2
votes
0answers
57 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
1
vote
0answers
49 views

Another version of the Poincaré Recurrence Theorem (Proof)

The task is to prove the following version of Poincaré's Recurrence Theorem: Let $(X,\Sigma,\mu)$ be a finite measure space, $f\colon X\to X$ a measurable transformation that preserves the ...
2
votes
2answers
73 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
3
votes
1answer
40 views

Convergence of measures — revisited

In this thread, I asked a question about the convergence of measures. The conjecture I posed there, which turned out to be false, was supposed to be a lemma that I wanted to use to prove a ...
1
vote
1answer
36 views

Proposed proof Lebesgue integration question

I just want to confirm the following proof: Consider a function $u: \Omega \rightarrow \mathbb{R}$ where $\Omega \subset \mathbb{R}^{n}$ and $u \in C^{2}(\bar{\Omega})$. Let $a_{jk}$ be smooth ...
2
votes
1answer
66 views

Show: $M\subset\mathbb{R}^n$ Jordan-measurable, iff $vol^*(\partial A)=0$

Show that a bounded subset $A\subset\mathbb{R}^n$ is Jordan-measurable iff and only if $\partial A$ is a Jordan null set, i.e. $vol^*(\partial A)=0$. Here Show some properties of the ...
1
vote
0answers
34 views

Prove that if the derivative $f'(x)$ of a function exists on the measurable set $E$, then $f'(x)$ is measurable on $E$.

Prove that if the derivative $f'(x)$ of a function exists on the measurable set $E$, then $f'(x)$ is measurable on $E$. We are told to only consider 1 dimensional spaces,that f is a measurable ...
2
votes
1answer
39 views

Properties of the distribution function.

Let $\Omega\subset\mathbb{R}^n$ be open. Let $f:\Omega\rightarrow\mathbb{R}$ be a measurable function and $g\in L^p(\Omega)$, for some $p\geq 1$. For a measurable function $f: ...
0
votes
0answers
62 views

$\lim\limits_{n\to\infty}\displaystyle\int_X n\log((1+(f/n)^{\alpha})d\mu$

suppose $\mu$ is a positive measure on $X$ and $f:X\to[0,\infty]$ is measurable with $\int_Xfd\mu=c$, where $0<c<\infty$ and let $\alpha$ be a constant, prove that; ...
3
votes
1answer
70 views

The “intersection property” of the symmetric difference metric

$\newcommand{\measure}{\operatorname{measure}}$ The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space: $$d(S,T)=\measure(S\oplus ...
1
vote
0answers
31 views

$f_n\text{ bounded }, f_n\to f\text{ uniformly}$, then $f$ bounded

Let $(X,\mathfrak{A},\mu)$ be a measurable space, $f_n,n\geqslant 1$ measurable and bounded functions with $f_n\to f$ uniformly. Would like to know if then $f$ is measurable and bounded, too. ...
1
vote
2answers
22 views

Using continuity of the measure in a proof…

I'm trying to understand the following proof: I don't understand how the conclusion came from the equation in the green box, did they use continuity of the measure?
1
vote
1answer
51 views

Jordan decomposition theorem (question to singularity)

Let $(\Omega,\mathcal{A})$ be a measurable space. (Hahn Decomposition Theorem) Let $\varphi\colon\mathcal{A}\to\mathbb{R}$ be a signed measure. Then there exist disjoint sets ...
2
votes
0answers
44 views

Show that $\mathcal{B}(\mathbb{R}^n)=\bigotimes_{i=1}^{n}\mathcal{B}(\mathbb{R})$

As the title says, I would like to prove that $$ \mathcal{B}(\mathbb{R}^n)=\bigotimes_{i=1}^{n}\mathcal{B}(\mathbb{R}). $$ (Here $\mathcal{B}(\mathbb{R}^n)$ is the Borel-$\sigma$-Algebra on ...
0
votes
0answers
19 views

Example of a kernel

Let $(\Omega_1,\mathfrak{A}_1), (\Omega_2,\mathfrak{A}_2)$ be two measurable spaces. A function $$ K\colon\Omega_1\times\mathfrak{A}_2\to [0,\infty) $$ is called kernel if (1) ...
2
votes
0answers
28 views

