Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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2
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0answers
34 views

Asymmetry in definition of regular measure

In a Borel measure space $(X, \mathcal{B}, \mu)$, $\mu$ is outer regular at $E$ if \begin{equation} \mu(E) = \inf_{U \textrm{ open}} \{\mu(U): U \supseteq E\} \end{equation} and ...
3
votes
2answers
80 views

Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
0
votes
2answers
68 views

Uniform integrability of a function in $L^1$

A collection of functions $(\phi_i)_{i\in I}\in L^1(\mu)$ is called uniformly integrable if given $\epsilon>0$ there exists $\delta>0$ such that : $$\int_E|\phi_i|d\mu<\epsilon~~~~\forall ...
2
votes
1answer
44 views

Why $m(B^n(0, r)) = c_nr^n$?

(Bear with me: I realize this is quite basic question, but I'm a little loss at how to search for an answer). Anyway: I came across a real analysis proof which uses a property that (as far as I can ...
0
votes
1answer
39 views

Measurable function that's defined almost everywhere

If $(X, \Sigma, \mu)$ is a complete measure space, and $f$ is a function that is defined almost everywhere, can I use the language that $f$ is measurable? What does it mean for this function that is ...
2
votes
1answer
26 views

Can we integrate a measurable function defined on a conull subset of a complete measure space?

Suppose $(X, \Sigma, \mu)$ is a complete measure space, and suppose $f$ is a measurable function with domain of $f$ the set $X \setminus N$ for a measurable set $N$ of measure $0$. Does it make sense ...
0
votes
0answers
56 views

Show that $\int_{\mathbb{R}^n}f_1f_2 …f_n dx_1 …dx_n ≤ (I_1 …I_n)^{1/(n−1)}.$

For $\quad k = 1,2,...n,\quad$ let $\quad\mathbb{R}^k = \mathbb{R},\quad f_k(x_1,...,x_{k−1},x_{k+1},\ldots,x_n)\quad$ be a nonnegative measurable function on $\quad\mathbb{R}_1\times\ldots\times ...
3
votes
0answers
64 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
0
votes
0answers
36 views

The restriction of an open bounded linear operator

I need some help with this question. Let $X$ be a Banach space and $T:X \to X$ be a bounded linear operator. Suppose that $T$ is open, and $X_0$ be a closed subspace of $X$. The restriction $T_0$ of ...
0
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0answers
24 views

Distribution of an upper limit of a sum of random variables

Let $\{X_{n,j},n\geqslant1,j\geqslant1\}$ be independent and identically distributed random variables. Denote $S_{n,k}=\sum_{j=1}^kX_{n,j}$. Let $\{Z_n,n\geqslant1\}$ be a sequence of random variables ...
2
votes
3answers
46 views

Example of sequence of measures?

Can you give me some examples of sequence of measure that converge to a measure? (I am reading the topic of weakly convergence of measure, and convergence of random variables in distribution)
1
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1answer
35 views

Existence of ergodic joining

Let $\underline{X}=(X,\mathcal{B},\mu,T)$ and $\underline{Y}=(Y,\mathcal{B},\mu,S)$ be ergodic measure preserving systems on Borel probability spaces. A joining of $\underline{X}$ and $\underline{Y}$ ...
1
vote
1answer
59 views

Sequence of measurable functions converging a.e. to a measurable function?

I understand if $(X, \Sigma, \mu)$ is a measure space, and we have a sequence of measurable functions $f_{n}$ such that $\lim \limits_{n \to \infty} f_{n}$ exists almost everywhere d$\mu$ (a.e. ...
3
votes
2answers
44 views

Convergence in $L^p$ by using Holder's inequality

Let $1\lt p \lt \infty$ and $f\in L_p[0,\infty )$. Show that a) $$\left\vert\int_0^x f(t)\,dt\right\vert\le\|f\|_px^{1-\frac{1}{p}},$$ for $x\gt 0$. b) $$\lim_{x\to \infty} ...
3
votes
1answer
137 views

Showing a certain sequence is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$

The problem is to show $$\left\{\frac{1}{\pi^{1/2}(i+z)}\left(\frac{i-z}{i+z}\right)^n\right\}_{n=1}^{\infty}$$ is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$ In another exercise, I have ...
2
votes
1answer
26 views

