1
vote
0answers
20 views

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 ...
0
votes
1answer
15 views

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 ...
0
votes
1answer
32 views

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 ...
0
votes
1answer
33 views

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 ...
0
votes
1answer
45 views

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 ...
2
votes
1answer
55 views

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 ...
0
votes
0answers
112 views

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 ...
0
votes
2answers
49 views

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 ...
1
vote
0answers
43 views

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 ...
2
votes
1answer
46 views

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 ...
1
vote
1answer
35 views

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 ?
5
votes
1answer
163 views

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 ...
0
votes
1answer
14 views

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 ...
1
vote
1answer
47 views

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 ...
2
votes
0answers
73 views

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 ...
0
votes
1answer
41 views

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 ...
0
votes
0answers
32 views

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 ...
0
votes
1answer
34 views

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. ...
2
votes
2answers
54 views

$C^0(\overline{\Omega}) \subset L^{\infty}(\Omega) ?$

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 $L^{\infty}$ ...
2
votes
0answers
29 views

Krylov-Bogoliubov theorem without continuity

This question is very closely related to: Continuity in the Krylov-Bogoliubov theorem. The standard counterexample, which is presented in Katok-Hasselblatt is the following: Let $f:[0,1]\rightarrow ...
1
vote
0answers
26 views

limit of limit superior w.r.t truncated set

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
0
votes
1answer
73 views

prove $\exists$ a continuous function g s.t. $|g(x)-f(x)|<\epsilon$ for all x in a subset E of (a,b) and $\mu[(a,b)-E]<\delta$

I am having trouble starting with ths problem. Let f be a Lebesgue-integrable function over a bounded interval (a,b). Prove that for any $\epsilon >0$, $\delta >0$, there exists a continuous ...
0
votes
1answer
56 views

Equivalence relation between measures $\nu$, $\mu$ is equivalent to $\nu = f \mu$ for a density $f$.

I'm working on an exercise that wants me to show that for $\sigma$-finite measures $\nu$ and $\mu$ the relation $\nu \sim \mu$ (defined by $\nu \ll \mu$ and $\mu \ll \nu$) is equivalent to $\nu = f ...
1
vote
1answer
125 views

$f$ 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 $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
3
votes
1answer
54 views

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 ...
2
votes
1answer
37 views

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 ...
0
votes
1answer
183 views

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 ...
2
votes
1answer
354 views

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 ...
1
vote
1answer
50 views

Moving boundaries for Ornstein-Uhlenbeck processes

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 ...
1
vote
0answers
99 views

Example of a function that has the Luzin $n$-property and is not absolutly 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 ...
1
vote
1answer
183 views

Is there any absolutely continuous function $f$ and a null set $X$ such that $f(X)$ is not a null set?

Here (wikipedia) there are some properties of absolutely continous functions. Some of them requires a closed interval to be the domain of $f$. So, I would like an example of an absolutly continuous ...
1
vote
1answer
174 views

Continuity of the Lebesgue function

If $x \in [0,1]$ has ternary expansion $(a_n)$, i.e. $x = 0.a_1a_2..$ with $a_n =0,1$ or $2$, define $N$ as the first index $n$ for which $a_n = 1$, and set $N = \infty$ if none of the $a_n$ are $1$ ...
6
votes
1answer
202 views

Example of a set $Y$ that has zero Lebesgue measure and a continuous function $f$ such that $f(Y)$ is not a set of zero Lebesgue measure.

Could someone give me an example of a set $Y\subset \mathbb{R}$ that has zero Lebesgue measure and a continuous function $f:X\subset \mathbb{R}\to\mathbb{R}$ such that $Y\subset X$ and $f(Y)$ is not a ...
-2
votes
1answer
351 views

It's problem 23 of the 2th chapter of the book “Real and complex analysis” by Rudin. Please correct my solution [closed]

Suppose $V$ is open in $R^k$ and $\mu$ is a finite positive Borel measure on $R^k$. Is the function that sends x to $\mu(x+V)$ necessary continuous? Lower semicontinuous?Usc? My proof: First we show ...
1
vote
1answer
340 views

Show that the upper envelope of a bounded function is upper semi continuous directly

Definition 1: A real valued function $f$ is said to be upper semicontinuous at a point $p$ if: $$f(p) \geq \limsup_{x \rightarrow p} f(x) $$ Definition 2: Let $f$ be a bounded real valued ...
4
votes
2answers
125 views

Showing $\mathcal{H}$ is a hilbert space.