Show: $\sum_{k=1}^n X_k$ convergent a.s. $\Leftrightarrow\lim_n \mathbb{P}(\bigcup_{j,k\geqslant n}\lvert S_j-S_k\rvert > \varepsilon )=0$

Let $(X_i)$ be a sequence of random variables. Show that $$ S_n:=\sum_{k=1}^n X_k \text{ converges a.s.} \Leftrightarrow\lim_n \mathbb{P}(\bigcup_{j,k\geqslant n}\lvert S_j-S_k\rvert > ...
2
votes
1answer
61 views

Show that $\frac{1}{n}X_n\to 0$ a.s.

Show that for any sequence $(X_n)_{n\in\mathbb{N}}\in (L_{\mathbb{P}}^2)^{\mathbb{N}}$ of identically distributed random variables it is $\frac{1}{n}X_n\to 0\text{ a.s.}$. The professor ...
1
vote
1answer
54 views

Show the equi-integrability of a finite set of $\mathcal{L}_{\mu}^1$-functions

Let $(\Omega,\mathcal{A},\mu$ be a measure space. A set $\mathcal{F}$ of measurable, numerical functions is called equi-integrable if for any $\varepsilon > 0$ it exists a nonnegative, ...
0
votes
1answer
48 views

$h_n:=\lvert f-f_n\rvert\to 0\implies\int h_n\, d\mu\to 0$?

Let $f,f_n\in\mathcal{L}_{\mu}^1$ with $f_n\to f\text{ a.s.}$. Define $h_n:=\lvert f-f_n\rvert$. Then $h_n\to 0\text{ a.s.}$. How can I prove that $\int h_n\, d\mu\to 0$? My idea ia to ...
2
votes
1answer
58 views

Show: $f_n\geqq 0$ with $f_n\nearrow f\text{ a.s.}\implies\int f_n\, d\mu\nearrow\int f\, d\mu$

Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $f,f_n\colon (\Omega,\mathfrak{A})\to (\overline{\mathbb{R}},\overline{\mathcal{B}})$ be measurable functions. Show: If $f_n\geq 0$ and ...
4
votes
0answers
20 views

Integration relating to transported measures

Let $(\Omega,\mathcal{A},\mu)$ and $(\Omega',\mathcal{A}')$ two measurable spaces and $T\colon (\Omega,\mathcal{A})\to (\Omega',\mathcal{A}')$ measurable. Show: If $f\colon ...
1
vote
0answers
30 views

Measure Theory - Series of functions

A question from my homework I wasn't sure about: Let $\{f_n\}$ be a sequence of integrable non-negative functions on a measure space $(X,F,\mu)$ s.t. $\int_Xf_n(x)\,d\mu=1$. Is one of the ...
0
votes
0answers
29 views

Show different properties of a distribution function (of a probability measure)

Let $F$ be the distribution function of a probability measure $\mu$. Show that (1) $x\leq y\implies F(x)\leq F(y)$, (2) $\lim_{x\to -\infty}F(x)=0, \lim_{x\to\infty}F(x)=1$ and (3) ...
3
votes
0answers
38 views

Show measurability for $\lvert f\rvert$

Let $f\colon (\Omega,\mathcal{A})\to (\overline{\mathbb{R}},\overline{\mathcal{B}})$ a measurable function. Show that then $\lvert f\rvert$ is measurable. In order to show, that $\lvert ...
0
votes
1answer
97 views

Counting measure $\sigma$-finite / not $\sigma$-finite for different sets

Let $\zeta\colon\mathcal{P}(\Omega)\to [0,\infty]$ be the counting measure. Show that for $\Omega:=\mathbb{R}$ it is not $\sigma$-finite but for $\Omega:=\mathbb{N}$. Hello and good ...
2
votes
1answer
27 views

Show that $\mathfrak{S}=\bigcup_{N=1}^{\infty}\mathfrak{Z}_N\cup\left\{\emptyset\right\}$ is a semi-ring

