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

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5
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1answer
121 views

Qual problem in Analysis

I am having trouble with the following qual problem. Some help would be awesome. Thanks. Let $f$ be a measurable function on $(0, ∞).$ Let $p > 1/2$ and define $g(x) = (x^p + x^{−p})f(x).$ Show ...
4
votes
1answer
69 views

Dense subspace of the space of measures on the torus $\mathbb{T}$.

Every measure $\mu$ on the torus $\mathbb{T}$ is the weak-$\ast$ limit of a sequence of absolutely continuous measures on $\mathbb{T}$ with $C^{\infty}$ densities. I'd like to see a proof of this ...
1
vote
1answer
51 views

Showing set is measurable in a product space w.r.t. the product $\sigma$-algebra

Statement: Let $(X,\mathcal{B}, \mu)$ be a $\sigma$-finite measure space, and $f:X\rightarrow [0,\infty]$ is $\mathcal{B}$-measurable. Together with $(\mathbb{R},\mathcal{B}[\mathbb{R}], m)$, the set ...
1
vote
0answers
63 views

LimSup of Random Variable

I have a seemingly trivial question. Why does $$\forall a\in\mathbb{R},\mathbb{P}(\limsup X_n>a)>0\Rightarrow \mathbb{P}(\limsup X_n=\infty)=1$$ Clearly, we don't have (at least trivially), ...
5
votes
1answer
77 views

Prove $\lim_{n \rightarrow \infty}$ $\frac{1}{n}$$\int_{0}^{\space n}xg(x)dx=0$

If $g$ is a Lebesgue integrable function in $E=\lbrack 0,\infty)$, prove that $$\lim_{n \rightarrow \infty}\frac{1}{n}\int_{0}^{\space n}xg(x)dx=0.$$ I want to use the absolute continuity of the ...
2
votes
0answers
68 views

A detail on Lusin's theorem

Suppose that $B$ is a ball of $\mathbb{R}^{m}$, $(m\geq2)$, and $f(x)$ a measurable function on $B$. According to Lusin's theorem, we can find a closed set $F\subset B$ whose complement has a measure ...
0
votes
0answers
56 views

Disjoint sets producing strict outer measure inequality

Can you produce a sequence of disjoint sets $\{E_i\}$ where $m^{*}(\bigcup E_{i}) < \sum m^{*} (E_{i})$? So I realize these sets cannot all be measurable, and must all have finite outer measure. ...
2
votes
1answer
81 views

Possible mistake in Safonov's lemma

$\newcommand{\measure}[1]{\lvert#1\rvert}$Let $B_{R_0}$ a ball in $\mathbb{R}^d$, $R_{0}>0, 0<\xi<1$ and $\Gamma \subset B_{R_0}$ be a measurable set such that $\measure\Gamma >0$. ...
2
votes
0answers
55 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 ...
1
vote
1answer
104 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
2
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1answer
138 views

Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$? I was thinking about using the ...
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vote
0answers
81 views

L2 norm and the Kullback distance

Let $P$ and $Q$ be two probability measures with densities $p$ and $q$ with respect to the Lebesgue measure on [0,1] such that: $0<a\leq p(x)\leq b$, $0<a\leq q(x)\leq b$ $\forall x\in $[0,1] ...
3
votes
1answer
48 views

function $L_p$ iff $1\leq p<2$

Let $X=<0,1>$, take the borel sigma algebra, and the lebesgue measure. Consider $g(x)=\dfrac{1}{x^{\frac{1}{2}}}$. Show that $g\in L_p$ iff $1\leq p<2$. I have done this: ...
1
vote
2answers
51 views

function that doesn't belongs to $L_1$, but belongs to $L_p$ for $1<p\leq\infty$

Working on Bartle's book The Elements of Integration I found this exercise: Take $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$, with $\mu$ as countable measure and define $f(n)=\dfrac{1}{n}$, prove that ...
1
vote
1answer
79 views

Product measure with a Dirac delta marginal

Let $(S,\mathcal F)$ be a measurable space, and let $\nu \in\mathcal P(S,\mathcal F)$ be a probability measure on $(S,\mathcal F)$. Fix some $x\in S$ and consider Dirac measure $\delta_x$. Would like ...
0
votes
1answer
84 views

bounded function continuous except for a set of measure zero

Let $f$ be a bounded real function on $\mathbb{R}^n$ and $P$ be a subset of $\mathbb{R}^n$ with Lebesgue measure zero. If $f$ is continuous on $P^c$, then $f$ is Riemann integrable. Is it true? my ...
1
vote
1answer
68 views

Identify the smallest sigma-algebra of subsets of $\mathbb{R}$ that contains the set [0, 1]

This is a past exam question which I've tried to do this myself, though I'm unsure of the solution. First of all, by the definition of a sigma algebra, I should include $\emptyset$. Then, ...
1
vote
1answer
31 views

Integrability and exponential integrability

I'm working on a paper, and I don't know if there is some kind of typo or if I just don't get what seems obvious to the author. Note : I'll be working with probabilities, but I guess this would be ...
0
votes
1answer
52 views

Does $L^p$ convergence imply convergence of integrals?

