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

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2
votes
0answers
24 views

Show that $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R}), \int_{\mathbb{R}} f dx=0\}$ is dense in $L^p(\mathbb{R})$

Show that $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R}), \int_{\mathbb{R}} f dx=0\}$ is dense in $L^p(\mathbb{R})$. Is the statement true if $\mathbb{R}$ is replaced by $[0,1]$? Also what can we say ...
0
votes
0answers
13 views

$E$ has finite exterior measure show it is measurable iff $E=(S\cup N_1)\setminus N_2$

Let $|E|_e< \infty$ then I want to prove $E\subset \mathbb R^n$ is lebesque measurable iff for each $\epsilon>0 $ we can write $$E=(S\cup N_1)\setminus N_2$$ where $|N_1|_e,|N_2|_e<\epsilon$ ...
1
vote
1answer
11 views

Show that $\mu(x\in X: \vert f(x)\vert\geq\alpha\})\leq\frac{1}{\varphi(\alpha)}\int{\varphi\circ\vert f\vert d\mu}$

Let $(X,\tau,\mu)$ a measure space and $\varphi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ a function no increasing (i.e., $f(x)\leq f(y)$ is $x\leq y$) and more than zero on $(0,\infty)$. If $f\in$ is such ...
0
votes
0answers
23 views

Inequality for integral of complex valued functions

assume that $f$ is a complex-valued function acting on some probability space $(X,m)$ and $g$ a non-negative function defined on a same space such that $$ \lvert \int_A f \, dm \rvert \le \int_A g \, ...
1
vote
0answers
13 views

Gaussian process via RKHS construction: joint measurability comes for free?

Billingsley's probability and measure and others show the joint measurability of the Brownian motion using the continuity of paths. Makes me wonder if we can say it before saying the continuity, if we ...
1
vote
1answer
13 views

Showing that $\mathbb{E}[ \frac{S'_n}{n \log_2 n}]$ converges to 1 for a problem related to geometric distribution

We define independent random variables $X_i$ which follow the law $P(X_i = 2^k)=\frac{1}{2^k}$. We set $S_n = X_1+ \cdots +X_n$. Since we cannot apply the law of large numbers to $S_n$, we define ...
0
votes
1answer
64 views

Shortest Proof of Lebesgue Dominated Convergence Theroem ( 5 lines) without using Fatou's lemma

If {$G_n$} is a sequence of bounded measurable functions and $ | G_n | \le M $ where M is a positive real number $\lim\limits_{n\mapsto \infty} G_n =F$ on a bounded measurable set E , $\epsilon> ...
0
votes
0answers
12 views

Outer measure of a nested sequence of non-measurable sets

Let $\bigcup_{n=1}^\infty E_n=E$ and $ E_{n} \subseteq E_{n+1} $ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) = \mu^*(E) $ even if each $E_n$ is a non-measurable set, where $\mu^*$ is outer ...
1
vote
1answer
14 views

Projection of measure with bowen - walters metric.

Given $X$ a compact metric space, $f:X\to X$ be a homeomorphism and consider the quotient space $Y^{1,f}=(X\times [0,1])/\sim$, where $(x,1)\sim(f(x),0)$ for all $x\in X$. Let $d^{1,f}$ be the ...
4
votes
2answers
49 views

Prove that $F(x,y)=f(x-y)$ is Borel measurable

Suppose $A$ is a subset of $\Bbb R$, let $s(A)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in A\}$. I already showed: If $A\in \Bbb B$ (Borel measurable set), then $s(A)\in \Bbb B \times \Bbb B$. I want ...
0
votes
0answers
20 views

Checking measurability on open sets

This is exercise 5 of section 53 in Halmos' Measure theory. Let $X$ be a locally compact Hausdorff space and $\mu^{*}$ an outer measure on the hereditary class of $\sigma$-bounded sets. Suppose ...
2
votes
1answer
52 views

If $f:\mathbb{R}\to[0, \infty)$ (uniformly) continuous and $f \in L^1$, then $\lim_{x\to\pm\infty}f(x)=0$?