Let $\Gamma$ be a finite set, $\Omega=\Gamma^{\mathbb{N}}=\left\{(x_1,x_2,\ldots):~\forall i\in\mathbb{N} x_i\in\Gamma\right\}$. For $a_1,\ldots,a_N\in\Gamma$ let $$ ...
1
vote
1answer
45 views

Show: The integral over a zero set is zero

From Show: $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$ was motivated: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space. Let $A\in\mathfrak{A}$ with $\mu(A)=0$. I would like to show that $\int_A f\, ...
5
votes
0answers
77 views

representation theorem on the path space

I'm working on a project and have done some work. However, there are some point where I'm unsure if my thoughts are correct. It would be appreciated if someone could share their thoughts about it. ...
3
votes
1answer
134 views

Inverse image of $\sigma$-algebra

Is this proof correct? Let $f$ be a function mapping $\Omega$ to $E$ with $\mathcal E$ a $\sigma$-algebra on $E$. Show that $\mathcal A=\{f^{-1}(B):B\in \mathcal E\}$ is a $\sigma$-algebra on ...
2
votes
2answers
41 views

Sub $\sigma$-algebra

Is my proof of the following correct? Let $\mathcal{A}$ be a $\sigma$-algebra on $\Omega$ and let $B\in\mathcal{A}$; then $\mathcal{B}=\{A\cap B:A\in\mathcal{A}\}$ is a $\sigma$-algebra on $B$ ...
5
votes
2answers
77 views

$f$ integrable $\Leftrightarrow f<\infty$ a.s.?

$f\colon\Omega\to\mathbb{R}$ measurable function on measure space$(\Omega,\mathfrak{A},\mu)$. I am interested to know if then $$ f\text{ is integrable }\Leftrightarrow f\text{ is finite a.s.}~~~. $$ ...
1
vote
1answer
776 views

Bonferroni inequality proof

Is this proof for $P(\bigcup_{i=1}^n A_i)\le\sum_{i=1}^nP(A_i)$ correct? Pf. By induction. For $n=2$, $$P(A\cup B)=P(A)+P(B)-P(A\cap B)\le P(A)+P(B)$$ Assume that the statement is true for $n-1$, ...
3
votes
1answer
59 views

Prove Intersection of $\sigma$-algebras is a $\sigma$-algebra and the powerset is a $\sigma$-algebra

Fix a set $\Omega$. A $\sigma$-algebra on $\Omega$ is a non-empty collection of subsets of $\Omega$ closed under taking complements and countable unions. I'd like to prove that (1) for finite ...
0
votes
0answers
52 views

Showing finite additivity of Lebesgue measure

I want to show that Lebesgue measure is finitely additive on the set of semi open rectangles of the form $[a,b)$ here is what I did ($\sqcup$ is disjoint union , $\lambda^n$ is n-dimensional Lebesgue ...
3
votes
0answers
46 views

Monotonicity of measures

Let $\mu$ be a measure defined on $\Omega$. Then $\mu(A)\le \mu(B)$ for all $A\subset B\subset \Omega$. pf. Let $A\subset B$, let $C=A^c\cap B$. Then $A\cap C=\emptyset$ and $A\cup C = B$. By ...
1
vote
1answer
149 views

Radon-Nikodým (chain rule and other properties)

(1) For three $\sigma$-finite measures $\mu\ll\nu\ll\eta$ it is $$ \frac{d\mu}{d\eta}=\frac{d\mu}{d\nu}\frac{d\nu}{d\eta}~~\eta-\text{a.s.} $$ (2) For two finite measures $\mu\sim\nu$ ...
7
votes
1answer
145 views

Counterexample to “Measurable in each variable separately implies measurable”

Some fellow classmates are preparing for a qualifying exam on real analysis, and asked me for help on the following question: Let $ \ f:[0,1]^2\longrightarrow\mathbb{R}$ be such that: (i) $\ ...
3
votes
0answers
57 views

Show $(h_1\mu_1)\otimes\ldots\otimes (h_n\mu_n)=h(\mu_1\otimes\ldots\otimes\mu_n)$ (and that this is a probability measure)

Let $\mu_1,\ldots\mu_n$ be $\sigma$-finite measures and $h_i$ probability densites on $(\Omega_i,\mathcal{A}_i,\mu_i),~i=1,\ldots,n$. Show that for $$ ...
0
votes
2answers
108 views

Is this a $\sigma$-finite measure?

