# 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.

131 questions
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### If $f$ is integrable in $\mathbb R$, Is it true $F$ is Absolutely continuous?

If $f$ is integrable in $\mathbb R$, define $$F(x)=\int_{-\infty }^xf(t) \, \mathrm dt$$ Is it true that $F$ is Absolutely continuous in $\mathbb R$?
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### Equivalence condition of Absolute Continuity

Let me restate my point. The intuition behind this construction is straightforward. Let $f$ be a continuous increasing function. Take out pieces of the graph of $f$ which is correspond to a collection ...
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### Absolutely continuous measure and equality to $0$

In the context of proving that the data processing inequality for $f$-divergences hold for any Markov kernel I am interested in the following statement If $\mu$ and $\nu$ are two probability ...
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### Absolutely continuous Banach space valued function

Let $X$ be a Banach space and $F:[a,b] \to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f \in L^1([a,b],X)$, is the ...
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### The function defined by the sum of non-decreasing absolutely continuous functions on $\mathbb{R}$ is continuous and differentiable a.e.

True or False? If $\{f_n\}_{n\ge 0}$ is a sequence of non-decreasing absolutely continuous functions on $\mathbb{R}$, and if $f:=\sum_{n\ge0}f_n(x), x\in\mathbb{R},$ is finite at each $x$, then $f$ is ...
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### $\operatorname{supp}(f_{X,Y})=\{(x,y)\in\mathbb{R}^2||x|+|y|\leq1\}$ then $X,Y$ are not independent

Let $Z=(X,Y)$ be a absolutely coninuous random variable such that $$\\\operatorname{supp}(f_Z)=\{(x,y)\in\mathbb{R}^2||x|+|y|\leq1\} \$$ Show that $X,Y$ are not independent. I don't have a good ...
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### Singular continuous

I am trying to construct a probability measure which is absolute continuous, singular constinuous and discrete. How can I do? I have not been able to find any example os such a measure. Might you help ...
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### If $\mu_F\ll\beta_1$ and $\mu_F$ is finite then $F$ is absolutely continuous

If $\mu_F\ll\beta_1$ and $\mu_F$ is finite then $F$ is absolutely continuous. Note: here $\beta_1$ is the Lebesgue measure restricted to the Borel $\sigma$-algebra in the real line. Im not sure if ...
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### Radon-Nikodym Derivative of a Limit of a Sequence of Measures

Let $\nu$ be a sigma-finite measure, let $\mu$ be a measure absolutely continuous with respect to $\nu$, and let $(\mu_n)$ be a sequence of measures such that $\mu_n(E)\uparrow\mu(E)$ for all ...
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### A counterexample to the epsilon-delta criterion for Absolute Continuity of Measures

Let $p>0$, and let $\mu$ be a Borel measure on $[0,\infty)$ defined by $\mu(E)=\int_Ex^pd\lambda$ where $\lambda$ denotes Lebesgue measure. Show that $\mu$ is absolutely continuous with respect to ...
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### Absolute continuity of increasing functions on an interval

I have been using Real Analysis by Royden and Fitzpatrick. I am currently stuck on problem 39 of chapter 6. The problem is stated as follows: "Use the preceding problem to show that if f is ...
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### Measure theory: on absolutely continuous measures

Let F be a random variable defined on a probability space $(\Omega,\mathcal F, \mathbb P)$ with values on $(E,\mathcal E, \lambda)$, where $E=(-1,1)$ and $\lambda$ is the Lebesgue measure. To prove ...
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