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

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0
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1answer
15 views

Real Analysis, Folland Theorem 1.9, extention of a measure to a complete measure

I have posted this theorem before but I am re-posting it again because I have a different question. Theorem 1.9 - Suppose that $(X,M,\mu)$ is a measure space. Let $\mathcal{N} = \{N\in M:\mu(N) = ...
0
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0answers
14 views

Real Analysis, Folland Proposition 1.7 elementary family

Definition - An elementary family is a collection $\varepsilon$ of subsets of $X$ such that i.) $\emptyset\in \varepsilon$ ii.) if $E,F\in \varepsilon$ then $E\cap F\in \varepsilon$ iii.) if $E\in ...
1
vote
1answer
25 views

$\mu$-completion of $\sigma$-algebra

Rudin shows that $\mu$ is well-defined on $\mathfrak{M}^*$. But this little bit confuses me since well-defined of function I understand in the following meaning: $f:X\to Y$ is well-defined if $x=y$ ...
1
vote
2answers
59 views

Question about proof extending measure to complete measure

I am looking through a proof in Folland, for Theorem 1.9, which states: Suppose that $(X, M, \mu)$ is a measure space. Let $N = \{N' \in M : \mu(N') = 0\}$ and $M' = \{E \cup F : E \in M' \text{ and ...
1
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1answer
23 views

The Second Hearts Problem

Intro: According to the last part of these lecture notes, if we have a standard deck of playing cards and turn cards until the first heart appears, the probability that the next card is a heart is ...
2
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1answer
69 views

Uniformly integrable implies integrable?

The term "uniformly integrable" sounds (to a layman like me) to be stronger than integrable. Just like how uniformly convergent is stronger than simply being convergent. However, from the definition ...
0
votes
1answer
155 views

Borel measurable functions

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a Borel measurable function and let $h:\mathbb{R}^2 \to \mathbb{R}$ be defined by $h(x,y)=f(x)+f(y)$. Prove that $h$ is Borel measurable
-1
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0answers
20 views

Measurable functions on characteristic functions? [on hold]

Let $E\subset\mathbb{R}$ a measurable set. We define the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ where $f(x,y)=\mathcal{X}_{y+E}(x)$. Prove that $f$ is measurable in $\mathbb{R}^2$.
1
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1answer
22 views

Examples of functions where the Lebesgue integral as a measure is complete.

Let $f\in\mathcal{M}(\mathbb{R})$ non negative. For each $E\subset\mathbb{R}$ measurable we define $\mu_f(E)=\int_{E}f$. Prove (a) $\mu_f$ is a measure in $\mathcal{M}$ (b) Give an example of a ...
8
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2answers
1k views

Stuck on existence proofs involving measurability and simple functions

Some classmates and I have been working through a sequence of problems in Royden's real analysis text, which are in the chapter on Lebesque measurable functions revolving around the Sequential ...
1
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0answers
8 views

stochastic variable constant on atoms = measurable?

Let $(\Omega, F, P)$ be a probability space, let $P_1,...,P_n$ be a partition of $\Omega$, let $F_0,...,F_n$ be a filtration of $F$, and let $P_t = \{ \text{atoms of } \ F_t\}$. I don't understand ...
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2answers
47 views

Infinite product probability spaces

Does the infinite product of probability spaces always exist (using the sigma algebra that makes all projections measurable and providing a probability measure on this sigma algebra)? I always ...
1
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0answers
24 views

Lebesgue dominated convergence theorem from RCA Rudin

Hello! This theorem from Rudin's RCA book. I would like to clarify some moments: $1)$ Since $\limsup\limits_{n} v_n\leqslant 0$ where $v_n=\int \limits_{X}|f_n-f|d\mu$. We get $0\leqslant ...
0
votes
1answer
19 views

A version of one-sided Chebyshev's inequality

Let $X$ be a real random variable with mean $\mu > 0$ and variance $\mu^2$. Does there exist a non-trivial upper bound on the probability $\Bbb P(X < 0)$ or is there a counterexample that shows ...
0
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0answers
38 views

Small filters are measurable

i want to show, that a filter $\mathcal{F}$ on $\omega$ (considered as a subset of $2^\omega$), which is small, is measurable. I found a lemma (without proof), that every small set is null. So, if ...
0
votes
1answer
32 views

Lebesgue integral, path connected and compact

Let $K \subseteq \mathbb R^d$ be path-connected and compact and $f:K\to\mathbb R$ continuous. How can I show that there is a $\xi\in K$ such that $$\int_Kfd\lambda^d=f(\xi)\lambda^d(K)$$ where ...
2
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0answers
42 views