Let $(\Omega,\mathcal{A},\mu)$ be a measure space and $\mu$ $\sigma$-finite. Let $h$ be a probability density on $(\Omega,\mathcal{A},\mu)$. Consider the measure $h\mu(A):=\int_{\Omega}h1_A\, ...
1
vote
0answers
103 views

Show for a set, that it is a $\sigma$-algebra

Let $T$ be any index set and $(\Omega_t,\mathcal{A}_t)_{t\in T}$ a family of measurable spaces. Furthermore consider $\mathcal{A}:=\bigotimes_{i\in T}\mathcal{A}_i$ and $$ ...
2
votes
0answers
35 views

Folner sequence: construct through semidirect product; proof understanding help

Sorry I have difficulty finding anything relevant on Folner sequence. I am reading this and come up against lemma 4. I have not studied much algebra, so I might interpret this wrong, but the lemma ...
1
vote
1answer
48 views

Convergence a.e. of a sequence of random variables (an equivalence is to show)

Show: For a sequence $(X_n)_{n\in\mathbb{N}}$ of random variables it is: $$ (X_n)_{n\in\mathbb{N}}\text{ converges ...
2
votes
1answer
69 views

Show that $f_n1_{A_n}$ convergences in mean

Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f,f_1,f_2,\ldots$ be measurable functions on that measurable space and $A,A_1,A_2,\ldots\in\mathcal{A}$. Let $(f_n)$ converge in ...
1
vote
1answer
144 views

Prove that if $f(x)$ is measurable function then $h(x)=…$ is also measurable function.

Prove: If $f: ( X, \mathcal{A}) \rightarrow \mathbb{R}$ is measurable function, $A \in \mathcal{A}$ then function $h:( X, \mathcal{A}) \rightarrow \mathbb{R}$ such that $$h(x) = \begin{cases} f(x) ...
3
votes
0answers
24 views

Weak law of large numbers ($\mathfrak{L^2}$-version)

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of pairwise uncorrelated random variables out of $\mathfrak{L}_{\mathbb{P}}^2$ on the probability space $(\Omega,\mathcal{A},\mathbb{P})$ with $$ ...
0
votes
0answers
75 views

Parameter integral and continuity (Theorem of Lebesgue)

I already kept myself busy with a proof concerning the Theorem of Lebesgue and differentiation of a parameter integral. Unfortunately I did not get an answer there yet. Now my task is nearly the ...
0
votes
0answers
74 views

Theorem of Lebesgue and differentiation of a parameter integral

Let $(a,b)\subset\mathbb{R}$ be an interval and let $\left\{f_t\colon\Omega\to\mathbb{R}\right\}_{t\in (a,b)}$ be a family of measurable functions on the measurable space ...
0
votes
0answers
44 views

$\int\lvert f_n\rvert\, d\mu\to\int\lvert f\rvert\, d\mu\Leftrightarrow\int\lvert f_n-f\rvert\, d\mu\to 0$

Consider a measurable space $(\Omega,\mathfrak{A},\mu)$ and let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions out of $$ \mathfrak{L}_{\mu}^1=\left\{f\colon (\Omega,\mathfrak{A})\to ...
3
votes
1answer
33 views

Show: $\mathcal{L}^p\subset\mathcal{L}^q$

Consider a measurable space $(\Omega,\mathcal{A},\mu)$. Let $\mu(\Omega)<\infty$ and $1\leq q\leq p$. Show that then $$ \mathcal{L}_{\mu}^p\subset\mathcal{L}_{\mu}^q. $$ Good ...
3
votes
1answer
120 views

Please check if my proof is correct of Monotone Convergent theorem

I was required to prove Monotone Convergent Theorem as a corollary of Fatou lemma,i.e using Fatou lemma to prove the MCT. The hint I was given is let $f_n$ be a sequence of increasing function, ...