0
votes
1answer
41 views

Show that if E is measurable set and f is continuous on E, then f(E) is measurable set

Please tell me how to prove or disprove it ! Show that if E is measurable set and f is continuous on E, then f(E) is measurable set
0
votes
1answer
25 views

How to interchange sum and integral when measure is in terms of Dirac measure?

Let $\{c_{k}\}_{k\in \mathbb Z} \subset \mathbb C$ such that, $\sum_{k\in \mathbb Z} |c_{k}| < \infty.$ Let $\delta_{k}$ is the unit Dirac mass at $k $, we note that $\mu = \sum_{k\in \mathbb Z} ...
0
votes
0answers
16 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
0
votes
1answer
39 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
0
votes
1answer
7 views

Question about outer regularity and inf

Let $\mu$ be a measure. Suppose for every $\varepsilon > 0$, there exists an open set $U \supset E$ such that $\mu(U) < \mu(E) + \varepsilon$. Then must $\mu(E) = \inf\{\mu(U): U \supset E, U ...
0
votes
0answers
11 views

Jordan decomposition of sum of two measures

Let $\mu$ and $\nu$ be finite signed measures. Then by the Jordan Decomposition Theorem, we can write $\mu = \mu^{+} - \mu^{-}$ and $\nu = \nu^{+} - \nu^{-}$ where $\mu^{\pm}, \nu^{\pm}$ are unsigned ...
1
vote
0answers
11 views

scale transformation is invariant for H_1

Consider the subspace $H_1$ of $C_0(0,\infty)$, where $\phi=\int_0^t\dot{\phi}(s)ds$ and $\int_0^{\infty}{\dot{\phi}}^2ds<\infty$. The transformation is $(T\phi)(t)=t\phi(\frac{1}{t})$. How to ...
2
votes
1answer
49 views
+50

Measurability of upper and lower derivatives of Radon measures

Let $\mu$ and $\nu$ be Radon measures in $\mathbb R^N$. Define their upper and lower derivatives by $$ \overline{D}_\nu\mu(x):=\limsup_{r\to0}\frac{\mu(B_r(x))}{\nu(B_r(x))},\qquad ...
0
votes
1answer
30 views

Change of variable (or measure)?

Hi Everyone: I am reading a book and there is a kind of "change of variable" they make that I do not understand fully. This is what they do: let $B(x,r)$ be a ball of $\mathbb{R}^{N}$ $(N>1)$, ...
1
vote
0answers
27 views

Lebesgue-Stieltjes measure

Is the following reasonment correct? There is a sort of duality between non-decreasing functions and Borel outer measures. In particular, given a non-decreasing function ...
0
votes
1answer
22 views

Everywhere continuous extension of a almost everywhere continuous function

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure. If $f$ is continuous outside a set $N$ of $\mu$-measure 0, does there exist an everywhere continuous $g$ such that $f = g$ on $X ...
1
vote
1answer
112 views

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
0
votes
2answers
45 views

How to show that this function is measurable?

$f$ is a monotone increasing real valued function defined on a interval. Let $g(x)=f(x)+x/n$ where $n$ is fixed natural number. Proof $g$ is strictly increasing function and is measurable.
1
vote
0answers
60 views

The Cantor set is nowhere dense

I am considering the so called Cantor ternary set $C$ on $[0,1]$. I have just proved that its Lebesgue measure is $0$. To show that $C$ is nowhere dense, is it correct the following reasoning? By ...
1
vote
0answers
41 views

Distribution function and decreasing rearrangement

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement ...
1
vote
0answers
24 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
1
vote
0answers
45 views

Integration question measure theory

For the function $$ f(x) = \begin{cases} \infty & \text{if $x=0$} \\ 1/x & \text{if $x \in \mathbb{Q} \smallsetminus 0$} \\ 0 & \text{Otherwise} ...
0
votes
0answers
20 views

Borel outer measure and Borel measure

I have these two definitions: Given $\ (X,\mathcal{M},\mu)$ measure space, the measure $\ \mu$ is Borel measure if $\ \mathcal{M}=\mathcal{B}(X) $ Given $\ \phi:X\to[-\infty,+\infty]$ be an outer ...
1
vote
0answers
16 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
2
votes
1answer
28 views