If $L^p-\lim_{t\rightarrow\infty} f_t = f$ ($p > 1$), is it the case that $\lim_{t\rightarrow\infty}\int f_t^p = \int f^p$?
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0answers
74 views

[Measure theory]. Proof of inequality of integrals of simple function

I have a question regarding a proof in my textbook. The theorem is as follows : if {$f_n$} is an increasing sequence of non-negative simple functions and $lim_{n \rightarrow \infty}f_n(x) \geq g(x), ...
2
votes
0answers
27 views

Meaning of a probability on a product defined by a transition probability

If $(\Omega,\mathbb{B},P)$ and $(\Omega',\mathbb{B}')$ are sets with sigma algebras and $P$ is a probability on $(\Omega,\mathbb{B})$. If $\nu(\omega,B')$ is a transition probability from ...
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1answer
181 views

Measure theory questions applied to Second Order PDE

Most of the questions are more measure theory and integration related but I need to give some context, so I will now. Consider the quasilinear 2nd-order partial differential equation ...
2
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0answers
76 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and ...
0
votes
1answer
38 views

measure theory problem

I am stuck at part d) of this problem. Do you see how to show that f is measurable? I must show that $f^{-1}[-\infty,r)$ is measurable for all r. I am not sure how to do it. I would assume that it ...
2
votes
1answer
35 views

Inequality $||f-g|| < \epsilon \Rightarrow |E[f] - E[g]| < \epsilon$

Let $C(X)$ be the space of continuous bounded functions on some metric space $(X,d)$. Can it be shown that if $||f-g||_\infty < \epsilon$ if follows that $| \int f \, \text{d}P - \int g \, ...
3
votes
1answer
87 views

Counter-example to $\{\omega\in\Omega|f(\omega)=g(\omega)\}$ measurable?

If $(\Omega,\sigma)$ and $(\Omega',\sigma')$ are sets with a sigma-algebras, if $f$ and $g$ are measurable functions $(\Omega,\sigma)\rightarrow(\Omega',\sigma')$, can one give a counter-example to ...
2
votes
1answer
102 views

Uncountable disjoint union of measure spaces

Let $(a,b)$ be an interval. Let $(A_i, \Sigma_i, \mu_i)$ be a measure space for each $i \in (a,b)$. Is it possible to put a measure space on the disjoint union $$\bigcup_{i \in (a,b)}\{i\}\times ...
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0answers
75 views

Show that the set $E \subset \mathbb{R}$ being measurable is equivalent to E cleanly dividing any closed interval I (w.r.t the Lebesgue outer measure)

This fact was stated without proof in a lecture, but I am struggling to justify why this is true. If a set E is measurable, then, for any $A \subset \mathbb{R}$ we have $$m^*(A) = m^*(A \cap E) + ...
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0answers
75 views

Sets which are open “modulo a nullset”

A set $A$ is said to have property of Baire there exists an open set $U$ such that $A\triangle U$ is meager. So this says that symmetric difference of $A$ and some open set is small (in the sense of ...
2
votes
2answers
39 views

Showing a set is measurable.

Let $X=Y=[0,1]$, equipped $X$ by giving Lebesgue measure on Borel sigma-algebra and equipped $Y$ by giving counting measure on the power set of $Y$. Define $D=\{(x,x):0\leq x\leq 1\}$ then how do we ...
2
votes
1answer
34 views

$\textbf{X}$-measurable set vs measurable set

According to Bartle's book, The Elements of Integration and Lebesgue Measure, it is written that "Any set in $\textbf{X}$ (measurable space) is called an $\textbf{X}$-measurable set, but when the ...
0
votes
2answers
41 views

Approximation by simple functions

Let $(X,S,\mu)$ a $\sigma$-finite measure space. Suppose that $f:X\to \mathbb{R}$ is measurable and $f\ge 0$, then there exists $s_n:X\to\mathbb{R}$ simple and measurable functions such that $0\le ...
1
vote
1answer
279 views

Proof that the Lebesgue measure is complete

My book states that the Lebesgue measure is complete, but does not give a proof. Is the proof difficult? What I know about the Lebesgue Measure_ It contains $\emptyset$ All finite half open ...
0
votes
1answer
77 views

Another Def. of Cantor set

I have a question about Cantor sets. In particular I want an elegant (especially avoiding as much construction as possible) proof of the existence of an uncountable subset of the reals with Lebesgue ...
2
votes
1answer
80 views

