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

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0
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6 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^* ...
3
votes
1answer
29 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 $N=\cup N_n$. ...
2
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1answer
17 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 ...
1
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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\ ...
1
vote
1answer
23 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 \ \ ...
6
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0answers
161 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
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1answer
32 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 $X$ ...
2
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0answers
80 views
+400

Integral $\int_0^{\pi/3}\log\bigg( \frac{1+2\cos\theta}{2}+\sqrt{\left( \frac{1+2\cos\theta}{2} \right)^2-1}\ \bigg)d\theta.$

Hi I am trying to calculate this integral I given by $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)d\theta. $$ ...
0
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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
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1answer
26 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 ...
0
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0answers
28 views

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

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
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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
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2answers
65 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 ...
1
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1answer
63 views

convergence in L^{1} strong

I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies ...
1
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1answer
28 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 ...
2
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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?
2
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2answers
42 views

Outer measure and Caratheodory's criterion

Suppose $m^*$ is an outer measure in Caratheodory's sense on the space $X$, which satisfies $m^*(\emptyset)=0$, $A\subseteq B\implies m^*(A)\le m^*(B)$, and $m^*(\bigcup_n A_n)\le\sum m^*(A_n)$. We ...
0
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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
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0answers
34 views

how many spots are enough to cover a 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 should be easy but I am stuck with it. A set ...
0
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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
2answers
34 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 ...
0
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1answer
27 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
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1answer
31 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 ...
2
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1answer
99 views

Another way to prove that $\sqrt{f}$ is Riemann integrable

[This is an exercise of Measure and Integration. I am repeating this in my vacation.] Define $f:[a,b] \rightarrow R$ a function such that $f(x) \geq 0$ over $[a,b]$ and f is R-Integrable in [a,b]. ...
2
votes
1answer
50 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 ...
1
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1answer
84 views

munkres analysis integration question

Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2 \to \mathbb{R}$ be defined by setting $f(x,y)=0$ if $y \neq x$, and $f(x,y) = 1$ if $y=x$. Show that $f$ is integrable over $[0,1]^2$.
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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? ...
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3answers
45 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.
6
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0answers
550 views
+200

Existence of a Strictly Increasing, Continuous Function whose Derivative is 0 a.e. on $\mathbb{R}$

This proof is almost done except for the step of showing that the function's derivative is $0$ a.e. Let $I = \{[p_n, q_n]\}$ denote the set of all closed intervals in $\mathbb{R}$ with rational ...
2
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1answer
254 views

Prove that set has zero Jordan content iff its closure has measure 0

Prove that set has zero Jordan content iff its closure has measure 0. I am having trouble with both directions , any tips would be great. THanks!
1
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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 $\mu$ ...
0
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2answers
45 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 ...
3
votes
0answers
38 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} ...
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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 ...
2
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0answers
35 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
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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 ...
0
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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 ...
0
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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} \ ...
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 ...
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1answer
27 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 ...
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3answers
65 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 ...
0
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1answer
55 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 $L^2(0, 1)$. ...
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0answers
15 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
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1answer
12 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 ...
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 \ \ ...
4
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0answers
206 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
1
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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
22 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
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1answer
21 views

Sufficient condition for equality of two radon measures

Let $ X $ be a locally compact Hausdorff space and let $ \phi_1 $ and $ \phi_2 $ be two Radon measures on X (outer measure means measure and the definition of Radon measure that I am assuming can be ...
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 ...