I'm learning about measure theory and need help with the following questions: True or False (justify): $(1)$ If $f:\mathbb{R}\to[0, \infty)$ measurable and $f \in L^1$, then ...
-4
votes
0answers
24 views

Measurability properties of functions

If f is measurable prove that any positive integral power of f is also measurable. Note: f is Lebesgue measurable. I wanted a proof from the definition of Lebesgue measurability of a function.
1
vote
1answer
21 views

If every borel measurable function continuous in compact metric space then metric space is finite

Let $(X,d)$ be a compact metric space. Suppose every Borel measurable function $f : X \to \mathbf{R}$ is also continuous. Show that X is a finite set. Thank you for your time
1
vote
0answers
21 views

Uniform integrability of a sequence of random variables defined by a recursive relation

I have an i.i.d sequence $(u_j)_{j\in \mathbb{Z}_+}$ with zero mean and finite variance, say $\sigma^2$. Furthermore, I have another random variable $X_0$ (defined on the same probability space) which ...
0
votes
1answer
16 views

Sum of non-finite measurable functions

Suppose $f(x) = \infty$ and $g(x) = -\infty$ for all real $x$. We know that both $f$ and $g$ are measurable. This is because the set $\{ f > a \}$ is measurable for all $a$. Same for $g$. However, ...
0
votes
1answer
22 views

Sequence of integrable functions that converge a.e but not their integral

I'm trying to find an example of a sequence of integrable functions $(f_n)_{n\in\mathbb{N}}$ such that $f_n\rightarrow 0$ a.e. (almost everywhere) but $\int f_n\nrightarrow 0$. Should be easy, but I ...
-2
votes
0answers
15 views

set of positive measure contains a point which lies in irrational distance from all points of $Q^n$ [on hold]

Let $A$ - measurable subset of $R^n$ having positive Lebesgue measure. Prove that $A$ contains a point which lies in irrational distance from all points of $Q^n$.
0
votes
0answers
30 views

Does absolute continuity imply no stochastic domination?

I have an interesting question which goes as follows: Let $F_0$ and $F_1$ be two (nominal) distributions defined on a measurable space $(\Omega.\mathscr{A})$, where $\Omega$ is continuous. ...
2
votes
0answers
27 views

Lebesgue-integrability of derivatives

Let $f:\mathbb R\to\mathbb [0,\infty)$ be a non-negative, twice-differentiable function. Suppose that $\int_{-\infty}^{\infty}f(x)\,\mathrm dx<\infty$, $\int_{-\infty}^{\infty}|f''(x)|\,\mathrm ...
-1
votes
0answers
16 views

Prove that polynomial functions are not Lebesgue integrable in $\mathbb{R}$ [on hold]

Given a polynomial function $f\neq0$, prove that it's not Lebesgue integrable in $\mathbb{R}$.
2
votes
0answers
29 views

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
0
votes
0answers
22 views

Necessary and sufficient condition for lebesgue measurability

I'm trying to determine whether the following statement is true or not: If $E\subseteq \mathbb{R}$ is a set with $\lambda^{*}(E)$ finite ($\lambda^{*}(E)$ is the lebesgue exterior measure of $E$), ...
2
votes
1answer
12 views

Exact value of Hausdorff measure of middle-third Cantor set

Is there any result about the exact value of $\log_3 2$-dimensional Hausdorff measure of the middle-third Cantor set? And is there any fractal (in $\mathbb R^n$) which is not contained in a ...
2
votes
2answers
42 views

Part of proof to show Lebesgue-lebesgue measurable

I want to prove the following: Suppose $E$ is a subset of $\Bbb R$, let $\gamma(E)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in E\}$. If $E\in \Bbb B$ (Borel/Lebesgue measurable set), show that ...
1
vote
0answers
29 views

