# Tagged Questions

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### Absolute continuity for non-measures?

Let $B$ be the collection of Borel subsets of $R^2$. A measure on $B$ is said to be absolutely continuous with respect to area if any subset with area 0 has measure 0. Is there a natural ...
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### Absolute continuity of two-dimensional measures

Absolute continuity has two different meanings: one for functions and one for measures. The Wikipedia page explains the relation between the two notions in the following way: A finite measure μ ...
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### Calculate Radon-Nikodym derivative in a point when it is continuous in that point

I can't solve the following exercise, even if I find it quite intuitive. Let $\nu, \mu$ be Radon measures on a metric space $(X,d)$. Suppose that: 1) $w\in L^1(X,\mu), w\geq 0$ $\mu$ a.e.; 2) $w$ is ...
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### Measure derivatives and the chain rule

Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$ such that $\mu << \lambda$. Prove that $\displaystyle \int D(\mu,\lambda,x)^2 d\lambda x= \int D(\mu,\lambda,x)d\mu x$. Is it ...
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### Prove if $E$ is a Lebesgue measurable set, there exists a continuous function $f$ differing from $\chi_{E}$ on a set of measure $< \epsilon$?

I am reviewing my analysis notes, and I don't really understand the proof given by my professor. He first proved if $E$ is a Lebesgue measurable set and $\epsilon > 0$, then there is an open set ...
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### Modifications of Weierstrass's continuous, nowhere differentiable functions

Recalling how nowhere continuous functions such as the Dirichlet function can sometimes be modified on a $\lambda$-null set of points (in this instance, a countable set) to become everywhere ...
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### Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
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### integral of a product of functions being $0$

Suppose we have a continuous function $f$ on $[a,b]$ such that for all integrable functions $g$ such that $\int_{[a,b]}g=0$, $\int_{[a,b]}fg=0$. Show that $f$ must be constant. Well, it's clear ...
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### Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
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### How to prove that a Lipschitz function is absolutely continuous?

$f:[a,b] \rightarrow \mathbb{R}$ is a Lipschitz function. How to prove that it is absolutely continuous on $[a,b]$? My attempt: Let $\epsilon> 0$. Set $d = \epsilon/M$. If $P = \{[x_i, y_i]\}$ is ...
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### Does bounded variation and continuous means total variation continuous

$F$ is of bounded variation and continuous. Is it true that total variation is continuous ? In case, $F$ is absolutely continuous it is trivial to see. But for the above case how to proceed ?
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### Are “most” continuous functions also differentiable?

Let $A$ be a nonempty open subset of $\mathbb{R}$. Consider a function $f : A \rightarrow \mathbb{R}$. Given that $f$ is continuous, what is the probability that it is differentiable? I suspect it ...
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### Continuity at $x$ of increasing function if certain sequences exist

I'm working through the first few chapters of Royden-Fitzpatrick to learn measure theory and I got stuck on this question. Let $f$ be increasing on $I$, an open interval. Then for $x \in I$, $f$ is ...
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### Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
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### Convergence of a sequence pointwise a.e. on a manifold given that it converges on a reference manifold

Let $\Gamma_t$ be a compact hypersurface for each $t \in [0,T]$. Let $Q=\Gamma_0\times(0,T)$ and $$Q_T :=\bigcup_{t \in (0,T)} \Gamma_t \times \{t\}.$$ For each $t \in [0,T]$, suppose that ...
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### Continuity of probability measure

Sorry, I just wanted to know whether I understand this correct. Let $(x_n)$ be an increasing sequence such that $x_n \rightarrow a$, then we have for the probability measure on an arbitrary ...
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### Prove the function is continuous on an interval [duplicate]

Consider the following function $f$ on $[0, 1]:$ $f(x)=\frac{1}{m}$ if $x=\frac{m}{n} \in Q$ $=0$ if $x \notin Q$ $Q$ is the collection of all rational numbers and assume m and n have no common ...
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### Follow-up regarding right-continuous $f:\mathbb{R} \to\mathbb{R}$ is Borel measurable

I have a follow-up to another question here on math.stackexchange, Are right continuous functions measurable?. The thread was a couple of years old, so I hope it's okay if I start a new question. ...
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### $C^0(\overline{\Omega}) \subset L^{\infty}(\Omega) ?$

In my functional analysis script there is an example that uses $$C^0(\overline{\Omega}) \subset L^{\infty}(\Omega)$$ where $\Omega \subset \mathbb{R}^n$ is an open subset and we take $L^{\infty}$ ...
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### $f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
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### is continuity preserved under Expectation?

Let's say I have a random function $X(t)$ that is continuous in $t$, almost surely. Is it true that $$\mathbb E(X(t_1)) = \mathbb E\left(\lim_{t\to t_1} X(t)\right)?$$ This seems incorrect to me ...
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### Continuity in the Krylov-Bogoliubov theorem

I have the following proof of the Krylov-Bogoliubov theorem, which asserts that given a compact metric space $X$ endowed with a continuous transformation $T \colon X \to X$ one can find a ...
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### example of a Riemann integrable function (on a bounded rectangle) that is discontinuous on a dense subset of the rectangle

Construct a nontrivial example of a Riemann integrable function (on a bounded rectangle) that is discontinuous on a dense subset of the rectangle. A (trivial) example would be to rede fine a nice ...
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### Generalization of absolute continuity with $f(x) = x^a \sin(1/x^b)$

As a generalization of Prove that $x^\alpha \cdot\sin(1/x)$ is absolutely continuous on $(0,1)$ : Let $f : (0, 1] \to \mathbb{R}$ be the function denoted by $f(x) = x^a \sin(1/x^b)$. Determine for ...
Let $\tau(X_t)$ be the first-passing time to the moving boundary $a(t)$ for an Ornstein-Uhlenbeck process $X_t$. I wonder how general an $a$ can be allowed in order to guarantee that $\tau$ becomes ...