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

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2
votes
1answer
12 views

$\exists E\in\mathcal{X}$ such that $\mu(E)<\infty$ and $\int_X|f|d\mu<\int_E|f|d\mu + \epsilon$

If $f$ is integrable on space X, then $\forall \epsilon >0$ $\exists E\in\mathcal{X}$ such that $\mu(E)<\infty$ and $\int_X|f|d\mu<\int_E|f|d\mu + \epsilon$ To prove this somehow I need to ...
1
vote
1answer
15 views

In the proof of that Complete convergence is equivalent to convergence a.s. under independence

Complete convergence is equivalent to convergence a.s. under independence In here, it uses the second Borel-cantelli lemma for the converse. But, it is necessary to verify that $(X_n - X)$'s are ...
1
vote
2answers
31 views

Let $D$ be a subset of $L^2[0,1]$ defined in the following way

any help with this problem, it gave me hard time: Let $D$ be a subset of $L^2[0,1]$ defined in the following way: A function $f$ belongs to $D$ if and only if f is equal almost everywhere to a ...
2
votes
0answers
32 views

the weak closure of $S$ is the closed unit ball $B$

I need some help understanding this problem: Let $H$ be an infinite or finite Hilbert space. Show the weak closure of $S$ is the closed unit ball $B$. $S=\{x \in H : ||x||=1\}$ and $B=\{x \in H : ...
3
votes
1answer
27 views

Proving hint given for this Lebesgue integration question

Suppose that $\{n_{k}\}$ is an increasing sequence of positive integers and $E$ is the set of all $x \in (-\pi, \pi)$ at which $\sin(n_{k}x)$ converges. Prove that $m(E)=0$. Hint: For every subset $A$ ...
1
vote
0answers
8 views

How to show: $l^*(A)\le l^*(G_\epsilon)\le l^*(A)+\epsilon $ , $l^* $:outer measure

If $A$ s a Lebesgue measurable subset of $\mathbb{R}$ and $\epsilon\gt 0$ How to show: $\exists$ an open set $G_\epsilon \supset A$ such that $l^*(A)\le l^*(G_\epsilon)\le l^*(A)+\epsilon $, $l^* ...
2
votes
1answer
19 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
3
votes
1answer
32 views

measure $\lambda(E)=0$ or $\lambda(E)=+\infty$

Let $\mu$ be a finite measure, let $\lambda<<\mu$ ($\lambda$ is absolutely continuous wrt. $\mu$) let $P_n$,$N_n$ be a Hahn decomposition for $\lambda-n\mu$. Let $P=\cap P_n$ and $N=\cup N_n$. ...
1
vote
1answer
24 views

What is the name of this measure property?

if we have a function $f \in L^p$ sucht that $||f||_p =1$ and $m$ being a finite measure. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ ...
1
vote
1answer
15 views

$r<s$ and $L^r\subset L^s$

I want to proved the following statement: Let $0<r<s\le\infty$. Then, $L^r(\mu)\subset L^s(\mu)$ if and only if $$ \exists\ \epsilon>0:\forall E\in\mathfrak M,\text{ either } \mu(E)=0\ ...
0
votes
0answers
40 views

Problem of BIG RUDIN: Chapter 3 , Q. 5 . last part [duplicate]

Suppose $\mu (X) = 1$ & $||f||_{r} < \infty$ for some $r>0$ . Show that: $lim_{p \to 0} ||f||_{p} =$ $exp. [\int_{X} {log|f|} d\mu ]$ . Now, there are arising lot of questions: 1) How the ...
1
vote
2answers
69 views

Boundary and closure of a measure zero set is not measure zero?

In $\mathbb{R}^n$, let $E \subset \mathbb{R}^n$ such that $E$ has measure zero. Prove that $\bar{E}$ and $\partial E$ need not have measure zero. I think I have a poor understanding of this. I ...
0
votes
1answer
26 views

$\lambda \ll \mu$, $\mu X <\infty$, then $\lambda X<\infty$

$\lambda \ll\mu$ : $\lambda$ is absolutely continuous w.r.t. $\mu$. and $\mu X \lt\infty$, where $X$ is a space how to show: $\lambda X\lt\infty$
1
vote
1answer
33 views

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ almost uniformly?