Specific Radon-Nikodym Derivative Interpretation

Suppose $(\Omega, \mathcal{F}, P)$ and $(\Omega, \mathcal{F}, Q)$ are two probability spaces. The Radon-Nikodym theory says that if $P$ is absolutely continuous with respect to $Q$, then there exists ...
2
votes
0answers
15 views

Approximate an integrable function using a simple function (Proving existance)

Let $f \in L^1(\mathbb{R})$, and let $\epsilon > 0$. Show that exists simple function $g=\sum_{k=1}^{n}c_k 1_{A_k}$, such that, $$\int_\mathbb{R} |f(x)-g(x)|dx \leq \epsilon$$,and such that $n \in ...
0
votes
0answers
39 views

Show that $\iint_{X \times Y}\varphi(x)k(x,y)\psi(y) d(\mu \times \nu)=\int_Y \Big[\int_X\varphi(x)k(x,y)d\mu \Big] \psi(y) d\nu$

Let $k(x,y)$ be a bounded Borel measurable function on $X \times Y$ and let $\mu$ and $\nu$ be Radon measure on $X$ and $Y$ i. Show that $\iint_{X \times Y}\varphi(x)k(x,y)\psi(y) d(\mu \times ...
1
vote
0answers
9 views

a Radon measure is determined by a linear functional on the space of continuous functions with compact support

Where can I read a proof of the following statement (if it is true). Let $X$ be a Hausdorff topological space (not necessarily locally compact) and let $\mu$ be a Radon (i.e. locally finite and inner ...
0
votes
1answer
31 views

Lebesgue integrals and polinomial functions

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non zero polinomial function. Prove that $f\notin\mathbb{L}(\mathbb{R})$.In other words is not Lebesgue integrable.
0
votes
0answers
5 views

Prove that the following measures are product measures in $\mathbb{R^2}$

Let $\mu : \mathbb{P}(\mathbb{R}) \rightarrow [0, \infty]$ be a measure over the power set of the real numbers given by $\mu (A) =0$ if $A$ is a countable subset of $\mathbb{R}$ and $\mu (A) = \infty$ ...
0
votes
0answers
9 views

If $| \int_A fdλ| ≤ λ(A)$ for $A \subset [-1,1]$. then range of f contained in [-1,1]

Let f : [−1, 1] → R be a continuous function. Let λ be the Lebesgue measure on [−1, 1]. Suppose $| \int_A fdλ| ≤ λ(A)$ for all measurable sets A ⊆ [−1, 1]. I want to show that the range of f is ...
0
votes
1answer
27 views

Inequality of Lebesgue integrals

Let $f,g\in\mathbb{L}(E)$. Suppose that $f\leq g$ and $A:=${$x\in E| f(x)<g(x)$}. Prove that $\int_{E}f<\int_{E}g$ if and only if $A$ has positive measure.
-2
votes
1answer
20 views

Integrable functions in $\mathbb{R}$? [on hold]

Let $f\in\mathbb{L}(\mathbb{R})$ integrable. If $a>0$, prove that $f^{-1}((a,+\infty))$ has finite measure.
2
votes
1answer
23 views

Continuous functions are locally integrable?

If $K\subset\mathbb{R}$ is compact and $f:K\rightarrow\mathbb{R}$ continuous then $f\in\mathbb{L}(K)$. In other words $f$ is integrable in $K$. So far i know that since $f$ is continuous then $f(K)$ ...
4
votes
1answer
38 views

What is a Dynkin system? ($\lambda$-system)

Until recently, all my knowledge of measure theory and Lebesgue integration are from Rudin's book, which focuses solely on the Lebesgue measure, its construction and nothing else. I have just put my ...
2
votes
0answers
57 views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
2
votes
1answer
26 views