How can higher dimension spaces have smaller unit balls? [duplicate]

I have recently been shown the gamma function and a few of its uses, and one of those is calculating the measure of the unit ball in $\Bbb{R}^n$. The formula shows the measure going to zero (rather ...
0
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0answers
22 views

Reference for function being a distribution function

Let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$. Let $\nu$ be a probability measure on $(\mathbb R, \mathcal B(\mathbb R))$. Lastly let $F:\mathbb R \rightarrow [0,1]$ ...
2
votes
2answers
33 views

Taking two times the sequential closure of the set of continuous functions in the topology of pointwise convergence?

Consider the unit interval $I=[0, 1]$ and assume that the function $f\colon I\to \mathbb R$ satisfies $$ f(t)=\lim_{n\to \infty} f_n(t), \qquad \text{for all }t\in I $$ where $$ f_n(t)=\lim_{j\to ...
2
votes
2answers
118 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. ...
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0answers
16 views

What does it mean for two probability spaces to be distributed like each other? [on hold]

Given two probability spaces $A = (\Omega,F,P)$ and $B = (\Omega', F',P')$ what does it mean for "A to be distributed like B"?
0
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1answer
28 views

Indicator function integral

Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space. Let $A, B\in\mathcal A$. Assume that $\mathbb P(A) = 0.5$, $\mathbb P(B) = 0.4$ and $\mathbb P(A\cap B) = 0.1$. Find the integral over ...
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0answers
35 views

When a limsup can enter inside an integral

In the book "partial differential equation in classical mathematical physics" by I.Rubinstein,L.Rubinstein at page 411 I found something that I can't justify. It seems that the author (between ...
0
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0answers
7 views

Understanding the posterior of the Dirichlet process

Draws from a Dirichlet process (DP) are discrete, and exhibit clustering behaviour. Suppose I draw $G_{1:5}$ distributions from a DP. Then the posterior probability for $G_6$ is given by (Blackwell ...
2
votes
1answer
39 views

Please check whether the proof is correct or not.

Please check my solving. I want to know where to be wrong or illogical, or where logical jumps are. Problem Let $y=Tx$ be a nonsingular linear transformation of $\mathbb{R}^n$. If ...
0
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0answers
21 views

Questions on measurable functions and $L^p $ spaces

I'm learning about measure theory and $L^p$ spaces and need help with the following questions: $(1)$ True or False (justify): If $f : \mathbb R \to \mathbb{R}$ is measurable on $(-n, n), \, ...
3
votes
1answer
28 views

Lebesgue-$\sigma$-algebras $\mathfrak L^{p+q}\neq\mathfrak L^p \otimes\mathfrak L^q$

I already know that for Borel-$\sigma$-algebras it holds that $\mathfrak B^{p+q}=\mathfrak B^p \otimes\mathfrak B^q$. Now I want to show that this is not the case for Lebesgue-$\sigma$-algebras ...
4
votes
1answer
46 views

Nonatomic measure space over set larger than the reals

Question: Does anybody know a non-trivial nonatomic measure space over a set larger of cardinality larger than the reals? By non-trivial I mean that no set exists of cardinality equal to that of the ...
-2
votes
1answer
22 views

Convergence pw if converges in Lp space

Let $p\in[1,\infty]$ be given. If $f$ and $g$ are non-negative analytic functions such that the following holds: \begin{equation} ...
0
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2answers
37 views

Sequence of Radon Measures $\mu_n$ on $\mathbb{R}$

Problem: Find a sequence of signed Radon Measures $\mu_n$ on $\mathbb R$ such that $\langle \mu_n, \phi \rangle \to 0$ for every $\phi \in C^1_c(\mathbb R)$, and $|\mu_n|([0,1]) \to +\infty$. ...
2
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1answer
161 views

Suppose that all functions ${f_n},f$ are integrable. Is $\lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Let ${f_n}$ be a sequence of continuous, strictly positive functions on $\mathbb{R}$ which converges uniformly to the function $f.$ Suppose that all the functions ${f_n},f$ are integrable. Is ...
1
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0answers
24 views

Converges of measures.