Absolutely Continuous measures and Hellinger integral

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
3
votes
0answers
100 views

Question about Feller's book on the Central Limit Theorem

My question concerns the proof of Theorem 1, section VIII.4, in Vol II of Feller's book 'An Introduction to Probability Theory and its Applications'. Theorem 1 proves the Central Limit Theorem in the ...
1
vote
1answer
40 views

Signed measure defined by an integral

Let $ (X,\mathcal{M},\mu)$ be a measure space and let $ f:X\to[-\infty,+\infty]$ be an integrable function (i.e. at least one of $ f_+ $ and $ f_-$ is integrable). I want to prove that $ ...
1
vote
1answer
39 views

Sequence of functions whose integrals converge to $0$.

I encountered the next problem and I'm having problems solving it. It says: Let $(X,\mathcal{S},\mu)$ be a measure space and let $\{A_n\}$ be a sequence of measurable sets satisfying $0< ...
2
votes
0answers
28 views

Measures and Outer Measures

I've a "little" doubt. From Carathéodory-Hahn Extension Theorem, we know that, starting from a measure space $\ (X,\mathcal{M},\mu)$, we are always able to obtain an outer measure $\ \mu ^* $. Then ...
1
vote
0answers
20 views

Two different ways to generate the topology of convergence in measure

Consider the measure space $(X, \mathcal{B}, \mu)$ where $\mu(X) < \infty$. Let $L(X)$ denote the space of measurable functions on $X \rightarrow \mathbb{C}$. Then one way to define the topology ...
1
vote
1answer
37 views

A Lebesgue measurable universal Borel function

In 1918 Sierpiński constructed a Lebesgue measurable real-valued function on $[0,1]$ which isn't bounded above by any Borel function (I couldn't find the original reference, but here is a pdf of a ...
3
votes
1answer
54 views

Continuous complex measures

I was wondering if following statements are true: If $\mu$ is the Lebesgue measure and $\mu(A)=\alpha_0$, then it's not difficult to verify that for any $\alpha<\alpha_0$, there exist $B\subset A$ ...
2
votes
0answers
79 views

Kolmogorov extension-type result

I would like to prove the following, using the standard Kolmogorov extension theorem (e.g. http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem): Let $(\Omega, \mathscr{F}, P)$ denote our ...
2
votes
0answers
47 views

Weak convergence in $L^1$

Does anyone have a reference for the following statement or similar ones? Let $U$ be an open bounded set in $\mathbb R^n$ and let $f\in C^0(U\times S^1)$. Then the sequence $f_m (x):=f(x,mx_i)$ ...
0
votes
1answer
29 views

Is $f$ integrable in $L(X,\mathcal{X},\mu)$

Is $f$ integrable $L(X,\mathcal{X},\mu)$ $\mu(E)=\sum_{n\in E\cap\mathbb{N}} |n^2+n-6|$ $f:\mathbb{R}\rightarrow \mathbb{R_+}\cup\infty$ $f=(x-2)^{-4}$
3
votes
1answer
56 views

Topology of weak convergence

Edited: Thanks to etienne. I start with a compact metric space $(X,d)$. Then I consider the collection of finite measure $\mathcal{M}$ on $X$ and I equip $\mathcal{M}$ with the topology of weak ...
0
votes
0answers
25 views

will which result Convergence in measure?

Suppose $(X,M,\mu)$ be a space measure with $ \mu(X)<\infty$ and $ f_n: X \rightarrow \mathbb{R} $ a sequence of measurable functions. for every $ n $ set $ E_n=\lbrace x\in X: f_n(x)\neq 0\rbrace ...
2
votes
3answers
119 views

help with showing completeness

Let $\left\{H_n\right\}_{n=1}^\infty$ be a sequence of Hilbert spaces and let $H=\left\{\left\{x_n\right\}:x_n\in H_n, \sum ||x_n||^2<\infty \right\}$. Define the inner product as ...
4
votes
1answer
62 views

Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
0
votes
1answer
46 views

Measurable vector bundles trivial

I hope you can help: If $E$ is a measurable vector bundle over a compact metric space $(X,\mu)$ then there is a subset $Y\subset X$ such that $\mu(Y)=1$ and $\pi ^{-1}(Y)$ is isomorphic to a trivial ...
0
votes
1answer
37 views

