure $\lambda$ on $\mathcal A$, for each $E \in \mathcal A$, by $\lambda(E)= \int_E fdm $, where…
bellcircle
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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 $q(x)$ : $H(p,q) = - \int_\mathcal{X}p(x)\log q(x)dr(x)$ The page mentions that " $p$ and $q$…
Que
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Regular conditional probability on Polish space and absolute continuity

Let $(\Omega,\mathcal F,\mathbb P)$ is a standard Borel space (i.e. $\Omega$ is Polish and $\mathcal F = \mathcal B(\Omega)$ ). Then $\mathcal F$ is separable and for every sub-sigma-algebra $\mathcal G \subset \mathcal F$ , then there exists a…
Cyril B.
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Prove that if $f:[0,1] \to \mathbb{R}$ is absolutely continuous, then $|f|^p$ is absolutely continuous for $p>1$

I am trying to show that if $f:[0,1]\to \mathbb{R}$ is absolutely continuous then $|f|^p$ is absolutely continuous for all $p>1$. Since the product of absolutely continuous functions is absolutely continuous, the result is obvious if $p$ is an…
Kyle Liss
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