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

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4
votes
0answers
99 views
+400

$\int_0^1 [ \frac{1}{x(x-1)} (2Li_2(\frac{1-\sqrt{1-x}}{2})-\log(\frac{1+\sqrt{1-x}}{2})^2 ) -\frac{\zeta(2)-2\log^2 2}{x-1} ]dx$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2Li_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) -\frac{\zeta(2)-2\log^2 ...
2
votes
0answers
61 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
votes
1answer
41 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
1answer
24 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
votes
0answers
26 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
votes
1answer
43 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
2answers
62 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
vote
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
vote
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
votes
1answer
18 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
votes
2answers
41 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
votes
1answer
25 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
0answers
33 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
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
2answers
33 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
votes
1answer
25 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
votes
1answer
30 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
votes
1answer
98 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
49 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
vote
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 $[0,1]^2$.
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
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.
5
votes
0answers
544 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
votes
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
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 $\mu$ ...
0
votes
2answers
43 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} ...
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 ...
2
votes
0answers
32 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
1answer
33 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
votes
0answers
30 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
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} \ ...
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
1answer
25 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
3answers
64 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
votes
1answer
29 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 $X$ ...
0
votes
1answer
53 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
14 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
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
votes
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
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
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
votes
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 ...
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 ...
-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: ...
4
votes
1answer
107 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 ...
1
vote
1answer
62 views

measure space problem.

Let $(\Omega,\mathcal{F},\mu) $ be a probability space. Let $\delta>0$ and for each $n\in \mathbb{N}$. Let $A_n \in \mathcal{F}$ satisfy $\mu(A_n)\ge\delta$. Prove that the set $A_\infty $ ...
0
votes
1answer
16 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 ...