So this is an early exercise in Conway's A Course In Functional Analysis. I'm trying to get to grips with this upto open mapping and closed graph to see if I want to do any more functional analysis. ...
1
vote
2answers
50 views

Identical to a continuous function a.e.

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function. Are the following two statements equivalent to each other? $f$ is continuous almost everywhere $f$ is identical to a continuous ...
1
vote
1answer
179 views

Preimages of Jordan-measurable sets

When is the preimage of a Jordan-measurable set Jordan-measurable? In particular, is continuity sufficient? Piecewise continuity with finitely many pieces?
3
votes
2answers
164 views

Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
1
vote
2answers
160 views

Definition of Lebesgue-Stieltjes measure on $\mathbb R$

Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure $$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$ ...
0
votes
1answer
71 views

Logical Relations Between Three Statements about Continuous Functions

(a) $f$ is continuous almost everywhere (b) there exists a continuous function $g$ such that $f = g$ almost everywhere (on every set of non-zero measure) (c) $f$ is nearly a ...
2
votes
1answer
113 views

Continuity of $\max$ of Lebesgue integral

Let $m$ be a probability measure on $Z \subseteq \mathbb{R}^p$, so that $m(Z)=1$. Consider a locally bounded $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, with $X \subseteq \mathbb{R}^n$, ...
1
vote
1answer
220 views

Continuous function: Borel and/or Lebesgue measurable?

If $f\colon [0,1]\to\Bbb R$ is a continuous function and differentiable on (0,1), then which of the following is correct? $f'$ is Borel and Lebesgue measurable. $f'$ is Borel measurable and is not ...
11
votes
1answer
411 views

Additivity + Measurability $\implies$ Continuity

A function $f:\Bbb R \to \Bbb R$ is additive and Lebesgue measurable. Prove that $f$ is continuous. I know that on $\Bbb Q$, $f$ comes out to be linear. So, if $f$ is to be continuous then $f$ must ...
1
vote
3answers
224 views

Continuity of the lebesgue integral

How does one show that the function, $g(t) = \int \chi_{A+t} f $ is continuous, given that $A$ is measurable, $f$ is integrable and $A+t = \{x+t: x \in A\}$. Any help would be appreciated, thanks
3
votes
1answer
197 views

Proving continuity of an integral

I have the following function: $$I_n(a)=\int_{-\infty}^{\infty}x^6e^{-x^2}\operatorname{sech}^n(ax)dx$$ where $\operatorname{sech}(x)=\frac{2}{e^x+e^{-x}}$ is the hyperbolic secant. Clearly, the ...
1
vote
0answers
57 views

Combined length of portions of a line segment

Suppose I have a continuous function $f : X \to \mathbb{R}^n$ (where $X \subseteq \mathbb{R}^n$) that is piecewise rigid, i.e. $X$ has a finite partition $\mathcal{P}$ such that for all $P \in ...
2
votes
0answers
142 views

Continuity in set functions

Let a function be defined as $f:(\Omega_1,\mathcal{F}_1)\rightarrow (\Omega_2,\mathcal{F}_2)$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are $\sigma$-fields in $\Omega_1$ and $\Omega_2$ respectively. ...
2
votes
1answer
120 views

Try to Understand Calculus Theorems, Real Analysis

I attempted some easy multiple-choice and T/F questinos to test if I am entirely clear about the topic before I do any proofy works. It's essential to be clear on these basic concepts and ideas. Let ...
1
vote
1answer
138 views

absolute continuity in trigonometric functions

I want to show the function $f:[0,1]\to \mathbb{R}$ $$f=\left\{ \begin{array}{ll} x^{3/2}\sin\left(\frac{1}{x}\right), & {x \in (0,1]} \\ 0, & x=0 \end{array} \right.$$ is ...