# 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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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 $f_{X_i}(x)... 1 vote 1 answer 125 views ### Conditional probability with absolute continuous random variable Given a probability space$(\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]$.... • 343 0 votes 1 answer 46 views ### If$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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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 ...
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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 ...
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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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### 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{...
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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 ...
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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 ...
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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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