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure How to show or give an counterexample: $f_n\rightarrow f$ almost uniformly. We believe it is false. Since both convergences imply there ...
0
votes
1answer
27 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
2
votes
1answer
22 views

$L^p$-limit and pointwise limit

For $p\ge1$, I proved that if $f_n\stackrel{L^p}{\to} f$ and $f_n\to g$ a.e then $f=g$ a.e. But, how about the case $0<p<1$? Is it also true?
0
votes
1answer
23 views

measurable function and composition of function

Show that if $f$ is a measurable function and $g$ is a continuous function on $\Bbb R$ then $g\circ f$ is measurable. please tell me how to prove it !
0
votes
1answer
43 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
2answers
35 views

$\int_2^\infty \left(\frac1{x\log^2x}\right)^p\mathrm dx$ diverges for p>1

I see this question and the answer by joriki. However I cannot understand joriki's argument that $$\int_2^\infty \left(\frac1{x\log^2x}\right)^p\mathrm dx$$ diverges for p>1. So I try to show that ...
2
votes
1answer
57 views

how many unit balls are needed to cover a unit sphere (1-dense set on a unit sphere)

There is an exercise in a geometry textbook to prove that "any $1$-dense set in the unit sphere $S^{n-1}$ has at least $\frac{1}{2}e^{n/8}$ points". It is supposed to be easy. A set $T$ is ...
0
votes
1answer
32 views

$f_n\rightarrow g$ in $L_1$ and $f_n\rightarrow h$ in $L_2$ .Then $g=h $almost everywhere

$f_n\rightarrow g$ converges in $L_1$ and $f_n\rightarrow h$ converges in $L_2$ how to show: $g=h$ almost everywhere Attempt: convergent in $L_1$ implies convergent in $L_2$. then by triangle ...
0
votes
1answer
28 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} ...
2
votes
0answers
19 views

A sebset of $\Bbb C^2$

Let $K\subset \Bbb R$ have Lebesgue measure $0$. Then I think that the set $$ \Omega:=\{(z, w)\in \Bbb C^2: |zw|\in K\} $$ has $4$ dimensional measure $0$. If so, how to prove(or shortly explain) it? ...
1
vote
3answers
46 views

What is the German word for “pre-measure”?

I'd like to know how to translate “pre-measure” to German. Unfortunately, the wiki article on pre-measure doesn't have a German version.
1
vote
1answer
12 views

Finitely additive function bounded by a measure…

have an elementary measure theory question here I can't seem to get. Suppose $\mu$ is a a measure, and $\nu$ is a finitely additive nonnegative set function such that $\nu(A)\le \mu(A)$ for all $\mu$ ...
4
votes
1answer
48 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
2
votes
0answers
37 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
0
votes
2answers
47 views

show a set is lebesgue measurable

For lebesgue measure, is it true that the union/intersection of measurable sets is also measurable (finite or infinite unions or intersections)? But it's not true for subsets? (i.e.,a subset of a ...
1
vote
0answers
18 views

Sequences of refining partitions of a measurable space

Let $(\Omega,\mathcal F)$ be a measurable space. For $k\in\mathbb N$ let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that each $\mathcal F_k$ is generated by a finite ...
-1
votes
1answer
44 views

equi integrablity

See page 5 here. Let $\Omega$ be an open subset of $\mathbb{R}^n$, and let $(f_n)$ be a sequence of measurable functions, $f_n \in L^1(\Omega)$, which is bounded in $L^1(\Omega)$ ($f_n \in ...
1
vote
1answer
28 views

Finite subcover of pairwise disjoint open intervals

I have the following exercise: Prove that if $X$ is a countable compact subset of $ \mathbb{R}$, then for any $\varepsilon>0$ there is a finite collection of pairwise disjoint open intervals ...
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
40 views

there is a measurable function $f$ on $X$ such that $|{f(x)}|=1$ for a.a $x \in X$ and $\nu(E)=\int_Efd|{\nu}|$ for any $E \in \mathfrak{M}$

any hints on this problem: Let $\nu$ be a finite signed measure on a measure space $(X, \mathfrak{M})$ and let $|{\nu}|$ be its total variation, prove that there is a measurable function $f$ on $X$ ...
7
votes
0answers
178 views
+400