Banach space under the Lip norm

Let $(X,d)$ be a compact metric space. A function $f:x\to \Bbb R$ is said to be Lipschitz continuous if $$\|f\|_d = \sup\left\{\frac {|f(x)-f(y)|}{d(x,y)}:x,y\in X,x\neq y\right\}< \infty.$$ Denote ...
1
vote
1answer
51 views

Expectation of the square of the minimum of iid positive random variables

Let $X_1, X_2$ be i.i.d., positive random variables with $E[X_i] < \infty$ (but $E[X_i^2]$ might be $\infty$). $Y := \min \lbrace X_1, X_2 \rbrace$. I want to show that $E[Y^2] < \infty$. The ...
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0answers
54 views

Question on Bounded Variation involving partitions

If $f\colon[a,b]\rightarrow \mathbb{R}$ and $P=(a=x_0, x_1, ...,x_n=b)$ is any partition of $[a,b]$ let $v_f(P)$ be defined as $$v_f(P) = \sum_{k=1}^n |f(x_k) - f(x_{k-1})| $$ and the total variation ...
2
votes
0answers
102 views

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
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0answers
55 views

measurable set = symmetric difference (NOT A DUPLICATE) [duplicate]

No it has not been answered before at all. Prove that: $E$ Lebesgue measurable $\Rightarrow E = D \Delta N = (D \cup N) \setminus (D \cap N) = (D \setminus N) \cup (N \setminus D)$ for $D$ Borel ...
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1answer
94 views

Question on Functions of Bounded Variation

We say that $f$ is of bounded variation over $[a,b]$ or $f\in BV[a,b]$ if $V_f[a,b] = \sup \sum_{k=1}^n |f(x_k) - f(x_{k-1})| < \infty, $ where the supremum is taken over all possible partitions ...
1
vote
1answer
106 views

$L^p$ continuous mapping problem

Let $f ∈ L_1 ∩ L_4$ (on some measure space). Prove that the function $[1,4] → R$ given by $p → ∥f∥_p$ is continuous. This is a qualifying exam problem and I am not sure what to use. All I can think ...
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1answer
104 views

An analysis qual problem

I am not really sure how to solve the following problem: It is a qual problem. I was thinking of invoking some sort of chang of variables and then holders but not sure it I can. Find all $q ≥ 1,$ ...
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0answers
22 views

Existance of an integer point in a convex set of measure over 2^d

I've been facing the following problem: Let $ A \subset \mathbb{R}^d $ be a convex set such that $ a \in A \Rightarrow -a \in A $ and $ \lambda(A) > 2^d $, where $ \lambda $ is the $ d ...
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votes
0answers
43 views

One doubt regarding measurable function

Let $\phi(x,y)$ be a random variable on the product space $(R,B) \times (S,E)$ and $E|\phi(X,Y)| < \infty$ where $X$ and $Y$ are random vectors taking values in $(R,B)$ and $(S,E)$ respectively. ...
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1answer
53 views

A function that is not in $L_2$

Let $\mathbb{Q}=\{r_1,r_2,\ldots\}$ and $\phi(x)=\dfrac{1}{\sqrt{x}}$ if $x\in (0,1)$, and $\phi(x)=0$ if $x\notin{(0,1)}$. Suppose $\displaystyle\sum_{n=1}^{\infty}a_n$ converges, where $a_n>0$ ...
0
votes
1answer
68 views

Showing $\sigma(X)= \{X^{-1}(B) , B \subset \mathscr{B}\}$

If $X:\Omega \to \mathbb{R}^n $ is any function. $\sigma(X)$ is the smallest sigma algebra generated by all the sets $X^{-1}(U)$, $U\subset \mathbb{R}^n$ open. I am confused as to how you show ...
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4answers
137 views

Some gamma function questions…

I have shown that $\Gamma(a+1)=a\Gamma(a)$ for all $a>0$. But I'd also like to show the following 2 things: 1) Using the previous fact, I'd like to show that $\lim_{a \to 0^{+}}a\Gamma(a) = ...
2
votes
1answer
61 views

“Duality” for weak $L^p$ spaces

Let $1<p<\infty$. Denote by $L^{p,\infty}$ the weak $L^p$ space in $\mathbb{R}^n$ and let $f\in L^{p,\infty}$ where we define the weak $L^p$ quasinorm as $$\|f\|_{p,\infty} = \sup_{\lambda ...
1
vote
1answer
35 views

Decoupling terms in integrals when measure is finite

I am reading about the Levy-Khintchine formula, and a particular result says that if $\nu$ is a finite measure, i.e. $\nu(\mathbb{R})< \infty$, then one can decouple the following integral like ...
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0answers
50 views

Does “$\lim_{s\rightarrow\infty}$ $f_s$ (x) exists a.e.” imply that for $x \in {R^n}$ a.e., all the functions $f_s(x)$ are defined?

I'm trying to solve problem 18 in chapter 6 from Jone's "Lebesgue Integration on Euclidean Space" When I read the problem first, I couldn't understand some condition in this problem. Some statements ...