Application of Dominated Convergence Theorem help finding a Dominating function

$$\lim_{n\to\infty}\int_0^\infty \frac{n\sin(x/n)}{x(1+x^2)}$$ I wish to use the Lebesgue Dominated Convergence theorem to solve this, but I'm having trouble finding a dominating function, $g(x)$. ...
0
votes
1answer
35 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$ a.s.? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
-3
votes
0answers
31 views

If $f = g +h$ then $\int_E f = \int_E g + \int_E h$ is independent of the choices of $g$ and $h$ [on hold]

Let $f$ be a measurable function on $E$ which can be expressed as $f = g +h$ where $g$ is a finite and integrable function over $E$ and $h$ is nonnegative over $E.$ Define $\int_{E} f = \int_E g + ...
2
votes
1answer
45 views

Proof that the Cardinality of Borel Sets on $\mathbb R$ is $c$ without using the ordinals .

I'm trying to prove that cardinality of Borel sets is $c$ without using the concept of Ordinal number ! I know that the Cardinal of Borel sets are greater than $c$ because of every point in $\mathbb ...
-1
votes
1answer
21 views

Discrete measure and piecewise function

Hi guys, can anyone please help me with why we can introduce a sectionally constant function that has support $\lambda_i, i \in \mathbb{N}$. I do not understand why we can do the part I marked with ...
0
votes
1answer
17 views

Holders Inequality: Suppose $\int_{0}^\infty x^{-2}|f|^5 dx < \infty$. Prove that $\lim_{t \to 0} t^{-\frac{6}{5}} \int_0^t f(x)dx = 0$

I discovered last night that I have an error in my proof to the following problem and I need help fixing it (or need a new solution) $$ \text{Suppose that} \int_{0}^\infty x^{-2}|f|^5 dx < \infty. ...
0
votes
1answer
32 views

Does finite expectation imply finite essential supremum?

I have a real valued function $f$ with the property that $$\mathbb{E}\big[f(X)\big] = \int f(x)\ d\mathbb{P}(x) \leq c$$ for some $c > 0$. Does this imply $$ \operatorname{ess sup } \|{f(X)}\|^2 ...
0
votes
1answer
13 views

Inequality regarding measure of function and integral of function

Let $(X,\Sigma,\mu)$ be a measure space. Let $f$ be a measurable function and $t > 0, t\in \mathbb{R}.$. Denote: $$C_f(t) = \mu \{x \in \Omega : |f(x)| \geq t \}.$$ In the first part of ...
4
votes
1answer
45 views

Question on proving tight sequences.

I was just wondering how you would go about showing that a sequence of random variables is a tight sequence. For example suppose $X_{n}$ is distributed Exponentially($\lambda_n$) how would I show that ...
0
votes
1answer
31 views

$C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$

Show that $C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$, where $\Omega$ is an open subset of $\mathbb{R^n}$. My try: Let ...
0
votes
0answers
22 views

Show that map is Borel

I was reading this paper of L. Ambrosio, S. Di Marino and G. Savare http://arxiv.org/pdf/1311.1381.pdf Under the definition of plans with barycenter in $L^q$ authors say that the map ...
0
votes
0answers
21 views

Minkowski Inequality when either $||f||_p = 0$ or $||g||_p = 0$.

I will recall that Minkowski Inequality says the following: Let $E$ be a measurable set and $p \in [1,\infty]$. If the functions $f$ and $g$ belong to $L^p(E)$, then so does their sum $f + g$ and, ...
0
votes
1answer
21 views

If $\{u_n\} \to u$ and $\{v_n\} \to v$. Show that $\{\rho(u_n, v_n)\} \to \rho(u,v)$

Let $(X,\rho)$ to be a metric space in which $\{u_n\} \to u$ and $\{v_n\} \to v$. Show that $\{\rho(u_n, v_n)\} \to \rho(u,v)$ Proof: Suppose $\{u_n\} \to u$ and $\{v_n\} \to v$. This means that ...