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

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0
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1answer
17 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 ...
0
votes
1answer
14 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
-4
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0answers
18 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.
0
votes
1answer
25 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
22 views
+50

Linear transform of a strictly stationary time series

First, let me clarify what I mean by a strictly stationary time series. Let $(X_t)_{t\in \mathbb{Z}}$ be a sequence of random variables on some probability space. If it holds that $$(X_t, ...
0
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0answers
22 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. ...
1
vote
0answers
18 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 ...
3
votes
2answers
479 views

Uniformly Continuous Like Property of the Integration on Measure Space

This is the Excercise 1.12 of Rudin's Real and Complex Analysis: Suppose $f\in L^1(\mu)$. Prove that to each $\epsilon>0$ there exists a $\delta>0$ such that $\int_{E}|f|d\mu<\epsilon$ ...
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 ...
0
votes
1answer
14 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, ...
-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 $Q^n$.
2
votes
0answers
25 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
15 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
27 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. ...
2
votes
0answers
54 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) ...
0
votes
0answers
18 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
2answers
36 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 ...
2
votes
1answer
11 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 ...
1
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0answers
23 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 there exists a function $f$ on $\Omega$ that satisfies $$ P(A) = \int_A f ...
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.
2
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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
26 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 ...
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 ...
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
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 ...
-2
votes
1answer
19 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
22 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
37 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
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
32 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 ...
1
vote
0answers
53 views

Prove that $(L[a,b], ||\cdot ||)$ is a normed linear space

I want to prove that $(L[a,b], ||\cdot ||)$ is a normed linear space with norm $$||f(x)|| = \int_a^bx^2|f(x)|dx.$$ First, let $\lambda \in \mathbb{R},$ then it is clear that by properties of ...
-3
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0answers
30 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 + ...
-1
votes
1answer
20 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 ...
3
votes
1answer
54 views

Part of proof of the set of continuous integrable functions is dense in $L^1(\Bbb R)$

I want to prove: If $g$ belongs to $L(\Bbb R, \Bbb B, \lambda)$ and $\epsilon\gt 0$, then there exists a continuous function $f$ such that $\Vert g-f\Vert_1=\int \lvert g-f\rvert \,\text{d}\lambda \lt ...
7
votes
2answers
119 views

How much larger is the $\sigma$-algebra than the algebra in Caratheodory extension?

Given a 'measure' $\lambda$ on an algebra $\mathcal{A}$ of sets, Caratheodory gives a way to extend this $\lambda$ to a $\sigma$-algebra. The idea is we define an outer measure (on all subsets) ...
4
votes
1answer
39 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

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
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. ...
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0answers
20 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
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1answer
68 views

Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

I am reposting a question from Math Overflow, because it seems it gets no attention. Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak ...
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 ...
0
votes
0answers
20 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
1answer
20 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 ...
3
votes
0answers
40 views

If $\{T_n\} \to T$ and $\{u_n\} \to u$, then $\{T_n(u_n)\} \to T(u)$.

Let $X$ and $Y$ be normed linear spaces. Define $$L(X,Y) = \{T:X \to Y \ \big | \ T \text{ is bounded}\}.$$ Let $\{T_n\} \to T$ in $L(X,Y)$ and $\{u_n\} \to u$ in $X$, then $\{T_n(u_n)\} \to T(u)$ ...
1
vote
0answers
32 views

Riemman-Stieltjes Integral Exercise

The truth is that I have no experience with the integral of Riemann-Stieltjes and developing a Bayesian inference problem in the book "Mathematical Statistics" by Shao, appears one of these steps, I ...
1
vote
1answer
26 views

Justifying the differentiation property of the Fourier transform

Let the Fourier transform of $f\in L^1(\Bbb R)$, denoted by $\mathcal{F}f$, be defined as $$ \mathcal{F}f(y) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ixy} f(x)\,dx.$$ An oft-quoted result ...
1
vote
1answer
23 views

Convergence of sequence of random variables 2

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
1
vote
0answers
20 views

Entropy of a 2-dimensional function versus 1-dimensionl function.

I am a novice in information theory so this is more of a question seeking pointers to ideas/references to think further on the thought. I want to make concrete the idea that a function of two ...
2
votes
2answers
43 views

If a measurable function $f$ has zero integral over every measurable set *of finite measure*, then $f=0$ a.e.?

Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a regular measure on $X$. Suppose that $g : X \to \Bbb C$ belongs to $L^{\infty}(X)$. My question is : Is it sufficient to assume ...