Indicator function of a level set

Consider a positive definite locally Lipschitz function $V:\mathbb{R}^2\to\mathbb{R}_{\geq0}$. Fix $c\in\mathbb{R}_{\geq0}$ and consider the sublevel-set $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
4
votes
1answer
50 views

Whether a set is closed or not

Denote by $C_{[0,1]}$ the ternary Cantor set on $[0,1]$. Now consider $[0,1] \setminus C_{[0,1]}$. It contains open intervals. Now define Cantor sets on all these open intervals by simply translating ...
19
votes
2answers
502 views

Lebesgue density strictly between 0 and 1

I am having trouble with the following problem: Let $A\subseteq \mathbb{R}$ be measurable, with $\mu(A)>0$ and $\mu(\mathbb{R}\backslash A)>0$. Then how do I show that there exists $x\in ...
2
votes
0answers
131 views

Show $\pi$ is a measure

Show that $\pi(E)=sup\lbrace \mu(A): A\subseteq E, A\in\mathbf{X} \rbrace$ is a measure on $\mathbf{X}$. $\mu$ is a charge on $\mathbf{X}$ ($\sigma$-algebra), let $\pi$ be defined for ...
1
vote
1answer
26 views

Geometric question involving integral of a function and its inverse.

I am given a function $\phi(s)$, continuous and strictly increasing with $\phi(0) = 0$, and want to show that for all $a,b \geq 0$, $$ab \leq \int_0^a \phi(x)dx + \int_0^b \phi^{-1}(x)dx.$$ I know how ...
0
votes
1answer
25 views

Integral Estimate Using a Function and its Inverse

I want to show the following: given a measure space $(X,\mu)$ and $f,g$ $\mu$-measurable functions on $X$, $$\int_X |f(x)g(x)| d\mu(x) \leq \frac{1}{2}\int_{|f(x)| \leq 1} |f(x)|^2 d\mu(x) + ...
2
votes
2answers
26 views

clarification on a question about showing that the closure of a subspace is a subspace

In a homework problem, I have been asked to prove the following "If $X$ is a normed linear space and $S$ is a linear subspace of $X$ then $\overline{S}$ is a linear subspace of $X$." ($\overline{S}$ ...
0
votes
2answers
64 views

Why is one set countable while the other uncountable?

This is an example from Chapter 1 of David Williams book - Probability with Martingales. Let $(S,\Sigma,\mu) = ([0,1],\mathcal{B}([0,1]),Leb)$. Let $V=\mathbb{Q}\cap [0,1]=\{v_n, n\in \mathbb{N}\}$. ...
1
vote
1answer
40 views

Oscilation Property of absolutely continuous functions

I have a question about absolutely continuous function $f:[0,T]\rightarrow \mathbb{R}$. First, as we know, as a function with finite variation, $f$ is almost everywhere differentiable. However, I ...
1
vote
1answer
88 views

Prove that $\int_{E}f =\lim \int_{E}f_{n}$

I'm doing exercise in Real Analysis of Folland, and got stuck on this problem. I try to use Fatou lemma but can't come to the conclusion. Can anyone help me. I really appreciate. Consider a ...
3
votes
1answer
58 views

$m(E)=0$ then $m(\lbrace x^2 : x\in E\rbrace$?

Let E be a subset of $\mathbb{R}$ with lebesgue measure zero. How can I prove that $\lbrace x^2 : x\in E\rbrace$ also has lebesgue measure zero?
1
vote
1answer
35 views

Borel measures on $\mathbb{R}$ questions

I am reading a textbook and need some help. First it mentions that we can find a Borel measure such that $\int_\mathbb{R} x^2 \mu(x)<\infty$ but $\int_\mathbb{R} x \mu(x)=\infty$. This seems ...
6
votes
1answer
47 views

How to determine measure from the integral equation?

Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...
0
votes
1answer
50 views

The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
2
votes
1answer
35 views

Continuous modification of functions with a given property

Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ with the following property: For all reals $x$, $\displaystyle\lim_{y \to x} f(y)$ exists. (In particular, note that its possible that ...