Good afternoon, we have the following: Let $(Y,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...
0
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2answers
69 views
+50

intersection of boundaries have Lebesgue measure 0

Suppose $A$ and $B$ are two sets in $R^n$ such that $\overline{A}\cap B \cup \overline{B}\cap A$ is empty then $\partial A \cap \partial B$ has $n$-dimensional Lebesgue measure $0$ (where $\partial ...
1
vote
4answers
35 views

Relation between counting measure and Tonelli theorem

This is from Rudin's RCA book. But I can't understand how he got Corollary. What he takes as $f_n, X$? If we consider counting measure how integral converts to sum? I can't show this after some ...
3
votes
1answer
53 views

How does the sum of the absolute values of the diagonal entries of a matrix change when the matrix is written in a random basis?

The set-up is as follows: I have a complex, Hermitian matrix $H$ with $\mbox{Tr }H=0$, and such that the trace norm $\|H\|_1=1$ (i.e. the sum of the singular values $=1$). Let me define the functiona ...
49
votes
2answers
16k views

$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
1
vote
1answer
27 views

$L_p(\mu)\subseteq L_q(\mu)$ [on hold]

Given a measure space $(\Omega,\mathfrak A,\mu)$ and $1≤q≤p$, how can I show that $$L_p(\mu)\subseteq L_q(\mu)$$ if the measure $\mu$ is finite, that means $\mu(\Omega)<\infty$?
3
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1answer
1k views

Constructing a strictly increasing function with zero derivatives

I'm trying to construct a fuction described as follows: $f:[0,1]\rightarrow R$ such that $f'(x)=0$ almost everywhere,f has to be continuous and strictly increasing. (I'd also conlude that this ...
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0answers
36 views

$f'$ Lebesgue-integrable

Let $f:[a,b]\to\mathbb R$ be differentiable and the derivative $f'$ bounded. How to show that $f'$ is Lebesgue-integrable on $[a,b]$ and $$\int_{[a,b]}f'd\mu=f(b)-f(a)$$ where $\mu$ denotes the ...
0
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0answers
8 views

measure concentrated on an axis

If I want to consider a Borel-measure $\mu$ in $\mathbb{R}^2$ concentrated on the x axis, how can it work? it could be $\mu(A) >0$, where $A \in \mathcal{B}(\mathbb{R}^2)$?
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0answers
5 views

Atomless measure space without measure preserving isomorphisms

Question: Could somebody give an example of an atomless measure space without measure preserving isomorphisms (except for the identity)? Background: A measure preserving isomorphism on a measure ...
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0answers
8 views

Subnormal Weighted shift and First order derivative

Let $\mathbb B^m$ denote the Eucledian ball in $\mathbb C^m.$ Does there exist a reinhardt measure $\mu$ supported on $\partial \mathbb B^m,$ the boundary of ball, so that the Hilber space $H^2(\mu)$, ...
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3answers
82 views

Prove that $\int \limits_{E}fd\mu=\int \limits_{X}f\chi_{E}d\mu$

Let $(X,\mathfrak{M},\mu)$ be measure space. Let $f\geq 0$ be measurable function. Prove the following equality: $$\int \limits_{E}fd\mu=\int \limits_{X}f\chi_{E}d\mu$$ I can show only that $\int ...
1
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1answer
27 views

Interchanging of order summation in proposition 1.25 [Rudin RCA]

Hello! This proposition from Rudin's RCA book. One moment confuses me, namely how he interchanges the order of summation in that double infinite series? Can anyone give a rigorous explanation of it? ...
0
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1answer
67 views

How can I prove the following equation.

For $0\le{}x\le1$ and for $0\le{}y\le1$, $f(x, y)$ satisfies that, for each $x$, $f(x, y)$ is an integrable function of $y$ and $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded ...
1
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1answer
28 views

Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$.

Problem: Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$. Attempt: Well, if $f \equiv 0$ we get 1. Provided some sort of goodness like $f \in C^1$ ...
0
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1answer
27 views

Product of Lebesgue-null-set and arbitrary Lesbesgue-set is a Lebesgue-null-set again

I already know that for Borel-$\sigma$-algebras it holds that $\mathfrak B^{p+q}=\mathfrak B^p \otimes\mathfrak B^q$. Now I want to show that this is not the case for Lebesgue-$\sigma$-algebras ...
2
votes
0answers
26 views

Why is the Newton quotient measurable when the conditions are like the following.

Let $f(x, y), 0 \le x, y, \le 1$, satisfy the following conditions: for each $x$, $f(x, y)$ is an integrable function of $y$. $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded ...
2
votes
2answers
29 views

Random variable independent of $\sigma$-algebra and conditional expectation

What does it mean to say that a random variable is independent of a sigma-algebra, and why then does this imply that $E(RV| \sigma) = RV$?. I have no clue what this independence stuff is about ...
4
votes
2answers
133 views

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, ...