$\int_0^1 [ \frac{1}{x(x-1)} (2\mathrm{Li}_2(\frac{1-\sqrt{1-x}}{2})-\log(\frac{1+\sqrt{1-x}}{2})^2 ) -\frac{\zeta(2)-2\log^2 2}{x-1} ]dx$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
0
votes
1answer
57 views

Let $\{f_n\}_{n=1}^\infty$ be non-negative functions and $f_n \to f$ then $f \geq 0$

I have trouble with this question: Let $\{f_n\}_{n=1}^\infty$ be a sequence of non-negative functions in $L^2(0, 1)$, and suppose that $f_n$ converges to a function $f$ in the norm of $L^2(0, 1)$. ...
0
votes
0answers
16 views

Absolute continuity of weak-* limit of measures.

Let $\{\mu_i\}_i$ be a sequence of measures on $\mathbb{R} \times \mathbb{R}^m$ such that $$ \int^u_0 \left(\int_{\mathbb{R}^m}\max(\mu_i\log(\mu_i),0)dx \right) ds < C $$ for all $i$. How can one ...
0
votes
1answer
13 views

Showing if functions are equal almost everywhere then

Say $f_n$ and $g_n$ are measurable $f_n=g_n$ almost everywhere for $n=1,2,3,...$ then how can we show that $\sup f_n = \sup g_n$ almost everywhere? I have tried to show that: $$m(\{\sup f_n \neq ...
0
votes
3answers
66 views

$∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$!

Hi I was thinking about a problem and have a question: we know that if $f∈C([0,1])$ for which $∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$! Now my question is: Do we still have the same when we ...
2
votes
1answer
18 views

Outer measure induced by a measure

Let $(X, \mathfrak{M}, \mu)$ be a measurable space. Let $\mu^* \ : \ 2^X \ni Y \rightarrow \mu^*(Y)= \inf \{\mu(A) \ | \ A \in \mathfrak{M}, Y \subset A\}$. Prove that $\forall Y \subset X \ \ ...
0
votes
0answers
31 views

Proving a theorem in Ergodicity

I read the theorem stated below on Wikipedia (http://en.wikipedia.org/wiki/Ergodicity#Formal_definition). But I do not understand how to prove the equivalence of these different definitions.Any hints ...
3
votes
0answers
36 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
1
vote
0answers
68 views

Proof that a function is measurable

Suppose $f$ is a joint probability density function of random variables $X$ and $Y$. $Y$ is integrable. I need to prove that the function $g(x) = \int_{\Bbb R} f(x,y)ydy$ is measurable function. I ...
2
votes
0answers
23 views

Set of measure zero and $C^{1}$ functions

Does a $C^{1}$ function map a set of measure zero into a set of measure zero?
0
votes
1answer
18 views

Certain property of convex functions…

I come to you with yet another qualifying problem we can't seem to solve... Let $f:$ $(0,\infty) \to \Bbb R$ be convex, and let $\lim_{x \to 0}f(x)=0$. Show that $g(x)$ = $f(x) \over x$ is increasing ...
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 ...
4
votes
1answer
110 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 ...
-2
votes
0answers
30 views

A basic question on absolute continuous measures [on hold]

suppose that $\nu$ and $\mu$ are $\sigma$-finite measures on $(\Omega, F)$ and $\nu \equiv \mu$ (i.e. they are absolutely continuous with respect to each other) consider the set $\{\omega: ...
0
votes
1answer
17 views

Finding the “most” continuous representative of a class of functions equal almost everywhere.

In measure theory, we consider functions to be basically the same if they are equal almost everywhere. It seems crazy, though, to choose any of these as the representative when doing calculations. Why ...
-1
votes
1answer
20 views

an example of almost uniformly convergence [closed]

$x^n\rightarrow 0 $ on [0,1]. does $x^n$ converge almost uniformly to zero. if we take out point 1 or if take out a small set $[1-\epsilon,1]$ why does not it a.u. converge when we take only point ...
0
votes
1answer
35 views

measurable subset of nonmeasurable set

show that if E is measurable and E⊂P where P is nonmeasurable set in [0,1), then m(E)=0. Can one please tell how to start .. and I have one more question: is the union of m'ble set and non-m'ble set ...