Confusion on statement of Fubini's theorem for characteristic function of measurable set

I'm having trouble understanding what this theorem is saying. Theorem. Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ be a complete measure space and suppose $E \in \overline{\Sigma ...
2
votes
3answers
40 views

Finite additivity in outer measure

Let $\{E_i\}_{i=1}^n$ be finitely many disjoint sets of real numbers (not necessarily Lebesgue measurable) and $E$ be the union of all these sets. Is it always true that $$ m^\star (E)=\sum_{i=1}^N ...
2
votes
0answers
25 views

Problem involving decomposition of measures

Let $\mu$ be a signed measure. We wish to prove that $$\left| \int{f} \> d\mu \right| \leq \int{|f|} \> d|\mu|.$$ (We are given the following defintion: $\int{f} \> d\mu = \int{f} \> ...
1
vote
1answer
42 views

How to interchange limit and integral?

Suppose $f_{n}, f\in L^{1}(\mathbb R)$ with the properties that, $f_{n}(x)\to f(x)$ point wise for each $x\in \mathbb R;$ $\|f_{n}\|_{L^{1}(\mathbb R)} \leq \|f\|_{L^{1}(\mathbb R)}$ for every ...
4
votes
2answers
34 views

Absolute continuity of pushforward measure

Problem: Let $\newcommand{\IR}{\mathbb{R}}\newcommand{\IL}{\mathcal{L}}\phi: \IR \times \IR^{n} \to \IR^n$ be $\IL^{n+1}$-measurable and satisfy for every $\IL^{n}$-nullset $A \subset \IR^n$ and ...
1
vote
0answers
21 views

Integration respect the Lévy measure

How I can compute numerically $$\int_{a}^{b}f(z)\nu(dz)$$ where $\nu$ is a Lévy measure and $f$ is a continuous function? Is it equivalent to $$\int_{a}^{b}f(z)d\nu(z)$$ Thanks
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vote
0answers
31 views

Closed support of a probability measure

I have proved that for every probability measure $\mu$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, there is a unique closed set $C$ such that $\mu(C)=1$ and for any closed proper subset $D\subset C$, ...
0
votes
1answer
31 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 ...
2
votes
0answers
21 views

Density and $\lambda$ - measurability of a Radon measure

Let $\lambda$ be a Radon measure on $\mathbb{R^n}$ and $A \subset \mathbb{R^n}$. Show that $\displaystyle \lim_{r \rightarrow 0} \frac{\lambda(A \cap B(x,r))}{\lambda(B(x,r))}=0 \ \ \ \text{for ...
1
vote
3answers
78 views

Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Suppose $(X, \mathfrak{A}, \mu)$ is a measure space. Let $\phi$ be a simple function with canonical representation $\sum^{k}_{n=1} a_{n} \chi_{E_{n}}$. I know we define the Lebesgue integral of $\phi$ ...
1
vote
1answer
22 views

Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.

Let $\Omega\subset\mathbb{R}^N$ be a open, unbounded and connected set ($N\ge 2$). Let $m$ and $\mathcal{H}^{N-1}$ denote respectively, Lebesgue and $(N-1)$-Hausdorff measures. Suppose that ...
0
votes
1answer
58 views

$P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$

Let $\mathfrak{F} = (A_n)_{n \in \mathbb{N}}$. Prove that $P(\limsup A_n)=1$ if $\forall A \in \mathfrak{F}$ s.t. $P(A) > 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$. (Side question 1 Is second ...
0
votes
1answer
20 views

Ito integrals and the Euler scheme

I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ...
3
votes
0answers
21 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
0
votes
2answers
49 views

Why we can not extend Lebesgue outer measure for all subsets of real line?

I wonder that why we can't extend the Lebesgue outer measure for all the subsets of real line? Why we can't define another such measure on subsets of reals? I can't image why this happen. Counter ...
3
votes
1answer
40 views

Help with conditional expectation

I need help finding a conditional expectation: Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the ...
0
votes
1answer
38 views

Rational number, dense but measure zero

When calculating the measure of Q in real number interval [0, 1], an interval $ (q_n-\epsilon, q_n + \epsilon)$ around each rational number $ q_n $ is defined to show the measure of Q is zero. Is ...
1
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0answers
24 views

A question concerning a complete measure and the outer measure induced by a measure

My question has to do with the very last paragraph of the top answer to this question. In this paragraph, the author chooses sets $G_1$ and $G_2$ in advantageous ways. How are we allowed to choose ...
1
vote
1answer
32 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
1
vote
1answer
50 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
2
votes
0answers
10 views

Pushfoward of measures Lipschitz continuous in total variation

Let $X,Y$ and $Z$ be metric spaces and $f:X\times Y\to Z$ be a measurable map. Suppose that we are given a probability measure $\mu$ on $Y$, and define a stochastic kernel $$ ...
2
votes
1answer
34 views

Calculating difference between two probability distributions.

What is a good measure of the difference between two probability distributions other than Kullback–Leibler divergence?
0
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1answer
43 views

A Set That Is “Precisely” Measure-Dense [duplicate]

This question asks for a set of real numbers that is measure-dense, whose complement is also measure dense. In terms of $[0,1]$, the question asks for an $S$ such that for every open interval $I$ we ...
8
votes
2answers
166 views

A Sperner type problem on infinite antichains

Let $\mathcal{A} \subset 2^{\mathbb{N}}$ be an antichain (with respect to containment). I want to measure the size of $\mathcal{A}$ in the following way: I create a set, $S$, by flipping a fair coin ...
0
votes
1answer
33 views

Limit of integral of L^p functions

Let $p\in (0,\infty)$ and $f\in L^p(\mathbb{R})$. Show that $\displaystyle \lim_{n\to\infty} \int_{\mathbb{R}} f(x) \chi_{[-n,n]}\frac{1}{n^{(1-1/p)}} dx=0$. I believe $f(x) ...
0
votes
1answer
18 views

Showing that the upper packing dimension is the packing dimension

I cannot see how the first inclusion in this proof works. $P$ is the maximum number of disjoint $B(\epsilon/2)$ with centres in $A$ and the following will help. Moreover I cannot see how it ...
1
vote
0answers
19 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
0
votes
0answers
36 views

Prove that the function $g(·)$ is twice continuously differentiable and that $g′′(α) ≥ 0$ for all $α ∈ \mathbb{R}$, i.e.

Let $f$ be a real Lebesgue measurable function on the interval $[0, 1]$ such that $∥f∥∞ < ∞.$ For $α ∈ \mathbb{R}$ define a function $g(α)$ by $g(α) = \log \int_0^1\exp[αf(x)] dx .$ (a) Prove that ...
0
votes
1answer
15 views

Equalities for the Upper and Lower Minkowski dimension definition

In a Geometric Measure Theory textbook the following was written: I cannot see how any of these equalities hold and dont believe they are obvious. If they are relatively obvious could someone ...
3
votes
1answer
32 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
1
vote
2answers
33 views

Are topologically well-behaved measure 0 subsets of $\Bbb R^2$ finite graphs?

Conjecture: If $X\subseteq \Bbb R^2$ is locally simply connected (hence locally path connected), compact and Lebesgue measure $0$ then $X$ is homeomorphic to a finite graph. It is clear that ...
0
votes
1answer
17 views

Properties of the Hausdorff measure

This comes from a book on geometric measure theory in a chapter introducing the Hausdorff measure $\mathcal{H^t}$. I cannot see in this proof how $\sum_i d(E_i)^s \leq \mathcal{H^s_{\delta}}(A)+1$ ...
2
votes
1answer
57 views

Showing that a set is not infinite in measure

Suppose $f_n \geq 0$ for all $n \geq 1$, $f_n \to f$ a.e. on $[0, \infty)$ and there exists a constant $M>0$ such that $$ \sup\limits_{n} \int_{E} f_n(x)dx \leq M \mu(E)$$ for each measurable ...
1
vote
1answer
68 views

$F(x)=\int^{x}_{a} f(y) dy$ continuous (Lebesgue Integral)

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be Lebesgue integrable and let $a \in \mathbb{R}$. We wish to show that $F(x)=\int^{x}_{a} f(y) dy$ continuous. I know, of course, what we need. We need to ...
0
votes
1answer
48 views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$ I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ...